# Seeking references for why it is good that students understand why mathematical rules work

I am currently advising a student at his final project (it is a graduation course for people who will become math teachers). We've chosen to pick some basic mathematical rules which are (or at least should be) known by students (like $$a^{-1} = \frac{1}{a}, \forall a \in \mathbb{R}^*$$) and explain the reasons why those rules actually make sense.

I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view, my personal opinion has no value. I need academic articles/books written by researchers of mathematical education supporting this idea. Could you please suggest me some articles/books along these lines?

From now, all I've got are some lines from G. Polya which resemble those ideas, but they are still not exactly what I want.

• When you say "it is good for students to understand why mathematical rules work," did you mean to say something like "it is a good idea"? It seems that some users here interpret what you said as something like "it is morally good." – Joel Reyes Noche Jan 24 at 15:56

The key term you are interested in is "conceptual knowledge" (more specifically, "conceptual understanding").

According to this document from the National Council of Teachers of Mathematics,

Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

I do not have a copy of the book "Concept-Rich Mathematics Instruction" by Meir Ben-Hur (Association for Supervision and Curriculum Development, 2006) (and I have not read it), but I suspect you might find what you're looking for there. You can read its first chapter ("Conceptual Understanding") here.

• The quoted document says things that are certainly true, but I think it's extremely wrong-headed to use that as a justification for wanting students to understand concepts. The purpose of conceptual understanding is not to improve fluency in calculation. Computers are fluent in calculation. We're humans. – Ben Crowell Jan 24 at 14:57
• @BenCrowell "The purpose of conceptual understanding is not to improve fluency in calculation." Where in my answer was it stated that the purpose of conceptual understanding is to improve fluency in calculation? – Joel Reyes Noche Jan 24 at 15:41

I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view, my personal opinion has no value. I need academic articles/books written by researchers of mathematical education supporting this idea.

Is there an agreed upon goal of mathematics education? It seems that you are assuming (correct me if I am wrong) that there is a "real goal" of mathematics education, and that "understanding the reasoning" is not one of those goals. Then your question is for evidence "understanding the reasoning" supports the "real goal".

I do not view things this way. I think that everyone has their own individual motivations for how they choose to engage (or not engage) with mathematics, and they are all equally valid. I have personal beliefs that engaging with mathematics in particular ways can enrich someones life, so I have built a career around sharing that perspective. I know that one's life can be enriched in this way because my own life has been so enriched. No academic justification needed.

For an analogy, I find it very enriching to forage for wild foods. When I am hiking and eating a pawpaw, and someone asks me what that is, I do not need a research study to justify sharing my enthusiasm for my hobby.

Now there are interesting math education research questions you could ask which are related to this question. It would be interesting to know the answer to questions like "Does having an explanation of a rule increase the chance that the rule will be applied correctly?". This is a question about objective reality, and hence has a chance at being answered by research. The value judgement that this would be a "good thing" is not answerable by research.

• This. There is no excuse for being ignorant of educational research or for intentionally ignoring educational research because it's more work or because the practices it supports aren't what's traditionally done. But educational research can't tell us the purpose of life. – Ben Crowell Jan 24 at 15:00

A while ago I was asking myself the same question: why $$a^{-1} = \frac{1}{a}$$. I haven't read anything on the topic. I just figured that this was engineered, based on existing and obvious definitions of exponentiation for positive integers, like $$a^{2} = a \times a$$ and $$a^{1} = a$$ and then you want to do some operations with them, like multiplying and dividing. So, if $$a^{1} \times a^{2} = a \times a \times a = a^{3}$$, that is, the exponents are added up, then it would be nice if the same happened when we go into the zero and then the negatives: $$a^{0} \times a^{2}$$ should be $$a^{2}$$, which means that $$a^{0}$$ should be 1. Likewise, $$a^{-1} \times a^{3}$$ should be $$a^{2}$$, which means that $$a^{-1}$$ should be $$\frac{1}{a}$$.

So, this rule works because it has been designed to work, not because it is a divine law.

• I’m sure you found the experience pleasant, but how does this answer the question? – Kevin Arlin Jan 25 at 13:30
• @KevinArlin It is really more of a comment on the question, but it makes a good point that (as educators and scholars) we should be very clear with our students about what is a definition, what is a convention, what is a motivation, what is an axiom, and what is a theorem. Since OP mentioned explaining $a^{-1} = \frac{1}{a}$ as an example of what he wants to do, it is worth pointing out that this is a definition: it is declared by fiat so that certain laws for exponents continue to work nicely. Until one makes the definition, it is meaningless. – Steven Gubkin Jan 28 at 22:48