A brief entry today. A number of my recent postings have been about brevity (whether anyone has noticed, I’m not sure.) The point I’m trying to make is that understanding is hard work. We can’t just watch a 4 minute video, or visit a few websites, or scan a news item and actually know very much about anything. This is a longstanding educational problem. I’ve pointed out Neil Postman’s arguments about TV culture, and I’ve suggested that internet “knowledge and understanding” are another species of the same problem. I will continue to provide bits and provocations, and will attempt to write  a longer, more coherent position some time in the near future.

The Doonesbury cartoon points to a longstanding problem. People think they can say something about world affairs, while simultaneously knowing very little about the world. The geography of Syria or Crimea are deeply relevant to understanding the ongoing conflicts in those lands. So are their histories, political economies, language, and culture.

Who among us knows enough to keep our politicians honest on these issues?

We can include anything and everything in mandatory education. But surely a deeper knowledge of the physical shape of the earth, and the main political divisions would greatly help to inform political dialogue.

Here’s a question for you. How many current heads of government can you name? (I mean presidents, prime ministers, chancellors, etc. not mayors and governors.) Have you got 20? 10? 5 at least?

While we’re at it, see if you can find Syria and Crimea on an unmarked map.





What you do is not what you understand

A very short entry today.

I’ve previously mentioned the work by Perkins et. al., as part of Project Zero at the Harvard Graduate School of Education  (from quite a while ago) in which it was suggested that understanding is a performance. I’ve tried in vain to get a full-bodied theory from this work; it just doesn’t seem to be out there. If I had to guess, I’d think that the authors figured out that they didn’t have a defensible theory and abandoned the project. They have ceased academic publishing on the idea, but continue to sell books and PD materials to schools…

Anyway, here’s a quick thought experiment to dispel the notion that understanding can be reduced to performance. Imagine watching two people play chess. Can you tell something–it doesn’t need to be a complete picture–about what the players understand by watching their moves?

Well, first, you’ll need to understand something about chess yourself to make this judgment (careful: this is in danger of circularity!). So now you, using your chess knowledge and understanding, try to form a judgment about what the players understand about the position on the board. Are they beginners? Are they good coffee-house players? Masters?

But wait. What about context? Who are these people? How do you know that they aren’t actors playing through a script? You don’t. Unless you have access to their internal processes–their thinking–you are in a position of significant uncertainty. We don’t even need the fanciful possibility of acting or fraud. Perhaps the players have memorized some moves from master games, and can play them so long as the opponent doesn’t stray from the memorized lines. (The longer the game goes on, the less likely this becomes.)

This is a real problem in modern chess tournaments. From time to time accusations of cheating come forth–perhaps the player is getting electronic assistance during bathroom breaks. How do you decide if the player understands the game well enough to make those moves? The standard approach is laughably simple. You give the player some sample positions in test conditions and get him/her to explain what moves to make and why.

Natural language dialogue appears to be the most obvious and sensible approach to the question of student understanding. This is not a theory but is a starting point. More to follow.

Teaching for Understanding—Harvard Disappoints

Back in October, 2015, I published brief look at some 20-year-old work on teaching for understanding by Harvard’s David Perkins.

Perkins, as part of Project Zero at the Harvard Graduate School of Education put down some preliminary ideas on teaching for understanding. Most provocative for me was the suggestion that understanding can be conceptualized as a kind of performance. Back in October, I promised to dig around to see if this seed grew into any kind of tree worthy of note, and whether the tree bore fruit.

I’m disappointed to report that the trail led me to nothing of significance. Perkins published a couple of articles back in the 90s wherein he began to explore his model of understanding, then seems to have abandoned the project. He continues to publish articles and books (many aimed at teachers, parents etc. with little eye to the scholarly community) on thinking and related matters, but nothing novel or interesting could I find in all this.

So I’m back where I started. I developed a few ideas about understanding in my doctoral thesis but have not had the time and energy to fully articulate my position, nor sufficiently to work out the details to the point where I have a theory worthy of the name.

Here’s a small thought experiment from that document.

An electronic calculator performs an arithmetic algorithm with greater speed and accuracy than I do in most cases. It does not appear to be the case, however, that the calculator is doing mathematics, because it does not have any concept of what it is doing. On the other hand, by entering the numbers into the calculator and pressing the buttons corresponding to the algorithm I wish followed, I am doing mathematics even though I am not performing the algorithm. What separates the human from the machine in this case is an interpretation of mathematical meaning. Let me clarify this with a more specific example. Imagine a right-angled plane triangle with legs of length 5cm and 12cm. How long is the hypotenuse of this triangle? You may simply recall the answer 13cm from previous experience, or you may be able to perform the relevant calculations mentally. Or you may use a calculator to come up with the solution. In each of these cases, it is clear that a person solving the problem is engaged in mathematics because that person’s thoughts are not only mathematical, but they are relevant to the problem. Alternately, someone who randomly punched numbers into the calculator then wrote down whatever showed up on the display would not seem to be doing mathematics of any kind, even if that person were to claim 13cm as the solution to the problem. Someone who failed to arrive at the correct solution, or perhaps even to arrive at any solution at all might still be said to be doing mathematics. Again, relevant thinking is what separates the mathematical solutions from the non-mathematical. The person who simply misperformed the calculation can still be doing mathematics, while the person who copies another’s solution without any understanding of the meaning of the symbols cannot.

My central arguments  are independent of views of mathematical ontology. Whether one holds a realist view that claims that mathematical objects are discovered as a mind-independent feature of reality, or one holds the view that mathematical objects are constructed through human practice, the arguments still hold. I defend a view that insists one must deal with mathematical objects as though they were independent of oneself in order to do mathematics. I remain silent on whether this pragmatic dealing corresponds to a true ontology. Further, I argue that once an appropriate pragmatic stance toward mathematical objects is taken, one must not only display certain mathematical performances, but one must also have mathematically relevant thoughts in order to do mathematics. To see that the display of mathematical performance is not necessary for the doing of mathematics, consider the possibility of a person reading the problem, mentally computing the length of the hypotenuse and going no further. Insofar as we can imagine such a situation, we can imagine mathematics being done without any trace of a publicly identifiable performance. In education, the teacher is obliged to assess the mathematical understanding of students, but has no direct access to their thoughts. It is not surprising, then, that student assessment is largely based upon performance, from which the teacher infers understanding. This has been complicated in recent years with the introduction of increasingly sophisticated calculators and computer programs. Student mathematics is often demonstrated through electronically mediated performance. The 5-12-13 triangle problem, for example, might be given to a student with access to computational technology. I will raise three possibilities, but will not discuss them deeply at this point. Rather, my purpose is to show the sort of thinking with which a fruitful theory of mathematics education must be able to deal.

Solution #1: The student has a pre-programmed right-angled triangle program. She runs the program and is prompted to enter the lengths of the two legs and the program types the output “13” to the screen. She then writes this number in her notebook.

Solution #2: The student uses a graphical program such as “Geometer’s Sketchpad”. The student uses the construction tools to create segments 5cm and 12cm long. With her mouse, she arranges the segments to be perpendicular at their endpoints. She constructs the third side of the triangle with the mouse. The program calculates the length of the third side to be 12.9999997, which she writes in her notebook.

Solution #3: The student enters the vector (5,12) in her calculator, and then pushes the button that calculates the modulus of the vector. She writes the output 13 in her notebook. (Macnab, 2006)

My question for you is this: what inferences, if any, can we make about the three students’ understanding of the mathematical problem?



Macnab, John S (2006). Epistemology, Normativity and Mathematics Education. University of Alberta.

Understanding as Performance–Part 1

My legs do better understand me, sir, than I understand what you mean.–Shakespeare. Twelfth Night 3:1.

Last year I wrote a bit about the idea of teaching for understanding. In that post, I tried to put forward two main points.
1. Understanding is a judgment. Perhaps I should have said verdict (with apologies to Leonard Cohen).
2. Understanding occurs across a continuum.
By the first point, I meant that we always use the term understanding to judge some fact about a person—even about ourselves. By the second, I meant that understanding isn’t usually a yes/no question: we can understand to differing depths.
I still hold to those points, but they aren’t my topic today. I’m going to dig all the way back to a 1993 article in American Educator, by Harvard’s David Perkins. The article Teaching for Understanding attempts to define “understanding” in an educational context. Perkins argues that understanding is a performative achievement.

My colleagues and I at the Harvard Graduate School of Education have analyzed the meaning of understanding as a concept. We have examined views of understanding in contemporary research and looked to the practices of teachers with a knack for teaching for understanding. We have formulated a conception of understanding consonant with these several sources. We call it a “performance perspective” on understanding. This perspective reflects the general spirit of “constructivism” prominent in contemporary theories of learning (Duffy and Jonassen, 1992) and offers a specific view of what learning for understanding involves. This perspective helps to clarify what understanding is and how to teach for understanding by making explicit what has been implicit and making general what has been phrased in more restricted ways (Gardner, 1991; Perkins, 1992).

Uh oh. Constructivism. The catch-all-while-meaning-nothing buzzword of the past few decades. I made a few general comments about constructivism here in the context of “whole language instruction”. In my view there are two types of constructivist theory: the trivial and the false. The trivial view is that learning takes place in the context of existing concepts, habits, ideas and so on (what Piaget called “schemata”). This seems to be obviously true, but not very deep. Other attempts have been made to deepen the theoretical scope of constructivism (von Glasersfeld “radical constructivism” is perhaps the most notorious of these). These theories attempt to explain the phenomenon of knowledge as a simple consequence of cognitive construction. They have all, thus far, failed to survive the charge of vicious circularity.
But things are not desperate yet. Perkins says only that he is working with the “spirit” of constructivism. He goes on.

This performance perspective says that understanding a topic of study is a matter of being able to perform in a variety of thought-demanding ways with the topic, for instance to: explain, muster evidence, find examples, generalize, apply concepts, analogize, represent in a new way, and so on. Suppose a student “knows” Newtonian physics: The student can write down equations and apply them to three or four routine types of textbook problems. In itself, this is not convincing evidence that the student really understands the theory. The student might simply be parroting the test and following memorized routines for stock problems. But suppose the student can make appropriate predictions about the snowball fight in space. This goes beyond just knowing. Moreover, suppose the student can find new examples of Newton’s theory at work in everyday experience (Why do football linemen need to be so big? So they will have high inertia.) and make other extrapolations. The more thought-demanding performances the student can display, the more confident we would be that the student understands.

Ok. This is a bit better. What concerns me are phrases like “really knows the theory” as if knowing (more appropriate for this context, understanding) were a simply yes/no judgment. The later comments about confidence in the judgment are more satisfactory. Sadly, this article does not provide enough detail for me to really understand what Perkins is getting to. He goes on.

Understanding something is a matter of being able to carry out a variety of “performances” concerning the topic–performances like making predictions about the snowball fight in space that show one’s understanding and, at the same time, advance it by encompassing new situations. We call such performances “understanding performances” or “performances of understanding”.

And now I’m really confused. Surely Perkins is conflating understanding with “evidence of understanding” here. The ability to carry out performances (such as predictions) is a consequence of understanding; it is not understanding in itself. We are able to use the consequences as evidence for our (always imperfect) judgments about understanding.
If you’re with me this far, you’re probably wondering how important these distinctions are. I know I’m wondering that too. Part of me is pleased with this old Harvard work, and part of me is perplexed by the gaping holes in the presentation.
Perhaps this isn’t surprising. The Perkins article is aimed at teachers, not other scholars. In this sense, I suspect that the article had virtually no impact on teaching and learning. Perkins’s writing simply does not provide sufficient guidance for teachers to move forward.
I’ll be looking into some scholarly back-issues to find out more about this Harvard theory and whether it ever made an impression on the community. Stay tuned.


NOTE (December 27, 2015). The link to the original article went dead. I have linked to a similar article by Perkins, published in Educational Leadership. The quotations are from the original (now lost) article.
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Teaching for Understanding

One can hardly turn a page of a teaching book or journal without reading a celebration of “teaching for understanding”. If we’re to buy the hype, teachers have traditionally just dumped information at students, who gathered and sorted as much as possible and then dumped it back at teachers. The teachers then took the student information dump, counted the bits deemed to be correct, then moved on. This often-repeated caricature has become something of a truism in educational criticism. Read some Paulo Freire for a view from critical theory, or listen to Sir Ken Robinson’s famous Ted Talk for a light and entertaining (but not convincing, IMO) version.

Caricatures notwithstanding, this does point to an important idea in education. It’s not that information is irrelevant to education, it’s that the understanding of information is more important. The internet is full of information; but without understanding it is somewhere between useless and dangerous. It also points to a tempting but woefully-wrongheaded view of education that pushes teachers to play the information banking scheme, depositing stakes test information into students’ lives, and insisting on equal withdrawals on exam day.

There is an opposite, but equally problematic push from this caricature of education. Some theorists, followed by some curriculum-makers have proclaimed that information is cheap and not worthy of a student’s mental energy. Who needs to know geographic (or historical, or scientific, or arithmetical…) facts, when they can be looked up in a few seconds on the internet? Why fill young brains, goes the argument, with information, when students can look at their monitors (or phones or calculators or whatever), get the information, then use their deep reasoning skills—the real business of education—to analyze, to predict, to create. I must say that this is tempting. But it turns out to be a false hope. Human short-term memory can only hold a very small amount of information at once. (There’s a good reason phone numbers are 7 digits long.) Pushing attention back and forth from a map, or calculator, to the work at hand is extraordinarily mentally taxing. The details are still very poorly understood, but we should be cautious about giving up on storing facts in long-term memory. It turns out that having number facts, scientific data, historical timelines, etc. in our minds makes the tasks of understanding and application much easier and straightforward.

Which brings me to the point of this entry: understanding. It’s easy to talk about understanding, but it turns out to be less clear what we are talking about. How can you tell that you understand something? What does it mean for a student to understand the theory of evolution by natural selection? Or to understand the causes of First World War?

It should be clear on a minute’s reflection that the recitation of facts does not indicate understanding. To demonstrate understanding, a student needs to provide an account that is acceptable to someone who already understands. (This sounds circular but it isn’t. The world is big and we’ve all been around long enough for it to be clear that many people have understanding; we acknowledge expertise in every field. We can safely leave the chicken-and-egg puzzle to the philosophers and just get on with teaching, learning and assessment.) This is why it is crucial that the teacher have disciplinary knowledge. The canons of argumentation and standards for evidence are different in each discipline—there are overlaps of course—and if the teacher is to judge that a student understands something, the teacher must be sufficiently accomplished in the discipline to make the call.

So that’s the first big idea for today: Understanding is a judgment.

What kinds of accounts are satisfactory? This is a very big and deep question. If you haven’t read my entries on Universal Design for Learning, now might be a good time. They don’t answer the question, but they do frame it in an important contemporary way. The general answer is that satisfactory accounts depend on the age and maturity of the student, the disciplinary achievements of the student and the depth that is demanded by the curriculum and/or the teacher.

Second big idea: Understanding occurs across a continuum.

This is fairly easy to see. If you ask a 12-year-old, a high school student, a history undergraduate and a university historian to explain the causes of the First World War, you should reasonably expect differing levels of detail, different sorting of the relevance of the details, differing understanding of competing views, and differing plausibility of causes and their effects. How could you not? The point is not that understanding is a “yes/no” judgment; it is that people can understand to differing depths.

So let’s get back to the classroom. What does it mean to teach for understanding? If what I’ve said so far is correct, it at least entails the following.

For students to understand something, they must

  1. Possess sufficient information to make inferences.
  2. Know how to assess new or “looked up” information in order to decide whether it is relevant.
  3. Understand what counts as relevant information for the question at hand.
  4. Know how to construct a logical argument.
  5. Be able to communicate the argument.
  6. Be able to respond to probing questions.

As I never tire of writing, for meaningful learning to occur, a great deal of stage-setting must be done first.

I intend to look more carefully at all the items in my list. I might have missed one or two, but this will do for now. Teaching for understanding is a good thing. It isn’t quite as simple as it first appears, though.