It is not uncommon to read articles about math education in the mainstream press, arguing that students must master basic skills *before* they can develop conceptual understanding. And moreover, that the road to such mastery is teacher explanation followed by repetitive drill. These essays frequently argue that it’s like learning to play the piano: you must practice scales before playing real music! When I mentioned this to a friend who is a piano teacher, he considered it to be an insult to his profession. He said that obviously these people are not piano teachers! Teaching piano is about music! Yes, students do need exercises, but if that’s all you had them do, you’d drive them away from music. The biggest motivator is the recital, when they play real music, not scales!

My friend is right: the authors of these op eds are not piano teachers, but they’re often not math educators either! Still, it is important to address their ideas, because they reflect a broad cultural consensus among many parents, administrators, students, and teachers. Some proponents of the “skills first” approach equate teaching for understanding to what they call “fuzzy math”, a flaky anything-goes sort of teaching, with no specific learning goals, no accountability, just feel-good teachers who allow students to wallow in their ignorance.

It behooves those of us who disagree with this caricature to clarify what we mean by understanding. That is the main purpose of this post.

To get a straw man out of the way, my position is that understanding cannot be divorced from the acquisition of skills. Without understanding, it is hard to develop an interest in the skills, or to retain them; without skills, understanding is out of reach. Good teaching requires a skillful weaving of those two strands. It is, in fact, like learning to play the piano! Skills are important, but it’s all about the music.

But on to the main point: what is understanding? This is a difficult question, and the true fact that experienced math teachers can “recognize it when they see it” is not a sufficient answer. Here is an attempt at spelling it out. **A student who understands a concept can:**

**Explain it.**For example, can they give a reason why 2(x+3) = 2x+6? Responding “it’s the distributive rule” is evidence that the student knows the name of the rule, but a better explanation might include numerical examples, or a figure using the area model, or a manipulative or visual representation. Therefore, we should routinely ask students to explain answers, verbally or in writing, even though many don’t enjoy doing that. It is a way for us to gauge their understanding, and thus improve our teaching, and more importantly, it is a way for them to go deeper and guarantee the ideas stick.**Reverse processes associated with it.**For example a student does not fully understand the distributive law if they cannot factor anything. More examples: can they create an equation whose multi-step solution is 4? Can they figure out an equation when given its graph? And so on. Reversibility is a both a test of understanding, and a way to improve understanding.**Flexibly use alternative approaches.**For example, for equation solving, in addition to the usual “do the same thing to both sides” for solving linear equations, students should be able to use the cover-up method, trial and error, graphs, tables, and technology. If they have this flexibility, they can decide on the best approach to solve a given equation, and moreover, they will have a better understanding of what equation solving actually is.**Navigate between multiple representations of it**. Famously, functions can be represented symbolically, or in tables, or in graphs. Making the connections between these three is a hallmark of understanding. I have found that a fourth representation (function diagrams) can also help deepen understanding, and be used to assess it. Multiple representations on the one hand offer different entry points that emphasize different aspects of functions, but making the connections between the representations is part and parcel of a deeper understanding.**Transfer it to different contexts**. For example, ideas about equivalent fractions are relevant in many contexts, such as similar figures and direct variation. Or, the Pythagorean theorem can be used to find the distance between two points, given their coordinates. If a student can only handle a concept in the form it was originally presented in class or in the textbook, then surely no one would claim they fully understand it.**Know when it does not apply**. When faced with an unfamiliar problem, students will tend to reach for familiar concepts, such as linear functions and proportional relationships. Sometimes, this makes sense, of course, but students need to be able to recognize situations where a given concept does*not*apply.

Clearly aiming for all this is a high bar, and it is tempting to just have students memorize some facts and techniques, and then test them to see if they remember those a few weeks later. (This is often how the “skills first” approach plays out.) But what good would that do? It would just add to the vast numbers who got A’s and B’s in secondary school math, went to college, and now tell us “they’re not math people”. Yes, teaching for understanding is ambitious, and it must be our goal for all the students.

Alas, there are obstacles. For one thing, understanding cannot be easily conferred by explanations. (A naive traditionalist once suggested that it was easy to teach about variables: patiently explain to the students that variables behave just like numbers! Would that it were that easy.) Moreover, understanding is not always valued by students, parents, and administrators, many of whom believe that everything would be so much more straightforward if we could just have the students memorize facts and algorithms. Finally, understanding is difficult to assess. (Actually, the list above is one way to improve assessments: each item on the list suggests possible avenues for authentic assessment. To reduce complaints that it’s “not fair” to assess students that way, such assessments can be ungraded. As the current jargon would have it: consider them formative assessments.) Being able to reproduce a memorized set of steps is a good test of memory and obedience. To test understanding, non-rote assessments are the most revealing.

The above list is also a tool in forward design. When planning a unit, ask yourself how you can incorporate reverse questions, alternate approaches, multiple representations, varied contexts, and so on. A tool-rich pedagogy is helpful, as different manipulative, technological, and paper-pencil tools provide a way to do this and avoid boring repetition. Of course, the implication of such planning is that it takes more time to teach any given concept. Because of the enormous pressure of coverage at all costs, it is generally necessary to take less time on less important topics, and approach the most important topics in as many ways as possible.

In any case, I hope you find this post helpful in your teaching, and also in your conversations with colleagues, administrators, parents, and students.

Good luck as you teach for understanding!

— Henri

[This post includes part of my Nothing Works article, updated and expanded.]

Thank you! This is so well said – it speaks my mind and articulates ideas that I have been having trouble saying.

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Greetings, Henry. Saw your request on Twitter for feedback on this post, so here are my thoughts on what I liked and where we disagreed in a (hopefully) agreeable manner.First off, I think you did a nice job of explaining your point of view. I believe Lockhart’s Lament uses the the piano scales as an an example of understanding versus skills, and I think it’s a good analogy.I think that students should be afforded every opportunity to learn and to wonder, be it in maths, literature or history class, to name three.However, I don’t think a full understanding is necessary to getting good grades, and at times is antithetical to that goal.I realize the above two paragraphs are almost contradictions, but that’s the rub: there’s an obligation to provide understanding but also a responsibility to prepare students for assessments that do not require demonstration of understanding.Overall, I thought this was an excellent read. It was well written, in an engaging style, and it caused me to pause multiple times to consider whether I agreed with a point and why. Very meta cognition rich! Best,apmTwitter: @autismplusmath

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We don't disagree. (In the post, I suggested that not all assessments need to be graded.) Grades often undermine learning. I wrote about this here:https://www.mathedpage.org/teaching/assessment/index.html(also follow the link to what the research says.)The challenge is to be clear on our priorities and to balance societal pressures with our commitment to authentic student growth.– Henri

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Thank you for the link to your deep dive on assessments. I devoured it and wanted to paste here some of your points that were especially powerful to me:- “Do not over-penalize students for small computational errors that could be eliminated by the use of technology such as calculators and computer algebra systems. Prioritize evidence of understanding, not nit-picking accuracy.”- “Use participation quizzes, during which you watch the class work and make notes on students' desirable behaviors. This is an amazingly effective technique to clarify what you consider the most productive ways to function in a math class. Students are being assessed on work habits, not math understanding, but one leads to the other.”I am a big believer that struggling learners often need to be explicitly taught and provided models of effective learning skills, including how to take effective notes in a maths classroom. Your participation quiz is spot on for providing that needed modeling. apmTwitter: @autismplusmath

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I didn't make up the participation quiz! I learned about it from Carlos Cabana, a legendary math teacher in the Bay Area. It is indeed a powerful technique!

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Henri,Have you by any chance read my recent blog post on conceptual understanding? https://davidwees.com/content/what-is-conceptual-understanding/I'm wondering what overlap there exists between our ideas here and to what extent we are focusing on different things?David

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I just read your post, and it leads me to add this to the list above:- *Recognize its connections with related concepts, and integrate it in a coherent mental framework*However what I was trying with the list was to suggest teaching, formative assessment, and curriculum development strategies. This additional item (at first sight) seems harder to put into practice. Still it adds some depth to the list, and it may work as a way to tie the other items together. Thanks for directing me to your post!– Henri

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Just to chime in on this most excellent conversation, I've done the participation quiz many times and my students and I love it. It sends a clear message that the focus is on doing math, not the “nit-picking accuracy”. I've even gone as far as to use the 8 Math Practice Standards as a “rubric” for what I look for in these participation quizzes: persevering in problem solving, construct and communicate viable arguments, reason abstractly and quantitatively, make models, use tools, examine structure, find patterns, be clear and “say what you mean”.Of course, we don't focus on all of these every time. We'll usually pick one or two to watch for. But it is neat to see the MATH that goes on.Thanks, Henri, for a great post. And thanks, apm, for the comment that lead to the posting of that link to the assessment post. 😀

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Wow! That is an advanced use of the participation quiz! I'm impressed.

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