A couple weeks ago, I wrote about my brother Jeff happily declaring women to be objects, and the philosopher Frege’s attempts to untangle concepts from the objects to which those concepts may apply. The point, apart from the self-indulgent pleasure of reminiscing and of pondering such things, was to raise the question of the meaning and the value of categorisation: how do we distinguish this from that and what is the value in doing so? The broader purpose of that post was to provide a (gratuitous and elaborate) introduction to two posts on Ashman and his dismissal of conceptual understanding. And, here we are, with Post Number 1.
ASHMAN
A few months ago, Ashman wrote a post, Conceptual Understanding is a Myth.1 Ashman attacks “the hocus pocus of conceptual understand” as practised by advocates of inquiry learning. His immediate goal is to poke good fun at some hilariously bad questions, which supposedly test “conceptual understanding”: the questions are a mix of: basically standard procedural questions needlessly caked with the muddy language of meaning; and left field questions that are not clearly testing anything. And yay to Ashman. Whatever “conceptual understanding” might mean and whether or not there are better tests of it, there is zero doubt that the inquiry learning cult has fetishised “understanding” with no clear positives and at a massive cost to procedural fluency, to the main highway business of simply teaching kids how to do things. The specific nonsense that Ashman is attacking is part and part parcel of the inquiry twaddle. But, while I am happy here with Ashman’s targeting, I have issues with his means of attack.2
To begin, and although I share Ashman’s contempt for the conceptual questions that he has chosen to discuss, Ashman is being a bit indiscriminate in his contempt. One of the questions asks for the number of fractions between 
and 
, and a second question asks for an explanation of the “=” sign. These are both very bad questions and, as Ashman notes, the expected and accepted answers are likely to simply be the recitation of a fact or definition. Which is utterly pointless; they are examples of what one might refer to as rote understanding. If the questions are both bad, however, the intent of the questions is very different and this should be appreciated. The fraction question, pointing to the infinite divisibility of fractions, is important but it is simply a property of fractions, and a property naturally exhibited and learned by standard procedural practice. The concept of equality, however, the fact, for example, that 3 + 2 and 12 – 7 and 
and 
and 4.999… all refer to the exact same thing, is much more fundamental, is much more difficult to grasp and follows much less readily from procedural practice. One may still argue that the best way, and perhaps the only way, to grasp the concept of equality is by procedure, by equating enough things. Nonetheless, a weak understanding of equality is a notorious tar pit. Not all concepts are equal and not all bad conceptual questions are equally bad, at least in motivation.
More fundamental to Ashman’s post, and a more fundamental concern of mine with his post, is, as indicated by his title, Ashman’s querying the very meaning of “conceptual understanding”. Reasonably enough, Ashman asks how one might test students’ “conceptual understanding” and he suggests that the proponents of conceptual understanding perform a subtle glide, to instead focus upon conceptual knowledge, which is supposedly more readily tested.
I don’t know, and my brain quickly glazes over with such nuancing. I may feel some slight difference in connotation between “conceptual understanding” and “conceptual knowledge” but nothing that would alter the fundamental questions: to what might the phrase refer, is it important, and how might one teach it and/or test for it? In any case, granting the change of terminology, Ashman is doubtful of the distinction between conceptual knowledge and procedural knowledge:
What does ‘understanding’ look like beyond being able to do and say certain things? Therefore, it is the doing and saying that are measured and these look a lot like knowledge. How, exactly, is this different to procedural knowledge?
I think it is a fair question, although Ashman is also engaging in a tautological gotcha: if we are measuring what students are doing then we are measuring what students are able to do, so all we are gauging is the “doing”. (The “saying”, which is not typically part of formal measurement, I will address below.) Still, there is point there, and Ashman then goes to town on nonsense questions of the type noted above, which supposedly test for conceptual understanding/knowledge/whatever. He then ponders what might be better tests of conceptual understanding: if it is not simply the pointless recitation of a fact or definition then what might it be? Ashman suggests, and I agree, that if conceptual understanding means anything that can be tested then there must be something novel in the test questions and/or the scenarios of those questions. But Ashman is not convinced that there is a there there:
If you think that conceptual questions only work if students haven’t seen those question types before and if you think training students in methods to solve those specific questions is a form of cheating, you are really not grasping for the conceptual knowledge but for the transfer of learning. There is nothing structural about a question that makes it conceptual rather than procedural because a procedure—and sometimes even the recitation of a definition—may be used to solve both. No, it is the fact students have not seen this question type before that is important.
In brief, Ashman’s argument is (I think), if it is the novelty that makes the question conceptual than why not simply eliminate the novelty and make the question procedural? It is then not really a distinction between conceptual versus procedural, but rather of familiar versus unfamiliar. Moreover, if the tests of conceptual knowledge/understanding amount to the testing of mastered procedures in novel situations then what one is really attempting to capture is not some superpower knowledge/understanding but, rather, the transfer of learning, about which Ashman appears to have parallel scepticism.
I have some sympathy for what Ashman writes here, and in particular I cannot stand novelty for novelty’s sake. I detest “challenging mathematics” that is challenging simply by cheating, simply by hiding some key element until the very end, like a bad detective writer who only reveals the location of the pistol in the final chapter. If the scenario is standard and there is a standard procedure for dealing with that standard scenario then just damn well teach it.
Nonetheless, I think Ashman gets some things badly wrong and is throwing out the baby with the inquiry dishwater. I think there is much more to mathematical thought, and thus to its teaching and learning, than Ashman is here suggesting.
1) Paywalled, because obviously Ashman’s insights are too valuable to simply be given away.
2) I am trying to present Ashman’s position and arguments as accurately and as fairly as possible, but I am happy and keen for readers to suggest any small or large corrections.
McCALLUM
Bill McCallum is an American-based mathematician who has been very active in the maths ed world for a very long time. Amongst many other endeavours, McCallum was one of the instigators of the Klein Project, and he was the cofounder of Illustrative Mathematics, a non-profit K-12 curriculum project.3 McCallum wrote a post, What is conceptual understanding?, a friendly but critical response to Ashman’s post.
I wasn’t thrilled with McCallum’s response. Mostly, he provides historical background, of prior attempts to capture the ideas at play, which may (or may not) be interesting but doesn’t seem to help much in nailing down these ideas. McCallum’s key conclusion, however, is that Ashman’s transfer of learning involves a repositioning of conceptual understanding, rather than its refutation:
… conceptual understanding is the connected cognitive structure … that lets you handle problems you haven’t seen before. Ashman is right to point out that it’s hard to measure, but wrong to treat the measurement difficulty as evidence that it doesn’t exist.
Ashman declares conceptual understanding a myth and says we should measure transfer instead. But his own sources say that transfer is conceptual understanding measured. Ashman’s remedy detects the thing whose existence he denies. He hasn’t refuted it, he has renamed it.
So transfer of learning is the measurable consequence of conceptual understanding? Maybe. But I’m not sure what is the important claim that Ashman or anyone should be considering here. Who, in the end, cares? The finally important questions are what is to be taught, how it is to be taught and how the subsequent learning is to be tested. I do not see that McCallum has done anything to address Ashman’s justified scepticism on the teaching and the testing of the whatever-the-hell-you-want-to-call-it.
There is another part of McCallum’s post, however, which I found very interesting. He begins his post with a description of what it feels like as a mathematician to come to “understand” a mathematical proof:
If I’m trying to understand the proof of a theorem, there’s a stage where I am working through the steps, checking them carefully. If the proof is complex it’s easy to get lost, but you keep plodding. Then there’s a stage where you assemble all the pieces in your head, see how they fit together, understand the essential ideas. That’s conceptual understanding.
Is it? I am not sure, and McCallum is seemingly not sure:
Ashman dismisses it as “a feeling or impression.” I might call it a cognitive structure. But I don’t really care what you call it: it is consequential.
McCallum is correct. Whatever you want to call it, it is something, and it matters. It is a feeling that will be immediately familiar to a mathematician, anyone, who has battled with and defeated a (for them) difficult, many-step mathematical proof.4 It is absolutely fundamental to the way mathematicians learn new mathematics and, eventually, discover new mathematics.
McCallum wants to use the above to frame his later discussion, although I don’t think he is very successful, and mostly it just raises new questions. How, if at all, does mathematicians’ “understanding” of complicated proof relate to the “conceptual understanding” one wishes school students to gain?5 How does it lead to something one might test (or does that not matter)? More fundamentally, is a mathematician’s “understanding” of a proof a fundamentally different thing from their understanding of the individual formal steps of the proof?
I think McCallum’s mathematician reflection is important. I don’t think McCallum made good use of it.
3) McCallum was also involved in a 1990s undergraduate calculus reform project, to shift away from the traditional Stonehenge-sized textbooking; the reform approach was kind of a CPA for grown-ups. That reform was pushed at Stanford while I still had some connection there, and I remember the push-back and bad feeling. At the time, I was interested and open-minded about the reform but, in sum and in the end, I decided I didn’t like it.
4) University students may wage such battles in a good first year “honours” mathematics subject or in a good third year pure mathematics subject or, more often than not these days, never. MathsCraft is Anthony Harradine’s very noble attempt to bring this type of mathematical conquering to the masses.
5) McCallum refers to and links to a story about Max, a grade four who “discovers” the how and why of the borrowing technique for calculating 145 – 98. This is worth discussing but it requires discussing.
ASHMAN 2
Ashman responded to McCallum’s collegial post with an uncollegial follow-up post, but his second post adds little and Ashman takes a long time getting there. True, McCallum did not write much that required addressing, but Ashman took the freedom to meander far afield. He gives the too-familiar spiel on cognitive load, takes a drive-by swipe at Illustrative Mathematics,6 takes a whack at “eminent mathematics professors” who, “cursed by knowledge”, lack the empathy to appreciate the value of children completing exercises with a pencil and paper, and generally lords it over anyone foolish enough to believe that “understanding” might be a thing. When he puts in the effort, Ashman can be a right charmer.
Before finally addressing McCallum’s post, Ashman demonstrates his profound understanding of understanding with a discussion of the equality sign:
A good example of this knowledge—and one that has been heavily researched—is the idea that the equality sign means ‘the same on both sides’ and not ‘put the answer here’. However, this is relatively straightforward to learn and recite in comparison to learning how to apply this knowledge to solve linear equations. And it is not qualitatively different from other forms of knowledge. It is simply knowledge of a different and simpler kind.
So, equality is easy: simply ‘the same on both sides’, and we’re done. The same what on both sides? It is remarkable how dismissive Ashman can be of the importance of meaning and the not infrequent difficulty of grasping it.
Finally, Ashman gets to McCallum’s post, which consists mostly of a victory lap that McCallum would “concede” the inability to measure directly conceptual understand. Which Ashman then extrapolates into some “big call” of McCaullum’s on this or that claim of the productive struggle advocates.
Ashman is happier that he and McCallum are on the more solid ground of transfer, although this does not stop Ashman returning to whine about “conceptual anything” being a “value-laden term”: Ashman really does not care for meaning. But even on the solid ground of transfer Ashman is still concerned, because “[transfer] is not straightforward [to] measure”, and Ashman has difficulty in accepting the reality of anything he cannot measure. Presumably it should not matter, however, since “transfer is notoriously difficult to achieve”. Which is a bit of a puzzle: if transfer is difficult to measure then I am not sure how one can also know it is notoriously difficult to achieve. I could have sworn Ashman was the evidence guy but I guess I was wrong.
Ashman never refers to McCallum’s description of a mathematicians’ sense of understanding of a mathematical proof but this is no surprise. Ashman is skilled at ignoring matters to which he cannot well respond.
6) I know absolutely nothing about Illustrative Mathematics and it may well be as bad as Ashman suggests. But the badness or otherwise of IM is entirely irrelevant to the counterarguments McCallum was making. Ashman is simply shotgunning.
CONCEPTION AND CONCEIT
A teacher can never be certain what is going on in her student’s head. She can have very good clues, of course, by observing and questioning the student, and by assessing him. Better testing and more empathetic observation can provide much better clues. These less or more formal measures, however, are simply a better or worse guide to what the student “knows” or “understands”; the student’s actual knowledge or understanding is not precisely knowable or understandable by anyone else. But that does not mean that what is going in the student’s head does not matter. In the end, and with the required nod to pragmatism, it is the only thing that matters.
This fundamental barrier is not an excuse to make up stuff about what might be happening on the other side of the wall, to wax lyrical about the mental wonderland there. It is also not an excuse, however, to trivialise the other side, to act as if a students’ mind is nothing more than a neuroelectronic device that will provide certain responses to certain stimuli. Our students are neither mystical minds nor lab rats.
As always, teaching must be based upon plain common sense and solid empathy. More than anything, we know about our students because of our shared humanity: our shared way of thinking and our shared manner of reacting to fear and doubt and knowledge and understanding. Is little Johnny able to add fractions, and to what extent does he understand the meaning of fraction addition? Any teacher worth their salt will be able to ascertain the answers, as good as can be gotten, with five minutes of gentle querying.
That does not mean we do not require formal assessment, and of course we do. Not all teachers are worth their salt and even those worth much salt can get things wrong, and how is anyone else to know what a given teacher might be scoring on the saltometer? Mass education requires at least broadly agreed upon standards and broadly agreed upon measures of attainment of those standards. Measurement is an artifice but it is a necessary artifice.
What does this mean for Ashman and McCallum and knowledge and understanding and conception? To begin, let’s try to get away from the idiotic word games.
In general parlance, we might say that knowledge refers to discrete facts and understanding refers to the connections between those discrete facts. So one might know about individual battles of WWII, for example, and then understand how those battles related to each other, of how one front of the war played out. Could the “understanding” of those relations also be referred to as knowledge? Of course it could. Go ahead. It would seem odd, flat usage, but whatever.
Lets’s be considerate of Ashman’s knickers and refer to it all as knowledge. The critical point, the only point, is that knowledge, like onions and like ogres, has layers. That is the way that everyone thinks, about everything: WWII or an AFL game or fractions or whatever. We take individual bits of knowledge and connect them and combine them to create a deeper and more solid bit of knowledge. In turn, those more solid bits can be connected and combined, and on and on.
That was the point, I believe, of McCallum’s discussing of a mathematician’s sense of a proof. She begins with the various steps of the proof and eventually knows how they work step by step, but eventually there is some aha-ing: she has obtained a deeper knowledge, beyond knowledge of the steps, beyond how the steps they fit together, to knowledge of why they fit together. McCallum’s point, I believe, is that a student’s knowledge of fraction arithmetic, for example, can be deeper in the exact same manner. I agree. I cannot understand (know?) how anyone could disagree.
Can such deeper knowledge be measured? In some sense no, because of Ashman’s tautological gotcha: Ashman will define anything that is measured as a measurement of doing, and thus it’s all just more knowledge, to be added to the pile of bricks. Tautology aside, such deeper knowledge certainly cannot be measured by asking inane “how many fractions between” questions. I think at least something can be done, that one can test for more than Ashman’s brickpile of knowledge; that will broadly be the topic of Post Number 2. But even if deeper knowledge cannot be tested it seems obvious that such deeper knowledge is a thing and it is utter craziness to act as if it does not exist or that it is not important.
Ashman is hell bent on focussing upon what can be measured, which makes this obviousness very difficult. The more one atomises, the more one treats this or that fact or procedure as an isolated thing to be tested, the more removed one is from how the student, anyone, is thinking. That is not to argue against the testing of isolated facts and procedures. Like Ashman, I am mega-strongly for it. Seemingly unlike Ashman, I believe that one can do much more and much better but there is no question that the atoms really matter. But again, measurement is an artifice. Even when measurement is good, it is an artifice. It is not the thing and the deification of measurement distracts from the thing.
Of course, in practice, this deification of measurement is much much worse than simply being a distraction because measurement in Australia is very very bad. The Australian Curriculum is appalling and NAPLAN is appalling and the VCE, together with its brothers in most or all states, is appalling, made much worse by utterly cynical ATAR gaming. This appallingness requires everyone to focus unceasingly upon “succeeding” on LCD junk. Ashman knows all this and at times he claims to care but I do not believe him. Ashman knows damn well that he and his N-E-T cult profits greatly from this destitute testing and he is very happy for his cult to so profit.
Ashman and his brain mates occasionally get accused of scientism, which mostly, although not always, seems to me unfair. But Ashman, and the entire N-E-T gang, now seem to be engaging in something much worse. They seem to be going all in on full-blown measurementism. It is absurd and it is destructive.
I remembered that under one of Ashman’s posts, I had asked why a student can consider both ‘ 1 + 2 = 3 + 4 =7’ and ‘1 + 2 + 4 =7’ correct simultaneously XD
i’m not quite sure of your point, but am also guessing that Ashman did not respond.
I think the question was asked when he claimed that the understanding of the equal sign is just telling students “two sides are the same”.
And yes, you are right. He did not reply.
Ah, and absolutely no surprise.
I want to talk around a few things here. Your analysis, Marty, is excellent, of course. Ashman’s is less impressive. Why?
First, Ashman talks about “transfer of learning”. In Psychology 101 (1967) I learned that “transfer of learning” was an attractive idea in an era when students asked “Why do I have to study Latin” when I want to be a … (choose any job that doesn’t need Latin?” The teacher’s attempt to answer encouragingly is that by studying Latin you learn how to learn, and this ability to learn can be transferred to any other subject. (I had a Vice-Chancellor who said with a smile on his face that doing an undergraduate degree in Law was an excellent preparation for … (fill in your choice)”. (John Hay, I hope, has retired, but the damage he did on his way there was immense. But that is another story.)
Clearly Ashman’s “transfer of learning” is not this idea of “studying any topic” to develop some kind of learning-muscle. What Ashman is talking about is learning a skill / idea / procedure that can GENERALISE to other topics. As an example, studying quadratics (important and practical as they are, encompassing Pythagoras’s Theorem amongst other vital topics) develops a skill with abstract algebraic computation (“Solve …”, “Simplify …”, “Show expression What is equal to Expression Which) that generalises to simultaneous equations, cubics, polynomials, coordinate geometry, and almost all those areas of mathematics where we have variables (X, Y, Z, …) and coefficients (a, b, c, …) and numbers (0, 1, 2, Pi, e, sqrt-2, …), even complex numbers.
Second, the word “rote” trips people up. For many, “rote” means brute force memorising. (Trigonometry definitions: tan = opp over adj, …; minus by a minus is a plus; divide by a fraction by inverting and multiplying.) For others “rote” is a mechanical process that is applied in routine situations. The better word in this case is “algorithm”. (Addition, etc., with multi-digit numbers, …)
My definition of “rote learning” is when a student learns and can recite but cannot explain.
In fact, this kind of rote learning is essential. We must reply on rote learning, for example, when we are starting out, so we know this — A — is a letter, and it’s the first in the alphabet, and it looks like a tent, or it can look like — a –, and its letter-name is (say it) “Aye”, and it is the sound at the start of “apple”, but can also make the sound in the middle of “was”, and so on.
People talk about learning our A, B, Cs precisely because untl we do we can only learn by talking and listening and we can’t read and write.
The same applies to the first steps into mathematics, typically, either arithmetic, or geometry. It is only by rote learning that we learn that 1 is a number, and has a name, and means an idea, and is the start of that invaluable process of counting (on fingers, or with blocks) and that leads to arithmetical computation.
(Or, in geometry, learning to fit Word-1 with Diagram-Q and Word-2 with Diagram-H, etc.)
Without that foundation of rote learning very little happens, UNLESS we live in a pre-literate ORAL knowledge society. And oral knowledge depends on its own rote learning. (Pre-literate oral cultures can be very clever. I am not putting them down.)
Later we can go back and study the history of the alphabet, or of the Hindu-Arabic numerals, and then we can EXPLAIN contextual details about matters that first we learned only by rote.
In short, be very careful when anyone is tempted to speak ill of rote learning. And be sure you known what “rote learning” means, and what other kinds of learning (explainable) are not rote.
Third, the difference between “familiar” and “novel” is the heart of problem solving.
A mathematical task, or challenge, is a problem for a particular student only if that student does not immediately know how to handle this specific task, or this type of task, but is willing to try.
Without being willing it is not a problem. The student walks away without attempting anything.
But here’s the catch.
When / if / helped? the student succeeds in completing that task (gets an “answer”) the next time the student sees another task that seems similar the student already has an idea of how to tackle this new task. It may be new but it is not novel.
That is, problem solving is what we do with unfamiliar tasks while we are learning how to handle those tasks.
With creativity, teacher’s help, answers-at-the-back-of-the-book, worked-examples-in-the-textbook-or-on-the-black-or-white-board, experience, what were initially novel tasks — problems — become familiar practice exercises.
Fourth, through all of this, and through all of Ashman’s attempt to define, and qualify, and distinguish “knowledge” from “practice” (etc.; something the Ancient Greeks did with their word “praxis” — current mathematics education researchers are not inventing anything properly new, only adding their veneer of seemingly new, seemingly helpful technical terminology, when earlier terminology already did a pretty good job), and all the talk of assessment (measurement of learning / knowledge / conceptual understanding / skill / procedures …), and all the talk about teaching (direct instruction, inquiry learning, …), the vital thing is the unavoidable need for a teacher to talk with a student, and for the student to hear and respond, and for the teacher to further respond to what the student says …
Teaching, and learning, are two sides of a coin we could call “conversation”. This conversation can include writing words, letters, numerals, and lines of logical argument, as well as drawing diagrams, and lots of pointing with fingers and hand-waving and head-nodding. (Socrates was basically pre-literate, oral, in a way that Plato wasn’t, writing it all down. But the emphasis on conversation was the heart of Socrates teaching, and solving the questions his listeners asked. The Ancient Greeks were doing this pretty well a long time ago. Plus Euclid and all his diagrams that, in a Euclidean classroom is supported by lots of talking, and manipulation of diagrams, and hand-waving.)
When we understand that we also see that assessment (measurement of learning / knowledge / understanding / procedures) is a version of conversation. The assessment item might be a question on a NAPLAN test or an item on a General Achievement / Ability Test (GAT: or Scholastic Aptitude Test, SAT, in the USA), but it is, in essence, the teacher saying to the student, “What can you tell me about this?” and the student’s performance on that assessment item is the student’s “reply” — this scribble-sketch-label-computation — is what I can tell you about the question you just asked me.
Good teaching, and good learning is careful negotiated conversation between teacher and student, mediate by common sense on both sides.
Thanks you very much, John. Brief thoughts in reply.
(1) I am not exactly sure what “transfer of learning” is supposed to be, although the vague sense is clear enough. I am less concerned with what comes under which title than figuring out what is to be taught and learned, and how this is to be done well. The measurement is also a concern, and obviously a great practical concern for the community, but secondary for me.
(2) I’m not quite sure why you are raising and discussing the “rote” issue. It is certainly relevant to Ashmnesque debates, but it seems a little off the point here. It is on my (long) list to write about “rote”, and the various meanings ascribed to it, and the knee-jerk hostility that the term inspires.
(3) I think your third point is key, and offline you have pointed to me to some of your very interesting writing on this. I think there is a *lot* to untangle but that it is critically important to untangle.
(4) I have nothing to add, except that I find Ashman’s approach a distorted reflection of reality and unhelpful.
Personally, I think the mathematical experience is wonderfully described in Andrew Wiles’ famous quote. “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of – and couldn’t exist without – the many months of stumbling around in the dark that precede them.”
I think the experience is roughly the same at every level, though in school one usually only focuses on a problem for a few minutes, and it is not until you’re well into university that you spend a whole day thinking about a single problem. I don’t see how this process can be taught. We can only teach students how to feel around in the dark. And hopefully, that mathematics is a worthy task. That at its highest level, it produces knowledge that is handed down for future generations as a central part of human inheritance. But even at its most modest levels each light switch turned on is a rewarding personal achievement.
Thanks, Grant. I strongly agree that “aha” can be part of all school students’ mathematics education, and that it is really bleak to be purposefully designing education to preclude it.
Your pondering that perhaps this “process” cannot taught is interesting and, similar to John Gough’s third point, above, I think this is worth a lot of thought. I’m hoping to write something on this soon (more likely, “soon”).
Perhaps the best way to cut the Gordian knot is to admit that we do not really understand learning or conceptual understanding or whatever. (cf AI and what constitutes consciousness, intelligence etc). I loved the mansion analogy – that was fantastic!
A bit of humility goes a long way……
Aboriginal songlines as the result of tens of thousands of years of oral culture transmitted through song enabled First Nations people to flourish and travel across the continent.
What humans can do is trial and error with the results of the learning transmitted over the generations. If the Aboriginals could do this with a very sophisticated oral culture, then surely a written culture can work out how to teach properly.
But then come crazy fashions, followed by a reaction that thinks that science can tell us more than it can, rather than science largely confirming and refining traditional knowledge.
I don’t think this is cutting the Gordian knot. I think it is simply saying, “Look. There’s a knot.”
Can you expand on that please Marty? I was trying to say that what has worked over the decades is the solution with gradually trying other things and it’s pointless trying to pull the bits apart too much as it’s too complex – what do you think?
Hi, JJ. I was responding to your first paragraph.
The teasing out of knowledge-understanding-conceptualising is either important to teaching or it is not. I think that it is not, that a teacher can teach very well without worrying any of this stuff, and that such a teacher probably has a better chance of teaching well than the worrier. But, if this stuff is important to tease out, I don’t think it is an effective answer to simply declare “we don’t really understand”.
If we are faced with a difficult and genuinely unavoidable question, about anything, then we have no choice but to answer the question in some functional manner. We do so with humility, as you are suggesting, but the humility is to be consistently tentative and thoughtful about our answers, not there to simply throw up our hands.
Thanks for this post Marty, it’s something I have thought about off and on for some time. I agree that “understanding” is a real and significant thing in high school mathematics that we want our students to gain. I have not read Ashman’s or McCallum’s articles, but I think there are clear things we want our students to develop that (i) deserve to be called “understanding”, and (ii) aren’t just simplistic definitions like “equals means the same on both sides”, or the transfer of concepts / knowledge / procedures to an unfamiliar case.
There are many things which distinguish stronger from weaker students. But if I were to try to pin down the specifically cognitive ways in which the stronger students are stronger, I would highlight:
(1) More fine-grained resolution of problems to procedures. Stronger students are more sensitive to the details of problems and can identify that it be more effective to choose significantly different procedures to solve superficially similar problems. Weaker students are more flat-footed and try to shoehorn all problems of a similar type into the same solution method. I think this sort of thing is often called “conditional knowledge” – it’s knowing *when* to use a method. But I think greater understanding is displayed to the extent this conditional knowledge is more fine-grained.
(2) Built in error-correction / sanity checking. Stronger students, like anyone else, occasionally make mistakes when using a procedure, or inadvertently use an inappropriate procedure. But they spot these quickly because they understand the concept behind a procedure, or its intended use in the problem, and they recognise when it is giving silly results. Weaker students just continue on because that’s the procedure that is meant to solve that problem (so they think).
(3) Being able to come up with suitable examples / counterexamples to illustrate a concept. Stronger students, when they want to clarify something or are confused about something, they will be able to come up with good examples they can use to illustrate their confusion. Weaker students default to generic cases which don’t neatly illustrate their confusion, or they just resort to examples already covered in notes / textbook.
(4) Long-term retention. This is perhaps the most consequential, because the more transient the knowledge is, the more difficult it will be for students to develop the kinds of cognitive achievement in (1), (2), (3). Stronger students will more readily reproduce routine procedures in response to standard questions even after a fairly long interval between using that procedure or answering those kinds of questions. Weaker students require more prompting, reteaching, or reviewing. Although, I think we tend to think of long-term retention as not really “understanding” itself. But do think that students improve their prospects at long-term retention by – in the short time – attempting to build the sorts of cognitive achievements in (1), (2), and (3). So while long-term retention might not *itself* be the sort of thing we think of as “understanding”, I think it’s often causally downstream from understanding.
(5) Abstraction / pattern recognition. This relates to McCallum’s point about understanding-as-cognitive-unification, but his example is about unifying the propositions in a proof to form a coherent logical apparatus. The unification I have in mind here is seeing multiple examples as instances of a single mathematical concept, and so seeing that all can be treated similarly. Even if stronger students require the aid of a teacher to notice the generalisation, they will more readily adapt to treating different examples as instances of the same concept. For instance, they will quickly adapt to solving equations like 4^x – 2*2^x + 1 = 0 and sin^2x – 2sinx + 1 = 0 once it’s pointed out they are both quadratic. Weaker students will struggle to apply their existing knowledge to different cases, even when they are told they are instances of the same concept – they tend to be more easily misled by surface level differences.
I’m sure there are many more. Perhaps we could re-express all of these behaviours as just students becoming more “procedurally fluent”. Whatever. I agree that ultimately what matters is (i) what do we want students to learn, (ii) how do we teach that effectively, (iii) how do we effectively check that students have learned it. What I would take issue with is the idea that our mathematics education should only aim at humdrum computational skills because “understanding” is not a real thing or too hard to measure.
Thanks, SRK. This is an excellent response. I will read carefully and reply properly tomorrow.
Thanks again, STK. It seems odd to not refer to all you have noted as “understanding” but, whatever the word one wishes to use, it is something.
Playing Ashman’s advocate, how does one teach this, how does a student learn it, and how does one test it? How would any of this differ, in either approach or outcome, for a teacher not so concerned, for a teacher who figures the “understanding “, whatever it is, will take care of itself?
Hi Marty, sorry for the delayed reply.
I think a necessary condition for almost all students to learn this is (unsurprisingly) that they have sufficiently many and sufficiently varied well chosen examples and exercises. I think this is one of the critical things that often gets missed in a lot of the discussion around explicit teaching and the kinds of routines we want in our classrooms (eg. discussion around the kind and frequency of checks for understanding, cold-calling, mini-whiteboards, etc, etc.) The details of the examples + exercises we give our students matters a great deal, and that’s where teachers with a good understanding of the material becomes critical.
However I don’t think it is sufficient. I think it is a form of discovery learning if we simply leave students to work through these exercises (and even checking that they have done them) and “expect” that they will make the sorts of cognitive achievements I mentioned in my first comment. Many students will not succeed. I think teachers also need to model some of those cognitive processes as they do worked examples (eg. how do I decide what method to use to solve this quadratic equation, what features of the equation do I look at…; how can I tell that this problem is of the same type as this other problem I did earlier…) Other things are more straightforward in concept – ie. long term retention is built by regular review at increasingly spaced out intervals – although logistically this can be challenging to implement.
Edit: I think for a teacher not so concerned, as long as they have exercises that students can do, and assessments which test that students can complete similar questions to those on the exercises, then it’s all hunky-dory.
I’m not sure about testing.
Thanks, SRK.
I agree very strongly with your first paragraph (of course). I’m less convinced by your second paragraph, and I am not sure I know what it means.
I think the solving of quadratic equations is a very good example to consider, since it is not mechanical: one definitely needs to establish the intuition to do this well. If the class is genuinely tackling the topic properly and is not simply teaching this technique for that “type” then the teacher has to say something. And, hopefully the teacher will be saying something more useful than Lovell’s ludicrous mind map. But I’m not sure what “model some of those cognitive processes” means, and I’m not sure that the teacher saying anything much helps. I’m open to the idea, and I certainly say stuff when I’m teaching quadratics, but I’d need clarification and convincing.
Sorry for being unclear. I’ll use solving quadratic equations as an example to (hopefully) clarify my point.
Let’s say we have taught students a few different methods of solving equations (simple factoring over integers, completing the square, quadratic formula), one after the other. I would say that a further goal is for students to not only show fluency with these methods, but also develop some good judgment for deciding which method to use. How can teachers promote this amongst our students? My point above was that I don’t think it’s sufficient to give our students a miscellaneous set of quadratic equations to solve and say “have at it!” If this were to happen, a good proportion of students will just default to always doing quadratic formula, while others will try to always factor over integers and then give up when that doesn’t work. Slightly more successful students will start to develop some sort of reasonable strategy (“check if I can factor over integers, if not quadratic formula”) but it’s likely to be rather flat-footed. Only the strongest students will develop the finesse and flexibility in choice of method that we want.
So I think that teachers need to take a more active and explicit role in trying to build that ability. In this case, I think the teacher “saying anything” – in the sense of doing worked examples and narrating their thinking as they decide which method to use – does help. *Of course* that needs to be supported by a good range of examples that illustrates the points the teacher is making, and of course that should be accompanied by exercises that students can use to practice those strategies.
Thanks again, SRK. I don’t think you were really being unclear. I think, rather, that it is unclear what the teacher should say or do. As such, I think quadratics is an excellent example of the lack of guidance, and misguidance, that can be provided by the “explicit” lodestone.
I agree completely that learning properly to factorise/solve quadratics/equations requires much more than a student being able to handle this or that type when instructed to do so, or when they are all-but-instructed to do so. I also agree that simply providing a good selection of exercises is necessary but not sufficient, that many students will simply see a particular image of a nail and employ their particular hammer. But …
I agree that teachers must take “a more active and explicit role”. The nature of that role, however, is in ways unclear to me. I assume your use of the adjective “explicit” here is coincidental, and was not meant to allude to Ashmanism, but I think it is important to state clearly that the teacher role I believe you are imagining, and certainly the role that I am imagining, while clear-headed and purposeful, is not “explicit” in the Ashman sense.
This is all interesting enough that I think it is worth a dedicated post, just on quadratics, and I’ve put it on the list. I will just say one thing, echoing your thoughts and echoing a point made in the post: any senior teacher worth their salt will be able to ascertain how well a kid has mastered quadratics with five minutes of gentle querying. Maybe ten minutes. But the point stands.
Last night I shared with my BEd students the clip of Tom Leher’s “New Math”.
Thanks for deleting the video. Is New Math relevant to this post?
Was just reading this Terence Tao article and noticed he mentions that a primary goal of doing mathematics is to gain conceptual understanding and insight.
Very good spotting, Craig. Thank you. Although, to be fair, Tao is an eminent mathematics professor cursed by knowledge.
ps. I have been trying to avoid writing on the AI horror but clearly I have to write something.
Naming of Parts (1942)
Today we have naming of parts. Yesterday,
We had daily cleaning. And tomorrow morning,
We shall have what to do after firing. But to-day,
Today we have naming of parts. Japonica
Glistens like coral in all of the neighbouring gardens,
And today we have naming of parts.
This is the lower sling swivel. And this
Is the upper sling swivel, whose use you will see,
When you are given your slings. And this is the piling swivel,
Which in your case you have not got. The branches
Hold in the gardens their silent, eloquent gestures,
Which in our case we have not got.
This is the safety-catch, which is always released
With an easy flick of the thumb. And please do not let me
See anyone using his finger. You can do it quite easy
If you have any strength in your thumb. The blossoms
Are fragile and motionless, never letting anyone see
Any of them using their finger.
And this you can see is the bolt. The purpose of this
Is to open the breech, as you see. We can slide it
Rapidly backwards and forwards: we call this
Easing the spring. And rapidly backwards and forwards
The early bees are assaulting and fumbling the flowers:
They call it easing the Spring.
They call it easing the Spring: it is perfectly easy
If you have any strength in your thumb: like the bolt,
And the breech, and the cocking-piece, and the point of balance,
Which in our case we have not got; and the almond-blossom
Silent in all of the gardens and the bees going backwards and forwards,
For today we have naming of parts.
Henry Reed
1914 – 1986
Always ask the poet…
This blog gets weirder and more eclectic by the day.
Thank you, Susan. My familiarity with and sense of poetry is weak, but it’s a great poem, and apt. It feels as if we should get onto the rectification of names, even if that is not what Reed was on about.