Mathematize and the CCSSM

Some time ago when I was learning about the Math Recovery program and reading the wonderful books co-authored by Bob Wright I came across the verb 'mathematize' and its noun counterpart "mathematization". I always thought it was a wonderful way of describing what math education at the elementary school is all about.

In his book, Developing Number Knowledge,  Wright defines the term (p15) this way;
                  Mathematization means bringing a more mathematical approach
                  to some activity. For example, when a student pushes some
                  counters aside and solves an addition  task without them,
                  we say they are mathematizing, since it is mathematically 
                  important to reason about relations independent of concrete
                  materials.

Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.

The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and  how it is sadly still taught in poorly taught math classes.

A classic example occurred in my math class yesterday when I asked students what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2. 

In other words people have not gone through the process of  mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like, from a concrete experience or idea  to a symbolic,  mathematical relationship. This happens all the time in math.

Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.

They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).

Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. Instead, we should be mathematizing them from concrete experience  to symbolic representation. This is what the Common Core State Standards for Mathematics is trying to achieve.

 

Thinking Fractions

If you think about what you are doing when working with fractions everything becomes crystal clear. The worst thing to do is to visualize the algorithm you learned in elementary school; something like 1/2 + 2/3 or 1/4 x 4/5 or, heaven forbid, 1/2 divided by 1/4. If we do this then we are confined to thinking instrumentally procedurally about something that is comprised of only procedural knowledge. We fall back on senseless rules like "find the common denominator" or "cross multiply" or "change the sign and flip the second fraction". learning tricks like this will probably get you a better score on a traditional math test (like those used to compile the TIMMS report)  but will do little to help you understand what fractions are all about.

So for 1/2 + 2/3 think about different ways you could make 1/2. You could say it's the same as 2/4, 3/6, 4/8 and so on. Now do the same with 2/3. It could be 4/6. Aha, no need to go any further if you know that you can add fractions that have the same denominator. So 3 sixths plus 4 sixths is 7 sixths. We can count sixths just like anything else. Now you can see how you can get sixths by multiplying 2 by 3 if you need to do more difficult ones. But at least you now understand why.

1/4 x 4/5 is more challenging because the x (multiplication symbol) and the whole concept of multiplication of fractions can be different from using it with whole numbers. Visualize 4/5 using the model in the picture above. Now take 1/4 of that 4/5 and you end up with 1/5. The key to understanding this is seeing how the size of the 1 to which each fraction refers changes. The 1 of the 4/5 is the red 1 above whereas the 1 being referred to by the 1/4 is the 4/5. Really we're asking what is 1/4 of 4/5? Once you start to see the pattern, the relationship between the numerator and denominator then things get easier. Try 1/3 x 3/5 or 1/2 x 2/7 or 1/4 x 4/9. If there is not relationship between the numerator and denominator you can do it another way. For example 1/2 x 3/8 is 3/16. This can be done conceptually by halving the size of the fractional pieces. 1/16 is half of 1/8 so 2/16 is 1/2 of 3/8.

1/2 divided by 1/4 is really asking how many 1/4s are there in a 1/2. This is easily conceptualized by thinking about a football game; how many 1/4s are there in the first 1/2? Clearly there are 2.

3/8 divided by 1/4 is a little more difficult because the referent of the fractions changes. Again, think how many 1/4s are there in 3/8 by visualizing the fractional pieces. Compare 1/4 (a yellow piece above) with 3/8 (three dark blue pieces and you'll see there are 1 1/2 quarters in 3/8. The referent for the 1 1/2 is the 1/4.

from this conceptual understanding it's easier to "mathematize", to use Bob Wright's term,  what is going on by working out how the procedural knowledge of operating with fractions works.