The Common Core Standards in Mathematics stress the importance of conceptual understanding as a key component of mathematical expertise. Alas, in my experience, many math teachers do not understand conceptual understanding. Far too many think that if students know all the definitions and rules, then they possess such understanding.
The Standards themselves arguably offer too little for confused educators. The document merely states that “understanding” means being able to justify procedures used or state why a process works:

But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.

A few of the “understanding” standards provide further insight:

Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction. [emphasis added].

Note what I highlighted: understanding requires focused inferential work. Being helped to generalize from one’s specific knowledge is key to genuine understanding.
Daniel Willingham, the cognitive scientist who often writes on education, offers a more detailed account of the nature and importance of conceptual understanding in math (along with the other two pillars of mastery, factual knowledge and procedural skill) in his article from a few years ago in the AFT journal American Educator.

A procedure is a sequence of steps by which a frequently encountered problem may be solved. For example, many children learn a routine of “borrow and regroup” for multi-digit subtraction problems. Conceptual knowledge refers to an understanding of meaning; knowing that multiplying two negative numbers yields a positive result is not the same thing as understanding why it is true.

…knowledge of procedures is no guarantee of conceptual understanding; for example, many children can execute a procedure to divide fractions without understanding why the procedure works. Most observers agree that knowledge of procedures and concepts is desirable.

Willingham discusses the poor results for basic content and procedural knowledge, as revealed by trends in testing. However, he also notes –

More troubling is American students’ lack of conceptual understanding. Several studies have found that many students don’t fully understand the base-10 number system. A colleague recently brought this to my attention with a vivid anecdote. She mentioned that one of her students (a freshman at a competitive university) argued that 0.015 was a larger number than 0.05 because “15 is more than 5.” The student could not be persuaded otherwise.

Another common conceptual problem is understanding that an equal sign ( = ) refers to equality—that is, mathematical equivalence. By some estimates, as few as 25 percent of American sixth- graders have a deep understanding of this concept. Students often think it signifies “put the answer here.”

Here is a lovely paper expanding upon the issue of misconceptions in arithmetic, from a British article for teachers (hence the word “maths” and the spelling of “recognise”):

1. A number with three digits is always bigger than one with two
. Some children will swear blind that 3.24 is bigger than 4.6 because it’s got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the ‘digits’ rule does work.

2. When you multiply two numbers together, the answer is always bigger than both the original numbers
. Another seductive ‘rule’ that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word ‘times’ we can always substitute the word ‘of.’ So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.

3. Which fraction is bigger: 1/3 or 1/6?
 How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.

4. Common regular shapes aren’t recognised for what they are unless they’re upright
. Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the ‘usual’ position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there’s no such thing as a diamond! It’s either a square or a rhombus.

5. The diagonal of a square is the same length as the side?
 Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.

6. To multiply by 10, just add a zero
. Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the ‘just add zero’ rule wherever it rears its head.

7. Proportion: three red sweets and two blue
. Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they’re comparing blue to red, not blue to all the sweets. Always stress that proportion is ‘part to whole’.

8. Perimeter and area confuse many kids
. A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.

9. Misreading scales. 
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.

From Teachers: January 2006 Issue 42 UK (alas, the link no longer works)

A definition of conceptual understanding. In light of the confusion about conceptual understanding and the pressing problem of student misunderstanding, I think a slightly more robust definition of conceptual understanding is wanted. I prefer to define it this way:
Conceptual understanding in mathematics means that students understand which ideas are key (by being helped to draw inferences about those ideas) and that they grasp the heuristic value of those ideas. They are thus better able to use them strategically to solve problems – especially non-routine problems – and avoid common misunderstandings as well as inflexible knowledge and skill.
In other words, students demonstrate understanding of –

1)   which mathematical ideas are key, and why they are important

2)   which ideas are useful in a particular context for problem solving

3)   why and how key ideas aid in problem solving, by reminding us of the systematic nature of mathematics (and the need to work on a higher logical plane in problem solving situations)

4)   how an idea or procedure is mathematically defensible – why we and they are justified in using it

5)   how to flexibly adapt previous experience to new transfer problems.

A test for conceptual understanding. Rather than explain my definition further here, I will operationalize it in a little test of 13 questions, to be given to 10th, 11th, and 12th graders who have passed all traditional math courses through algebra and geometry. (Middle school students can be given the first 7 questions.)
Math teachers, give it to your students; tell us the results.
I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.

1)   “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)

2)   “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

3)   You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

4)   Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002

5)   “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

6)   A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

7)   Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

8)   Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.

9)   As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.

10)  Why were imaginary numbers invented? [EXTRA CREDIT for 12th graders: Why was the calculus invented?]

11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)

12)   “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.

13) “In geometry we assume many axioms.” What’s the difference between valid and goofy axioms – in other words, what gives us the right to assume the axioms we do in Euclidean geometry?

Let us know how your kids did – and which questions tripped up the most kids – and why, if you discussed it with them.
Thanks to reader Max Ray for pointing out a few TEACHER answers to the test!

A handful of math teachers & mathematicians (so far) have taken up the challenge posed by your 13 questions, answering them for ourselves before asking students to dive in, so that we have a sense of what we might want to hear from kids.

Here are the ones I know of so far:

And here is a nice commentary from one of our AE math consultants, Rita Atienza: Atienza math comment.
And here is a great summary as to the ability to use the lack of definition of points, lines, and plane to make valid hyperbolic proofs that reflect Euclidean assumptions (hence, the validity of hyperbolic geometry:
PS: I also had a nice phone conversation with friend and former HS student(!) Steve Strogatz, the celebrated mathematician-author about the test. He reminded me of two test questions that I should have asked (and that he and I have previously discussed):
True or false: .999999… = 1
Explain why a negative times a negative = a positive.
Steve also pointed me to a cool example of the value of undefined terms (beyond the one I taught him years ago by Poincare, in which a plane is imagined as an enclosed circle, used to prove the relative validity of one branch of non-Euclidean geometry) using the children’s’ game Spot It.
A postscript for geeky readers of my blog, and for fans of E D Hirsch’s work who have been critics of mine in the past re: Knowledge:
I have been surprised to discover that there are a whole bunch of smart, literate, and learned teachers who seem to deny that (conceptual) understanding even exists as a goal separate from knowledge – and by extension that my work and the work of many others is without merit. To them – as to E D Hirsch, it seems – there is only “Knowledge.” This, despite the fact that the distinction between knowledge and understanding is embedded in all indo-European languages, has a pedigree that goes back to Plato and runs through Bloom’s Taxonomy; the National Academy of Science publication How People Learn; and is the basis of decades of successful work in understanding by Perkins, Gardner, the research in student misconceptions in science, and the research on transfer of learning.
Some of my critics regularly cite Willingham’s summary of educational research, and a paper by Clark, Kirschner, and Sweller (discussed below; Dan Meyer has a link to all the key papers and rebuttals here. And thanks to a blog reader, I was led to the articles related to the debate on the USC web page (Clark’s University); scroll to the bottom) to make clear that direct instruction leading to knowledge is the only way to frame the challenge of both aim and means in effective education. As I will show, I believe they overstate what the research actually says and have little ground for suggesting that there is no meaningful difference between knowledge and understanding.
Wilingham on conceptual understanding in math. First, let’s look more closely at what researcher Daniel Willingham has to say about conceptual understanding in mathematics. His article is based on the idea that successful mathematics learning – presumably generalizable to all learning – requires three different abilities that must be developed and woven together: control of facts, control of processes, and conceptual understanding. And throughout the article he discusses not only the importance of understanding – and how it is difficult to obtain – but also notes that instruction for it has to be different than the learning of basic skills and facts. I quote him at length below:

Unfortunately, of the three varieties of knowledge that students need, conceptual knowledge is the most difficult to acquire. It’s difficult because knowledge is never acquired de novo; a teacher cannot pour concepts directly into students’ heads. Rather, new concepts must build upon something that students already know. That’s why examples are so useful when introducing a new concept. Indeed, when someone provides an abstract definition (e.g., “The standard deviation is a measure of the dispersion of a distribution.”), we usually ask for an example (such as, “Two groups of people might have the same average height, but one group has many tall and many short people, and thus has a large standard deviation, whereas the other group mostly has people right around the average, and thus has a small standard deviation.”). [emphasis added]

This is also why conceptual knowledge is so important as students advance. Learning new concepts depends on what you already know, and as students advance, new concepts will increasingly depend on old conceptual knowledge. For example, understanding algebraic equations depends on the right conceptual understanding of the equal sign. If students fail to gain conceptual understanding, it will become harder and harder to catch up, as new conceptual knowledge depends on the old. Students will become more and more likely to simply memorize algorithms and apply them without understanding.

Yet, for some reason, critics fail to accept this distinction – or see the inherent paradox, therefore, in education (discussed below). Novices need clear instruction and simplified/scaffolded learning, for sure. But such early simplification will likely come back to inhibit later nuanced and deeper learning – not as a function of “bad” direct teaching but because of the inherent challenge of unfixing earlier, simpler knowledge.
Perhaps part of the problem are the either-or terms that some researchers have used to frame this discussion. The essence of the false dichotomy is contained in Clark, Kirschner, and Sweller. Here is the introduction to the paper:

The goal of this article is to suggest that based on our current knowledge of human cognitive architecture, minimally guided instruction is likely to be ineffective. The past half-century of empirical research on this issue has provided overwhelming and unambiguous evidence that minimal guidance during instruction is significantly less effective and efficient than guidance specifically designed to support the cognitive processing necessary for learning.

The authors suggest, in other words, that evidence-based research shows that so-called “constructivist” i.e. “discovery” views of teaching are wrong on two counts:

  1. The authors claim that those who use discovery/problem-based/project-based learning – all unhelpfully lumped together as one thing by the authors – have confused the cognitive meaning “constructivism” (a correct psychology theory of how minds make sense of data) with “constructivist teaching” (an unsubstantiated theory of how people best learn).
  2. The authors claim that this inappropriate view of inductive pedagogy confuses the needs and traits of the expert with that of the novice:

Another consequence of attempts to implement constructivist theory is a shift of emphasis away from teaching a discipline as a body of knowledge toward an exclusive emphasis on learning a discipline by experiencing the processes and procedures of the discipline (Handelsman et. al., 2004; Hodson, 1988). This change in focus was accompanied by an assumption shared by many leading educators and discipline specialists that knowledge can best be learned or only learned through experience that is based primarily on the procedures of the discipline. This point of view led to a commitment by educators to extensive practical or project work, and the rejection of instruction based on the facts, laws, principles and theories that make up a discipline’s content accompanied by the use of discovery and inquiry methods of instruction. The addition of a more vigorous emphasis on the practical application of inquiry and problem-solving skills seems very positive. Yet it may be a fundamental error to assume that the pedagogic content of the learning experience is identical to the methods and processes (i.e., the epistemology) of the discipline being studied and a mistake to assume that instruction should exclusively focus on methods and processes.

In sum, those who promote “discovery” or “unguided” learning make two big mistakes, unsupported by research, say the authors: effective learning requires direct, not indirect instruction. And the needs of the novice are far different than the needs of the expert, so it makes little sense to treat novice students as real scientists who focus on inquiry. (Even though the authors offer the aside that “a more vigorous emphasis on the practical application of inquiry and problem-solving skills” is a good thing.)
But: huh? In 30 years of working with teachers I know of no teacher – secondary school or college – who rejects the teaching of “scientific facts, laws, and principles.” Indeed, science classes in HS and college universally are loaded with instruction, textbook learning, and testing on such knowledge.
Here is what the Clark et al. say in a follow-up article in American Educator:

Our goal is to put an end to the debate (about direct vs discovery learning). Decades of research clearly demonstrate that for novices (comprising virtually all students) direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn. [emphasis in the original]

What a curious definition of “novice”! The “novice” category is stretched to include “virtually all students.” This is surely a sweeping overstatement – much like the sweeping categorization of all non direct-instruction pedagogies as “discovery” that has been so criticized by others. We quite properly expect older middle and high school students, never mind college students, to do extensive self-directed and inductive work in reading, writing, problem solving, and research because they are no longer novices at core academic skills. Indeed, here is research with college science students that counter their argument.
Indeed, later in the article, the authors strike a somewhat different pose about the complete repertoire of pedagogies needed by good teachers:

…[T]his does not mean direct, expository instruction every day. Small group work and independent problems and projects can be effective – not as vehicles for making discoveries but as a means of practicing recently learned skills. [emphasis in the original]

Though this properly expands the list of effective instructional moves, their framing is odd – and telling. The purpose of non-routine problem-solving, making meaning of a new text, doing original research, or engaging in Socratic Seminar they say is to “practice” recently learned “skills.” Hardly. These approaches have different aims, understanding-related aims, that are never addressed in their paper.
Indeed, this is just how conceptual and strategic thinking for transfer must be developed to achieve understanding: through carefully designed experiences that ask students to bring to bear past experience on present work, to connect their experiences into understanding. As Eva Brann famously said about the seminar at St. John’s College, the point of student-led discussion is “not to learn new things but to think things anew.” Indeed, Willingham’s warning about “not pouring concepts into a student’s head” when understanding is the goal is the important advice that is constantly overlooked by the authors and their supporters as the focus is overly-narrowed to teaching skill via direct instruction.
The authors even tacitly acknowledge this later in the article, in discussing why what works for novices doesn’t work for “experienced learners” in a subject – and vice versa:

In general, the expertise reversal effect states that “instructional techniques that are highly effective with inexperienced learners can lose their effectiveness and even have negative consequences when used with more experienced learners.” This is why, from the very beginning of this article, we have emphasized that guidance is best for teaching novel information and skills. This shows the wisdom of instructional techniques that begin with lots of guidance and then fade that guidance as students gain mastery. It also shows the wisdom of using minimal guidance techniques to reinforce or practice previously learned material.

Well, which is it, then? Are “virtually all” students “novices” or not? When does a gradual-release-of-responsibility kick in? Just when is a student “gaining mastery” enough to use more inferential methods? We know the answer in reading: in middle school, based on the “gold standard” controlled research of Palinscar and Brown – that the authors mention in the citations!
Willingham in fact concludes his article by questioning the very novice-expert sequence laid out by Clark, Kirschner, and Sweller when the goal is conceptual understanding. After describing the “caricatures” in the math-wars debate of “process” vs. “conceptual” knowledge, he says:

Somewhat more controversial is the relative emphasis that should be given to these two types of knowledge, and the order in which students should learn them.

Perhaps with sufficient practice and automaticity of algorithms, students will, with just a little support, gain a conceptual understanding of the procedures they have been executing. Or perhaps with a solid conceptual under- standing, the procedures necessary to solve a problem will seem self-evident.

There is some evidence to support both views. Conceptual knowledge sometimes seems to precede procedural knowledge or to influence its development. Then too, procedural knowledge can precede conceptual knowledge. For example, children can often count successfully before they understand all of counting’s properties, such as the irrelevance of order.

A third point of view (and today perhaps the most commonly accepted) is that for most topics, it does not make sense to teach concepts first or to teach procedures first; both should be taught in concert. As students incrementally gain knowledge and understanding of one, that knowledge supports comprehension of the other. Indeed, this stance seems like common sense. Since neither procedures nor concepts arise quickly and reliably in most students’ minds without significant prompting, why wouldn’t one teach them in concert?

Indeed. Sequence in learning is not at all settled, as Clark et al profess, when the aim is understanding as opposed to basic skills to be learned the first time.
The key to understanding understanding: the ubiquity of persistent misunderstanding. Ultimately, a key lacuna in the everything-is-knowledge-through-direct-instruction view is its inability to adequately explain student misconceptions and transfer deficits that persist in the face of conventional direct teaching in science and mathematics. A glaring weakness in the Clark, Kirschner, and Sweller paper is their one-sentence treatment of student misconceptions: they suggest that misconceptions are the likely result of allowing students to discover concepts and facts for themselves!
This is surely a slanted view. There is a 30-year history of research in science and math misconceptions that shows conclusively that traditional high school and college direct instruction leads unwittingly to persistent misconceptions, and that a more interactive concept-attainment approach works to overcome them.
Multiplication is not repeated addition. The equal sign does not mean “find the answer.” Then, why is this a near-universal misunderstanding of these ideas? Presumably as a result of teachers not teaching for conceptual understanding and failing to think through the predictable misunderstandings that will inevitably arise when teaching novices the basics in simplified ways. Teaching a concept as a fact simply does not work, as Willingham notes.
The paradox of education. What these examples beautifully indicate is the paradox of teaching novices that so many knowledge-centric educators seem to overlook. Yes, we must simplify and scaffold the work for the novice and make direct instruction clear and enabling – but in so doing we invariably sow the seeds of misconceptions and inflexible knowledge if we do not also work to attain genuine understanding of what the basics do and do not mean.
Indeed, the success of Eric Mazur’s work at Harvard and with other college faculties, and the Arizona State Modeling project in physics, both backed by more than a decade of research in college and high school science, cannot be understood unless one sees the connection between conceptual understanding and transfer, and the failure of transfer to occur when there is just factual and procedural instruction.
In fact, a telling comment made by Barak Rosenshine, a leader in direct instruction, that DI has a more limited use than Clark et al acknowledge:

Rosenshine and Stevens concluded that across a number of studies, when effective teachers taught well-structured topics (e.g., arithmetic computation, map skills), the teachers used the following pattern:

Begin a lesson with a short review of previous learning.

    • Begin a lesson with a short statement of goals.
    • Present new material in small steps, providing for student practice after each step.
    • Give clear and detailed instructions and explanations.
    • Provide a high level of active practice for all students.
    • Ask a large number of questions, check for student understanding, and obtain responses from all students.
    • Guide students during initial practice.
    • Provide systematic feedback and corrections.
    • Provide explicit instruction and practice for seatwork exercises and monitor students during seatwork.

[emphasis added]

Rosenshine is far more careful than Clark et al to clarify the meaning of the term “direct instruction” which he claims has five different meanings that need to be sorted out. In fact, he notes that reading comprehension is a different kind of learning task than developing straightforward skills, and thus requires a different kind of direct instruction – instruction in cognitive strategies:

Even though the teacher effectiveness meaning was de­rived from research on the teaching of “well-structured” tasks such as arithmetic computation and the cognitive strategy meaning was derived from research on the teach­ing of “less-structured” tasks such as reading compre­hension, there are many common instructional elements in the two approaches.

In most of these studies students who received “direct instruction” in cognitive strategies significantly outper­formed students in the control group comprehension as assessed by experimenter-developed short answer tests, summarization tests, and/or recall tests.

(Note, therefore, that DI offers no justification for the kind of “direct instruction” done by ineffective high school and college teachers – i.e. too much teacher talk. DI is a method for learning and applying skills.)
Here we see the paradox, more clearly: no one can directly teach you to understand the meaning of a text any more than a concept can be taught as a fact. The teacher can only provide models, think-alouds, and scaffolding strategies that are practiced and debriefed, to help each learner make sense of text. Otherwise we are left with the silly view that English is merely the learning of facts about each text taught by the teacher or that science labs are simply experiences designed to reinforce the lectures. As I noted here, Willingham argues that teaching cognitive strategies are beneficial in literacy – in contrast to Hirsch’s constant and sweeping complaints about the lack of value in teaching such strategies and asking students to use them.
Interestingly, in an interview Rosenshine seems a bit insensitive to the problem of inflexible knowledge in less able students who need to rely on initial scaffolds for a long time:

Rosenshine: “Cognitive strategies” refers to specific strategies students can use to provide a support in their initial learning. For example, in teaching writing there is a cognitive strategy called the five-paragraph essay. The format for this essay suggests that students begin with an introductory paragraph containing a main idea supported by three points. These points are elaborated in the next three paragraphs, and then everything is summarized in the final paragraph.

After describing a lesson on Macbeth in which the essay template and DI are used, Rosenshine says:

The teacher told me he used this same approach with classes of varying abilities and had found that the students in the slower classes hung on to the five-step method and used it all the time. Students in the middle used the method some of the time and not others, while the brighter students expanded on it and went off on their own. But in all cases, the five-step method served as a scaffold, as a temporary support while the students were developing their abilities. [emphasis added]

I find this an ironic comment since I have often written about the English test item in Massachusetts in which 2/3 of all 10th graders could not identify a reading as an essay because “it didn’t have 5 paragraphs.” It is precisely the paradox of the inflexibility and over-simplification of well-scaffolded novice knowledge that has to be aggressively addressed if understanding (what an essay is as a concept) and transfer (recognizing that this is an essay, even though its surface features are unfamiliar) are to occur.
How hard would it be to show weaker students a 3- and a 9-paragraph “essay” as well as a 5-paragraph essay, all on the same topic; and then ask them to explain what makes an essay an essay, regardless of surface structure? Indeed, this is just the kind of scaffold for inferring a concept that lies at the heart of teaching for understanding: concept attainment and meaning-making via examples, non-examples, and guided inferences – mindful of prior learning experience (and likely misunderstanding). Not at all the same as “discovery learning” and hit or miss “projects.”
Yes, the research is clear: direct instruction is better than “discovery learning” when the aim is brand new unproblematic knowledge and skill and when contrasted with “students discovering for themselves core facts and skills.” But this is a very cramped argument. And it simply does not follow from it that all important learning occurs through direct instruction or that knowledge = understanding. Indeed, as Plato said 200 years ago, learning for understanding is “not what is proponents often say it is, that is the putting of sight into blind eyes. Rather, it is more like turning the head from the dark to the light…”
PS: Rosenshine offers a very different take on the issue that so motivated Clark, Kirschner, and Sweller, i.e. the link between the practice of experts and the pedagogy that supports developing expertise. He laments our failure to pursue the pedagogical question of how novices become experts:

Rosenshine: One very promising area of teaching research has been to compare the knowledge structures of experts and novices. For example, the experts might be professors of physiology and the novices might be interns or graduate students. Or the experts could be experienced lawyers and the novices were first-year lawyers.

What the researchers consistently found was that the experts had more and better constructed knowledge structures and they had faster access to their background knowledge. These findings occurred in diverse areas such as in chess, in cardiology, chemistry, and law. They also compared expert readers with poor readers and found that the expert readers used better strategies when they were given confusing passages to read.

A lot of expert-novice research was done from the mid-1980s until about 1992, but then it stopped. I would have hoped they would have gone on to ask questions such as, “What sort of education should novices go through in order to become like experts?” and “What does creating expert knowledge mean for classroom instruction?”

But, unfortunately, the research was never used to develop an instructional package for training experts. It was never used to establish instructional goals for classes to teach all children to be like the experts. Our goal should be to develop experts, and we’re not doing it.

A postscript to the initial critics of the post. No, I have NOT made a category mistake. Knowledge is necessary but not sufficient for understanding; understanding is not a direct function of knowledge. Understanding is the result of a deliberate attempt to make meaning of and connect one’s discrete experiences, effects, as well as knowledge and skill. Similarly, performance is more than the sum of skill; it requires judgment and strategy. That’s why there are three types of performance achievement, not two – declarative, procedural, and conditional. Some students (and players), with limited knowledge, have great understanding; some students (and players) with extensive knowledge and skill have little understanding (as reflected in questions/tasks that demand transfer). All of us have experienced such contrasts. You explain them, then. And also please explain the transfer deficit and misconception literature while you’re at it. Then we’ll talk further.



61 Responses

    • Right you are – I’ll fix. PS: I am not making a category mistake. Knowledge is necessary but not sufficient for understanding – period. And since it is possible to have great understanding with limited knowledge and no understanding with great knowledge, we are talking about two different cognitive processes and outcomes.

      • I would reject the notion that you can have great understanding with limited knowledge. There is a strong argument against this proposition in chapter 2 of “Why don’t students like school,” by Dan Willingham.

          • geology, though, was the quote. And if you read the chapter it’s clear what he meant and it relates to Kuhn’s paradigm. The geologists could not/would not see the implications of their own knowledge in terms of developmentalism.

          • Fair enough. I stand corrected. However, I also suspect that Darwin’s knowledge of geology was better than most too. He probably had a good idea of the different rock types and and what the word ‘strata’ means etc. whilst not being a world expert. It is exactly these sorts of facts that are required to think within a discipline and it is just these sorts of facts that are often disparaged. We expect children to somehow think deeply without them. It is surprising just how keen educators are to diminish the role of knowledge. I see endless quotes misattributed to Einstein on similar lines.

  1. Can I offer an example of evidence that conceptual understanding exists?
    My son, who is 7, with no formal procedural instruction in fractions, is able to add familiar fractions with a like denominator together, and decompose those fractions into components. He understands what it means to add together objects, and so is able to generalize his understanding of addition to adding fractions together.
    See this transcript of a conversation he and I had together a year ago when he was 6.
    One thought I have after reading your post is that in my work with teachers I need to check for understanding of the phrase “conceptual understanding.” I’m curious about what works to build a different model of what it means to know something for teachers who are fixated on a ‘there is only procedural knowledge’ mindset.

    • David, in my experience, much of the conceptual understanding model prevalent throughout much of Common Core is intimidating to many teachers. After all, it means admitting that they have not been teaching with the best approach and will probably need further support in reaching the conceptual level themselves using their own content.
      In short, depth is tough.

  2. The citing of the 2006 Kirshner, Sweller, Clark article has become a good barometer for identifying folks with naive understandings of educational research and learning theories. They may not even be aware that several rebuttals were published in the very same journal, or that the same authors are also against the use of videogames in education, too:
    Unfortunately one of those folks was very active on Wikipedia, biasing several educational articles:
    Interpreting educational research and critiques is itself a difficult skill to learn. I’ve noticed grad students tend go through phases, first believing anything they read, then rejecting everything else, and finally trying to take a more balanced point of view and look for the advantages and limitations of different research findings and theoretical frameworks. If you’re not aware of ANY criticisms or limitations of something or any positives, then maybe you don’t have a good conceptual understanding, then maybe you don’t understand it well yet:

  3. Hattie makes mention of the seven characteristics of experts vs novices in Visible Learning and the Science of How We Learn. He also notes that students can spend years performing activities successfully without increasing their skills to expert levels.

  4. I’m a bit confused, although flattered to be called “learned”. I do not reject your work at all. I may have confused things by getting into the middle of the discussion. And certainly, I conflate the difference between “constructivist teaching” and “constructivism” in my writing, because I’m lazy, but not in my head, because my head’s better at keeping things straight.
    I do not in any sense think that conceptual understanding exists as a goal separate from knowledge. In my experience, kids forget most of what they are taught no matter the method, so I want them to have the sense of understanding, which can only come if they engage with the concept and tasks directly, not just because I told them.
    Perhaps this is unnecessary, but I’m bothered enough that you think I don’t respect your work that I’ll try to explain my pov:
    1. Picture a debate between Jo Boaler and Harry Webb. I’d be on Harry’s side all the way, even though we don’t have similar teaching styles or values.
    2. In a discussion between, say, Dan Meyer and Harry Webb, I’d come down in the middle but nearer Harry than Dan. Dan’s lessons are far too open-ended for my tastes, and if you examine the reasons why, I’ll sound a lot like Harry. But I actually have a lot in common with Dan. I just don’t think he’s spent enough time with low ability kids to understand their issues, and isn’t (in my view) cognizant that his lack of experience is relevant.
    3. In a debate between you and Harry, I’d come down much nearer you. My big issue with you, as I’ve written, is that your examples are all very Jo Boaler squish, but your reasoning is all very solid. So I have to ignore your examples and just focus on the substance. It took me several years to figure this out.
    I think conceptual understanding is essential to my teaching. However, for many of my students, remembering the procedures is *not* the simplistic task that many portray it. For example, multi-step equations–many of my algebra I kids can explain and understand distribution, combining like terms, and isolation with a fair degree of conceptual knowledge. But put all the steps together and they become disoriented. Focusing on procedure to help them put the concepts in action is essential. So yes, as you say, both are needed. But worked examples and lots of practice doesn’t mean that teachers are overly focused on procedures, nor does it mean they’ve ignored concepts.
    As for your test, my kids would do pretty well. We go through division by 0, I teach proofs through algebra, not geometry, I don’t even know the answer to the 13th one, though. But then, I’m an English major. I’ll give it to them and report back.

    • Thanks for this. I appreciate your candor and clarity. I’ll have more to say, i think, when the dust settles. First, I want to see some test results!
      PS: The answer to the 13th question – and that was a VERY inside joke to those who have played music with me (Seatrain, 70s) -runs something like this: we have the axioms we do in order to prove the theorems we want. Famously, the parallel postulate had to be added to make all the key theorems work. Then, it was discovered that if we try out alternative postulates of parallelism, we get VALID non-Euclidean geometries….
      It’s like baseball. Once we decide how the game should be played, we make the rules fit the “spirit” of the game. Hence, the famous pine-tar bat ruling by the AL Commissioner, overturning the umpires’ decision against George Brett’s home run….
      It’s like the first 10 Amendments to the US Constitution…
      A goofy axiom would lead to nonsense or internal contradictions. Famously, mathematicians who denied the parallel postulate as part of investigating its validity decided prematurely to end their investigations because they thought the new theorems were absurd (e.g. no similar figures, only congruent; no 180 degrees in all triangles, etc.) But it took Gauss, Bolyai, and Lobascevsky to say: hey just ’cause the results are weird doesn’t mean they are nonsense – and so, non-Euclidean geometry was born.
      Great history of this and the importance of this line of argument in Morris Kline’s book on the Loss of Mathemtical Certainty about 30 years ago.

  5. Came across this post via
    More discussion, over several posts, on the Kirschner, Sweller & Clark (2006) paper on my blog here
    My main problems with the KSC paper are that they over-extrapolate the conclusions that can be drawn from the data, overlook the reasons why some students fail to learn even with direct instruction, and, as Grant points out above, make sweeping generalisations about the nature of both direct instruction and ‘minimal guidance’ pedagogies.
    They also, for reasons I haven’t yet fathomed, omit any reference to the Baddeley and Hitch model of working memory that’s dominated the field for 4 decades. Harry Webb doesn’t think that matters, but I do – even if only on the basis of good scholarship.

  6. Thank you for the list of misconceptions. I am happy to learn that I teach “conceptually” as I anticipate possible misconceptions prior to teaching a new concept and then ask plenty of questions. However, I realized I had a misconception such as the the diagonal of a square not being the same size as its size! Oops 🙂 I also liked the step by step procedure of how to go about having a productive maths class. Excellent resource which I have shared with our maths teachers.

  7. This blog really hit “close to home.” We have just recently had this very conversation in my school district. Early primary teachers think the math curriculum is “too easy” and does not offer a “challenge” for their little guys. We are having a very difficult time convincing these teachers how important it is to engineer many learning opportunities for these young learners in order to build lifelong conceptual understandings around foundational number patterns and the intricacies of how numbers work together. Teachers are still in the mindset that they need to “cover” lots of ground in early math or parents think their children are not learning anything new. This blog made me realize the importance of educating both teachers AND parents about math being more than just memorizing formulas and rules.
    Thank you!

  8. Grant, a few disagreements:
    1). Regarding your “but, huh?”: some teachers think the primary evidence of learning is not in the knowledge obtained, but in, for example, the <a href= questions generated.
    2). Regarding your incredulity that the “novice” category is stretched to include “virtually all students”: I think the author’s intent is much more limited than how you’ve read it. They’re simply saying that almost all students are novices in what you’re currently teaching, though hopefully not in what you’ve already taught. They’re not saying that most students are novices for their entire lives. Would you disagree?
    3). I’m unconvinced by your test of conceptual learning. The division-by-zero question is a good example. Even if a student can state a good reason that division by zero is impossible, that doesn’t mean the student understands it. They could simply be paraphrasing what they’ve been told. Isn’t it easy to imagine a student who correctly explains why you can’t divide by zero, but is not able to tell you a number that makes y=3/(x +1) come out as undefined…or even, a student who gives a good explanation and then says that 3/0=0 a few days later? The only observable behavior that will show us whether the student understands it is whether the student transfers that knowledge correctly or efficiently in new situations. If transfer is the measure, then conceptual understanding gets entangled with memory, because a student’s probability of correctly transferring a concept is partly determined by her ease of recalling it.
    Similarly, to test whether Eric Mazur’s techniques are working, wouldn’t it be better to give students the Force Concepts Inventory test, rather than ask them to explain Newton’s laws? This doesn’t mean that asking students to explain things isn’t a good idea–generating explanations is great for learning, because the Generation Effect seems to be a valid cognitive principle.
    I realize I’m making a pretty sweeping claim here, but let me make it explicitly: asking students to explain things is great for their learning, but it doesn’t provide much evidence of their level of conceptual understanding. Please note that I’m not saying that conceptual understanding doesn’t exist.

    • Fair enough on the test – it was merely suggestive. Obviously, to take the Mazur example, the FCI makes good sense. But my demand for explanation was to link it to the C Core and to do what far too many math teachers do – use non-multuple-chcoei questions. But I wasn’t trying to design a valid & reliable test, just a suggestive one.
      As for the ‘novice’ point, I think their argument is ludicrous – so ludicrous they back off it later in the article.
      Something got lost in #1…

  9. Interesting that you think their novice point is ludicrous, because I see it as self-evident, which again makes me think you’re not interpreting their statement correctly. You ask, “Are ‘virtually all’ students ‘novices’ or not? When does a gradual-release-of-responsibility kick in? Just when is a student “gaining mastery” enough to use more inferential methods?” In cognitive load theory, the answer is biological: you are no longer a novice when the information you’re processing has made its way from your working memory to your long-term memory. Humans have very high limits on how much information they can process simultaneously from their long-term memory, and very low limits for how much they can process from working memory. As Kirschner, Sweller, and Clark (2006) say, we lose what’s in our working memory after about 30 seconds of inactivity, and we can only hold between 4-7 pieces of information there (like the 7 digits of a phone number).

    • But by hs every student should be able to read and write, so the cognitive load is diminished on those core skills.The 7-item rule about memory is also being misused. We have learned to chunk those items in reading and writing after many years of doing it. By your argument, no one should be bale to understand complex text when they read independently. Again, their argument is surely a stretch. An experience HS and college student is not a novice except if you think of the ‘content’ – which, ironically, is not the focus of their argument; their argument is based on research involving skills.

      • Ah, well in my reading, they ARE talking about ‘content’, not reading complex texts. For example, the biggest example they describe in the 2006 paper is medical students being trained to make diagnoses in a lab setting with minimal guidance–a task with little reading involved.
        Their argument is that the costs of inquiry instruction come from the way the student must search the scenario for the relevant information while simultaneously figuring out how to put that information together. It’s not about the demands that core academic processes like reading or listening impose on a student; it’s about the demands involved in asking, “What information here is relevant? What information is extraneous? What do I still need to find out?”.
        Here is the central quotation of their paper as I understand it: “Inquiry-based instruction requires the learner to search a problem space for problem-relevant information. All problem-based searching makes heavy demands on working memory. Furthermore, that working memory load does not contribute to the accumulation of knowledge in long-term memory because while working memory is being used to search for problem solutions, it is not available and cannot be used to learn. Indeed, it is possible to search for extended periods of time with quite minimal alterations to long-term memory (e.g., see Sweller, Mawer, & Howe, 1982)” (p. 77).
        If you’re interested, last night I wrote up up how I think Clark/Kirschner/Sweller 2006 can be applied to improve Dan Meyer’s Shipping Routes lesson on least common multiples. I think the post lays out why I think Sweller et al have made a useful contribution. My blog post is here:
        If you don’t have time to read it, I’ll just say here that my conclusion isn’t that inquiry is always bad, but that cognitive load theory helps us make inquiry lessons like Dan’s more effective.

        • I appreciate your comments and the citations. But cognitive load is not the issue; engaged, focused, and quality learning is the issue. I have worked with 4th graders who can answer the questions you pose. Nothing about the brain mechanics addresses the fundamental question: what is a good education?

          • When you say “I have worked with 4th graders who can answer the questions you pose”, I assume you’re talking about the questions, “What information here is relevant? What information is extraneous? What do I still need to find out?” If so, please let me clarify…those were just examples. Other similar questions include “Where am I right now in the problem-solving process? If I take this step, will I get closer to a solution?” Even when these are easy questions, the point is just that considering them takes up working memory, which inhibits the formation of long-term memories related to the content you’re studying. It has nothing to do with how easy the questions are.
            I get that you’re saying education is terrible when it’s reduced to accumulating skills. And that’s true: lessons that teach students to master a procedure don’t even expose students meaningfully to the conceptual understanding your blog post is about. So if your learning objective is deeper understanding, you have to do something richer. But within the universe of possible richer tasks that *do* engage students in conceptual thinking, being vigilant about cognitive load means your students will develop that understanding more successfully. Cognitive Load Theory is not just for skill development. For conceptual learning, it says that you have to stop students frequently along the way to help them make meaning out of what they’re doing and help them encode that meaning into their long-term memories. Being dismissive about cognitive load theory tends to produce inquiry activities in which students are either don’t make the desired discovery or do make it but are unable to remember it (or the logic underlying it) by the next class. The blog post I linked to above gives an example of restructuring an inquiry activity in light of cognitive load, not (I hope) turning it into crappy drill practice.

  10. Reblogged this on principalaim and commented:
    If you are following principalaim, I have shared some of my favorite educators, innovators, and creative thinkers. Among my favorites is Grant Wiggins, co-author of Understanding by Design. Wiggins, an authority on subjects dealing with assessment, student engagement, and the Common Core, he is constantly being sought out to answer many of the toughest question concerning the best practices in education. In Wiggin’s latest blog, he asked us to think about “what conceptual understanding in mathematics means and how best to use it to help students understand which ideas are key (relevant). Ultimately, it is critical that we understand how best to prepare our students to become mathematical thinkers, independent, and innovative creators.

  11. A handful of math teachers & mathematicians (so far) have taken up the challenge posed by your 13 questions, answering them for ourselves before asking students to dive in, so that we have a sense of what we might want to hear from kids.
    Here are the ones I know of so far:

    • Great! I’ll check it out and offer any comments that seem helpful. I agree that making it practical and more test-like is the next step (just from having read your intro to your attempt). As I noted in other comments, my examples were meant to be suggestive of issues – why does seomtthing work? What are common misconceptions? What assumptions need to be explained/justified in the system? etc.

  12. Have you seen the journal “Mind, Brain, and Education” from imbes – vol 6, #3, September 2012? The articles in this volume on math support your claims regarding teaching math for conceptual understanding. Good examples for the concept of fractions and variables.

  13. Again, thanks for the post. Since your post on “It’s time to retire E D Hirsch’s tired refrains” my colleagues and I have been discussing this rather ironic controversy and the hurdles involved for both students as well as teachers. I’ve found that there are two main disconnects for teachers who advocate knowledge and understanding are one in the same. Either they have the same misconceptions as our students due to their own learning experiences, or they have understanding, but have not reflected on how they have may arrived at this point, and so they have what you have called “the expert blind spot” in UBD. To be honest, I was one of the former before I became involved with the ASU modeling program where they gave me the FCI and exploited my own misconceptions. From there, I was able to build a new framework that relied less on schema, and more on experience; “to see how it operates or functions, what consequences follow from it, what causes it, what uses it can be put to” (Dewey) It was only then that I was able to move forward and advocate for my students understanding. I have found that those teachers in the latter category that have been closed to understanding can be opened when they see the impact of strategies that elicit understanding from their students.
    To this end, I have also been trying to figure out how to elicit students understanding to a higher degree. Of course, student discourse will naturally bring it out effectively. But on an independent level it was the use of your understanding rubric that really helped me clarify my expectations to students when they were writing independently. Although knowledge and understanding are two different things, the knowledge piece, I’ve found, can actually be a hindrance. Because so much emphasis has been placed on content acquisition students have been programed in schools to give vague, procedural descriptions based on content because that is all what knowing content requires. Once I used an explanation rubric (from the six facets of understanding) that was framed in the context of a content based question, but with understanding expectations for their explanation, I was able to get the responses from students that showed understanding of concepts. Showing these off a bit has, in turn, helped me advocate for the very point you are making in your blog to teachers in that latter group. I’m still honing this practice, but I wish to thank you for all your insights which have helped get me and my students here.

    • Keith, like you, have a description rubric really helped me clarify how to assess understanding. Knowledge and understanding are both necessary and with the dawn of the age of the internet, hopefully people will gain greater insight as to the differences between knowledge and understanding. Let’s hope that metacognition will increase and we will all realize how to dig deeper.

  14. Interesting article! It is indeed quite difficult to get concepts through to the younger minds, and educators need to find original ways to reach out to them. The issue is that many academic systems around world will focus on pure theoretical exercise, which might attract some pupils, but others will be much less able to follow. We interviewed Kalid Azad (, a math enthusiast who loves to create ‘simplified’ (again depending on the target) explanations. Conceptual understanding has to do with intrinsic interest of the subject, not just the work put into it.

  15. Our school has recently embraced the AIW framework to looking at teacher tasks, instruction and student work. Much of our interdisciplinary conversations involve talking about teaching about concepts, not topics. It appears that a subject like math can easily overlap those 2 words.
    Any suggetions on what true math concepts might be 7-12?

    • A concept is a model, theory, general principle – an idea, an inference that is used to explain and connect facts. Mathematics contains many, some of which are crucial for understanding: congruence, equality, linear or non-linear relationship, function, derivative, imaginary number, etc. 2 more complex concepts: internal consistency (which is why you can’t divide by zero but you can multiply by zero, etc.) A key concept in problem-solving: finding simpler equivalences. In short, many core concepts that kids often do not understand. So, the question becomes: what must be understood about these concepts for understanding to advance (vs. just calling them topics to be covered)? What is often misunderstood about these concepts that impedes understanding? You might also want to check out the article that Randy Charles wrote for the NCSM journal on big ideas in math about 5 years ago.

  16. your explanation on conceptual understanding is very rich but let me think critically i will get back to you later. thank you.

  17. Good article ! I loved the analysis – Does anyone know where my business would be able to get access to a blank a form form to work with ?

  18. I feel the need to negotiate with such an amount of knowledge as would facilitate conceptual understanding. It’d require delimitation of knowledge to such an extent where the onset of conceptualisation would seem coming in sight.

  19. Wow that’s true. I am a student and I feel that math is being wrongly thought. You don’t get to see the beauty and usefulness of a subject when you are told to solve questions in the beginning of class, for cramming formulas, methodology of solving etc… Its such a shame on the name of education.. Really, I wish I could get a teacher who had a conceptual understanding and respect for the subject he teaches, rather than seeing teaching as a source of “income”…

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