Only for geniuses, eh?

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

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.

Exponents (or pOwers)

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.


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.


Meyer and Stokke (again)

Since I’ve looked at a bit of Dan Meyer’s and Anna Stokke’s recent comments on mathematics education, it might be fun to see a popular press discussion of their positions.

From Metro News.

TORONTO – Don’t get math teachers started on best teaching practices.

The discussions are emotional, heated and they don’t agree on much — except that Canadian kids are falling behind their peers in other countries, and there’s no clear solution.

There are generally two camps: those in favour of the old-school method to lecture kids with a “drill-and-kill” format that preaches practice, and another, ever-growing group that believes a more creative approach is needed to engage students.

At a recent event in Toronto, dozens of teachers waited in line to take selfies with math-teaching celebrity Dan Meyer, delaying his keynote talk at the Ontario Association for Mathematics Education conference. He is part of the new-school camp.

His approach is simple, Meyer says on the phone from California, where he’s a math education researcher at Stanford University.

He presents a problem at the start of class, and lets the students try to figure it out. Hopefully, he says, the students will struggle.

“That initial moment of struggle prepares them for what they’ll learn later,” he says.

Meyer cites several studies that back up his ideas, including one from Manu Kapur, a professor at Nanyang Technological University in Singapore. Kapur’s study shows students who are given a problem to solve on their own — before instruction from a teacher — outperform students who are given the traditional lecturing style.

The technique is in the early stages of implementation across Ontario, according to Sheena Agius, a math coach who helps teachers with the new method in the Dufferin-Peel Catholic District School Board.

Just like all other boards in Ontario, it is moving away from rote learning to try to get students to understand math at a deeper, more conceptual level.

“Just because we’re doing it, doesn’t mean we’re doing it well yet,” she says. “But it’s a learning process for teachers and that will come.”

Meyer has many acolytes, such as Paul Alves, president of the Ontario Association for Mathematics Education and a high school math teacher at Fletcher’s Meadow Secondary School in Brampton, Ont., northwest of Toronto.

“Teachers are really engaged by the way (Meyer) teaches math because when they try it they see the same thing — the excitement students have to do the math — and it changes the classroom. It invigorates it and energizes it, which wasn’t the case before,” Alves says.

That engagement is priceless, Alves says.

He says a teacher at another school dove headfirst into the new-school method for his Grade 9 applied math class. The class, he says, jumped from 40 per cent on the provincial tests using the old method to 70 per cent after implementing the new one.

Yet both Meyer and Alves say they aren’t advocating abandoning the classic “chalk-and-talk” style.

“At some point I need to know that kids can factor a quadratic equation, and sometimes you have to practise that skill to get good at it,” Alves says.

On the other side of the dividing line, old-school math teachers are just as vociferous.

Anna Stokke, a math professor at the University of Winnipeg, is a staunch defender of lecturing and practice.

She recently published a report with the C.D. Howe Institute that showed Canadian students’ math performance in international exams declined between 2003 and 2012.

Stokke blames the decline on the style promoted by Meyer, which she dubs “discovery-based learning.”

“A direct method is a more effective way to teach,” she says.

“So guys like Dan Meyer will say, ‘We’re going to spend the next week building a birdhouse and you’ll need to use measurements to figure out dimensions,’ and the kids will learn about area and volume and all that. Then a week goes by and what have you learned? How to build a birdhouse.”

Meyer fires back, calling Stokke’s research simplistic.

“The best teaching is some shade of grey, where before the teacher talks about what to do, the teacher gives students some reason to care and some background on how to care,” he says.

“None of this suggests teachers shouldn’t explain or lecture.”

Stokke does offer Meyer and his disciples an olive branch in her report, saying 20 per cent of math teaching time can be used for these “alternative methods.”

“I’m trying to be objective and I don’t want to tell teachers they can’t use a particular method at all, but I want to be clear on which methods have been shown to work and which haven’t.”

Having said all that, Stokke admits that her research can’t conclusively pinpoint discovery-based learning as the reason for Canada’s faltering math scores.

Her research reinforces an assessment by the Organization for Economic Co-operation and Development in 2012, where 65 countries took part in the Program of International Student Assessment that examined math skills of 15-year-olds. Canada fell in those scores as well.

Her report found every province declined in math scores except for Quebec.

Annie Savard, a math education professor at McGill University, said her research indicates the difference may be rooted in training.

In Quebec, students go to teachers’ college for four years, as opposed to a one-year program that follows a bachelor’s degree in the rest of the country. Ontario is set to move to two years in the fall.

And if you read my posts on the two, you’ll find that I agree with some of what each of them say, and disagree with some. And I would find the whole “debate” laughable if it weren’t for the sobering fact that children are drawn unwillingly into ideology.

Cathy Bruce, a math education professor at Trent University, is tired of the so-called “math wars.”

“It takes away from figuring out what is happening to Canadian students. The solution is likely somewhere in the middle.”

I suspect that the solution lies in paying close attention to your students. They cannot possibly go from A to B if you have no sense where A is. (Or, as Anna Stokke would likely point out, if you don’t understand B yourself.)

Returning to the classroom

After 20 years of teaching, I took a foray into bureaucracy. For five years, I conducted and coordinated research and evaluation in my school district, and for the past two years, I’ve been on secondment to the government, working in various areas of curriculum development.

And now it’s time to return.

This September I’ll be teaching high school mathematics (plus International Baccalaureate Theory of Knowledge). And I have to say that I’m stoked.

I remain convinced that high school mathematics is grounded in technique: arithmetic and symbolic manipulation are to high school mathematics as scales and arpeggios are to musical development. And I remain convinced that technique by itself is uninteresting, anti-motivational, and useless. Students need to deploy their technique to make their mathematical experiences come alive. This can be explored through practical application, through stereotyped problem solving and through exploration of abstract ideas that have nothing to do with the physical world.

The art of teaching is to find the right balance, to match students to the experiences that will matter the most to their developing technical skill, mathematical understanding and, yes, their aesthetic sense of learning mathematics.

Being out of the classroom has the wonderful benefit of giving you time to think, to reflect on what it is you do for a living. I have enjoyed this immensely. Being out of the classroom, I’ve had time to think about student assessment, about motivation, and about understanding.

But being out of the classroom has a tendency to lead to romanticized notions. When I’m thinking about all the wonderful things my imaginary students and I will do, well it’s pretty amazing. We’re brilliant together. But I will have to live with the uncomfortable friction that will arise when theory and reality collide in the classroom.

It’ll be fine. No. It’ll be great. I’m excited to be back.

Evidence, Anna. Where’s the Evidence?

Earlier this month, the C.D. Howe Institute published a polemical paper by University of Winnipeg mathematician Anna Stokke. The paper explores the politically-charged question of What to Do about Canada’s Declining Math Scores. It’s one of those funny papers, written by an academic wandering outside (or to the fringes) of her field. Stokke has some insightful things to say, followed by some absolute howlers. Let us begin with the “problem”: on the mathematics components of the Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA) tests, most Canadian provinces have shown a statistically significant decline over the past decade or so. The first two questions that come to mind should be

  1. Is it true?
  2. Does it matter?

Stokke states that #1 is true and provides some reasonable evidence to back this up. I agree with her basic position that Canadian scores have declined on these tests. I haven’t seen a compelling case for why this is the case, but it definitely is borne out by the data. It is worth noting that Canada has declined from being near the top of the international list in 2003 to being near the top of the list in 2012. The decline is real, but it’s from height to height. Stokke assumes the answer to #2. I won’t go into #2 deeply, but I will note that it is far from obvious that the tests measure the things we value in mathematics education. There are countless possibilities for what we could value in school (computation, reasoning, geometry, algebra, number, problem solving, etc.) No curriculum can do full justice to all of them, so choices have to be made. I have no quarrel with the material emphasized by PISA and TIMMS, but I believe that any political action based on the scores must take such niceties as these into account. Let’s also note that K-12 education is a provincial matter in Canada: each province creates its own curriculum, assessment standards and teaching professional standards. Nation-wide decline becomes a problematic issue to study because each province is teaching a different curriculum. There are, of course, similarities because change doesn’t occur in a vacuum. Provincial ministries of education do talk together and have a pretty good idea of what each other are doing, and American publishers and academic trends tend to influence Anglophone curricula. I am uncertain of the influences on Francophone curricula. So far, we have a few quibbles, but the argument is fine. International scores are declining. Next comes the bogey du jour: “Discovery-based Instruction” whatever that is. The issue that Stokke pushes is the difference between structured “direct instruction”[1] and “discovery learning[2]”. If we accept Stokke’s crude dichotomy here, the empirical evidence is pretty strong. In terms of developing skills, teacher guidance is critical. This is why teachers are necessary. Well-structured instruction, with appropriate questioning, feedback and revision to assist student understanding are crucial to student achievement. So if the goal is to teach students to perform long division, or to multiply polynomials, or to solve quadratic equations, clearly worked examples under the direction of the teacher are clearly and decisively shown to be the best choice for student instruction. Stokke makes reference to some of the empirical literature on page 4 of her report, and I have no quarrel with her summary or with her interpretation of the results. (On a side note, Stokke’s footnote 5 on that page is an absolute joke. The “well-informed journalist” is David Staples, who is well-intentioned but hilariously ill-informed on the issue. The claim that phrases are interchanged “to avoid criticism” is at best a wild guess, and at worst yellow journalism.) So where’s the complaint? It is undeniably true that inquiry forms the basis of what is valued in K-12 education in Canada today. Science classes should and do have laboratory experiments. Social studies classes should and do include source-document analysis and open inquiry into relevant issues. How could one study literature without inquiry? The same goes for mathematics. Students need to acquire the basic tools of mathematics, but they also need to learn how to mobilize these skills to explore questions both practical and theoretical. If there were no room for inquiry in mathematics, Dr. Stokke would be out of a job. Which leads us to the million dollar questions.

  1. Which skills must be mastered?
  2. Which skills should be developed to the point of acquaintance?
  3. How much inquiry is desirable at each stage of the student’s development?
  4. What do we ultimately hope that our students will gain from their mathematical education?

Notice that these questions are only slightly empirical. At their hearts are values. What do we think is important and why do we think it’s important? Stokke shows some sensitivity to this problem, but she effectively abandons her project and jumps into speculation and assertion. Starting on page 9 she begins spelling out what she thinks is important for each grade of the curriculum. Stokke provides no justification for this, she makes no appeal to research literature on child development; she somehow just knows what every child should learn, and when they should learn it. Frankly, this is just spitting in the wind. She’s one voice in a multitude. I agree with some of her suggestions, but not all. (But then, I only have 20 years of classroom experience and a doctorate in education; what do I know about the matter?) And then it starts to get weird. Stokke writes

One way to redress the balance between instructional techniques that are effective and those that are less so would be to follow an 80/20 rule whereby at least 80 percent of instructional time is devoted to direct instructional techniques and 20 percent of instructional time (at most) favours discovery-based techniques.

Where does this come from? For which students? At what grades? Where’s the evidence? Stokke is relying on her intuition to make an 80/20 prescription. Is she joking? On the positive side, Stokke recognizes that there’s more to the issue than the simple-minded “discovery vs direct” dichotomy suggests. But to make an ad hoc recommendation of this type is to be irresponsible. It gets worse. Stokke generalizes from an American study of first and third grade teacher knowledge, suggesting that Canadian teachers should all be given regular licensure examinations. Seriously? You’ll need some evidence that

  1. The American study applies to Canada.
  2. What’s true for first and third grades applies to all grades.
  3. Licensure exams will actually improve classroom practice.

Ultimately, Stokke’s report reeks of the well-intentioned dilettante. That Stokke loves mathematics and is a believer in quality mathematics education is evident. Unfortunately, it’s also evident that she jumped on a bandwagon, read a small amount of research in the area and picked up a contract from a “think tank” to join a public debate. As an insider, I’m not thrilled with the current state of mathematics education in my country. But also I am convinced that single-cause arguments that fail to acknowledge differences between students—not even considering that age matters—do nothing to clarify our issues, but serve only to support loud and ill-informed public shouting.

[1] Not to be confused with an American product that uses this name. “Direct instruction” should be read generically as any teaching in which the teacher strongly guides or directs learning throughout the process.

[2] Despite the claims of Stokke and a few other public protesters, you simply won’t find this term or anything like it in current curriculum documents in Canada. Individual teachers are undoubtedly using “discovery” methods to some degree or other, but there is nothing in the curriculum to mandate them.

Beware the Instant Expert

In my home province of Alberta, a certain Dr. Nhung Tran-Davies has made a crusade of ending what she sees as mistaken educational reform and of returning to some notion of “the basics” in our classrooms. If nothing else she has demonstrated that by being aggressive and persistent an individual citizen can get the attention of local and national media, get invited to speak to public organizations and even get the ear of the Minister of Education. It’s been a fascinating PR exercise. Dr. Nhung Tran-Davies If you’d like to know more or to sign her petition you can catch her main points here. This whole odyssey began with concern over her daughter’s math homework. The issues are the basic “math wars” battles that I outlined in post in December, 2014. In short, students are often being taught a wide array of techniques—some efficient, some not—in an attempt to develop understanding of arithmetic and not just computational fluency. As I pointed out in the other post, the issues are deep and interesting, and are nowhere near resolved in the research literature. In short, no one knows how much computational fluency is needed or useful for students in the 21st century, and no knows how much students benefit from the attempts to teach for understanding in computation. Finally, the legitimate goals of elementary education in mathematics are not clear: what, exactly are we trying to have students do with their mathematics education? So back to Dr. Tran-Davies. Her daughter was apparently frustrated, Dr. Tran-Davies is a parent and a medical doctor, so she must know how to reform (deform?) mathematics education. Right? Yeah, right. Sure and armed with Google and Wikipedia, I can be a medical doctor too. Let’s be honest. As a parent, Dr. Tran-Davies has a legitimate interest in this discussion. As an educated woman, she has a particular interest and experience in education (but no more than someone who has had a different experience). As a citizen, Dr. Tran-Davies has a legitimate voice in the discussion. All absolutely without question. But why would the Minister or anyone else give her a privileged voice in this discussion? Here’s where the practical politics come into play. Dr. Tran-Davies found a sympathetic ear at the Edmonton Journal, in columnist David Staples. Staples doesn’t know crap about education or mathematics, if one is to judge from his columns. But that’s not his responsibility. Staples’s job is to attract readers, and by setting up the battle “upset Dr. Mom versus the mindless, ideological government ‘educrats’”, he found a winning combination. Thoughts and reasons make for boring commentary (which is probably what’s wrong with this blog) but anger and hyperbole sell papers. And of course papers influence voters. Which brings us to the real problem. The Edmonton Journal has privileged one person’s point of view. It’s an interesting point of view, but it’s uniformed and inexpert. But it’s good-looking, credentialed, articulate and ANGRY. So the minister listens. So the Ministry spends the past year disarticulating itself as it bends backwards to try to minimize the damage—acknowledge nothing, while working like mad to show that you care. It’s a pathetic sight. So what can we take away from this? IMO, Dr. Tran-Davies is rather like the anti-vaccine movement, the sincere climate change deniers, the churches opposed to evolution. It’s a democratic society: have at ‘er. But God help us if the uninformed change public policy just because they have a platform handed to them.

Math Wars?

If we’re to believe the press, mathematics education is in the midst of a war. Education saw the reading wars a couple of decades ago, and academia still seems to be in the middle of a science war. At least according to the press.

Before getting into the ideas, let’s just stop for a moment and say something about the metaphor. Real wars are violent, disgusting affairs that result in death, destruction and social instability. Nothing of the sort is going on. The math wars are blood sport only insofar as political careers can be destroyed by standing on the wrong side of an angry press.

Division by repeated subtraction. From

In short the problem is simple. Elementary school children are currently being taught a number of pencil and paper methods that are clearly

  1. Not the same as the methods their parents were taught;
  2. Not mathematically efficient;
  3. Reportedly frustrating for students, parents and teachers.

So what’s going on?

Traditionally (for the last century or so, anyways) elementary school mathematics has largely been devoted to developing knowledge about numbers and to developing facility in computation. You begin by learning to count, learning the names of numbers, learning to add, subtract, multiply and divide. Students typically began with whole numbers and after developing knowledge and skill, they extended their work to fractions. Later on (typically Junior High, or Middle School), this arithmetic knowledge and skill expanded further to integers and real numbers. High School typically saw further extension to computation with radicals and polynomials. Depending on the location and political climate, complex numbers were added to the mix. Sure there were other aspects to the study of high school mathematics—a bit of geometry, theory of functions and so on—but school mathematics has traditionally been about taking arithmetic from whole numbers in the early grades to polynomials in high school. The central mathematical and educational idea was that this progression allowed for increasing mathematical generalization and increasing computational sophistication, leading to the development of some rudimentary mathematical understanding. (Before you get too excited about the “depth” of high school trigonometry and calculus, remind yourself that at the university level, first and second year mathematics textbooks invariably contain the words “introduction” or “elementary” in their titles. There’s a reason for this.)

As a general plan, I like the sketch I’ve given. Mathematics is largely about generalizing and abstracting the particular context away from particular problems, calculations, and whatnot. What’s the evidence that the “traditional” system gave students the payoff that was planned? To be sure, most English-speaking countries do not enjoy a population that by and large loves mathematics and has fond recollection of school math studies. Worse yet, the great majority of people do not need more than the most rudimentary mathematics to fulfill their life-plans, including education, employment and personal satisfaction. It would be nice to know if people were somehow better off because of their school mathematical experiences, but alas, we have nothing to indicate that this is so. Maybe we were on the wrong path. Hold that thought. I’ll return to it in a couple of days. There is some uncertainty whether the math education of the past few decades adequately prepared academically competent students for further study, and whether it provided much of value to those students whose futures were not dependent on mathematical achievement.

A second line of thought looks at the basic achievements of students past. Computation is a very valuable thing. You can’t do your taxes without it; you can’t take care of a significant amount of daily business without it. Accurate computation keeps the trains on time, and the books balanced. Nobody doubts this. But the days of accurate manual computation are gone, and there is no sign of them returning. A $5 calculator is better at all basic computation than anyone would ever want or need to be. Better, in fact, than any human ever could be (if you include the capacity to compute trigonometric ratios and logarithms). So why bother teaching children to compute? Why indeed?

There are two schools of thought on this question. One says that we should abandon manual computation because machines are all we need. Perhaps that’s too strong. Computation should be learned, says this camp, only insofar as it can be demonstrated to increase student understanding of mathematical concepts that are worth learning—whatever those turn out to be. Not so fast, says the other school. Computation is valuable in and of itself. It not only is a worthwhile thing to learn, it is the key to unlocking number sense in young minds, and it is the best way to increase student understanding of deeper mathematical concepts.

How to adjudicate? To date there is absolutely no compelling research evidence that is useful in addressing this matter. We have two ideological positions, each grounded in respectable intuitions, but neither having the weight of empirical evidence to support it. This saddens me deeply. With all the educational research published every year, why in the world is no one pursuing this question in a responsible way? I’m sorely tempted to give up my position in bureaucracy to get my feet back into a research institution just to answer this question.

In the meantime, what is happening in schools? Not surprisingly, two things. In some jurisdictions, you’d never notice that the times are changing and students are computing away like it’s 1969. In others, we have (yet another) “new” math. Students are no longer given extensive drill on basic number facts, and are learning a number of pencil and paper algorithms that look very strange to the students’ parents. These algorithms are not the tidy methods that were taught decades ago; they are often long-ago-discarded methods that involve a lot of extra writing and computing, but which are supposed to reveal the deeper structures that hide inside arithmetic. (E.g. Division by repeated subtraction.) Are these students gaining deeper understanding into arithmetic than their peers that learn the same algorithms as their parents? Nobody knows. To the best of my knowledge, there are no reliable evaluations of these new programs.

So here we are, stuck between two ideological positions, with a distressing lack of evidence. We’ll look at a bit more detail in a few days.