What is the difference determining success or failure?

Now we should think about following questions:

“When children encounter a mathematical concept or notation against their existing understanding, how can they overcome it, understand the meaning of it and rebuild their existing understanding?”
“What is the difference between those who can do it and those who can’t?”

To come to the point, the answer is,
Whether they can flexibly recognize mathematical concepts or notations as both “Process” and “Object” depending on situations or not.

Source left: https://codeopinion.com/self-descriptive-http-api-in-asp-net-core-object-as-resource/; right: https://www.usfsp.edu/academic-affairs/2016/12/08/change-of-process-for-new-programs/

That means any mathematical concepts and notations can be interpreted both as “Process”, which is a series of procedures of some actions or operations, and as “Object”, which is a product generated by the process.

It would be better to think specifically, so I’m going to take decimals as an example.
For instance,
0.2 can be interpreted
as Process like: “To divide 1 by 10, and then take two pieces of them”
and at the same time
as Object like: “two pieces of size equal to 1 divided by 10” generated by the process above.

Another example is negative numbers.
Let’s say -3.
-3 can be interpreted
as Process like: “To subtract three” or “To shift seven units in the opposite direction along the number line”
and at the same time
as Object like: “the concept of the negative number –3” produced by the process above.

It might become a little more complicated, but I give you more precise explanation about it.

How do humans acquire concepts?

How human acquire any kind of (abstract) concepts originally follows this way:
By perceiving same actions or situations repeatedly, humans can get to recognize them as a set of process. And after or at the same time of that, by giving the process its name with words or symbols, they can recognize the whole of it as one entity and get to think with it.

In fact, this fact is not limited to only mathematics, but also it can apply any abstract concepts used in daily life.
Take your time to think about the concept of “Love”, “justice”, or something like that.
You can realize that you could explain the meaning of “Love” as a process of some actions or situations and on the other hand you could say like “there isn’t love between us” as if you deal with the concept of “Love” as one entity.

Back to Math, learning the concept of number, which is the entrance of learning mathematics, starts with the action of “counting”.

For instance, children count five objects pointing each of them with their finger and saying “one, two, … , five”. After counting in this way a lot of time, they come to notice that the last word (five) is always same even if a type of objects, an arrangement of them, or an order of counting them are changed. At this stage, the word of “five” or the symbol of “5” would work to compress the whole process of counting five objects to the concept of five and children could recognize the concept as an entity using the word.

If you acquire a concept in this appropriate way, you can flexibly “see” the concept as Process and as Object depending on situations.

Although children should get any mathematical concepts in that way, educators sometimes teach Math without considering that fact, which is exemplified by teaching decimals or fractions.

Success or Failure in multiplication of decimals

We discuss “multiplication of decimals” again.
Decimals are introduced as fractional magnitudes, which could encourage children to recognize decimals only as magnitudes (which means Object) without taking attention to its Process that “divide 1 by 10, 100, … , then take some pieces of them”
In this case, because you see decimals only as Object, “0.4 sets” from 0.3×0.4 can’t make sense.
But if you can see decimals as Process, you could interpret “0.4 sets” as like “divide 1 set by 10, then take 4 pieces of them”. That would be make sense.

This is the difference deciding whether you can understand the meaning of multiplication of decimals.
It might not be so problematic to learn the rule of computation by rote, if concerned only with multiplication of decimals.
However, if you learn fractions without being able to recognize them as Process (that means you can’t see 3/4 as “divide 3 by 4”), you would have difficulty to understand the meaning of various kind of operations on fractions like multiplication and reduction of fractions, or transforming fractions to decimals.

Conversely, if you can recognize decimals and fractions as Process and Object flexibly, you can infer the meaning of relating operations and understand them with organic connections.
It is obvious that the gap would rapidly expand as the study progresses.

Success or Failure in variable expression

Let’s take another example.
You can find out opposite situation of multiplication of decimals in the difficulty of variable expression.

Source:https://www.udemy.com/math_mojishiki/

Children who can’t understand 3+4x are much likely to recognize it only as Process.
For them, 3+4x is just a procedure like “add some number multiplied by 4 to 3” and so they can’t recognize that the variable expression itself indicates a certain number or amount.
Therefore as given 3+4x as answer, they would feel strange and hardly find it making sense.

On the other hand, if you can see 3+4x as Object (a number generated by the process of operation), you can accept 3+4x indicating a certain number understandably.
And once you can recognize variable expression as Process and Object, you can handle operations effectively by switching recognition of variable expression flexibly like the following:
3x+2+y, y=2x+1 → 3x+2+(2x+1) → 3x+2x+2+1 → 5x+1

In this way, to be able to recognize mathematical concepts or notation as both Process and Object could determaine whether you can understanding the meaning of computation and operation and carry them out efficiently.

if you face multiple concepts whose duality you can’t recognize at the same time, they would push you into the deep abyss.
For children who don’t grasp the concept of fractions and variable expression, mathematical expression like “(2+3/4x)×2” would be so messed up.

That means if you failed to grasp duality of mathematical concepts in early stages, your confusion could greatly accelerate.

Source: https://www.moneymarketing.co.uk/brokers-slam-confusion-around-help-buy-isa/

In Summary

To sum up, the reason many children have difficulty to learn mathematics is due to the characteristics of Math itself and lacking for sufficient consideration of how human acquire concepts in teaching and leaning Math.

It is an essence of leaning mathematics to get to understand mathematical concepts against existing understanding acquired in previous learning by recognizing concepts in dual way and rebuild existing understanding to better-rounded one.
This is the process to develop mathematical thinking, but many children failed on the way due to various factors like insufficient study, or ineffective teaching, and so on.

This finding gives a great insight to why children fail to learn Math and what we should encourage them to learn.
However, it is actually not enough to answer completely a question, how we should teach Math.

“How to encourage children to acquire mathematical concepts with recognition of the duality?”

I will talk about this issue in next post.
See you again!

ReFerence

David Tall(2013). How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. Cambridge University Press