tag:blogger.com,1999:blog-38242854016981214262018-03-05T22:43:55.294-08:00Elementary ThoughtsRichard Fullernoreply@blogger.comBlogger12125tag:blogger.com,1999:blog-3824285401698121426.post-91374910416052342632016-03-23T08:27:00.000-07:002016-03-26T11:14:39.789-07:00What is Mathematics for STEM? What is a Mathematics of Solution?<h2>Why all the Fuss about Multiplication</h2><div> We only consider multiplication of non-negative integers. (multiplication in the natural number system). When we say "multiplication" the operands and the product are natural numbers.</div><div><br /></div><div>I think it's because multiplication is the first encounter with arithmetic that is more than just elaborated counting. Students do not bring applicable intuition: multiplication must be taught and learned in an academic setting. The traditional approach skirts its difficulty one, or both, of two ways:</div><div><ul><li>it says multiplication is (multiple) addition so there is really nothing new, and/or</li><li>it says its too difficult to understand it, remember the multiplication tables.</li></ul><div>I assume each of these approaches contributes to an efficient route to worthy goals, but their pursuit short circuits the opportunity to see what multiplication really is. In the past it could be argued that students only had a need for vocational math, hence it wasn't helpful to know how arithmetic worked; it was just a tool, learn how to do it fast. That was probably a reasonable argument 50 years ago. It's a lousy argument for the world where today's student will work. Understanding how things work is the way to be effective with and around computers/processors, and that is where today's students will be. One doesn't compute with computers, one tells computers <u>how</u> to compute. To do that you need to know how computation works</div><h3>Why is it Hard to Understand Multiplication?</h3><span style="font-weight: normal;">I think there is a simple explanation: the definition of multiplication involves multiplication itself; it also involves addition. We all know what is being defined can not also appear in its definition; but it turns out there is an escape clause. We are interested in the definition of the multiplication operation, and, although the multiplication operation does appear in its own definition, one of its operands is different, and that allows the operation to work itself out as a process. This is best seen in an example. C</span><span style="font-weight: normal;">onsider 6 x 4 with the natural number multiplication definition: </span></div><div><ul><li>6 x 4 = 6 x (3 + 1) + 6 = 6 x 3 + 6 =</li></ul><div>Read this as "(six times four) equals/is (six times three) plus six". The multiplication operation being defined does appear on both sides of the equals sign, but what happens when we substitute the same definition for 6 x 3, and continue in this manner:</div></div><div><ul><li>6 x 4 <span style="color: red;"> </span>=</li></ul><ol><li>6 x (3+1) = 6 x 3 + 6 x 1 = 6 x 3 +<span style="color: red;"> <span style="color: red;">6</span></span></li><li>6 x (2+1) + <span style="color: red;">6</span> = (6 x 2 + 6 x 1) + <span style="color: red;">6</span> = (6 x 2 + <span style="color: red;">6</span>) +<span style="color: red;"> <span style="color: red;">6</span></span></li><li>(6 x (1+1)) +<span style="color: red;">6</span>) + <span style="color: red;">6</span> = ((6 x 1 + 6 x 1) + <span style="background-color: white;"><span style="color: red;">6</span></span>) + <span style="color: red;">6 </span>= (( <span style="color: red;">6</span> + <span style="color: red;">6</span> ) + <span style="color: red;">6</span> ) + <span style="color: red;">6</span></li><li>(12 + <span style="color: red;">6</span>) + <span style="color: red;">6</span> = </li><li>18 + <span style="color: red;">6</span> = </li></ol><ul><li>24</li></ul></div><div>The definition specifies a process for multiplication. When you expand the definition for an actual multiplication, you write down a process:<br /><ul><li>multiplication <i>is</i> a process. </li></ul><div>In fact it is a concatenation of two processes, one for multiplication and the other for addition. Each step in a multiplication(addition) process is followed by another multiplication(addition) step until the process terminates; each of these processes is recursive. </div><br />And finally we have spoken the name of a process that refers to itself. The idea of a recursive process can be hard to grasp; the best way to get a feeling for recursion is to do recursions. Laying out the process as the one above is, reveals its dynamic as a pattern: it first lays out the decomposition of the multiplication one sub-multiplication at a time and then it collects these terms into one sum by one addition at a time. It first goes out and then it comes back:<br /> multiplication is an process executed as<br /><ul><li>a recursive multiplication process followed by</li><li>a recursive addition process. </li></ul><div>A two step process is a general architecture for problem solution:<br /><ol><li>Formulate an expression for the solution that can be evaluated.</li><li>Evaluate the expression.</li></ol><div>The first step is often neglected in school mathematics. Multiplication is an example of that neglect; it is often defined as the multiple addition produced by the first step, and then it is said "multiplication is multiple addition". This approach takes for the solution an expression where the results of the multiplication in the first step have already been incorporated; it is easy to overlook in natural number multiplication where n x 1 = n. In word problems, problems that involve quantities in the real world, the product is not the same thing as either of the operands: the product is not just a different number of what an operand is for natural numbers.<br /><h3>And Your Point Is?</h3></div><div>Its slogan could be "multiplication is multiplication plus addition":</div><div><ul><li>Multiplication is a process, </li><li>the process terminates when it produces a product.</li><li>To understand multiplication is to understand that process.</li><li>It is a recursive process:"multiplication is multiplication plus addition" </li></ul><div>Multiplication as a solution is dynamic. It is not just the inert answer to a problem. In this, it is an exemplar of the solutions that productive members of society construct. </div></div><div><br /></div><div>For the most part school mathematics is now taught as if the problems of mathematics education itself had been solved. Success is just figuring out how to get students, teachers, and the rest of us to buy in to a program that says: we lay it all out for you, all you have to do is retrieve it. It is a long way from here to "the use of this stuff is in the understanding of it". I want to participate in that journey. </div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><br /><div></div></div><br /><br /></div><div><br /></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-74216550817555451212016-03-15T09:24:00.000-07:002016-03-21T07:49:59.239-07:00Values of One<h2>Numbers Keep the Score. They Don't Play the Game.</h2><div>I started this blog to force myself to look for the reason(s) I found it difficult to do problems in freshman university physics. Many of the friends I talked to experienced the same thing when they first confronted problems with their school mathematics. It does appear to be a common problem in US math education: students are fine with straight forward arithmetic, but have a big problem seeing how it is applied.<br /><ul><li><i>one of the greatest deficits in U.S. students learning mathematics is in their ability to solve problems</i>( page 138 in <u>Adding it Up</u>, National Research Council, 2001). </li></ul>I think I can finally see what could cause the problem, and I even think there is a practical way to address it: little is offered in school arithmetic to help a student focus attention on the problem itself, to first think what the problem is about; what objects are involved? how are they related? and from this perspective, work toward an expression of the problem. In my experience and reading, school mathematics is taught the other way around: the numbers drive the strategy. If this is the perspective taken in formulating problems, it could be a reason they often feel contrived, more like puzzles than problems.<br /><br />"Expression" can suggest it is necessary to wait for students' exposure to school algebra before they have the tool to express problems. This is the conundrum: until algebra there appears to be no way to express problems, but students get to algebra already expecting numbers to guide them to solution of word problems. Before students get to algebra they need a low barrier approach to learning what the objects in a problem are, and how they act and how they are related.<br /><br />To be a successful word problem solver a student needs to able to call upon a conceptual framework for solving word problems. Such a framework is not a memory file of formulas. It is the concept structure built from knowing that words contain the problem:<br /><ul><li>words convey the information that must be understood in order to formulate the problem;</li><li>a general formulation of the problem exists that does not depend on the numbers.</li></ul><div>Successful problem solvers must conceive of themselves as solution formulators, not arithmetic evaluators. To do that, a student needs a concept that connects a problem's expression to the construction of the expression of its solution. </div><h3>Values of One</h3><div>School mathematics problems concern quantities resulting from making a measurement or from counting, actually or conceptually. In either case the quantity is expressed in "units", where a unit is one of something. That "something" is the "value of one". It determines how the quantity for which it is the unit participates in the expression of the solution to the word problem. The numerical value of a quantity expresses the magnitude, not the manner, of its problem participation.<br /><br />Ignoring <i>values of one, </i>what we will generally call "unit values" from now on, has resulted in misunderstanding the multiplication introduced in early elementary education. The product of two natural numbers is a natural number; multiplication that happens in the natural number system stays in the natural number system. Within the natural number system the unity "1" is itself the unit value, and for any natural number n, n x 1 = 1.<br /><br />A number system whose unit value is not simply "1", say (1 rabbit) or (1 foot), does not have a unity value. The product (1 rabbit) x (1 rabbit) is not in the same number system as (1 rabbit) {or should we say set/collection}. Addition can only be performed within the same number system, and this holds for both natural numbers and and non trivial unit values</div><h4>Examples of Unit Values</h4><div>For an object to participate in a analytic problem it must have its own unit, which serves to state the terms of its participation: are the apples participating as</div><div><ul><li>individual apples with a unit of 1-apple?</li><li>as individual pieces of fruit with a unit of 1-fruit?</li><li>as a collection of 12 apples with a unit of 1-doz-apples?</li><li>as a pound of apples with unit of 1-lb-apples?</li><li>as a fruit in a collection of apples and oranges?</li></ul><div>Is time participating as a </div></div><div><ul><li>1 second,</li><li>1 week</li><li>1 average-squirrel-lifetime </li></ul><div>Is distance participating as </div></div><div><ul><li>1 mile,</li><li>1 mm</li></ul><div>Are rabbit cages participating as </div></div><div><ul><li>1 rabbit-cage</li><li>1 doz rabbit-cages </li></ul><div>And for rabbits</div><div><ul><li>1 rabbit</li><li>1 pair of rabbits</li><li>1 buck rabbit</li><li>1 doe rabbit</li></ul></div>Numbers for counted quantities generally behave numerically as whole/natural numbers; units for measured quantities, as real numbers. Measured quantities are associated with a continuum; a number for a measurement is often expressed as an integer value embedded in a continuum . More generally we can assume measurements are expressed as rational numbers embedded in that continuum; this makes it possible to formulate measurement problems amenable to school arithmetic without worrying too much about number systems.<br /><h4><b>Arithmetic and Units</b></h4></div><div>* I assume problems addressed to the elementary level can only rely on evaluation of an expression, not on the algebraic manipulation of variables across an equal sign.</div><div><br />A major difference between addition and multiplication is easily expressed in terms of the unit value of their operands:<br /><ul><li><u>Addition</u>: there is only one unit value; each operand, as well as the sum have the same unit valuee.</li><li><u>Multiplication</u>: there can be three distinct unit values, and there generally there are. </li></ul>This difference does not arise when you are simply multiplying two natural numbers where it can appear that multiplication is a variant of addition. If addition were the only available operation in arithmetic, the universe of addressable problems would be small indeed: addition can only deal with augmentation by a like quantity. Its single unit value factors out of the addition expression, and hence there is essentially only one type/kind of arithmetic addition problem, and it makes sense to teach/drill natural number addition without reference to a specific problem. If you conclude addition is the operation for a problem, then you will have identified the unit value, and can proceed to execute the numerical addition and then append the unit to the sum.</div><div><br /></div><div>If you do not see augmentation as the problem, then you are facing different issues: what are the objects? their unit values? and how do the unit values fit together to express the problem? </div><h3>Multiplication Problems</h3><div>The post before this one considered natural numbers with unit values. In fact it derived the arithmetic for them. The work there shows how the difficulty students now have with word problems creeps into their education at its earliest stage: they are taught the basis of multiplication is a table of small-natural-number results that can stand alone with no consideration of unit values. From the very beginning, mathematics is presented as separable from the problems it addresses; that it is somehow self contained within the numbers it considers. Instead of telling students "multiplication is xxx", should not we be putting ourselves in position to say "the multiplication we do works like this: xxx"?<br /><br /></div><div>First some notation anguish; what would better serve young learners? Something that carries the notion of context beyond the usual "3 feet". Let us try "3:ftW" for 3 feet of width. It is not standard, but it better expresses the close coupling of number to unit value than the standard stand-off notation; for a given length, the number would not be 3 unless it referred to feet.<br /><h4>An Area Calculation </h4></div><div>Let's review some natural number work. With the notation, "[ ]" for successor introduced in the last post we have</div><div><ul><li>1:ftW = [0:ftW] = 0:ftW + 1:ftW,</li><li> </li><li> </li><li>4:ftW = [3:ftW] = 3:ftW + 1:ftW</li></ul><div>indicating 1:ftW is the unit for Natural Width Numbers. Now we can construct and evaluate the expression for a 6 foot by 3 foot rectangle:</div></div><div><ul><li>6:ftL x 3:ftW = 6:ftL x [[[0:ftW]]] = </li><li>6:ftL x ( 2:ftW + 1ft:W) = 6:ftL x 2:ftW + 6:ftL x 1:ftW =</li><li>(6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW = </li><li>((6:ftL x 0:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) =</li><li>((6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) </li></ul><div>The multiplication is now the sum of 3 terms, each the product 6:ftL x 1:ftW; the multiplication of length by width is now the sum of the area of three sub rectangles:</div><div><ul><li>6:ftL x 1:ftW = (6 x 1):(1:ftL x 1:ftW) = (6 x 1) x 1:ftftA = 6:ftftA,</li></ul><div>where the product of unit values is the unit value of the unit value product: 1:ftftA for "feet squared" or "square feet". We resume the evaluation:</div></div><div><ul><li>(( 6:ftL x 1:ftW + 6:ftL x 1:ftW ) + 6:ftL x 1:ftW ) = </li><li>(( 6:ftftA + 6:ftftA ) + 6:ftftA ) =</li><li>( 12:ftftA + 6:ftftA ) =</li><li>18:ftftA </li></ul><div>The full decomposition is provided to show how the multiplication expression is first expanded into a sum of terms from the decomposed product, in this case three 6:ftL x 1:ftW sub areas each of area 6 square feet. </div></div></div><div><br /></div><div>This is the way of multiplication: the multiplication itself is sequentially reduced by taking one unit value from one of the multiplicands in the multiplication to make a "1-unit-value multiplication by the other multiplicand" contribution to the product. Multiplication includes recursion process, not just the final assembly of the sum, and it is necessary for it to do so in order to obtain an expression that only involves multiple multiplications by 1 unit value; all we know coming into this calculation is multiplication by one unit value; this multiplication is constructed, not retrieved from memory. </div><div><br /></div><div>This is an instance of confusion arising from conclusions drawn from the multiplication of unitless Natural Numbers. There we have,</div><div><ul><li>n x 0 = 0 and n x [m] = n x m + n, </li></ul><div>If we take that over into unit natural number multiplication, we would be led to the error: </div></div><div><ul><li>6:ftL x 3:ftW = (6:ftL + 6:ftL) + 6:ftL.</li></ul><div>For the natural number n, n x 1 = n, while 6:ftL x 1:ftW = 6:ftftA. Unit values matter. In fact the solution to the problem follows from understanding area and its relation to the perpendicular sides of the rectangle, and that understanding is contained in the unit value relation, 1:ftL x 1:ftW = 1:ftftA, of the product of unit values for the sides and the unit value of the resulting square's area.<br /><br /></div><div>Word problems encountered in school mathematics often appear to be reverse engineered from arithmetic problems. For an area calculation like the one above, an academic word problem is often stated as if the point is "6 x 3 =18". In real word problem solving, the battle has been won by the time you get to the arithmetic. Attending to unit values focuses attention on the factors important to problem solving.<br /><br /></div><div>The unit values in the area problem are unit values of length measurement: unit values of a measurement of a physical continuum. They are real numbers, in this case numbers with integer values embedded in a continuum. For counting problem, unit values are equally significant. We will return to some examples in the next section. </div><h4>Ratios and Rates in Multiplication</h4><div>Understanding rates and ratios provides access to a large fraction of the interesting problems involving multiplication. </div><div><br /></div><div>A ratio is a comparison of two quantities in a multiplicative context just as a difference is a comparison of two quantities in a additive context. In either case the significance of the quantities is determined by their unit values. Multiplication is the interesting case.<br /><br />The distance and duration of a trip can be related by their ratio, say 60:mileDis/2:hourDur. This is equivalent to the average speed 30:mile/1:hour = 30:mphSpeed. At this average speed, a 6:hour trip covers<br /><ul><li>30:mphSpeed x 6:hour = (30:mile/1:hour) x 6:hour = 30:mile x (6:hour/1:hour) = </li><li>(6 x 30):mile = 180:mile.</li></ul><div>We now consider a problem based on counted quantities. A rabbit breeder has 18 rabbits and 6 cages. Take a single-rabbit unit value "oneR", and "oneC" for a single cage as the cage unit value, then the rabbit-to-cage ratio is</div><div><ul><li>18:oneR/6:oneC = (18/6) x (1:oneR/1:onrC) = 3:oneR/1:oneC.</li></ul><div>which says the uniform rabbit cage occupancy rate is 3 rabbits per cage.</div></div><div><br /></div><div>Rates can be useful in decomposing a problem where the issue is not obviously a rate. A teacher has 3 books and would like to know how many ways they could be paired with 6 students. Now 1:book can be paired with any one of 6:student: the book pairing rate is 6:B-S per book. The total number of book-student pairings is this rate times the number of books:</div><div><ul><li>( 6:B-S /1:book ) x 3:book = ( 6 x 3):B-S x (1:book/1:book) = 18:B-S</li></ul><div><br /></div></div><div>We return to the rectangle area problem to see the rate at which area accumulates. It is slight variation on the accumulation of chunks of area previously discussed. The rate of area increase per unit width for the 3 foot wide, 18 square ft area follows from<br /><br /><ul><li>area per width unit value ratio = 1:ftftA/1:ftW</li></ul><div>for the rectangle under consideration as </div><div><ul><li>18:ftftA/3:ftW = (18/3):ftftA/1:ftW = 6:ftftA/1:ftW.</li></ul><div>This says the area increases by 6 square feet per 1 foot of width, which is the length 6:ftL, providing an interesting variation of the rectangle area problem<br /><br />The point here is: a problem can be addressed through an expression involving only unit values, a general expression for the common kernel in a collection of problems that differ only in their number values.<br /><h4>The Significance of Unit Values in School Mathematics </h4></div><div>For the most part today's students are asked to approach word problems though numerical values and remembered arithmetic operations. They are not provided with any conceptual framework to address the problem on its own terms. They are often left with the idea that each word problem is special unto itself, a creature of the numbers in it, each problem requiring its own sorcery.</div><div><br />Unit values and their relations are the bricks and mortar of a conceptual framework to organize the terms of a word problem, and those terms are stated in words. The words alone describe a general problem, and that is the problem addressed by unit values in what be called a unit expression. <br /><br />I maintain there are even larger issues at stake here. Now students are assigned an essentially passive role in their mathematics education. They see a mathematics that is essentially a definitional subject where their learning role is largely passive. Mathematics is presented to them as a corpse they are invited to pick at, whose parts they are required to remember. Elementary school students are invited to learn mathematics the way medical students learn anatomy. By way of an example:<br /><ul><li>a number line is used to demonstrate numbers are already "there" to be laid out for inspection,</li><li>multiplication is what is in multiplication tables,</li><li>for larger numbers multiplication is what an ancient algorithm does.</li></ul><div>Unit values support the construction of a solution to word problems. Students are introduced to a tool, and thereby given agency. They construct a solution to a word problem. A previous post provided a way to construct numbers, as well as addition and multiplication. Constructive mathematics addresses students as potential producers, not as mere consumers, and this is true on several levels. The realistic hope for STEM in elementary school is students will be given the opportunity to see themselves as producers. </div><br />Is this practical? Can it be taught? I see nothing in a constructive approach that is intrinsically more difficult than the current approach for either students or teachers. On the other hand it differs greatly from what a deeply embedded system currently offers. A lot of discussion would need to take place, and much work anticipated to complete a story that speaks to the concerns of educators<br /><br /><br /></div></div></div></div></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /><br /><br /> </div><div><br /></div><br /><br /><br /></div><div><br /></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-43219532133888007612016-03-01T12:14:00.000-08:002016-03-01T12:14:18.246-08:00Constructive Elementary Arithmetic<br /><h2>Need More Learning-to-Understand in School Mathematics</h2>For all intents and purposes, the basis for early school arithmetic is memorized small number addition and multiplication. This can be an efficient way to learn an arithmetic for calculation. It also may have a significant downside: (1) it sends a message that it is good enough to know how to do without understanding, and (2) it forgoes the opportunity to start learning how to understand. Sharply focused tools address immediate problems; learning to understand is the base on which to move forward.<div><br /></div><div>To some students mathematics education can be a series of ever higher barriers, and they often appear to come as surprises, as if the student is seeing something for the first time. It should not be this way. Mathematics at any level can reveal modes of thought and understanding that lower barriers to next level.</div><div> </div><div>This post intends to show how to understand early arithmetic with a constructive approach. The slogan might be: arithmetic is built, not memorized. It is based on the assertion that speed is not of the essence, understanding is necessary to get you further along. </div><div><br /></div><div>Constructive mathematics is close to the programming of arithmetic. It is much like writing in a computer language. I understand this might be a good thing now.</div><div><br /></div><div>Of course there is nothing new here. It is the result of trying to follow up simple questions about a foundation for school arithmetic. </div><h3>Constructing Natural Numbers</h3><div><div>A Natural Number is one of the non-negative integers, the numbers with which we all started our mathematics education. Historically they were considered as "natural" and hence without need of further explanation. This view changed in the 19th century when concerns were raised about the foundations for arithmetic and Euclidean geometry. For arithmetic this eventually lead to the understanding of calculation which plays an important foundation role in Computer Science. Foundation work for the natural numbers provided a simple way to construct them, and this leads to simple construction of arithmetic with them. I think it could be a practical way for beginning students to understand their arithmetic; and to begin learning how to understand mathematics.<br /><br /></div> (mostly from Halmos, <i>Naive Set Theory)</i><br />Define a natural number as a set/collection that contains exactly that number of elements/units. By analogy think of a meter as being defined by a selected object, that on intuitive or practical grounds deserves to be called one meter long. Then "one-meter-ness" is attributed to another object just in case it has the same length as the selected object. Our number is such an object for number-ness; e.g. for 2 we take a set/collection with exactly 2 elements, or a set/collection with<br />2-number-ness. Where do we get an element with number-ness? We need a scheme that creates elements as we proceed in the construction of numbers, a process that starts with a small seed and then proceeds on its own with a simple repetitive process, a bootstrap with a small seed that brings itself into being.<br /><br />Let's say we already have a number, say 7 defined as a set/collection with 7 elements. Clearly the number 7 itself can not be one of those 7 elements, so we can take it as the eighth element. We can think of the number 7 as the name of the set that defines it.<br /><br />Now we are in position to specify the step in the repetitive process that constructs all the natural numbers. Before any step the process has at its disposal all the names of all the sets/collections produced up to this step. For any set n the next or <i>successor</i> set [n] is given by<br /><br /> [n] = n U {n} ,<br /><br />where U specifies the union of n, the set/collection of the elements of n; and the singleton set {n} whose element is the name of n. That is every natural number is to be the set of (the names of) its predecessors. This is more straight-forward than it might at first appear. It is easiest to see in the construction of natural numbers shown below.<br /><br />0 is set/collection with no elements, the empty set/collection:<br /><br /> 0 = { } .<br /><br />For each number to be the set of its predecessors:<br /><br /> 1 = [0] = 0 U {0} = { } U {0} = {0},<br /> 2 = [1] = 1 U {1} = {0} U {1} = {0,1} ,<br /> 3 = [2] = 2 U {2} = {0,1} U {2} = {0,1,2} ,<br /> etc.<br />Here etc. has its usual meaning: you get the drift, and the process can now be left on its own to produce any natural number we should need. "[-]" is the successor of the number between the brackets. With this construction in mind, it is easy to picture the<br /><br />Axiom of Infinity: There is a set containing 0 and containing the successor of each of its elements.<br /><br />Where did those numerals 1, 2, 3, ... come from? We used the familiar ones, but we could have made up our own; all a numeral does is name a set/set/collection that is filled with names we supplied previously. The problem with custom names: this scheme is designed to construct natural numbers that do what the natural numbers did before the scheme was invented. Mathematics did not invent them, but it did make this scheme to build them so they can be trusted because they can be understood.<br /><br />All this is elegant and efficient; we have a process to construct all of the natural numbers in terms of the natural numbers themselves that is ordered to generate natural numbers before they are needed to generate succeeding natural numbers. On the other hand it seems like a little too much; we never really use all all this structure for much beyond constructing the numbers themselves. It is like scaffolding: it allows us to construct what we need, but then what it allowed us to build can provide what it was designed for on its own. The good news: you can construct the numbers you will use yourself, you understand how they work, and you no longer have to fall back on the authority of "naturality". Just remember what 0 is, and what it means for each natural number to have a successor:<br /><br /><ul><li>0</li><li>1 = [0]</li><li>2 = [1] = [[ 0 ]]</li><li>3 = [2] = [[[ 0 ]]]</li><li>4 = [3] = [[[[ 0 ]]]] </li><li> </li><li><br /></li><li> </li></ul><h3>Defining Constructive Arithmetic on Natural Numbers</h3>We have numbers, how we do arithmetic with them? What is addition, and what is multiplication, with Natural Numbers? Here's the plan:<br /><ul><li>Define addition on Natural Numbers. Then show [] and "+1" have the expected relation [n] = n+1 : we can count to construct a natural number. </li><li>Introduce the "unit" for the element being counted; i.e, for the unit "ball", .</li><li> Given a unit, we can construct a number of units, or count out the "number of units"; e.g. 4 balls. A "number of units" is constructed in the same way a natural number is constructed.</li><li>We show: a "number of units" can only be added to a "number of the same units" .</li></ul><div>The definitions for addition and multiplication found in the Natural Number Wiki appear to be the common ones. Addition must be defined before multiplication; addition is contained in multiplication's definition. </div><h4>Definition of addition, "+" , in n + m</h4><h4><span style="font-weight: normal;">There are two clauses:</span></h4><h4><ul><li><span style="font-weight: normal;">n = n + 0,</span></li><li><span style="font-weight: normal;">n + [m] = [n + m] = [n] + m </span></li></ul><div><span style="font-weight: normal;">The second clause states the equivalence of the successor of different combinations of the operands:</span></div><div><ul><li><span style="font-weight: normal;">n + ( the successor of m) is the same as the successor of ( n + m) is the same as (successor of n) + m. </span></li></ul></div><div><span style="font-weight: normal;">This will lead to a strong uniformity constraint on, and among, operands and sum; this will come later.</span><br /><span style="font-weight: normal;"><br /></span><span style="font-weight: normal;">With 1 = [0] : </span><br /><br /><ul><li><span style="font-weight: normal;">n + 1 = n + [0] = [n + 0] using the second clause.</span></li><li><span style="font-weight: normal;">[n + 0] = [n] using the first clause, giving</span></li><li><span style="font-weight: normal;">n + 1 = [n] for any n, its successor is equivalent to n + 1.</span></li></ul><div><span style="font-weight: normal;">With addition we can give a second construction for natural numbers:</span></div><div><ul><li><span style="font-weight: normal;"> 0</span></li><li><span style="font-weight: normal;"> 0 + 1 = [0] = 1,</span></li><li><span style="font-weight: normal;"> 1 + 1 = [1] = 2,</span></li><li><span style="font-weight: normal;"> 2 + 1 = [2] = 3,</span></li><li><span style="font-weight: normal;"><br /></span></li></ul><span style="font-weight: normal;"> </span><span style="font-weight: normal;">Let's look at the pattern of an addition example working itself out. But first we consider just a single number working itself in:</span><span style="font-weight: normal;"> 3 = [2] = [[1]] = [[[0]]], </span><span style="font-weight: normal;">showing the expression of a natural number as 0 clothed in the number of its successors that take it back to, in this case, 3. Since 0 is not the successor of any natural number it it not possible to go further. </span><span style="font-weight: normal;">Consider the the addition: 9 + 3:</span></div></div></h4><h4><ul><li><span style="font-weight: normal;">9 + 3 = </span></li><li><span style="font-weight: normal;">9 + [ [ [ 0 ] ] ] = </span></li><li><span style="font-weight: normal;">[ 9 + [ [ 0 ] ] ] = </span><span style="font-weight: normal;">using the second clause in the definition of addition </span></li><li><span style="font-weight: normal;">[ [ 9 + [ 0 ] ] ] = and again</span></li><li><span style="font-weight: normal;">[ [ [ 9 + 0 ] ] ] = and yet again, and for the third time.</span></li><li><span style="font-weight: normal;"> [ [ [ 9 ] ] ] = using the first clause ) </span></li><li><span style="font-weight: normal;"> [ [ 10 ] ] = 9 + 1 10 is the successor of 9 ,</span></li><li><span style="font-weight: normal;"> [ 11 ] = (9 + 1) + 1 11 of 10.</span></li><li><span style="font-weight: normal;"> 12 = ( (9 + 1) + 1) + 1 = (10 + 1) + 1 = 11 + 1 </span></li></ul></h4><h4><div><div><div><span style="font-weight: normal;">If we are counting a set/collection of "these" where "these" could be balls, days, elephants, nails, ... then 0 of "these" is an empty set/collection of and for "these", and the successor of 0, [0] is 1 of "these". Let's take ball for 1 of "these". We can now construct the set of ball enumerators:</span></div><ul></ul></div></div><div><ul><li><span style="font-weight: normal;">0 ball</span></li><li><span style="font-weight: normal;">[0 ball] = 0 ball + 1 ball = 1 ball,</span></li><li><span style="font-weight: normal;">[1 ball] = 1 ball + 1 ball = 2 ball,</span></li><li><span style="font-weight: normal;">[2 ball] = 2 ball + 1 ball = 3 ball,</span></li><li><span style="font-weight: normal;">[3 ball] = 3 ball + 1 ball = 4 ball.</span></li><li><span style="font-weight: normal;"> </span></li><li><span style="font-weight: normal;"> </span></li></ul><div><span style="font-weight: normal;">4 ball = </span><span style="color: red; font-weight: normal;">((((</span><span style="font-weight: normal;">0 ball + 1 ball<span style="color: red;">)</span> + 1 ball <span style="color: red;">)</span> + 1 ball<span style="color: red;">)</span> + 1 ball<span style="color: red;">)</span> and etc. All those parenthesis are there because we are describing a process, not just describing a relationship, and we only know how to add two numbers. In any case 4 ball can be constructed by 4 consecutive additions of 1 ball to 0 ball.</span><br /><span style="font-weight: normal;"><br /></span><span style="font-weight: normal;">4 ball can also be constructed by 4 successor steps of 0 ball. </span><br /><ul><li><span style="font-weight: normal;">[[[[ 0 ball ]]]] = </span></li><li><span style="font-weight: normal;">[[[ 1 ball ]]]] = </span></li><li><span style="font-weight: normal;">[[ 2 ball ]] = </span></li><li><span style="font-weight: normal;">[ 3 ball ] = </span></li><li><span style="font-weight: normal;">4 ball</span><span style="font-weight: normal;"> </span></li></ul><div><span style="font-weight: normal;">And now we have to confront a basic question: what does "4 ball" mean? . We construct "4 ball" by four successor steps from "0 ball" ( we know this is equivalent to four consecutive additions of "1 ball" ). The sense of "4 ball" is it is the thing we constructed in four steps, each step incrementing by "1 ball"the count that started with 0. If we think of "Ball" as the name of a set/collection of balls from which one ball is removed and added to the set of counted balls, just as the above process does, until the ball set/collection is empty, then by doing that process we will have <i>proved,</i> by construction, there were 4 balls in Ball. In this case "4 ball" is a sentence, something like" there are four balls in Ball", and the above process is a proof this sentence is true. And the purpose of mathematics is to write true sentences about significant mathematical objects like numbers, triangles, a space, ... We will return to the notation for a "number with unit" after we have defined natural number multiplication.</span></div><div><span style="font-weight: normal;"><br /></span></div><div><span style="font-weight: normal;">An elementary school student knows or learns, in an informal way, this proof and it is the sense of this proof that is the student's earliest number sense. It is their number sense before they learn arithmetic, but when they turn to arithmetic a strange thing happens, or so it seems to me. Instead of building out their intuitive, informal understanding of constructing numbers in order to have the foundation for a constructive arithmetic, school education turns to an arithmetic of natural numbers that is contained in tables that are to be memorized. This is all well, if not good, if the student is just being prepared for speed with simple arithmetic with numbers, but this is not the case; students are expected to apply their arithmetic to problems beyond numbers themselves, to word problems that can ask ask for multiplication of rabbits by cages.</span></div><div><br /></div><div><span style="font-weight: normal;">A rabbit and a cage are examples of units, just as ball was. </span></div><br /></div></div><div>Multiplication: n x m</div></h4><div>for n and m any Natural mumber. Again there are two clauses in the definition:</div><div><ul><li>n x 0 = 0,</li><li>n x [m] = n x (m + 1 ) = n x m + n.</li></ul><div>Note, the multiplication definition uses addition and the relation between addition of 1 and successor: [n] = n + 1. We also note that n x 1 = n x [0] = n x 0 + n = n. This is expected, but for multiplication of numbers with units will have to be reconsidered.<br /><br />Here is an example of multiplication of natural numbers:<br /><br /> 9 <span style="color: lime;">x</span> <span style="color: red;">3 </span><span style="color: lime;">=</span><br /><ul><li>9 <span style="color: lime;">x [ [ [</span> <span style="color: red;">0</span> <span style="color: lime;">] ] ] =</span> (using clause 2 and [m] = m + 1) </li><li>9 <span style="color: lime;">x ( [ [</span> <span style="color: red;">0</span> <span style="color: lime;">] ] +</span> <span style="color: red;">1</span> <span style="color: lime;">)</span> <span style="color: lime;">= </span><span style="color: lime;">9</span> <span style="color: lime;">x [ [</span> <span style="color: red;">0</span> <span style="color: lime;">] ] +</span> 9 <span style="color: red;"> </span> <span style="color: lime;">= </span></li><li>(9<span style="color: lime;"> x [0] + 9)</span><span style="color: lime;"> +</span> 9 <span style="color: red;"> </span> <span style="color: lime;">= ( (9 x 0 + 9 )</span> <span style="color: lime;">+</span> 9<span style="color: lime;"> ) + </span>9 <span style="color: red;"> </span> <span style="color: lime;">= </span></li><li><span style="color: lime;"><br /></span></li><li><span style="color: lime;">( (9 + 9) + 9) =</span></li><li><span style="color: lime;">(</span> 18<span style="color: red;"> </span><span style="color: lime;">) +</span> 9 <span style="color: red;"> </span><span style="color: lime;"><span style="color: red;"> </span>=</span> </li><li><span style="color: magenta;">27 </span>. </li></ul><div>This example again shows the expansion of the expression as the multiplication is layed out, followed by contraction as the expression is evaluated by one addition at a time. A multiplication is both expansion and contraction, it is both "a taking apart" and a "putting together". The taking apart uses step-wise dis-assembly of one operand to produce a multiple addition that is evaluated, as it must be, one addition at a time. What else could it involve? To make multiplication we only have what we have already constructed, or at least that is all we have for defining multiplication, and multiplication is defined. </div><h3>Unit and Multiplication</h3><div>We return to the question of notation; what does "1 ball" mean? OK it means "one ball", but then we say "2 ball" is its successor. It feels like we are thinking of it as one word, in fact we are, so from now on we use "1_ball for "1 ball" and "n_ball" for "n ball". Remember numbers can only refer to one "thing"; "one thing" can be "one thing or another thing", but that really counts as "one thing". The two operands and the sum in addition are numbers for the same "thing".<br /><br />Multiplication is altogether a different matter. The numbers in each operand and the product can be numbers of different "things". For multiplication we must learn how to work with this difference. An addition problem deals with one "thing", and the arithmetic for addition could be considered independently of that "thing". This is the not the case for multiplication.<br /><br />We need to conform with the common nomenclature: "1_ball" is the counting unit for balls, and etc. To to do useful work with multiplication a student needs unit sense, number sense is insufficient. The secret for solving real problems is to get the units straightened out first, the operations second, and the numbers last.<br /><h4>There are Three Different Multiplications ,</h4><h4><ol><li>Only one operand has a unit. <span style="font-weight: normal;">These are multiplications where "x" is often pronounced "times". Examples: ** An example is: n x (m_cages) = m x (n_cages) = (n x m)_cages. These can be derived by arguments you must be tired of. ** 3 x (6_rabbit) = 3_rabbit x 6 = (3 x 6)_rabbit = 18_rabbit. </span></li><li><span style="font-weight: normal;">Each operand has its own unit, but the two units can be related to produce one:</span><span style="font-weight: normal;"> <span style="font-weight: normal;">O</span></span><span style="font-weight: normal;">ne unit is a for a rate or ratio that acts on the unit of the second to produce a third unit from the rate unit. Examples: ** 3_(rabbits/cage) x 6_cage = 3_(rabbits/cage <span style="color: red;">x</span> cage) x 6 = 3_rabbit x 6 = (3x6)_rabbit = 18_rabbit, ** </span><span style="font-weight: normal;">3_(miles/hour) x 6_hour = 3_(miles/hour <span style="color: red;">x </span>hour) x 6 = 18_mile. </span></li><li><span style="font-weight: normal;">Each operand has its own unit, but now multiplication ties them together. <span style="font-weight: normal;">In the previous two cases the product is the number of "thing"s. Now the product is the number of pairings of one operand's unit with the other operand's unit. The meaning of the pairing varies with the problem. Here multiplication</span></span><span style="font-weight: normal;"> acts on the "things" themselves and not just on the number of things; unit sense is central to the problem; e.g. two perpendicular lengths, one dimensional figures, can be multiplied to produce a two dimensional, rectangular-shaped surface. This is not elementary mathematics, but the sense of it is. Now we have a multiplication that goes beyond multiple adding. Examples: </span><span style="font-weight: normal;">** 6_cage <span style="color: red;">x</span> 3_rabbit = (6 x 3)_(cage,rabbit) = 18_(cage,rabbit) , is a pairing of a cage unit with a rabbit unit. In this case 1 rabbit with 1 cage. Each of the 18 pairings can not exist at the same time, there are not enough rabbits, or there too many cages. There are 18 distinct ways to pair a cage with a rabbit. If we think of a rabbit as well as a cage as points, then a (cage,rabbit) is a pair of points that locates the pair. This is much like a two dimensional Cartesian coordinate system, and indeed the pair is the result of a Cartesian product of sets, in this case one of cages and another of rabbits. Let me end with an example that shows multiplication producing a measure of a two-dimensional object, the area of a rectangle, from two one-dimensional objects, the Length side, 6_ftL, and the Width side, 3_ftW. The Length side is perpendicular to the Width side. Each side length has the measure unit, foot(ft). The area is given by ** 6_ftL x 3ftW = (6 x 3) x( 1_ftL <span style="color: red;">x </span>1_ftW) = 18_square-ft. </span></li></ol></h4></div></div></div><div><h3>Summary </h3></div><div><ol><li>You can construct the Natural Numbers you need, including a Natural Number count of Things.</li><li>On the system of these numbers, the Natural Number system, you can define a process for addition and one for multiplication that constructs a Natural Number for the sum or the product. This also works for Natural Number counts of "Things". In later posts, the Integers, the positive and negative whole numbers, will be defined in terms of the Natural Number system. Integer arithmetic, including subtraction is also defined in terms of Natural Number arithmetic. Finally the Rational Number system along with division is defined in terms of the Integer Number system; and now we have the complete picture of elementary school arithmetic based on the Natural Number system.</li><li>Multiplication can be multiple addition for either Natural Numbers of Natural Numbers of Things. It can also be a Cartesian Product that produces pairings. </li></ol></div><h3>Conclusion</h3>There is a straight forward constructive path through school arithmetic based on Natural Numbers.<br /><h3></h3></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-71426015153369371752016-01-29T09:04:00.000-08:002016-02-07T07:11:20.288-08:00DRAFT Natural Number Arithmetic<span style="font-size: large;">I try to construct a rigorous and relatively simple path from natural numbers to arithmetic for rational numbers. It is pieced together from Wiki articles on the natural, integer, and rational number systems. I would like to think it shows the existence of an accessible, and rigorous basis for all of school arithmetic, accessible for teachers to enrich and inform their instruction.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">[I did not intend the font change. In published form it does not follow the word processing instructions.]</span><br /><span style="font-size: large;"><span style="font-size: small;"><br /></span><span style="font-size: xx-small;">Rote learning can be put in its proper place: it can make simple, practical calculation possible. But at the end of the day we can rest assured there is firm support for what we are doing A little exposure to foundation plants the seed to grow mathematical sense; and besides that, seeing how thought leads to understanding is the unique pleasure of mathematics.</span></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><br /></span><span style="font-size: x-small;">I am not qualified to present material in a ready-to-teach-young-students form. I do try to get at the roots of elementary arithmetic. What is presented is certainly sufficiently straight-forward to be inform teachers on the foundation of the material they teach. This material should put teachers in position to evaluate for themselves claims like "multiplication is really addition". It looks to me like arithmetic proceeds in a straight line from a foundation well within the grasp of young students, and certainly well within the grasp of their teachers. </span></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><br /></span><span style="font-size: x-small;">The mathematical approach can only be an abstraction of what? To use the result of that abstraction it is not necessary to know "what" it is, or could be. Teaching mathematics to beginners, on the other hand, requires taking a stand, conscientiously or not. Since I am trying to teach myself, I eventually had to face up to having no plausible story for how the bootstrap could have started for me. This is not cognitive science, it is simply what I see as a possibility when I look at the mathematics: what I could have been doing before I started learning school mathematics?</span></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><br /></span><span style="font-size: x-small;">Here is the story I imagine. I realized one thing came after another thing, it wasn't all one long hum. I heard sound from my most significant other; it came in bursts, one after the other, sometimes little bursts inside big bursts. Eventually I got the hang of this after, after, ... business. At the same time I noticed the leg going out in front of me followed my other leg going out out in front of me, sort of like one sound burst followed another. "after" was like "following". My feet made marks in the sand, one after the other. When I took a step from a rock there was a mark in the sand, when I continued walking more marks appeared. Eventually I came up with a song to hum when I was walking with a word for the first step followed by another word for this next step. I noticed the tree close to the to the rock was ug1-ug2-ug3-ug4-ug5, that is an ug5 walk from the rock. </span></span><br /><h4><span style="font-size: large;">Counting</span></h4><span style="font-size: large;">Now I would describe counting as the attribution of "1 more", call it "+1" to the count of the pile before I put a ---- onto it. It is the same pile that was empty, call its count "0"(zero), before I put a --- onto it: Counting produces a sequence of counts:</span><br /><ul><li><span style="font-size: large;">starting with: old-count = 0 </span></li><li><span style="font-size: large;">proceeding with: new-count = old-count +1</span></li><li><span style="font-size: large;"> old-count = new-count</span></li><li><span style="font-size: large;">and continuing new-count = old-count +1</span></li><li><span style="font-size: large;"><br /></span></li><li><span style="font-size: large;"><br /></span></li><li><span style="font-size: large;"> </span></li></ul><span style="font-size: large;"><span style="font-size: x-small;">Counting evaluates the sequential growth/augmentation of the pile in the process of adding a --- to the pile that was empty to begin with. It is a sequential process, it does carry the idea of progression, of "step after step ..." but it also</span><span style="font-size: x-small;"> carries a record of augmentation, the pile is getting bigger. We attend to the progression part, or rather we let Peano and his axioms for the sequence of Natural Numbers take care of it. There is a problem with calling them numbers at this stage because Peano's axioms alone are only requirements that a sequence of words must satisfy in order for</span></span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">the sequence to start with a unique word, </span></li><li><span style="font-size: large;">for each word in the sequence to be different from all other words and for each word to have a unique place in the sequence, and </span></li><li><span style="font-size: large;">for there to be a simple way to prove something for every word in the sequence.</span></li></ul><div><span style="font-size: large;">The set of all the words is written as N = {0,1,2,3, ...}. Each one of the words is called a natural number, but at this stage we need to be careful to remember they are only what we say about them, and we say they are words that Peano calls natural numbers. Peano says:</span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">0 is a natural number.</span></li><li><span style="font-size: large;">Every natural number has a successor.</span></li><li><span style="font-size: large;">0 is not the successor of any natural number.</span></li><li><span style="font-size: large;">if the successor of x equals the successor of y. then x equals y.</span></li><li><span style="font-size: large;">if: [<i>a</i>] something (since we are talking about words that "something is a statement) is true about 0, and [<i>b</i>] if the truth of that statement for a number implies (means) it is true for the successor of that number, then [<i>c</i>] the statement is true for every natural number. </span></li></ul><span style="font-size: large;">Here is a place you have to be careful not to assume "1" or "2" mean what you remember they meant before we decided to get at the truth of things. "2" is not the successor of "1" is not the successor of 0 until we show how Peano's axioms let that be so. We have to prove what we say so we can know, really know the truth of what we are talking about. We are talking about natural number arithmetic. Is it true, and exactly what is it?</span><br /><span style="font-size: large;"><span style="font-size: small;"><br /></span><span style="font-size: small;">First we need to define addition. We do not know what is the successor of what. All we know is </span></span><br /><ul><li><span style="font-size: large;">that special word 0 is not the successor of any natural number, 0 is a natural number, and therefore 0 is not a successor of itself.</span></li><li><span style="font-size: large;">Every natural number has a successor. </span></li></ul><div><span style="font-size: large;"><span style="font-size: small;">If we use the usual numerals in th</span><span style="font-size: small;">e following, we are likely to use the properties we have come to associate with them in the proof that all those properties follow from just the Peano's axioms. For now we take the usual alphabet for the natural numbers. Use "+" for addition, then we define "+" by</span></span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">a + the-succ-of b = the-succ-of ( a + b), and</span></li><li><span style="font-size: large;">a + 0 = a.</span></li></ul><div><span style="font-size: large;">First notice that we are trying to define + on the left hand side of = by an expression on the rhs that contains + ; rather like the definition of a word including the word itself. It works in this case because the b is the-succ-of c in a + b , and then a + b = a + the-succ-of c = the succ-of (a + c), and we continue in this "recursive" manner until we inevitably arrive at a + 0. 0 is not the successor of an natural number. That is why setting a + 0 = a is part of the definition of addition. And finally we have + on the lhs but not on the right hand side, and the recursion stops. This is a general definition, it works for all natural numbers a and b, but recursion is slippery; we will provide some numerical calculation later to show how it all hangs together</span><br /><span style="font-size: large;"><span style="font-size: small;"><br /></span><span style="font-size: small;">We are free to name the-succ-of 0: let's say 1 := the-succ-of 0. </span></span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">b + 1 = b + the-succ-of 0 = the-succ-of ( b + 0 ) = the-succ-of b,</span></li></ul><span style="font-size: large;">b + 1 is the successor of b, where b can be any natural number. Using only: [<i>a</i>] 1 defined as the succ of 0, and [<i>b</i>] the defined addition for natural numbers, we can name/define them in accord with our expectations for them:</span></div><div><ul><li><span style="font-size: large;"><b>0</b> is not the successor of any natural number</span></li><li><span style="font-size: large;"><b>1</b> := the-succ-of 0 , definition of 1</span></li><li><span style="font-size: large;"><b>2</b> := the-succ-of 1 , definition of 2</span></li><li><span style="font-size: large;"><b>3</b> := the-succ-of2 , definition of 3</span></li><li><span style="font-size: large;"><b>4</b> := the-succ-of 3 , definition of 4</span></li><li><span style="font-size: large;"><b>5</b> := the-succ-of 4 , definition of 5</span></li><li><span style="font-size: large;"><b>6</b> := the-succ-of 5 , definition of 6</span></li><li><span style="font-size: large;"><b>7 </b>...</span></li><li><span style="font-size: large;"><b>8 </b>...</span></li><li><span style="font-size: large;"><b>9 </b>...</span></li><li><span style="font-size: large;">...</span></li></ul><div></div><div><span style="font-size: large;">From here on we will denote "the-succ-of" by "succ". The defined addition gives:</span></div><div><ul><li><span style="font-size: large;">6 + 3 = </span></li><li><span style="font-size: large;">6 + succ (2) = succ (6 +2) = </span></li><li><span style="font-size: large;">succ ( 6 + succ (1)) = succ(succ(6 + 1)) =</span></li><li><span style="font-size: large;">succ(succ(6 + succ(0)) = succ(succ(succ(6 + 0))) = succ(succ(succ(6))) =</span></li><li><span style="font-size: large;">succ(succ(7)) = succ(8) = </span></li><li><span style="font-size: large;">9</span></li></ul><div><span style="font-size: large;">This is a general pattern for recursion; construct the "program" to produce the result as a sequence of steps, for addition each step is a "succ"; until it comes to a step that can be executed; and then it runs back up the program, executing each step to produce the input for the next until it runs out of steps. Also note the associated "shape" of the program: the width starts out small, reaches a peak, and returns to small corresponding to first building out a counting program and then consuming it as each counting step is executed. </span></div></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">How is this useful?</span></div><div><ul><li><span style="font-size: large;">It shows how mathematics works: it proves things about words that are specified only by simple relations among them. For the set of natural numbers it shows how the definition of an addition of two of these words/numbers leads to a relation between them that can be expressed with the numbers themselves.</span></li><li><span style="font-size: large;">It shows how addition of natural numbers is true: it is the rigorous working out of the implications of a small number of requirements for the set of all the natural numbers.</span></li><li><span style="font-size: large;">It displays the elegance and economy of mathematics. </span></li><li><span style="font-size: large;">It will turn out that all of school analysis can be related back to natural number arithmetic. </span></li></ul><ul></ul><div><h4><span style="font-size: large;">Multiplication</span></h4></div></div><div><span style="font-size: large;">Given that addition has been defined, multiplication can be defined in an analogous fashion by setting</span></div><div><ul><li><span style="font-size: large;">a x 0 = 0, and</span></li><li><span style="font-size: large;">a x succ b = (a x b) + a.</span></li></ul><div><span style="font-size: large;">For the experience represent the set of natural numbers as N = {0, #, ##, ###, ...} with # = succ 0, etc. Then</span></div></div><div><ul><li><span style="font-size: large;">###### x ### </span></li><li><span style="font-size: large;">(###### x succ ##) = (###### x ##) + ###### =</span></li><li><span style="font-size: large;">(###### x succ #) + ###### = ((###### x #) + ######) + ###### =</span></li><li><span style="font-size: large;">(( ###### + ######) + ######) =</span></li><li><span style="font-size: large;">( ############ + ###### ) =</span></li><li><span style="font-size: large;"> ##################</span></li></ul><div><span style="font-size: large;">We used</span></div><div><ul><li><span style="font-size: large;">(###### x #) = (###### x 0) + # = 0 + ###### = ###### .</span></li></ul><div><span style="font-size: large;">Sorry for any problems reading this. Actually there is an advantage in being limited to a small number of symbols that can only be displayed in a straight line, but the advantage is mine; I am allowed to look for a scheme that makes sense without facing diversion by pretty notation. It is not good for you though.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Eventually I noticed word processing a multiplication of natural numbers in the # notation was mostly a matter of copying and pasting. And then it occurred to me that was true because the # notation for natural numbers is a self representation; e.g. pasting a # on to the end of natural number constructs its successor, and that leads to the realization a natural number can be representing by the thing that constructs it: a string of successor "operators" we have denoted by "succ" that extends to the left ( this is just the way readers of mathematics are used to seeing it) and originates with a 0 on the right. It is best to think of succ as a small machine that pulls in the natural number on its right, makes the successor of that natural number, and puts it directly under itself, and then goes away. Starting from the right it is then easy to see that 6 can be represented by the construction</span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">6 := succ succ succ succ succ succ 0</span></li></ul><div><span style="font-size: large;">In the # representation of natural numbers: we can have</span></div><div><ul><li><span style="font-size: large;">for # = succ 0, </span></li><li><span style="font-size: large;"># + ##... = ###... for any natural number; i.e,</span></li><li><span style="font-size: large;">succ ##... := prepend # to ##...</span></li></ul><div><span style="font-size: large;">We can go back and forth between the mathematical construction of a natural number with a string of succ operators, and a displayed representation of them made by executing a string of prepend instructions. In either case we could write a program to produce</span></div></div><div><ul><li><span style="font-size: large;">natural numbers as prepend(prepend( prepend( ... prepend(# to 0) ...))). There are n prepends to create an n length string of #s.</span></li><li><span style="font-size: large;">n + m as prepend(m to n). where the natural numbers n and m are represented as n and m length strings of #s as shown in first bullet. The a correct program that implements the bullet above would produce a string of (n + m) #'s.</span></li><li><span style="font-size: large;">n x m as prepend(prepend(...prepend(n to 0)...))). There are m prepends and again n is a natural number produced as the first bullet indicates. Now a correct program would generate a n x m length string of #s.</span></li></ul><div><span style="font-size: large;">This is just an outline. You could work entirely in the # representation by implementing counting as picking a # off a string off the string of #'s that represents a natural number until there were no more. If this doesn't relieve you of some worries, forget it.</span></div></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">I did not start with the intent to show all natural number arithmetic, at bottom, is just fiddling with strings of #'s or succ's, by whatever names these may be known; it just turned out that way. All integer and rational number arithmetic can be expressed as natural number arithmetic, and hence all school arithmetic is, at bottom, just more of this fiddling. At the end of the day that is why computers can do it. What we learn in school is the syntax of working with the names of these strings. For computers this is called syntactic candy. For us, the time we must spend on syntax it is the price we pay for being slow calculators. Still I think we could write computer programs that showed young students how it all really works, we could call it the "The String Theory Game".</span></div></div><h4><span style="font-size: large;">Windup </span></h4></div></div><div><span style="font-size: large;">What have we learned?</span><br /><br /><ul><li><span style="font-size: large;">There is a foundation for arithmetic.</span></li><li><span style="font-size: large;">A little of how mathematics works.</span></li><li><span style="font-size: large;">A good basis for going on to develop the foundation for the integer number system which incorporates negative integers, and the rational number system where fractions/ratios live. I hope to tackle this in the next post.</span></li></ul><br /><div style="color: black; font-family: 'times new roman'; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><div style="margin: 0px;"><span style="font-size: large;">Currently there is a belief, or at least a strong wish, that natural number multiplication is really addition. This is the sort of issue that is difficult to address from the perspective of school arithmetic itself, where apparently a thing can get defined by the practices associated with the signs in it; if an expression only has plus signs it must be addition. The plus signs that appear in an intermediate stage of multiplication do reflect addition in the definition of multiplication; addition must be defined before multiplication can be defined, but the definitions of addition and multiplication are not the same: multiplication is not addition. For one thing thing, with a slight abuse of definition, addition counts by 1 from n, while multiplication counts by n from 0. There should be, at the least, someone available to every teacher who can explain questions at the level of: what is addition? what is multiplication? what is the difference? </span></div></div></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">I left high school thinking math was mostly a matter of retrieval of small facts put in memory by rote learning and repetitive manipulation. This may have produced some number sense, but it is not number sense that underlies success in later work in mathematics. It is more a matter of mathematics sense. I came away with very little. Some exposure to the foundation of arithmetic would have been helpful.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"><br /></span></div><div><br /></div></div></div></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-21172783707668225132015-12-24T16:01:00.000-08:002015-12-25T08:13:37.384-08:00Arithmetic Symmetries, Understanding <h3><span style="font-size: large;">Arithmetic Symmetries and Dualities</span></h3><div><span style="font-size: large;">From the way we usually come at arithmetic, subtraction and division can appear to be subordinate to, and even derived from, addition and multiplication. Why is it not the other way around? And why are positive numbers "natural" and negative numbers not? The answer is straightforward: we bootstrap our way to negative and rational numbers from the whole numbers, the non-negative integers, which, strictly speaking, can only be added and multiplied: we start from numbers intended for counting which itself is then elaborated as addition and multiplication, and end up with negative numbers, and division. </span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This would present no problem if mathematics instruction looked back from the results of the bootstrap, and cleaned up the mess. With the full set of integers and rational numbers, there is nothing special about "subtraction of (-M) is the same as addition of (+M)", or "division by M is the same multiplication by the M^(-1)".</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">For each of addition/subtraction and multiplication/division there are symmetries that allow us to pass back and forth from one member of a pair to the other if that is convenient. This can all be derived by "removal" of the term in an expression for N that is responsible for M "being in" N, thereby mathematizing the concept of "removal":</span><br /><ul><li><span style="font-size: large;">N remove (+M) == </span><span style="font-size: large;">N - (+M)</span></li><li><span style="font-size: large;">is obtained by removing the term + (+M) from</span></li><li><span style="font-size: large;">N = N + 0 = N + (-M) + (+M), to obtain</span></li><li><span style="font-size: large;">N - (+M) = N + (-M).</span><span style="font-size: large;"> </span></li></ul></div><div><span style="font-size: large;">This shows "subtracting (+M) is the same as adding (-M)". Since we are discussing symmetries it is convenient to use explicit notation for signed values. Interchanging (+M) and (-M) in the above gives</span><br /><br /><ul><li><span style="font-size: large;">N - (-M) = N + (+M), </span><span style="font-size: large;"> </span></li></ul></div><h3><span style="font-size: large;"><span style="font-weight: normal;">the notorious "minus-a-minus is plus-a-plus". Here it is merely one member of a symmetric result: subtracting a positive(negative) is the same as adding a negative(positive), nothing special here. We could have approached this problem as "addition" is the removal of a subtraction operator; there really is nothing special about subtraction (or addition).</span></span></h3><div><span style="font-size: large;">We can approach multiplication/division with the same "removal" approach. Now we start with</span></div><div><ul><li><span style="font-size: large;">N = N x 1 = N x M(^-1) x M(^1),</span></li></ul><div><span style="font-size: large;">and remove the operation "x N(^1)" from it to obtain division:</span></div></div><div><ul><li><span style="font-size: large;">N remove "x M</span><span style="font-size: large;">(^1)" == N / M = N x M(^-1),</span></li></ul><div><span style="font-size: large;">With this we have "N divided by</span><span style="font-size: large;"> M( the inverse of M) is the same as N multiplied by the inverse of M( M)". And again the same result could have started with multiplication as the removal of a division operation.</span></div></div><h3><span style="font-size: large;">Trying out the Two-Number-Line</span></h3><div><span style="font-size: large;">It is always a good idea to confront any </span><span style="font-size: large;">process/procedure/scheme</span></div><div><span style="font-size: large;">you are trying to understand with the simplest problems you can find. I would like to see how the Two-Number-Line provides the values of the addition expressions:</span></div><div><ul><li><span style="font-size: large;">(N + 0) : The relative coordinate system for 0 is the same as the fixed coordinate system for N except its origin is at the far-end of the N-number-segment. The sum can be read off as the coordinate of this origin point.</span></li><li><span style="font-size: large;">((-N) + 0) : The same as above except the sum is on the other side of the origin of the fixed coordinate system.</span></li><li><span style="font-size: large;">(N + (-N)) : In the relative coordinate system for the first problem, the far end of (-N)-number-segment is at the origin of the fixed coordinate system. The sum is the coordinate of this point in the fixed coordinate system, 0. </span></li></ul></div><h3><span style="font-size: large;"><span style="font-weight: normal;">and the multiplication problems:</span></span></h3><div><ul><li><span style="font-size: large;">(N x 1) : In multiplication the relative and fixed coordinate systems have the same origin point. The far-end of the N-number-segment is the far-end of the unit-segment in the relative coordinate system, ie the point at the far-end of the 1-number-segment in the relative coordinate system has the coordinate N in the fixed coordinate system. This is the value of the product </span></li><li><span style="font-size: large;">((-1) x N) : The point at far end of the (-1)-number-segment in the fixed coordinate system is the end of unit-segment of the relative coordinate system. Now the relative coordinate system is oriented oppositely to the fixed one, and the far-end of the N-number-segment is on the opposite side of the origin and has the coordinate (-N) in the fixed coordinate system. </span></li></ul><div><span style="font-size: large;">It is good to know the value of each of these expressions, because we can then concentrate on how that value arises from the appropriate process on the Two-Number-Line( may be called the TNL in what follows). See the immediately prior post for a detailed discussion of the TNL.</span><br /><h3><span style="font-size: large;">The Difference between Knowing the Answer and Understanding the Problem.</span></h3><span style="font-size: large;">We can see a process on the Two-Number-Line(TNL) that produces a representation of the answer. In playing that back in our minds, we are visualizing a process that evaluates arithmetic expressions; we have something to hang an understanding on, a way to see how the mathematics works. For those of us who use mathematics, this is the sort of understanding we need; we have discussed this before. Arithmetic on a TNL is just one way to see arithmetic. Expect to find others or maybe just pieces of others, and to eventually discover that you are working in a web of understanding. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">There is another reason to think about understanding as a necessary part of the foundation that supports learning. From what I can tell it is generally accepted, unquestioned as far as I can tell, that memorization of the facts of small number arithmetic is an absolute requirement for future success. It may be necessary for executing the algorithms taught to do arithmetic, but that is an academic issue. When and if you come to do calculation, my experience is that you will put it all together for your own purposes with no great anxiety about what you have forgotten from elementary school. You may regret not coming to understand the significance of understanding at an earlier stage, I certainly did.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Let me state this baldly, and wait for enlightenment: it is fraud to state that there is only one route to any of the mathematics any of us is likely to use, and pernicious fraud at that. It somehow suggests mathematics is this dainty, fragile flower that drops its pedals if you approach it with anything less than immaculate preparation. Relax. Mathematics is robust. break it into pieces, put pieces together, understand what happens. </span><br /><br /></div><div><span style="font-size: large;">Any structure presented to have single points of failure is scary; it is rational to be anxious if you think success requires you retrieve the one fact of arithmetic or the one piece of nomenclature that can solve that problem. Math anxiety is not pathological.</span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;">Let me back up a bit on one-digit whole-number arithmetic. I do not think you should rely on a calculator for small jobs. That is silly. I do think small number arithmetic should be thought of as part of our everyday language, not something esoteric that exists only for mathematics. A smart phone has a good dictionary, but you do not look up every word. </span><br /><span style="font-size: large;"><br /></span></div></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-79021712720095106392015-11-24T10:47:00.000-08:002015-12-08T11:04:04.373-08:00Geometrical Arithmetic on a Two-Number Line.<h2><span style="font-size: x-large;">A Geometry of Arithmetic</span></h2><span style="font-size: large;">We'll work toward a geometrical description of arithmetic. In the last Post a number-segment on a Straight Line Segment(SLS) represented a number. A <u>coordinate system</u> is <u>located</u></span><span style="font-size: large;">, </span><u style="font-size: x-large;">oriented</u><span style="font-size: large;">, and <u>scaled</u> the segment. Representation of arithmetic on a SLS requires the representation of the second number in the arithmetic expression. How does the number segment for the first segment locate, orient, and scale the number segment for the second number? That requires a second coordinate system, and it is built with the first number's number segment.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This may get a bit tedious, but I have to lay the basis out once and for all to make sure it works. At the end we will have a foundation to build subtraction on addition and division on multiplication. We also will have a strong footing for the physical model we introduced in a previous Post; w</span><span style="font-size: large;">e want a way to "see" arithmetic as an active process. </span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Here is the outline of the geometrical process:</span><br /><ul><li><span style="font-size: large;">Represent one number using a coordinate system picked for its convenience.</span></li><li><span style="font-size: large;">For addition, take the outer end point of this number segment for the origin of a second coordinate system. The scale and orientation of the two coordinate systems are the same. </span></li><li><span style="font-size: large;">For multiplication, </span><span style="font-size: large;">take the outer end point of this number segment for the outer end point of a second coordinate system's unit number-segment . Here the two coordinate systems have the same origin. </span></li><li><span style="font-size: large;">Represent the second number as a number-segment with the second coordinate system.</span></li><li><span style="font-size: large;">Read out the coordinate of the outer end of this number-segment. It is the value of the sum or product.</span></li></ul><span style="font-size: large;">What is traditionally executed as a numerical calculation is now a geometrical process in a one-dimensional space. It produces a numerical result from a geometrical process. Although somewhat disguised, this approach takes the number line apart to get at the tool used to construct it: a coordinate system. The geometric process then attaches a second coordinate system to the SLS. The result is then retrieved from this Two-Number-Line. </span><br /><br /><span style="font-size: large;">For addition our approach superficially looks like the usual one on a number line, but ours is a unified treatment of addition and multiplication that follows from the two possible ways to represent the combination of two numbers in a superposition of two number systems on one line. I would argue it is the inevitable extension of the ideas underlying the traditional number line to incorporate both addition and multiplication. It is the representation of </span><br /><br /><span style="font-size: large;">A more intuitive interpretation sees arithmetic as taking the representation of one number to make another representation of the other number. It is arithmetic on a Two-Number-Line.</span><br /><br /><span style="font-size: large;">There are many representations of addition and multiplication on the Number-Line. In all of them that I know about, multiplication relies on a representation of addition. This suggests addition is somehow intrinsic to multiplication, confusing what multiplication does how computational algorithms work. It is a confusion of mechanism with the meaning and purpose.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">On the other hand, on a Two-Number-Line, addition and multiplication are independent variants of one process; and together they exhaust the possibilities of that process; they neither overlap each other, nor do they leave room for any other independent variant in the process: it is not an accident that we learn two arithmetic operations, and that those operations are addition and multiplication. Arguably, once we want the one number that represents the possible results of the commutative interaction of two, it is inevitable. And finally it is easy to draw a picture of the process, and even to make a physical model of it. </span><br /><h3><span style="font-size: large;">Notation</span></h3><div><span style="font-size: large;">From the Post <i>Semantics of Elementary Arithmetic Expressions</i></span></div><div><span style="font-size: large;"><i><br /></i></span></div><div><span style="font-size: large;">Addition: </span></div><div><span style="font-size: large;">(to-this-Operand + add-this-Operand = Sum =</span></div><div><span style="font-size: large;">(to-this-N1 + add-this-N1) x common-unit =</span><br /><span style="font-size: large;">N3 x common-unit.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Multiplication: </span></div><div><span style="font-size: large;">(by-this-Operand) x (multiply-this-Operand) = </span><span style="font-size: large;">Product = </span><span style="font-size: large;"> </span></div><div><span style="font-size: large;">(by-this-N1 x multiply-this-N2) x product-unit = N3 x product-unit.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">N1 and N2 are number operands.</span><br /><br /><ul><li><span style="font-size: large;">For addition N3 is the number result of evaluating the addition expression (to-this-N1 + add-this-N2).</span></li><li><span style="font-size: large;">For multiplication N3 is the result of evaluating the number expressions: </span><span style="font-size: large;">(by-this-N1 x multiply-this-N2)</span><span style="font-size: large;"> .</span></li></ul></div><div><span style="font-size: large;">In short we are now addressing the number side of the problem.</span><br /><span style="font-size: large;">We are going to represent a number by a segment. For this we have to set a relative scale, the ratio of the length imposed on the Straight-Line-Space SLS) by coordinate system to the common-unit for addition or the product scale for multiplication. We shall assume it is 1; e.g. 1 inch on the SLS corresponds to 1 square foot, and etc.</span><br /><h4><span style="font-size: large;">Cartesian Coordinate System</span></h4></div><div><span style="font-size: large;">For <span style="color: red;">Co</span> a Cartesian Coordinate System(CCS), with <span style="color: red;">N</span> the coordinate of <b>P</b> on a Straight Line Space <b>Sp</b>:</span><span style="font-size: large;"> </span></div><div><br /><ul><li><span style="font-size: large;"><span style="color: red;">Co(</span><b>P</b><span style="color: red;">)</span> = <span style="color: red;">N</span>, </span></li><li><span style="font-size: large;"><span style="color: red;">invCo(N)</span> = P .</span></li></ul><span style="font-size: large;">That's it for what a coordinate system does.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">For a CCS the points on <b>Sp</b> whose coordinates differ by a fixed amount are evenly spaced. </span><br /><br /></div><div><span style="font-size: large;"><span style="color: red;">Co</span> is fixed on </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;"> </span><span style="font-size: large;">by two points: </span><b style="font-size: x-large;">P0</b><span style="font-size: large;"> and </span><b style="font-size: x-large;">P1 </b><span style="font-size: large;">with coordinates </span><span style="color: red; font-size: large;">0</span><span style="font-size: large;"> and </span><span style="color: red; font-size: large;">1</span><span style="font-size: large;"> respectively.</span><br /><ul><li><span style="font-size: large;"><b>P0</b> </span><span style="font-size: large;">is <span style="color: red;">C</span>'s origin,</span></li><li><span style="font-size: large;">One of: (a) <b>P1</b> is to the right of <b>P0</b></span><span style="font-size: large;">, the points to the right of the origin have positive coordinates, those to left, negative. (b) <b>P1</b> </span><span style="font-size: large;">is to the left of <b>P0</b></span><span style="font-size: large;">, it is oppositely oriented.</span></li></ul><ul></ul><ul></ul></div><div><span style="font-size: large;">The distance from <b>P0</b> to <b>P1</b> is the scale of the coordinate system: e.g if <b>P1</b> is one centimeter from the origin then any two points separated by one centimeter have coordinates that differ by <span style="color: red;">1</span>.</span><br /><h4><span style="font-size: large;">A Number-Segment</span></h4></div><div><span style="font-size: large;">A representation of a number, N1, on <b>Sp</b>, a SLS, is a geometrical figure that relates properties of the <b>Sp</b> t</span><span style="font-size: large;">o the number. With </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;"> a CCS on </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;">, the number can be represented as the geometrical figure: the line segment on </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;"> with its inner end at the origin P</span><b style="font-size: x-large;">0</b><span style="font-size: large;"> of </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;">, and its outer end at the point </span><b style="font-size: x-large;">PN1</b><span style="font-size: large;"> on </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;"> whose coordinate in </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;"> is </span><span style="color: red; font-size: large;">N1</span><span style="font-size: large;">. Refer to it as a "number-segment" </span><b style="font-size: x-large;"><u>N1</u></b><span style="font-size: large;">:</span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;"> : the representation of the N1 on </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;"> built with </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;">. It relates a number to the oriented distance of a point from the origin of a coordinate system.</span><br /><h3><span style="font-size: large;">The Geometrical Process</span></h3></div><div><span style="font-size: large;">Three steps are needed in the process to evaluate (N1 op N2) for op one of "+" an "x". The first and last steps are common, the second is the isolated step particular to op.</span></div><div><span style="font-size: large;"> </span></div><div><span style="font-size: large;">1) For one of the two numerical operands, say N1, with a preexisting coordinate system </span><span style="color: red; font-size: large;">C0</span><span style="color: red; font-size: large;"> </span><span style="color: red; font-size: large;">and </span><span style="font-size: large;">the coordinate <b><span style="color: red;">N1</span></b> = N1, construct the number-segment </span><b style="font-size: x-large;"><u>N1</u></b><span style="font-size: large;">:</span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;"> on a SLS, </span><b style="font-size: x-large;">Sp.</b><span style="font-size: large;"> Its outer end is the point <b>PN1</b></span><span style="font-size: large;"> in </span><b style="font-size: x-large;">Sp</b><span style="font-size: large;">.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">2) One variant for this step is for addition; the other, for multiplication. In both cases it produces a point <b>PN2</b> at the outer end of a number-segment <b><u>N2</u></b>:<span style="color: red;">COPo</span>, This number-segment is constructed with a coordinate system <span style="color: red;">COPo</span> particular to the operator op.</span></div><div><span style="font-size: large;"><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><span style="font-size: large;">3) The coordinate of <span style="color: red;">N2</span> of the point <b>PN2</b> in <span style="color: red;">Co</span> is the Sum or Product N3 = <span style="color: red;">N2</span>.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Step 2 for addition.</span><br /><ul><li><span style="font-size: large;">construct the coordinate system </span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;"> for </span><span style="color: red; font-size: large;">N2</span><span style="font-size: large;"> = N2 as</span><span style="font-size: large;"> the translation of </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;">;</span><span style="font-size: large;"> ie, the origin </span><b style="font-size: x-large;"><u>0</u></b><span style="font-size: large;">:</span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;"> = </span><b style="font-size: x-large;">PN1 </b><span style="font-size: large;">with the </span><span style="font-size: large;">scale and orientation unchanged.</span></li><li><b style="font-size: x-large;">PN2</b><span style="font-size: large;"> is the outer end of the number-segment <b><u>N2</u></b>:<span style="color: red;">COPo</span>.</span></li></ul><div><span style="font-size: large;"><br /></span></div><span style="font-size: large;">Step 2 for Multiplication.</span><br /><ul><li><span style="font-size: large;">the origin of </span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;"> and </span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;"> are the same point </span><b style="font-size: x-large; text-decoration: underline;">P0</b><span style="font-size: large;">.</span></li><li><span style="font-size: large;">the scale and orientation are fixed by N1 as </span><b style="font-size: x-large;"><u>1</u></b><span style="font-size: large;">:</span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;"> = </span><b style="font-size: x-large;"><u>N1</u></b><span style="font-size: large;">:</span><span style="color: red; font-size: large;">Co</span><span style="font-size: large;">; ie, the scale for </span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;"> is N1 multiplying the scale that N1 was represented by in step 1. The relative orientation is determined by the sgn of N2.</span></li><li><b style="font-size: x-large;">PN2</b><span style="font-size: large;"> is the outer end of the number-segment <b><u>N2</u></b>:<span style="color: red;">COPo</span>.</span></li></ul><h3><span style="font-size: large;">Wrap Up </span></h3><div><span style="font-size: large;">It all worked out pretty much as advertised. Now that we have the notation, we can show another way addition and multiplication diametrically differ from one another.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">For addition, the relation of the coordinate systems is a difference:</span></div><div><ul><li><span style="color: red; font-size: large;">Co(</span><b style="font-size: x-large;">PN1</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> - </span><span style="color: red; font-size: large;">COPo(</span><span style="font-size: large;"><b>PN1</b></span><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> = </span><span style="color: red; font-size: large;">N1</span><span style="font-size: large;"> = N1 , <b>PN1</b> is the origin of <span style="color: red;">COPo</span></span></li><li><span style="font-size: large;"><span style="color: red;">C</span></span><span style="color: red; font-size: large;">o(</span><b style="font-size: x-large;">P</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> - </span><span style="color: red; font-size: large;">COPo(</span><span style="font-size: large;"><b>P</b></span><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> = </span><span style="font-size: large;"> N1 , for all <b>P</b> , <= </span><span style="font-size: large;">Cartesian CS </span></li><li><span style="font-size: large;"><b><span style="color: red; font-weight: normal;">Co(</span><b>PN2</b><span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> - </span><span style="color: red; font-weight: normal;">COPo(</span><span style="font-weight: normal;"><b>PN2</b></span><span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> = </span><span style="color: red; font-weight: normal;">Co(</span>PN2<span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> - N2 = N1 ,</span></b></span></li><li><b style="font-size: x-large;"><span style="color: red; font-weight: normal;">Co(</span>PN2<span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> = N1 + N2 .</span></b></li></ul><div><span style="font-size: large;"><br /></span></div><span style="font-size: large;">For multiplication, the relation is as a ratio:</span></div><div><ul><li><span style="color: red; font-size: large;">Co(</span><b style="font-size: x-large;">PN1</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> / </span><span style="color: red; font-size: large;">COPo(</span><b style="font-size: x-large;">PN1</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> = </span><span style="color: red; font-size: large;">N1</span><span style="font-size: large;"> = N1 , </span><span style="color: red; font-size: large;">COPo(</span><b style="font-size: x-large;">PN1</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> = <span style="color: red;">1</span> ,</span></li><li><span style="font-size: large;"><span style="color: red;">C</span></span><span style="color: red; font-size: large;">o(</span><b style="font-size: x-large;">P</b><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> / </span><span style="color: red; font-size: large;">COPo(</span><span style="font-size: large;"><b>P</b></span><span style="color: red; font-size: large;">)</span><span style="font-size: large;"> = </span><span style="font-size: large;"> N1 , for all <b>P</b> , <= </span><span style="font-size: large;">Cartesian CS ,</span></li><li><b style="font-size: x-large;"><span style="color: red; font-weight: normal;">Co(</span><b>PN2</b><span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> / </span><span style="color: red; font-weight: normal;">COPo(</span><span style="font-weight: normal;"><b>PN2</b></span><span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> = </span><span style="color: red; font-weight: normal;">Co(</span>PN2<span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> / N2 = N1 ,</span></b></li><li><b style="font-size: x-large;"><span style="font-weight: normal;"><b><span style="color: red; font-weight: normal;">Co(</span>PN2<span style="color: red; font-weight: normal;">)</span><span style="font-weight: normal;"> = N1 x N2 .</span></b></span></b></li></ul><div><span style="font-size: large;">Difference and ratio are the two ways numbers are compared in elementary education. One provides the additive comparison and the other is the multiplicative comparison. These are the two ways coordinates are compared in the Two-Number Line. Further,t</span><span style="font-size: large;">hey reflect the two ways two Cartesian Coordinate Systems relate the points that define them.</span></div></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">For addition:</span></div><div><ul><li><b style="font-size: x-large;">PN1</b><span style="font-size: large;">, the outer end of the number segment for N1 in the preexisting coordinate system <span style="color: red;">Co</span>, is the origin of coordinate system <span style="color: red;">COPo</span> used to construct N2's number-segment.</span></li></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-snvxfuWX8J8/VmW52zxNVWI/AAAAAAAAAxE/54Vj0bCo48Y/s1600/IMG_0311.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="425" src="http://2.bp.blogspot.com/-snvxfuWX8J8/VmW52zxNVWI/AAAAAAAAAxE/54Vj0bCo48Y/s640/IMG_0311.JPG" width="640" /></a></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">For Multiplication:</span></div><div><ul><li><span style="font-size: large;">The origin points of Co and COPo coincide.</span></li><li><span style="font-size: large;">Now </span><b style="font-size: x-large;">PN1</b><span style="font-size: large;">, the outer end of the number segment for N1 in the preexisting coordinate system <span style="color: red;">Co</span>, is the outer end of the unit number-segment <u><b>1</b></u> in <span style="color: red;">COPo</span>. This fixes both the scale and the orientation of </span><span style="color: red; font-size: large;">COPo</span><span style="font-size: large;">. </span></li></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-alz_FRLidW8/VmW960qPdDI/AAAAAAAAAxY/18cPGEZrmpw/s1600/IMG_0312.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="426" src="http://4.bp.blogspot.com/-alz_FRLidW8/VmW960qPdDI/AAAAAAAAAxY/18cPGEZrmpw/s640/IMG_0312.JPG" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">A coordinate system is determined by its origin, its orientation, and its scale. These three possibilities are exhausted by addition and multiplication. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">We conclude:</span></div><div><ul><li><span style="font-size: large;">two-operand operations can be represented on the Two-Number Line</span></li><li><span style="font-size: large;">the two two-operand, operations, addition and multiplication, exhaust the possibilities;</span></li><li><span style="font-size: large;">addition and multiplication are independent of one another; ie. either one can be defined independently of the other.</span></li><li><span style="font-size: large;">subtraction and division can respectively be defined in terms of addition and multiplication. </span></li></ul></div></div><div><span style="font-size: large;">It is not an accident that addition and multiplication, along with their variants subtraction and division, command so much attention. Understand them. </span></div></div></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-27102019895098334082015-11-17T17:25:00.001-08:002015-11-24T10:36:24.303-08:00 What Does Arithmetic Do?<blockquote class="tr_bq"><blockquote class="tr_bq"><span style="font-size: x-large;">One Way to See What Addition and Multiplication Do</span><span style="font-size: large;">, </span><span style="font-size: x-large;">and </span><span style="font-size: x-large;">Why that is Important </span></blockquote></blockquote><span style="font-size: large;">Early education teaches you to do arithmetic. One way or another, traditional rote or Common Core understanding, the emphasis is calculation by you. In real (or word) problem solving, before you can calculate, you must first construct/design a mathematical expression (call it a formula ) whose value, obtained by calculation, is what you want. This construction/design is the hard part; you must understand what the arithmetic does before you can use it in a formula. That understanding is the part of problem solving left for us; calculation is more a craft, and like many traditional crafts is now largely mechanized; there really has been a sea change.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This Post develops a representation of arithmetic on a Straight Line Space(SLS). It presents the mathematical model; an earlier Post initiated a discussion of a concrete model that uses Tape Measures to insert and read number segments off a "slotted-slat arithmetic platform" that models the SLS. </span><span style="font-size: large;">[ Refer to the Post<i> Number Lines, Straight Line Spaces, and Coordinate Systems</i> for a discussion of the SLS and a CCS. ] </span><span style="font-size: large;">We will return to the concrete model in a later Post.</span><br /><span style="font-size: large;">Our motto might be: </span><i style="font-size: x-large;">See arithmetic happen - Understand what arithmetic does</i><br /><br /><span style="font-size: large;">Of potentially greater interest is an attempt to separate numerical and symbolic components in the expression of a problem. It is just possible that the inability to separate them is the major barrier to problem solving. For a real problem the symbolic issues come before the calculational, in early education it is the other way around; looks to me like a recipe for immense frustration. It is not a question of checking or labeling the results of a numerical calculation; it points the way to expression of the problem, to the formula that expresses the problem. We will be brought back to all this, but first lets try to make a geometric approach to calculation show us how numerical addition and multiplication work.</span><br /><h3><span style="font-size: large;">How to Make a Number You Can See - Make It Big.</span></h3><span style="font-size: large;">At bottom, a number is a (measure of a) deviation from zero. In a SLS the deviation from zero can be represented as the directed segment from the origin to the point whose coordinate is the number. </span><br /><h4><span style="font-size: large;">Construction of a Coordinate System on the SLS</span></h4><div><span style="font-size: large;">A number is represented by a special segment in an SLS. For the number N, the number-segment is denoted by </span><u style="font-size: x-large;">N</u><span style="font-size: large;">. Construction of that representation requires a Cartesian Coordinate System(CCS), call it C, on the SLS;</span></div><div><ul><li><span style="font-size: large;">C locates any point on the SLS with a coordinate-number address, and </span></li><li><span style="font-size: large;">for any point in the SLS, C provides a coordinate. </span></li></ul></div><div><span style="font-size: large;">One end of a number-segment is at the point whose coordinate is zero: the origin. Zero is a number, it is represented by a zero-length number-segment, C:</span><u style="font-size: x-large;">0</u><span style="font-size: large;">, both of whose end points have the coordinate zero: C:<u>0</u> is both a point and a number-segment. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">For C a Cartesian coordinate system, equally spaced points have equally spaced coordinates, and it is only necessary to specify one segment, say the unit number-segment C:<u>1</u> with near end at the origin and far end at the point whose coordinate is 1. For C a CCS </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">C:<u>0</u> = {C:<u>0</u> , C:<u>0</u>} ,</span></div><div><span style="font-size: large;">C:<u>1</u> = {C:<u>0</u> , invC(1)} , </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">[ notation for C a particular CCS:</span><br /><br /><ul><li><span style="font-size: large;"> invC(N) takes the coordinate N of a point P and points at P.</span></li><li><span style="font-size: large;">C(P) works the other way around, shown the point P in the SLS, it provides its coordinate. </span></li></ul><br /><span style="font-size: large;">This is all coordinate systems do.]</span><br /><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">For C a particular CCS:</span><br /><br /><ul><li><span style="font-size: large;">positive coordinates locate points on the C:</span><u style="font-size: x-large;">1</u><span style="font-size: large;"> side of </span><span style="font-size: large;">the</span></li><li><span style="font-size: large;">origin at C:</span><u style="font-size: x-large;">0</u><span style="font-size: large;">. In addition to providing the orientation of C</span></li><li><span style="font-size: large;">C:</span><u style="font-size: x-large;">1</u><span style="font-size: large;"> is the scale of a number-segment because its length is the "unit" in which the length of any segment is expressed; ie, it relates a difference in coordinates to a length in the SLS.</span></li></ul><br /><span style="font-size: large;">A drawing to illustrate the relation of a Cartesian coordinate system:</span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/--SxXIKxukqc/VkyJxDUQtvI/AAAAAAAAAuc/uiIRmyAosq4/s1600/IMG_0296.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="213" src="http://3.bp.blogspot.com/--SxXIKxukqc/VkyJxDUQtvI/AAAAAAAAAuc/uiIRmyAosq4/s320/IMG_0296.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Black line is the SLS. Red line is the coordinate syustem</td></tr></tbody></table><span style="font-size: large;">This is a good time for me to vent on a problem I have with the name commonly given to a plane with a coordinate system on it. A "coordinate plane" seems to imply a plane with a coordinate system defined on it is a plane with numbers in it. It is not. The numbers in a picture of a plane are, for a particular coordinate system, coordinates of the points that compose the plane, they are not part of the plane. A coordinate system on a plane; Cartesian, polar, or whatever, is a construct built on the plane, not a property of it; a plane, or line, exists independently any coordinate system(s) placed on it, and more than one coordinate system can be placed on a plane, line, or etc. </span></div><div><h4><span style="font-size: large;">Construction of a Number Segment in an SLS</span></h4></div><span style="font-size: large;">The representation of a number N on the SLS is a number-segment located, oriented, and scaled by a coordinate system C:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">C:<u>N</u> def<u>n</u> {<u>0</u> , C:invCCS(N)} , [ "def<u>n</u>" says that what follows is the definition of what preceded ] </span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">is a segment in the SLS extending from the origin to the point whose coordinate is N. The number makes a representation of itself in the SLS using a coordinate system on the SLS.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The point whose coordinate is N in the coordinate system C is defined by </span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><span style="font-size: large;">C:<u style="color: red;">N</u> def<u>n</u> </span>C:invCCS(N) .</span><br /><br /><span style="font-size: large;">"You" made the coordinate system C, that is what you do. The number made the segment that represents itself using your coordinate system (or, as it will turn, out one made made for it within some process), that is what a number does on a SLS. The figure below shows number segments for positive and negative numbers.</span><br /><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-TX38N8oBi4o/Vkye3bZIljI/AAAAAAAAAu0/KXs22A9o9a0/s1600/IMG_0297.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://2.bp.blogspot.com/-TX38N8oBi4o/Vkye3bZIljI/AAAAAAAAAu0/KXs22A9o9a0/s320/IMG_0297.JPG" width="213" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Positive Number and Negative Number Segments.<br />The coordinate system is oriented with positive coordinates<br />to the right.<br /><br /></td></tr></tbody></table><span style="font-size: large;">We now have constructed a representation of a number as a realized segment on a SLS. First "you" built a coordinate system using your instructions that takes coordinates provided to it to locate points in the SLS. A number, in arithmetic an operand, you provide then represents itself on the SLS as a segment anchored at the origin, with a length equal to its magnitude using the scale the coordinate system provides, along with an orientation that reflects the sign of the number.</span><br /><span style="font-size: large;"><br /></span><br /><span style="color: red; font-size: large;">An active view could picture the number as an entity carrying its identity as a signed number, emerging into the SLS from a hole at the origin, looking in each direction along the SLS for the direction a number with the sign of its ID number should go; then, pulling its tail along behind it, proceeding in that direction until the coordinate of the point on the SLS at the tip of its nose is its ID number. This picture will carry over to the two active numerical operands in arithmetic; the action of arithmetic is their action on one SLS, guided by the different views of the world in their own coordinate system. </span><br /><br /><h3><span style="font-size: large;">Construction of Arithmetic in an SLS</span></h3><div><span style="font-size: large;">[ from the Post <i>Semantics of Elementary Arithmetic Expressions</i> ]</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Our format for arithmetic expressions:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><u>addition</u></span><br /><span style="font-size: large;">(to-this Operand + add-this Operand) = Sum</span><br /><span style="font-size: large;">(OP1 + OP2) = (N + M) where OP1 is one of N or M, OP2 the other .</span><br /><span style="font-size: large;"><u>multiplication</u></span><br /><span style="font-size: large;">(by-this Operand x multiply-this Operand) = Product</span><br /><span style="font-size: large;">(OP1 x OP2) = (N x M) where OP1 is one of N or M , OP2 the other.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">In the discussion that follows we use the above convention and nomenclature for an arithmetic expression, an Operand, and a Sum or Product.</span><br /><br /><ul><li><span style="font-size: large;">"to-this", "add-this", "by-this", and "multiply-this" are adjectives modifying operands.</span></li><li><span style="font-size: large;"><u>Any calculation starts by picking OP1.</u></span></li><li><u><span style="font-size: large;">(N + M) is the Sum,</span><span style="font-size: large;"> </span><span style="font-size: large;">(N x M) is the Product.</span></u></li></ul><h4><span style="font-size: large;">The Unit in Evaluation of Addition and Multiplication Expressions</span></h4><span style="font-size: large;">In addition each of the operands and the sum have the same unit: each's unit is one of the same thing, or at least one of the thing being added is the same:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">(OP1 + OP2) = Sum = </span><br /><span style="font-size: large;">(N1 + N2) x common-unit = N3 x </span><span style="font-size: large;">common-unit</span><br /><span style="font-size: large;">The unit common-unit is common to each term and hence</span><br /><span style="font-size: large;">The addition, (N1 + N2), to produce N3 is a numerical </span><span style="font-size: large;">calculation, but the meaning of (OP1 + OP2) includes the way the "unit" works it way through its evaluation; in short, the meaning of the evaluation is more than just the result of a numerical calculation. Following the units through an adition evaluation provides the semantics, the meaning, of the evaluation. Consider an addition problem: </span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">If there is 1fruit-per-apple and 1fruit-per-orange, how many fruits are there in 6apples and 3oranges? </span><br /><span style="font-size: large;">Since </span><br /><span style="font-size: large;">1fruit = (1fruit/1apple) x 1apple = (1fruit/1orange) x 1orange = 1fruit,</span><br /><span style="font-size: large;">and</span><br /><span style="font-size: large;">6apples =</span><span style="font-size: large;"> 6 x 1apple, and 3ornages = 3 x 1orange :</span><br /><span style="font-size: large;">(1fruit/1apple) x 1apple x 6 + (1fruit/1orange) x 1orange x 3 =</span><br /><span style="font-size: large;">6 x 1fruit + 3 x 1fruit = </span><span style="font-size: large;"> (6 + 3) x 1fruit = 9fruit</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Of course this is overly elaborate, but it is just a careful working out of the implications of the simple formule</span><br /><span style="font-size: large;">1fruit = (1fruit/1apple) x 1apple, etc. </span><br /><span style="font-size: large;">for the relation between an 1apple and 1fruit. The lesson: to express the meaning of a problem is to understand the meanings of the problem's components and the relations among the meanings. Numerical calculation, itself, is a separate issue and generally a simpler one. </span><br /><br /><span style="font-size: large;">In multiplication each operand and the product are generally different from one another: each's unit differs from that of the others; e.g. (speed x time = distance), or (L x W) = A where each of L and W may have a length unit as foot, but for a rectangle they denote lengths along different perpendicular lines, and A has has its own unit "1foot x foot" pronounced as "foot squared" or "square foot". In either case; however, the unit of (Operand1 x Operand2) is the same as that for Product:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">(OP1 x OP2) = Product =</span><br /><span style="font-size: large;">(N1 x unit-O1) x (N2 x unit-O2) = N3 x unit-Product =</span><br /><span style="font-size: large;">(N1 x N2) x (unit-O1 x unit-O2) = N3 x unit-Product where:</span><br /><span style="font-size: large;">(unit-O1 x unit-O2) = unit-Product; ie.</span><br /><span style="font-size: large;">the unit of a Product is the product of the units of its Operands. We now have</span><br /><span style="font-size: large;">(N1 X N2) x unit-Product = N3 x unit-product, or finally</span><br /><span style="font-size: large;"><br /></span><span style="color: magenta; font-size: large;">(N1 x N2) = N3 and </span><br /><span style="color: magenta; font-size: large;">(unit-OP1 x unit-OP2) = unit-Product</span><br /><span style="color: magenta; font-size: large;">for</span><br /><span style="color: magenta; font-size: large;">OP1 = N1 x unit-OP1 and etc.</span><br /><span style="color: magenta; font-size: large;"><br /></span><span style="font-size: large;"><span style="color: magenta;">We have separated a single equation with numbers and units entangled into two equations that isolate numerical and symbolic issues. The symbolic equation contains the meaning of the problem; it leads to a meaningful mathematical expression of it. I think this separation is the key to a principled approach to problem solving. At the level of mathematics in elementary education it looks easier to me than calculation.</span></span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">For the distance evaluation we could encounter 1mile-per-hour for the unit of speed which multiplies 1hour for the unit of time to make the product 1mile for a unit of distance. The symbolic equation is</span><br /><span style="font-size: large;">(1mile/hour) x 1hour = 1mile. </span><br /><br /><h3><span style="font-size: large;">Wrapup</span></h3><div><span style="font-size: large;">This is getting too long. Representation of a number as a number-segment is the foundation of a geometrical calculation, and I set out to develop that</span><span style="font-size: large;">, but I let myself get diverted by the separation of a problem into numerical and symbolic parts</span><span style="font-size: large;">. Then and now that seemed more interesting. Geometric calculation will be in the next Post.</span></div><span style="font-size: large;"><br /></span><span style="font-size: large;"> </span><br /><h4></h4></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-5179942809843672112015-11-10T12:50:00.001-08:002015-11-14T06:26:26.311-08:00Why Make Arithmetic Real<h2>Why Look for Active Arithmetic?</h2><div><span style="font-size: large;">Mathematics is powerful. It gets at the core of the matter by removing as many of the particulars as possible in order to get at bare essentials. Even at the elementary level, the resulting abstraction is embraced; those whole numbers stand alone awaiting their interaction to produce an inevitable result that has long ago been abstracted from what it described. Alone, without further meaning, an arithmetic result holds for everything it has been abstracted from, and that is its power; we can reliably use the facts of arithmetic because they abstract facts of the world. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">We all stand on the shoulders of giants who themselves stood on the shoulders of giants. We are a long way from the ground, and that is where the problems that confront us are, even those pesky word problems we remember from elementary school. How do we help students climb down from mathematics to to level of real problems?</span><br /><span style="font-size: large;"><br /></span></div><div></div><div><span style="font-size: large;">A young student's goal is not the ability to do mathematics, it is rather to have the ability to use it; not to create abstractions for the advance of mathematics, but to relate an understanding of mathematics to problems in their world.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">In first becoming competent in arithmetic, students learn how to evaluate mathematical expressions before they know where they could have come from, more significantly, before they know how they they relate to the real world? They can do arithmetic problem that stands in for a large class of real world problems before they can solve any one those. Eventually they must "de-abstract" the mathematics, get at particulars, and solve the one problem sitting before them</span></div><h3><span style="font-size: large;">Make it Real. Real is Active</span></h3><div><span style="font-size: large;">At least that is how I remember reality in my youth, the fun and useful parts of it anyway. This is not the way I remember my early formal education. I do remember beginning to see reality in undergraduate mathematics and science. I also remember the stark realization in my freshman year that I just did not get the relation of mathematics to real problems. Up to that point I could more or less fake it since the problems appeared next to the presentation of the mathematics used to solve them: extract the numbers, plug them in, and turn the crank. Although </span><span style="font-size: large;">I worked on and around lots of farm machinery, </span><span style="font-size: large;">I never made the connection between it and problem solving. I did not see the corruption inherent in my faked learning; I pretended to learn and school pretended to teach something useful.</span><span style="font-size: large;"> </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Most of my early formal education was self referential: it taught what you needed at its next stage; and you could never see outside of it. Problem solving was in the service of learning academic mathematics. Real problem solving, even for "real" problems in an academic setting, was never addressed. </span></div><div><br /></div><div><span style="font-size: large;">Have things improved, or has the current testing regime made them worse? I do not see how it could make early education less inward looking. My fear is that school education has chosen rigor to aid in the prediction of results it itself determines. How much time does it spend confronting problems as problems where you have to figure out how to understand because you can not fall back on what you have done before? </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Whatever it is, if it is does not work, little room has been left for students to learn and create on their own. Creativity has historically been our strength; I do not think we ever knew if it resulted from the time schools took from children, or from the time it left to them.</span></div><div><br /></div><div><span style="font-size: large;">Lest I give the wrong impression: I think I learned a lot in my childhood. My early formal education was not pretentious; it did not act like it was the only way to learn and therefore should occupy most of my time. I had time to work, play, and read on my own, and my teachers gave me most of what I ultimately needed from them. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Who knows, perhaps it was just that there was no TV in my home town until I was in the sixth grade. Or maybe it was because my mother had been a teacher, and my father read in much of his free time. I do know my parents thought learning was important. They did not think sports were, or at least they did not do much to encourage my participation. They did let me solve many of my own problems.</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">I think I could ultimately relate mathematics to real problems because I had the opportunity to live where problems were active, where imagination could play on an active stage based on the real world. I am not sure that world is still commonly available. I want to present some results of thinking about making arithmetic active, to render active something all students now experience. Failure is an option. </span></div><div><br /></div><div><br /></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"> </span><span style="font-size: large;"> </span></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-59800232284387889262015-11-03T08:34:00.003-08:002015-11-11T08:06:45.388-08:00Number Lines, Straight Line Spaces, and Coordinate Systems<div class="separator" style="clear: both; text-align: center;"></div><h2></h2><span style="font-size: large;">A Number Line is widely used to represent a dense number system as an image of uniformly spaced numbers at tick marks on a straight line, think of a long ruler. Students are invited to think of the numbers as operands participating in arithmetic operations related to the geometry of the line, an analytic geometry of the straight line figure.</span><br /><br /><span style="font-size: large;">I want to do arithmetic, both addition-subtraction and multiplication-division, by arrangement and measurement of two line segments. The location, orientation, and scale of the second operand's segment can differ from those for a (it's our choice) first operand. The difference is determined by the segment for the first operand whose location, orientation, and scale are ours to fix. For addition-subtraction the two segments are constructed on the same Straight Line Space (SLS). For multiplication-division they can be in different SLSs.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">In what follows it is necessary to distinguish numbers-as-coordinates from numbers-as-operands, from numbers as results. ( see page 92 in <i>Adding it Up, </i>National Academy of Sciences, 2001 for a different approach to addition on the Number Line where this distinction is made. I got much of my understanding of arithmetic on the number line from <i>Understanding Numbers in Elementary School Mathematics </i>by Hung-Hsi Wu, 2010). In what follows I will try to to refer to numbers as one of: "operands", "coordinates", or "results" ( or sums or products, ideas for something better that "results"?).</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">I like to think of a SLS as (an idealization as of) a structure in the world, waiting there for our investigation of it with tools like coordinate systems. Of course the power of our tools increases together with our understanding of the SLS. Children can see the mathematics they develop helping them understand their world; and, conversely, come to a deeper understanding of mathematics by seeing it act in the world. </span><br /><h3><span style="font-size: large;">Straight Line Space:</span></h3><div><span style="font-size: large;">a collection of points densely arranged in an otherwise structureless</span></div><div><span style="font-size: large;">straight line that extends forever in both directions. No point can come to know where it is, and there is no way to distinguish one point from another.</span></div><h3><span style="font-size: large;">Cartesian Coordinate Systems</span></h3><div><span style="font-size: large;">A coordinate system names and locates any point by naming it with its address expressed as a coordinate (one number for a one dimensional line).</span></div><div><span style="font-size: large;">A Cartesian Coordinate System(CCS) for the SLS is the entire collection of ordered, uniformly spaced pairs of coordinate, tick-marks. It performs two functions:</span><br /><br /><i style="font-size: x-large;"></i><br /><ul><i style="font-size: x-large;"><li><i>You put the tick-mark at the point that you chose on the SLS, the CCS provides its associated coordinate.</i></li><li><i>You provide the coordinate, the CCS provides the tick-mark pointing at the SLS point whose coordinate it is.</i></li></i></ul><i style="font-size: x-large;"></i></div><div><span style="font-size: large;">Our calculation scheme requires b</span><span style="font-size: large;">oth these CCS capabilities, so does traditional Number Line use.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Three choices are made to construct a coordinate system for the SLS. You or the calculation (but of course ultimately you) choose:</span></div><div><ul><li><span style="font-size: large;"><i>the point on the SLS whose coordinate will be "0", it locates the origin of the CCS at that point</i><i>. This sets the CCS </i><u><i>location</i></u><i>.</i></span></li><li><span style="font-size: large;"><i>the direction away from the origin where points will have "positive" numbers for their coordinates. This sets the CCS </i><u><i>orientation</i></u><i>.</i></span></li><li><span style="font-size: large;">the length of a segment on the SLS, the coordinates of whose end points differ by</span><span style="font-size: large;"><i> "1". It sets the "unit" of the coordinate system. Since the coordinates are uniformly spaced, it is sufficient to specify the unit as the distance from the origin to a point whose coordinate is "1". The unit sets the CCS </i><u><i>scale</i></u><i>.</i></span></li></ul></div><div><ul></ul></div><div><span style="font-size: large;">Having specified a Cartesian coordinate system, any point on the SLS has a unique coordinate; inversely any coordinate designates a unique point on the SLS. The same can be said for any other coordinate system, and there is no restriction on allowed coordinate systems; since the SLS is an infinitely extended space the location of an origin is arbitrary, so is the direction from it of the half-line of points with positive coordinates, and so is the unit of length from this origin. </span><br /><span style="font-size: large;"><br />To reiterate, a different one of the three coordinate system parameters</span><br /><ul><li><span style="font-size: large;">locates the origin of a CCS in front of us, another</span></li><li><span style="font-size: large;">sets a scale to accommodate the range of operand values, that interest us, and the third</span></li><li><span style="font-size: large;">sets a convenient orientation. </span><span style="font-size: large;"> </span></li></ul><br /><h3><span style="font-size: large;">Make it Real</span></h3></div><span style="font-size: large;">We can model a CCS with a tape measure</span><span style="font-size: large;"> and lay out a SLS along a long, narrow, and flat slat. </span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/--953mv6oZbU/VjukscLOzHI/AAAAAAAAAsY/2jWh3QROvFM/s1600/IMG_0260.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://4.bp.blogspot.com/--953mv6oZbU/VjukscLOzHI/AAAAAAAAAsY/2jWh3QROvFM/s320/IMG_0260.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="background-color: #fff2cc; font-size: large;">My slat SLS: 8 feet long, slat top is a 2.5 inches wide surface 26.5 inches off the floor.</span></td></tr></tbody></table><span style="font-size: large;"><br /></span><span style="font-size: large;">The provides a section of an SLS where a tape measure can serve as a CCS. Here addition(multiplication?) will be performed by location and measurement of segments by tape measures on the slat. </span><br /><h4><span style="font-size: large;">Tape measure CCS</span></h4><div><span style="font-size: large;">Uniformly spaced numbers and their associated tick marks are imprinted on the tape, which is single ended and has a zero tab at the end withdrawn from the case. A Tape Measure(TM) is "handed", the commonly available ones are "right-handed":</span></div><div><ul><li><span style="font-size: large;">intended to be worn on the user's right hip, </span></li><li><span style="font-size: large;">retrieved from there with the right hand holding the case, </span></li><li><span style="font-size: large;">left hand pulls the zero tab out to clip it over an edge of the object being measured, </span></li><li><span style="font-size: large;">tape extended by pulling the case with the right hand , and </span></li><li><span style="font-size: large;">opps! you now have your left hand to mark the measured object at the intended location on the tape. </span></li></ul></div><div><span style="font-size: large;">At least you would if your intent was to locate a point on the SLS from its coordinate. Given the limitations of a right handed TM, you might guess someone is making a "left handed" model , and you would be right. Search for one on the internet, and look at the photo:</span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-wV81nIF1lRM/Vju_ic5UL7I/AAAAAAAAAtA/99i8Ud09kPI/s1600/IMG_0258.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://2.bp.blogspot.com/-wV81nIF1lRM/Vju_ic5UL7I/AAAAAAAAAtA/99i8Ud09kPI/s320/IMG_0258.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: large;">Lower TM is right handed: its coordinates are right-side-up when the tape is extended to the right. Upper TM is left handed: its tape is the mirror image of the lower's with right-side-up coordinates when extended to the left.</span></td></tr></tbody></table><br /><ul><li><span style="font-size: large;">The TM on the right(left) provides the coordinate system for points in the SLS to the right(left) of the origin.</span></li><li><span style="font-size: large;">A transverse slit in the slat SLS receives the zero tabs of the left and right located the CCS.</span></li><li><span style="font-size: large;">The numbers are the coordinates for the points their associated tick marks point to.</span></li></ul><span style="font-size: large;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-7nkar6ce7GQ/Vjv-74ffi8I/AAAAAAAAAtU/i6lXs1nEgmQ/s1600/IMG_0275.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="213" src="http://4.bp.blogspot.com/-7nkar6ce7GQ/Vjv-74ffi8I/AAAAAAAAAtU/i6lXs1nEgmQ/s320/IMG_0275.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="background-color: #cccccc;"><span style="font-size: large;"><span style="background-color: #eeeeee;">Numbers, tick marks, and zero tab on a left-handed tape measure. </span></span></span></td></tr></tbody></table></span><br /><span style="font-size: large;">For the TMs shown the scale unit is one inch.</span><br /><h3><span style="font-size: large;">Discussion</span><span style="font-size: large;"> </span></h3><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-VcT2YARaLaY/VjwHWazcQGI/AAAAAAAAAto/5Y5geJ3uIoo/s1600/IMG_0277.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://4.bp.blogspot.com/-VcT2YARaLaY/VjwHWazcQGI/AAAAAAAAAto/5Y5geJ3uIoo/s320/IMG_0277.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="background-color: #fce5cd; font-size: large;">Looking down on the middle of the slat SLS. Postive coordinates to the right, negative to the left. The zero tabs of both TMs mark the origin.</span></td></tr></tbody></table><div><span style="font-size: large;">The figure shows positive coordinates in their traditional position on the right. </span><span style="font-size: large;">The </span><span style="font-size: large;">0 coordinate is neither positive nor negative. </span><span style="font-size: large;">At this stage "positive" and "negative" are just words we use to distinguish coordinates on different sides of the origin. </span><br /><span style="font-size: large;">At this stage they have no other meaning.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The figure can be seen as a number line; or perhaps better as a number line under construction with the numbers and their tick marks not yet imprinted on the line</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The numbers (or is it "numerals" here?) are coordinates. Our discussion of the semantics of addition says their sum should also be a coordinate; but there is no simple sense to be made of that. And what could a product be? It will turn out that sense can be made of arithmetic with segments in different coordinate systems; this will be the topic of another post </span></div><h3><span style="font-size: large;"> </span></h3><span style="font-size: large;"><br /></span></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-5627950391855510642015-10-26T13:10:00.000-07:002015-11-10T12:33:03.422-08:00Semantics of Elementary Arithmetic Expressions<h2><span style="font-size: x-large;">Musing on the Semantics of Arithmetic Expressions</span></h2><div><span style="font-size: large;">What is meant by the arithmetic expressions that we read and write? I certainly do not intend to present anything new, just report some observations made while working out what arithmetic can mean, with an eye toward the possibilities when numbers denote more than just themselves. I want to look at elementary mathematics as an active endeavor where you start by writing an expression for something you want in terms of what you know, and whose evaluation plays as a movie in your mind. </span></div><div><span style="font-size: large;">For addition and multiplication what actions flow out from the form of their </span><span style="font-size: large;">expression? In the infix notation used at this level, expressions are of the form: "(operand operator operand)"; the parentheses are just my notation to improve blog readability. It has long been noted that the semantics are ambiguous: do we read (2 + 3) as instruction to calculate, or is it just another way to write (5). If the meaning is calculation, </span><span style="font-size: large;">we use a process that takes two numbers and produces a third. If the expression is to just to be rewritten, then </span><br /><span style="font-size: large;">we look in a big table </span><span style="font-size: large;">we carry in our heads. T</span><span style="font-size: large;">his is what most of us do for small numbers</span></div><div><span style="font-size: large;">These are real questions when you are trying to instruct a computer; they are also real questions whenever their answer is not clear from the context. It was clear in elementary education before we tested students on material long after it was taught. For small numbers elementary math instruction first teaches a calculation process and later assumes a student has the big look-up table.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">For our purposes, the arithmetic expression shall mean the operator denotes calculation with operands: the expression (2 + 3) means "add 3 to 2 " and produce the sum. How do we know it does not mean "add 2 to 3"? In a sense we do not care, the sum produced by any process that we know about produces the same result, and in fact we could have chosen the reverse order, but the two processes are different: they treat their two operands differently, and we must choose one. This is reflected in our usual interpretation of the expression for addition as "add N2(1) to N1(2)" and not "produce the sum of the two operands"; or for multiplication: "multiply N1 by N2" and not "produce the product of the two operands".</span></div><div><span style="font-size: large;">The classical words for the two operands in both addition and multiplication are different from each other and are suggestive of the calculation process:</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"><i>math.stackexchange.com/.../what-are-the-formal-names...</i></span></div><div><span style="font-size: large;"><i><br /></i></span></div><div><span style="font-size: large;">addition: </span><span style="font-size: large;">Augend<i> + Addend = Sum</i></span></div><div><span style="font-size: large;"><i>subtraction: Minuend - Subtrahend = Difference</i></span></div><div><span style="font-size: large;"><i><br /></i></span></div><div><span style="font-size: large;"><i>multiplication: Multiplicand x Multiplier = Product</i></span></div><div><span style="font-size: large;"><i>Division: Dividend / Divisor = Quotient</i></span></div><div><span style="font-size: large;"><i><br /></i></span></div><div><span style="font-size: large;"><i>...contain a wealth of Latin. If you happen to know Latin, you will understand these meanings more deeply. For example "minuend" comes from a form meaning "about to be lessoned" and "subtrahend" comes from a form meaning "about to be taken away". In general, "-nd" will carry the meaning "about to be --ed".</i></span></div><div><span style="font-size: large;"><i><br /></i></span></div><div><span style="font-size: large;">Interesting! What appear to be exotic labels suggests a practical way to look at the calculation process. I choose to read "Augend" as the operand about to be augmented, with "Addend" about to be the one doing the augmenting. The "Multiplicand" is the operand about to be multiplied, and "Multiplier" is the one that will do the multiplying</span><span style="font-size: large;">. </span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">This is consistent with the operational meaning: choose one operand to be acted on and the other to act on it. For addition one operand plays the passive role, waiting to be augmented by the second operand. In multiplication, again one operand plays the passive role, waiting to be multiplied ( stretched either lengthwise or crosswise in an amount determined) by the second operand. The classical nomenclature suggests actions and actors.</span></div><div><br /></div><div><span style="font-size: large;">Any suggestions for different words for the operands? Surely you can improve these: ("abt" is "about to be", "at" is "about to")</span></div><div><span style="font-size: large;"><br /></span><span style="font-size: large;">1:</span></div><div><span style="font-size: large;">Addition: (atbAdded-to + atAdd) </span></div><div><span style="font-size: large;">Multiplication: (atbMult-ed + atMultiply)</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">or</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">2:</span><br /><span style="font-size: large;">Addition: (atbAdd-edTo + atbAdd-ingTo)</span><br /><span style="font-size: large;">Multiplication: (atbMultipli-ed x atbMultipli-er) .</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">These position-specific "words" for "operand"</span><span style="font-size: large;"> </span><span style="font-size: large;">are all adjectives, each one now modifies an unvoiced "operand"; e.g. "[the] atbAdd-edTo [operand]" in</span><br /><span style="font-size: large;">"(the atbAdd-edTo 2 + the atbAdd-ingTo 3)" for the expression (2 + 3).</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">A more practical notation might forgo the "about to be":</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">3:</span><br /><span style="font-size: large;">Addition: (To 2 + Add 3)</span><br /><span style="font-size: large;">Multiplication: (Mult 2 x By 3)</span><br /><br /><span style="font-size: large;">and this suggests the more elegant</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">4:</span><br /><span style="font-size: large;">Addition: (To 2 , Add 3) means "add 3 to 2", or "with 3 augment 2".</span><br /><span style="font-size: large;">Multiplication: (Mult 2 , By 3) means "multiply 2 by 3", or "with 3 multiply 2".</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">None of these feel quite right; but </span><span style="font-size: large;">the real "problem" is now apparent. "add" and "multiply" are not symmetrical. "Add" refers to what is is going to be added, not to what is about to be added to; "multiply" refers to what is about to be multiplied, not to what it is going to be multiplied by:</span><br /><ul><li><span style="font-size: large;">(2 + 3) means add 3 to 2, </span></li><li><span style="font-size: large;">(2 x 3) means multiply 2 by 3.</span></li></ul><div><span style="font-size: large;">In short, one adds "to", while one multiplies "by".</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Things might be less confusing if we could use symmetric terminology for (operand , operator, operand). For addition:</span><span style="font-size: large;"> </span></div><div><ul><li><span style="font-size: large;">the name of the operator "+" could be "augmentation":</span></li><li><span style="font-size: large;">the left operand is the "augment"-this-operand,</span></li><li><span style="font-size: large;">the right operator is the do-it-"with"-this operand; ie,</span></li></ul></div><div><span style="background-color: #cccccc; font-size: large;">Addition expression: </span></div><div><span style="background-color: #cccccc; font-size: large;">(augment-this-operand + by-this-operand)</span></div><div><span style="background-color: #cccccc; font-size: large;"><br /></span></div><div><span style="background-color: #cccccc; font-size: large;">Multiplication expression: </span></div><div><span style="background-color: #cccccc; font-size: large;">(multiply-this-operand x by-this-operand).</span></div><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">Do you think the order of the operands should be reversed? In the order shown above the operands on the right are those most likely considered prior to those on the right.</span></div><div><br /></div><span style="font-size: large;">For both addition and multiplication there is generally the connotation that after the calculation the left operand has been changed, and the right one did the changing; however, neither the sum nor the product depend on the operand order: addition and multiplication commute. To reduce confusion in subsequent work, however, I will try to keep the changed operand on the left.</span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;">There is a major difference between addition and multiplication: In multiplication we must first consider the changing operand because Multiplication distributes the change it specifies over the entire changed operand while, for addition, the augmentation can be effected by one "attachment". In a sense multiplication is melding; and addition, an add-on. At this juncture all this is premature; perhaps you can see it. It will come up in later posts, and I may go back and change the first post where I think some of it appears more naturally. </span><br /><span style="font-size: large;"><br /></span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;"><br /></span></div><div><br /></div><div><span style="font-size: large;"><i><br /></i></span></div><div></div><div><br /></div><div><br /></div><div><br /></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0tag:blogger.com,1999:blog-3824285401698121426.post-45569088781888340222015-10-20T13:34:00.000-07:002015-11-11T07:16:18.423-08:00Semantics in Elementary Math<h2>Where I'm Coming From</h2><div><ol><li>Have not taught mathematics.</li><li>Did theoretical physics, and taught some.</li><li>Did engineering at Bell Labs. Retired from there.</li><li>Did research in the Dept of Civil and Env Eng, Princeton.</li><li>Started to wonder why most of us eventually have trouble applying the mathematics we learned in elementary and even secondary school; the initial exam results in my first undergraduate physics course indicated I did not not have a clue.</li><li>Read, talked, thought, and wrote over the past year.</li></ol><h2>Where I Am Now</h2><div>I assert: </div><div><ol><li>The mathematics taught in elementary school is both too abstract and too concrete. It is largely based on teaching an arithmetic of abstract numbers which is then used in concrete algorithms whose utility is largely superseded by electronic computation. </li><li>Common Core is more a matter of rationalizing the traditional core, less a matter of getting at a core of problems in it.</li><li>Within the framework of current instruction questions arise that can not be addressed because there is no explicit consideration of the<i> semantics</i> in the mathematics taught; what is the <i>meaning</i> of the numbers and expressions that appear? In the recent discussion of <i>Multiplication is Addition, </i>it was necessary to go way beyond elementary mathematics to argue that multiplication is not addition. Consideration of the semantics of addition and multiplication is a path all of us can follow to understand why it is not a good idea to teach that it is.</li><li>Semantics is the bridge between numerical calculation and word problem solving.</li><li>Those who teach the use of mathematics in subjects like engineering, physics, etc., should be in the discussion. Mathematics is too useful, too meaningful, to leave its explanation entirely in the hands of mathematicians. </li><li>Manipulables in support of an active approach to simple arithmetic can be used to make a concrete semantics. I will discuss this in follow-on blogs.</li></ol><div>My experience suggests it is more effective, and usually more efficient, to assert and apologize than to hedge and explain. Those for whom I am writing know the subject far better that I do; I can only hope you work the tension between cutting me some slack and giving your best shot. I also know I have a responsibility to argue for what I assert. That will follow in this or subsequent blogs.</div></div><div><br /></div><h2>Semantics of Addition and Multiplication</h2><div>Outside the classroom and aside from pure mathematics the semantics of operands and expressions is the primary point of mathematics, or at the least understanding the semantics is the primary value added by us. Computers can take care of both numerical and algebraic calculation; i.e. reducing to a useful value or form the numerical and algebraic expressions that encode our understanding of the semantics of the problem.</div><div>Of course we also must be able to do simple calculation problems. We must learn how to do simple and easy syntax because it can be easier to do it than it is to set it up for a computer to do; computers do the complex or onerous syntax.<br />In any case bringing consideration of meaning into the elementary learning of calculation by addition and multiplication shows how mathematics deals with real problems.</div><h3>Addition</h3><blockquote class="tr_bq">N apples + M oranges = (?). As apples and oranges, a member of the sum collection N+M is not defined; what is an (apple AND orange). If instead of apples and oranges we consider a member of each of the collections N, M, and (N+M) to be an (apple OR orange ), or perhaps just a fruit, then the addition is well defined: (?) is (N+M) fruits.The point is simple: only a common, additive, identifying characteristic that adheres to each element of the operands is summed by addition. Since it is common to all of the terms in the addition expression, it may be factored out and then the numerical expression alone, N+M= (N+M), describes the addition of the N individuals and M individuals. For this reason ignoring semantics in elementary addition education is not greatly problematical. After all, any rewriting of a numerical addition expression as another numerical addition expression does not change the meaning of the terms in the resulting expression. At some level the semantics of any addition problem is the same as that of any other: you might as well leave the semantics implicit. And finally, from what we have said, it is not possible to chose to draw an apple(orange) from the sum; the element drawn is either an apple or an orange. I think this is the case for a sum in Type Theory, but not for Category Semantics or Set Theory. Obviously this is not elementary issue, but I would appreciate some help here. </blockquote><h4>Multiplication</h4><h4><span style="font-weight: normal;">Unlike addition, semantics is an active participant in multiplication. Generally the meaning of one operand differs from that of the other, and the meaning of the product is the result of the interaction of the meanings of the operands; in a real sense the meaning of the product is the product of the meanings of the operands Ignoring the semantics of multiplication and just relying on syntax is dangerous: for the whole number M = 3 consider</span></h4><div>3 x N = (1+1+1) x N = (1 x N) + (1 x N) + (1 x N).</div><div><br /></div><div>If one ignores any meaning of 1 (or 3) beyond its property as a whole number and takes (1 x N) = N, then </div><div><br /></div><div>3 x N = N + N + N,</div><div><br /></div><div>and one is led to infer multiplication by a whole number is equivalent to multiple addition. This can be true for an N that only carries the meaning of a whole number, but most of us have little truck with such problems, and it misleading to suggest equivalence in our problems where multiplication is not addition because what is being added is not the same as what is being multiplied. We illustrate the issue with exemplars of two broad classes of multiplication problems: rate/ratio and area calculations.<br /><h4>Rate as Speed</h4></div><div>The distance D traveled in a Time T at a constant speed R for T = N hours and R = M miles/hour is given by the evaluation of the expression M miles/hour x N hours. For M =3 and N = 2, D follows from</div><div><br /></div><div>3 miles/hour x 2 hours = </div><div>(1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) =</div><div>2 miles + 2 miles + 2 miles.</div><div><br /></div><div>The semantics is apparent in 1 mile/hour x 2 hours = 2 miles; it is not the case that three Ms = M plus M plus M because 1 x M does not equal M. In the real problem "1" is a speed that acts over time that results in a distance. ( I would go further and say it is not a good idea to decompose 3 miles/hour because it suggests there are 3 parallel trips at speed 1 mile/hour while the original problem concerned one trip at 3 miles/hour. If a problem in elementary education is the application of mathematics to word problems, then the mathematics should be taught as following from words not numbers. Your thoughts?)</div><h4>Area of a Rectangle</h4>For length L feet and width W feet, say L = 3 feet and W = 2 feet, the area A is<br /><br />3 feet x 2 feet = 6 square feet.<br /><br />Here each operand is a length measured in a unit of 1 lineal foot, think of a foot-long ruler; while the area is measured in a unit of 1 square foot, think of a square tile one foot on each of its sides. This is the common refrain in multiplication problems: the meaning of a product is the product the meanings of the operands.<br /><h2>Wrapup</h2><div>Multiplication is not addition. Addition is not equivalent to multiplication: it has different semantics, and the addition process is not equivalent to the multiplication process. The product resulting from multiplication can always be rewritten as a multiple addition, but each of its terms the product in an already performed multiplication: the multiple addition is simply a rewrite of the product, a different way to express it, not an equivalent way to obtain it.<br /><br />It is the case that virtually all algorithms for multiplication start by decomposing the operands into parts that can be easily multiplied with each other to produce numbers whose sum is the desired result. Now that computers can do the actual arithmetic, teaching the multiplication algorithm can be taken as an opportunity to show/see how big problems are broken up into smaller ones, and the results of the small ones are then swept up into the solution of the big one. Eureka! they have the keys to the kingdom.</div><h2>Looking Ahead</h2><div>Here I've written mostly about the semantics of problems using arithmetic. I want to turn to some ideas on making school mathematics a subject with a more active approach. If semantics/meaning is to go anyplace, ways have to be found to make mathematics an active endeavor involving things you can touch, feel, and name while at the same time raising good mathematics questions. The useful heart of mathematics holds meaning not manipulation. I believe two strands of science, physics and computer science, have developed useful methods and concepts that can be suggest ways to go forward in the effort. Your nominations? Of course my dream is to entice you to join.</div><div><br /></div><div>I want to discuss ideas for a hands-on coordinate system to make calculation on a number line self-contained, calculation by measurement if you will. It should be in a follow-on blog.<br /><br /><h3>Addendum</h3></div><div><span style="background-color: lime;">I want to revisit the semantics of addition and multiplication to look for better, or at least other, ways to encapsulate their meanings. This discussion is intertwined with the one in the next post. It looks for visceral understanding of the difference between addition and multiplication.</span><br /><span style="background-color: lime;"><br /></span></div><div><span style="background-color: #d9ead3;"><span style="background-color: lime;"><u>Multiplication</u> spreads the effect of one of its operands over the other to produce a third that generally differs qualitatively from each. In a process of multiplication, and here I picture the physical process: speed acting over time (or time over speed ) to produce distance: 1 mile/hour x 2 hour = 2 miles, extending along a line produced by velocity converting</span><span style="background-color: lime;"> time into distance.</span></span><br /><span style="background-color: #d9ead3;"><span style="background-color: lime;">Multiplication of length by width produces area, it distributes width(length) along length(width) to produce area by extending the entire the one dimensional length (width) by the one dimensional width(length) </span><span style="background-color: lime;">to produce a two dimensional area. 1 foot-length-rectangle x 2 foot-width-rectangle = 2-square-foot-rectangle. Here the extension by one operand is transverse or crosswise to the other operand, sweeping out an area, not "extension along" as it was in the previous conversion calculation.</span></span><br /><span style="background-color: #d9ead3;"><span style="background-color: lime;"><br /></span><span style="background-color: lime;"><u>Addition</u> produces a sum with the same qualitative features as those of the operands whose addition produced it. There is no connotation of extension, rather addition augments or appends one operand to the other with no conversion of units or change in dimension.</span></span></div><div><span style="background-color: lime;"><br /></span></div><br /><h4></h4><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><br /><div><br /></div><h2></h2><br /></div>Richard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com1tag:blogger.com,1999:blog-3824285401698121426.post-49113720520221336022015-10-20T13:09:00.002-07:002015-10-20T13:09:16.944-07:00Elementary ThoughtsRichard Fullerhttps://plus.google.com/113941782082000016414noreply@blogger.com0