Values of One

Numbers Keep the Score. They Don't Play the Game.

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.

• one of the greatest deficits in U.S. students learning mathematics is in their ability to solve problems( page 138 in Adding it Up, National Research Council, 2001). 

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.

"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.

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:

• words convey the information that must be understood in order to formulate the problem;
• a general formulation of the problem exists that does not depend on the numbers.

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.  

Values of One

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.

Ignoring values of one, 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.

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

Examples of Unit Values

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

• individual apples with a unit of 1-apple?
• as individual pieces of fruit with a unit of 1-fruit?
• as a collection of 12 apples with a unit of 1-doz-apples?
• as a pound of apples with unit of 1-lb-apples?
• as a fruit in a collection of apples and oranges?

Is time participating as a 

• 1 second,
• 1 week
• 1 average-squirrel-lifetime 

Is distance participating as 

• 1 mile,
• 1 mm

Are rabbit cages participating as 

• 1 rabbit-cage
• 1 doz rabbit-cages  

And for rabbits

• 1 rabbit
• 1 pair of rabbits
• 1 buck rabbit
• 1 doe rabbit

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.

Arithmetic and Units

* 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.

A major difference between addition and multiplication is easily expressed in terms of the unit value of their operands:

• Addition: there is only one unit value; each operand, as well as the sum have the same unit valuee.
• Multiplication: there can be three distinct unit values, and there generally there are. 

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.

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? 

Multiplication Problems

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"?

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.

An Area Calculation 

Let's review some natural number work. With the notation, "[ ]" for successor introduced in the last post we have

• 1:ftW = [0:ftW] = 0:ftW + 1:ftW,
•  
•  
• 4:ftW  =  [3:ftW]  =  3:ftW + 1:ftW

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:

• 6:ftL x 3:ftW  =  6:ftL x [[[0:ftW]]] = 
• 6:ftL x ( 2:ftW + 1ft:W)  =  6:ftL x 2:ftW + 6:ftL x 1:ftW   =
• (6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW  = 
• ((6:ftL x 0:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) + 6:ftL x 1:ftW)  =
• ((6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) 

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:

• 6:ftL x 1:ftW = (6 x 1):(1:ftL x 1:ftW) = (6 x 1) x 1:ftftA = 6:ftftA,

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:

• (( 6:ftL x 1:ftW + 6:ftL x 1:ftW ) + 6:ftL x 1:ftW ) = 
• (( 6:ftftA + 6:ftftA ) + 6:ftftA ) =
• ( 12:ftftA + 6:ftftA ) =
• 18:ftftA 

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. 

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. 

This is an instance of confusion arising from conclusions drawn from the multiplication of unitless Natural Numbers. There we have,

• n x 0 = 0 and n x [m] = n x m + n,    

If we take that over into unit natural number multiplication, we would be led to the error: 

• 6:ftL x 3:ftW = (6:ftL + 6:ftL) + 6:ftL.

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.

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.

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. 

Ratios and Rates in Multiplication

Understanding rates and ratios provides access to a large fraction of the interesting problems involving multiplication. 

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.

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

• 30:mphSpeed x 6:hour = (30:mile/1:hour) x 6:hour = 30:mile x (6:hour/1:hour) = 
• (6 x 30):mile = 180:mile.

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

• 18:oneR/6:oneC = (18/6) x (1:oneR/1:onrC) = 3:oneR/1:oneC.

which says the uniform rabbit cage occupancy rate is 3 rabbits per cage.

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:

• ( 6:B-S /1:book ) x 3:book  =   ( 6 x 3):B-S x (1:book/1:book) = 18:B-S

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

• area per width unit value ratio  =  1:ftftA/1:ftW

for the rectangle under consideration as 

• 18:ftftA/3:ftW  =  (18/3):ftftA/1:ftW  =  6:ftftA/1:ftW.

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

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.

The Significance of Unit Values in School Mathematics  

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.

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.

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:

• a number line is used to demonstrate numbers are already "there" to be laid out for inspection,
• multiplication is what is in multiplication tables,
• for larger numbers multiplication is what an ancient algorithm does.

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. 

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