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.