Only for geniuses, eh?

As regularly as the spring rain, memes of the following sort show up on social media.

sevens
As a math teacher, I don’t know whether to laugh or to cry. Adults—sometimes hundreds or thousands of them—argue about the correct answer to this “problem” and others like it.
Let’s be clear: this is a question about the order of operations in arithmetic. In the Province of Alberta (my home) this is expected of all Grade 6 math students. So, Facebook is clogged with adults working below 6th grade in basic number sense. Ok I do know. It’s cry, not laugh.
What sense can we make of this? Do adults not remember their elementary-school arithmetic? Apparently many do not. But some of the comments are also telling. It appears that many were taught methods that almost work, but don’t quite.
The acronyms BEDMAS or BODMAS are often taught to children.

Brackets
Exponents (or pOwers)
Division
Multiplication
Addition
Subtraction

If you follow BEDMAS, you’ll be right most of the time. It’s fine for the above problem. There are no brackets or exponents, so you divide 7/7 and multiply 7×7, turning the problem into 7+1+49-7=50. But there is a problem with BEDMAS/BODMAS and that is that the acronym suggests that division has priority over multiplication and that addition has priority over subtraction. This is not true.
Once brackets and exponents are cared for, you work left to right. If you come to a multiplication or division, do that before continuing with the addition or subtraction. Schematically, the problem above simply becomes 7+(7/7)+(7×7)-7, which is pretty easy mental arithmetic.
Even calculators can make errors.

calculator-god

If you’re not working left to right, you run the risk of making the error on the left.
Regardless, what am I on about here?
First, order of operations is elementary school arithmetic. It should not pose a problem for adults. But it does. This points to a serious educational deficiency—for the adults. This is not a problem of “new math” or “constructivism” or “Common Core”. The people getting it wrong online are, by and large, from earlier generations of failed arithmetic education.
But it’s clearly a problem, and I think I know why. It’s a problem of assessment. You see, if students (in any generation) get most of the questions right, they get a good grade. I suspect that the adults who can’t solve simple order of operations problems never could do it well. But they got all the easier questions right, so no one bothered to dig deeply into their failure on the one or two harder items on the test. Yes, this is just speculation on my part, but I’m willing to bet that it explains a good deal of the problem.
But there is a positive note to all this. Adults are arguing about math in their spare time.

They care. And that’s encouraging.

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?

 

Reference

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