How to correct visualization of mathematical expressions?

Question

This happens a lot when I try to explain the commutative property, mostly in elementary grade levels. I say

2 + 3 = ?

and then the student usually replies with 5. Albeit they're not wrong, it's not the idea of commutative property. I feel that students so conditioned early on to see that 2 + 3 is necessarily 5 as opposed to understanding that 2 + 3 can just be that: 2 + 3. I had this problem myself growing up.

How can I address this issue?

2 + 3 = 3 + 2

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more examples,

I was explaining cross-cancellation and this is the visualization

$\frac{2}{7} \cdot \frac{7}{3} = \frac{2 \cdot7}{7 \cdot 3} = \frac{2 \cdot \require{cancel} \cancel{7}}{\require{cancel} \cancel{7} \cdot3}$

instead, they visualize the following

$\frac{2}{7} \cdot \frac{7}{3} = \frac{14}{21}$ {we are stuck here since we cannot see the cancellation working out}

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so what I mean is that students cannot comprehend that it isn't necessary to always work out the arithmetic right away but can keep them as expressions to observe patterns. I hope this is a bit more clear.

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Addition of a story: A famous example, the story of Gauss. Once upon a time, there was a teacher who was bored and asked the class, included Gauss, who asked students to sum up from 1 to 100. While the rest of the class struggled to solve it, Gauss got it with ease. This is the idea I was trying to convey.

Instead of sitting down and computing hard, compute smarter kind of idea.

$ \sum_{i=0}^n i = \dfrac{n\times(n+1)}2$

$ \sum_{i=0}^{100} i = \dfrac{100 \cdot 101}2$

Answer 1

The problem is due to imprecise specification of the intended result. Here's a more precise way.

$\text{Recall that the }{\bf commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true for all reals } X,Y$.

$\text{Use the above law to $ $ simplify }\ 2\, +\, \color{#c00}{\pi}\, +\, \color{#0a0}3\ \text{ to the form }\, n + \pi\,\text{ for some integer }n$.

Update $ $ In case it wasn't obvious, the idea is to choose a sum where it is clear that performing the commutation simplifies the addition. If you can't use $\,\color{#c00}{\pi}\,$ (or $x)$ then it is clear how to tweak it to use "simpler" numbers, e.g. $\ 9 + 1/123 - 9\ $ or $\ 1/11+123+10/11,\,$ etc. But these forms have the disadvantage that they don't prohibit the student from diving head-first into computation, i.e they might try computing $9 + 1/123$ before commuting - something they can't do with $\,2+\pi\,$ or $\,2+x$. Hence using a transcendental forcefully guides the student along the correct solution path.

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Update $ $ You added a new problem exhibiting the utility of lazy (vs. eager) evaluation (of arithmetic) in order to help apply simplifying transformations. This idea should be explicitly taught using multiple types of examples to better lend intuition, e.g. besides your example of delaying multiplication in order to exploit cancellations, it would help to give other examples where one can exploit innate arithmetical structure before diving head-first into brute-force arithmetical computation. Below are some further complementary examples.

Special Polynomial structure $ $ If we notice that the arithmetical expression has the form of a well-known polynomial formula then applying that first may simplify the arithmetic. For a simple example consider differences of squares

$$123^2 - 122^2 = (123-122)(123+122) = 245$$

This is simpler than brute-force arithmetic computation, i.e. squaring both $123$ and $122$ then taking their difference. It will be much simpler for analogous examples with much larger numbers.

Reflection symmetry $\ $ Sums symmmetric about a midpoint can be rearranged as follows

$$\begin{align} 1 + 2 + 3\\ +\ 6 + 5 + 4\\ \hline = 7 + 7 + 7 \end{align}$$

which yields an easy proof that the sum is divisible by $7$. If we view the above sum $\!\bmod 7\,$ then it is an additive form of the cancellations in your fractions since $\ 6\equiv -1,\ 5\equiv -2,\ 4\equiv -3$ so the sum is $\equiv 1 + 2 + 3 -1 -2 - 3$. So here the reflection is negation and the key idea is to preprocess the sum by pairing each summand with its negation in order to simplify the arithmetic (this is the key idea behind one proof Wilson's Theorem).

Generally, before diving head-first into brute-force solution methods, it is wise to first perform some "meta level" preprocessing - searching for interesting innate structure that may help simplify it or shed further intuition on the heart of the matter. With that idea in mind I am sure you can come up with many interesting examples appropriate to the level of your class.

Answer 2

One way to see if the student understands the commutative property of addition is to have "fill-in-the-blank" questions such as $$2+3=3+\_\_$$ $$2+3=\_\_+2$$ $$2+\_\_=3+2$$ $$\_\_+3=3+2$$

Answer 3

One way to address this

students cannot comprehend that it isn't necessary to always work out the arithmetic right away but can keep them as expressions to observe patterns.

is to do (or have the students do) a problem in as many different ways as possible, then take the time to compare the solutions (to see that they agree) and the methods. Some will be faster, some more conceptual.

Often considering a few problems in depth is more useful than a list of many where you just check for correctness.

Answer 4

You have not stated the grade level here, but one way to approach this is by giving problems where commuting actually makes the computation easier. For example $97+26+13$. You could have students do "number talks" on things like this. This also gives them an opportunity to showcase their understanding of place value, which is another major goal at this age.

For some resources on "number talks" I suggest watching the following video by Sherry Parrish. She also has some books on number talks which might be helpful to you.

https://www.youtube.com/watch?v=twGipANcIqg