*Click here to read Part 1: Understanding Math: A Cultural Problem.*

Educational psychologist **Richard Skemp** popularized the terms **instrumental understanding and relational understanding** to describe these two ways of looking at mathematics. It is almost as if there were two unrelated subjects, both called “math” but as different from each other as American football is from the game the rest of the world calls football.

Which of the following sounds the most like your experience of school math? And which type of math are your children learning?

### Instrumental Understanding: Math as a Tool

Every mathematical procedure we learn is an instrument or tool for solving a certain kind of problem. To understand math means to know which tool we are supposed to use for each type of problem and how to use that tool — how to categorize the problem, remember the formula, plug in the numbers, and do the calculation. To be fluent in math means we can produce correct answers with minimal effort.

**Primary goal:** to get the right answer. In math, answers are either right or wrong, and wrong answers are useless.

**Key question:** “What?” What do we know? What can we do? What is the answer?

**Values:** speed and accuracy.

**Method:** memorization. Memorize math facts. Memorize definitions and rules. Memorize procedures and when to use them. Use manipulatives and mnemonics to aid memorization.

**Benefit:** testability.

Instrumental instruction focuses on the *standard algorithms* (the pencil-and-paper steps for doing a calculation) or other step-by-step procedures. This produces quick results because students can follow the teacher’s directions and crank out a page of correct answers. Students like completing their assignments with minimal struggle, parents are pleased by their children’s high grades, and the teacher is happy to make steady progress through the curriculum.

Unfortunately, the focus on rules can lead children to conclude that math is arbitrary and authoritarian. Also, rote knowledge tends to be fragile, and the steps are easy to confuse or forget. Thus those who see math instrumentally must include continual review of old topics and provide frequent, repetitive practice.

### Relational Understanding: Math as a Connected System

Each mathematical concept is part of a web of interrelated ideas. To understand mathematics means to see at least some of this web and to use the connections we see to make sense of new ideas. Giving a correct answer without *justification* (explaining how we know it is right) is mere accounting, not mathematics. To be fluent in math means we can think of more than one way to solve a problem.

**Primary goal:** to see the building blocks of each topic and how that topic relates to other concepts.

**Key questions:** “How?” and “Why?” How can we figure that out? Why do we think this is true?

**Values:** logic and justification.

**Method:** conversation. Talk about the links between ideas, definitions, and rules. Explain why you used a certain procedure, and explore alternative approaches. Use manipulatives to investigate the logic behind a technique.

**Benefit:** flexibility.

Relational instruction focuses on children’s thinking and expands on their ideas. This builds the students’ ability to reason logically and to approach new problems with confidence. Mistakes are not a mark of failure, but a sign that points out something we haven’t yet mastered, a chance to reexamine the mathematical web. Students look forward to the “Aha!” feeling when they figure out a new concept. Such an attitude establishes a secure foundation for future learning.

Unfortunately, this approach takes time and requires extensive personal interaction: discussing problems, comparing thoughts, searching for alternate solutions, and hashing out ideas. Those who see math relationally must plan on covering fewer new topics each year, so they can spend the necessary time to draw out and explore these connections. Relational understanding is also much more difficult to assess with a standardized test.

What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.

The subject matter of relational and instrumental mathematics may be the same: cars travelling at uniform speeds between two towns, towers whose heights are to be found, bodies falling freely under gravity, etc. etc.

But the two kinds of knowledge are so different that I think that there is a strong case for regarding them as different kinds of mathematics.

### For Further Reading

**The difference between instrumental and relational understanding****Richard Skemp’s Relational Understanding and Instrumental Understanding****Instrumental vs. Relational: Student Response**

*Click to read Part 3: Is There Really a Difference?*

**CREDITS:** “Humphreys High School Football” photo (top) by USAG- Humphreys via Flicker (CC BY 2.0). “Performing in middle school math class” (middle) by woodleywonderworks via Flicker (CC BY 2.0). “I Can Explain My Thinking” poster by Nicole Ricca via Teachers Pay Teachers. “Snow globe” photo by Robert Couse-Baker via Flickr (CC BY-SA 2.0).

This is the second post in my Understanding Math series, adapted from my book *Let’s Play Math: How Families Can Learn Math Together—and Enjoy It,* available at your favorite online book dealer.

This is not quite the dichotomy I see the world divided into and actually I think its bit more complex than just 2 alternatives anyway. There’s a family of thinking that does think of math like a tool and that kids should learn to be interpreters, modelers and technology orchestrators without ever necessarily doing much direct computation. This is the Conrad Wolfram view of the world. Computer software removes much of the need to emphasize any computation be it via traditional or constructivist methods.

Then there are traditionalists who view computation as a skill that needs to be directly taught. But I don’t think most members of that camp would claim its prime benefit is testability or that direct instruction is all about memorization. Even in these world views one is shown models and explanations and then taught algorithms. Let’s call this the Singapore model. I think to be fair to this camp one should acknowledge that there is a progression towards higher level problem solving built in. While basic skills are taught very directly as you progress there is increasing amounts of more complex tasks. Proofs were traditionally introduced first in Geometry.

Then there is constructivist math. There are two camps even here. The first also de-emphasizes computation but prizes conceptual understanding. An example like TERC assumes that we don’t really need standard algorithms since such problems will mostly be solved by machines and concentrates on discovering principles. A lot of other newer approaches have oddly gone the other way and while not emphasizing standard algorithms have tried to build numeracy where students can solve problems mentally or on paper but via a number of different approaches like number bonds, landmarks etc. In middle school, the real divide seems to be on instructional approach and how “real world” problems should be.

Oddly enough the farther one goes in the sequence the less alternatives remain. By the time, we reach Calculus most instruction still follows a textbook approach.

Anyway thanks for the piece, it definitely made for an interesting read.

True, nothing in this world ever fits perfectly into a dichotomy. But I find Skemp’s classification helpful in recognizing things I like or don’t like about particular curricula, math articles, workshops, books, etc. And it helps me make sense of the political storm that surrounds the Common Core.

Many people misunderstand this dichotomy as having to do with surface-level, visible differences in instructional approach, but it’s not that at all. It’s not about whether you use a textbook, whether you lecture, or whether you use “real world” projects. It’s about the goal behind the teaching. That goal — often below the level of conscious thought — shapes how a curriculum is used, what a teacher emphasizes or skips, how tests are written, how grades are assigned and interpreted. It shapes what a student pays attention to, cares about, and remembers.

To take Singapore math (because that’s a curriculum that I’ve used, so I’m familiar with it) — it can be taught from either of these worldviews. It can be taught with a goal of training kids to produce answers, or it can be taught with a goal of building a connected framework of concepts. It was

designedfor the latter. But teachers do not use a curriculum according to the writer’s worldview, they use it according to their own.Denise, your first two entries on these issues have been very concise summaries of many important issues. Of course, there are variations and more camps than two, but in the actual world of K-12 teaching, it’s really not all that nuanced. Those who control what goes on in classrooms now – the high-stakes test/rigid standards crowd, have tried to and to a very large extent succeeded in rigging the game so that most public school administrators and classroom teachers are forced to march to the beat of the testing drum. And it’s 100% clear which of Skemp’s types of mathematics will dominate in such a world.

As a teacher coach, I can tell you that even where teachers believe in promoting more relational understanding, they are extremely hard-pressed to structure classroom time and curricular content in ways that jibe with that sort of view of mathematics. There’s just too much to be “covered” (think about various meanings of “covering” a curriculum), too much that will be on The Test than there is time to explore and discuss in any sort of depth at all. Kids must be prepped for The Test, must be convinced that doing well on it matters to them as individuals (for the most part, it does not, particularly in lower grades and in high-poverty, low-achieving districts), because the teacher’s (and administrator’s) job depends on student test scores. Promotions, raises, tenure, the whole she-bang hangs upon how kids do on a test that often is antithetical to the other sort of mathematics Skemp is thinking of.

I must take issue with one point you raised in the first part of this series: the Math Wars would be alive and sick regardless of Common Core, and they will flare up again, openly and violently, when the Common Core and the testing mania that comes with it goes into decline (a day that cannot come too soon for me). If you keep abreast of some of the long-time educational conservatives and traditionalists in the Math Wars since 2008, they hate the Common Core for various reasons, but one of the biggest is the Standards for Mathematical Practice, a clear successor/heir to the NCTM Process Standards, and repository of all the “fuzzy” progressive ideas about mathematics teaching, learning, and curriculum content in K-12 that these self-appointed Math Warriors loathe. On the surface, some of the opposition, like that from R. James Milgram, seems to be about the CCSS-M Content Standards not being “competitive” or “rigorous” enough at the high end (never mind that authors of the CCSS-M made clear that they weren’t going to write calculus standards because of the widespread acceptance of the AP Calculus test standards & curriculum). But that’s mostly a sham. The old Mathematically Correct/NYC-HOLD crowd just despises anything that attempts to make K-12 mathematics more appealing, more relevant, less forbidding, more inclusive, and in any way different from the teacher-centered, calculation-intensive mathematics classrooms in which they learned. As one notorious Math Warrior has written many times, “There’s nothing new under the sun that is good, and nothing good under the sun that is new.” And he’s not kidding.

I agree, the Math Wars are not going away in any conceivable near future. There have been debates about how best to teach math going back at least to the last century, and probably to the Ancient Greeks. What is different in recent years, I think, is how political it all has become. It’s toxic to learning.

Then again, arguments about education got pretty toxic in Ancient Greece, too — just ask Socrates. So maybe there is nothing new…

May I please point out something that really bothers me? You are quoting an “educational psychologist” to explain how to “understand math” and calling it a “worldview” which would be referring to philosophy!

What you don’t realize is that philosophy and psychology are different fields.

https://www.psychologytoday.com/us/blog/view-runaway-trolley/201307/psychology-and-philosophy-irreconcilable-differences

In Ancient Greece, mathematics was taught as philosophy. In college, I took philosophy 101 and philosophy of mathematics at the same time, separate classes. The philosophers we’re the same people as the scientists and mathematicians.

Here’s another link to get you thinking some more:http://www.storyofmathematics.com/greek_plato.html

You are really writing about teaching, not learning. The teacher has a worldview. A child is either learning from, or in spite of, the teacher.

I wish you would relate more to homeschool settings where there’s a family- a mom and/or dad, helping their kids learn as a family, or someone tutoring one-on-one.

When you decide to send your children to school( let’s assume the case where you have a choice), you are giving the school the authority to teach and discipline your children. Then, you must do what the school says. We give up some of our rights and have to do what the school says to do. At least in the more lenient states, the parents can homeschool and use whatever methods they deem best. Parents love their children and know them well. I think homeschool makes it a lot easier to help children learn. I like reading how you interact with “kitten”. I enjoy hearing how you help her.

When I say “worldview,” I don’t mean it in a philosophical, metaphysical sense, but just the way one looks at things — in this case, limited to how one sees mathematics. We all have a point of view, a way of thinking, a perspective that colors our understanding of math.

Perhaps “perspective” would have been a better word?

In the homeschool setting, both the parents and the students have a perspective on math. They each have assumptions about what math means and expectations of what “learning math” should look like. Those assumptions and expectations will shape their family’s experience of math, even more than the curriculum they use.

My goal in this series of articles is that people will realize there is more than one way to understand math. And then for them to consciously choose which type of understanding they want to help their children develop.

Of course, the articles are slanted by my own point of view. I believe that a relational understanding will provide children a better foundation for future learning and for applying math in their later life — and is more enjoyable, to boot.

Ok. I understand better that your worldview is how one looks at learning math. I see their was a part before this one. I will reread it and this one.

Is this the book you read by Richard Skemp?:

https://www.maa.org/press/maa-reviews/the-psychology-of-learning-mathematics

The reviewer includes a #3 for a level of understanding which was:

“Formal (logical) understanding is the ability to connect mathematical symbolism and notation into chains of logical reasoning (as in mathematical proof).

I know it may seem that the formal understanding is not achievable in grade school, middle school age or high school age. However, I just wanted to say it makes me happy that he has a third point and that is logic, the abstract understanding of math. I think it helps to let them read and explore all kinds of living math books- just like you list on your blog.

I haven’t read Skemp’s book, but I think it grew from the article I linked to above. Oops! I see that link has gone bad — try this one: https://astro.temple.edu/~madman/ruiu.pdf (And I’ll go fix the blog post.)

The article explored the differences between instrumental and relational understanding, but it didn’t go into formal, logical understanding.

Hi Denise, I just downloaded the article. I read the beginning of it. It looks very interesting.I will read the whole thing.

Thank you.