# How to differentiate between mathematical skills and understanding of mathematical concepts?

How would my Colleagues here on Math Educators differentiate between mathematical skills and understanding of mathematical concepts?

I'm a community college instructor and high school instructor looking for examples from my colleagues. I have researched several examples (one here: http://aft.org//sites/default/files/periodicals/wu.pdf) but I'm wondering if my colleagues here had any more.

• This seems like a prompt for an essay. Stack Exchange is not designed to do people's homework for them. Can you revise your question to indicate your understanding of the issue? Are there specific topics that you are having trouble teaching or learning, because you are not sure if the topic is a "skill" or a "concept"? – Jasper Jan 21 '15 at 2:06
• @Jasper I'm long past homework. I'm a community college instructor and high school instructor looking for examples from my colleagues. I have researched several examples but was wondering if my colleagues here had more aft.org//sites/default/files/periodicals/wu.pdf – Gerardo Jan 21 '15 at 2:20
• Try searching google scholar for teaching and learning for understanding (sample). – Benjamin Dickman Jan 21 '15 at 4:01
• @Gerardo You've already accepted an answer. In the future, if you wait longer it leaves open the chances that more, possibly greater answers may come forward. – Chris C Jan 21 '15 at 14:55

I agree with Jasper that your first paragraph is prompting for an essay, but is your second paragraph asking for examples, meaning references that discuss the difference?

Here are three:

1. Richard Skemp's article on Relational Understanding and Instrumental Understanding.

2. The five strands of mathematical proficiency described in Adding It Up. The first two strands are conceptual understanding and procedural fluency.

3. Phil Daro's 10-minute video on Four Levels of Learning.

• When i was reading this question, the first thing that came to mind was Skemp's article, +1 – celeriko Jun 14 '15 at 16:33

I use the term skills to refer to procedural knowledge and the term understanding to refer to conceptual knowledge.

There is a lot of literature about this. One good starting place is Conceptual and Procedural Knowledge: The Case of Mathematics by James Hiebert (Hillsdale, NJ: Lawrence Erlbaum Associates, 1986). It contains ten chapters by different authors.

Chapter 1 of this book (by James Hiebert and Patricia Lefevre) define procedural knowledge as "made up of two distinct parts. One part is composed of the formal language, or symbol representation system, of mathematics. The other part consists of the algorithms, or rules, for completing mathematical tasks" (p. 6). They define conceptual knowledge as "knowledge that is rich in relationships" (p. 3).

Let me show how I understand this distinction. Let's say the domain is the comparison of fractions. When I say that $\frac{4}{7}>\frac{3}{7}$ because of two fractions with the same denominator, the one with the larger numerator is larger, or when I say that $\frac{3}{7}>\frac{3}{8}$ because of two fractions with the same numerator, the one with the smaller denominator is larger, I am using conceptual knowledge.

But when I say that $\frac{4}{7}<\frac{5}{8}$ because I used the standard procedure of multiplying both sides by the product of the denominators ($7\cdot 8=56$) and got $32<35$, I am using procedural knowledge.

• (Adapted from Math Fireworks.) A conceptual solution to the procedurally solved problem at the end: Let us think about both of these proper fractions in terms of proximity to $1$ (the "missing parts" strategy). Looking at $\frac{4}{7}$ and $\frac{5}{8}$, the former is $1$ missing $\frac{3}{7}$ and the latter is $1$ missing $\frac{3}{8}$. Already JRN shows conceptually that $\frac{3}{7} > \frac{3}{8}$, so the former has $1$ missing a bigger part than the latter, hence the former is less than the latter. – Benjamin Dickman Jun 14 '15 at 11:12
• @BenjaminDickman, you make a good point. I'll edit my answer to make things clearer. – Joel Reyes Noche Jun 14 '15 at 11:14
• No criticism whatsoever -- just a supplementary comment to your nice answer! – Benjamin Dickman Jun 14 '15 at 11:15
• Thanks for the reference. – Joel Reyes Noche Jun 14 '15 at 11:18