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.

Hannah’s Sweets—Anatomy of a Math Problem

The following problem has the internet all abuzz.

hannahs sweets

The problem comes from the Edexcel Maths GCSE paper—which, as I understand it is an examination for UK 16-year-olds with aspirations of studying at a university.

The buzz is fascinating for a number of reasons: some positive; some not so positive. But all are worth thinking about. First off, many students found the problem to be very difficult. I’m not sure why this is, but I’ll offer some speculations as we move through this post.

(I’ve had to convert my Word document into graphics because the equation objects wouldn’t paste into the WordPress format.)

Why I Like the Problem

There are a few very nice things about this problem. First, it asks students to demonstrate something, rather than to find a single solution. As a math teacher, I am always looking for demonstrations of understanding, rather than the consequence of a search that may or may not simply be a matter of “getting lucky”. On that note, I am pleased that this question requires a human marker. No multiple choice or numerical response item could replicate this. To demonstrate understanding, students must use natural language (i.e. English) and structured symbolic representation (i.e. numbers and letters with mathematical connectives) to convince another human mind that they understand. This is good.

Second, there are at least four straightforward methods of solution available to an average post-secondary-bound high school student. No one is bound to a single method. If I were to use this question in class, I would have choices for how to approach it, depending on the goals of the lesson.

Solution #1—Elementary Probability

This is probably the most straightforward way to approach the problem. We use two pieces of basic knowledge.


This is not hard, and it involves very easy manipulation. I’ll go over the next two solutions quickly, as they are very similar in form, but not in conception.

Solution #2—The Fundamental Counting Principle

This is a close cousin to #1. The idea is that if you have n ways to do one thing, and m ways to do another distinct thing, then you have n×m ways to do both. So if you have 3 pairs of pants and 4 shirts, you have 12 different outfits you can wear. We now see that Hannah has n(n-1) ways to pick two sweets out of n, (She ate the first sweet, so she only has n-1 left for her second choice.) By similar reasoning she has 6×5=30 ways to pick and eat two orange sweets. From here, the arithmetic and algebra are the same as #1.

Solution #3—Combinatorics


Solution #4—All Guts no Math


The Downside

One of the standard complaints about this problem (and others like it) is that it is so artificial. Sure it gives us a protagonist, a storyline and some talk of yummy sweets, but this is a million miles from real-world mathematics.

I hear this kind of objection all the time, and frankly I don’t get it. One of the other complaints about mathematics is its coldness and distance from human creativity. Well this is a silly story. It’s a flight of fancy; it’s a goofy narrative that paints a picture. Yes, very few people will do math when eating candy; we all know that. But no one ever complains that fiction is unlike real life, because it reminds us of real life and gives us something to think about. As do silly problems like this one.

Now it is often easy to create a problem that can be stated with or without the narrative. Indeed, it is often the case that the narrative obscures the mathematics. That isn’t the case here. A scenario free statement of this question would be quite difficult to write without absolutely obscuring the clear and simple question being asked.

So why did students find it hard? My guess is because the question is surprising. Students are often lulled into calculating numbers, solving equations, concretely answering problems. In this case, they were given a concrete problem and were asked to show that the problem leads to an unexpected equation. The work isn’t difficult, but the interpretation and strategy-selection were surprising. At least that’s my supposition at this time.

Whew. This is more concrete mathematics than I aim for in this blog. Hope it brought some thought and (dare I say it?) some fun to your day.

If any students who wrote the paper happen to stumble onto this blog, I’d love to hear your reflections on the problem.

Puzzles, puzzlement and learning

Here’s a bit of an odd thing.

Everybody knows that the two circles are of different size. We might even remember the right words to say that they have different circumferences. Heck we might even remember the fact that the circumference is proportional to the radius.

The outside circle is bigger than the inside circle.

But in the video, they both look exactly the same.

What is going on?

The answer, I’m sure you know, is mathematical. Things are not what they seem.

Imagine that you were a student, and your teacher showed you this video. What would your reaction be?

If you were engaged in your work, or if you were engaged in physical work (maybe you’re a bicycle enthusiast), you’d be pretty amazed, I think. Because if this were true, wheels couldn’t work. Of course, if you were not interested in the material, but cared about your grade, you might ask, “Is this on the test?” Or maybe you’d not care at all, and turtle up for a bit. (Yes, I’ll return to the question of engagement at a later date.)

It seems to me that the worst thing a teacher could do now is to reveal the “solution” to the conundrum. To answer the question now would be to kill it. The answer is important, but it’s not important right away. What is necessary is to feel the question. What does the video purport to show? Could I replicate it somehow? Why does this crash my common sense to the ground?

So I won’t answer it today. Maybe tomorrow. Maybe the day after…

Dan Meyer part 2

To see the video in question, drop back here.

Meyer is a high school math teacher, and I believe at the time of this talk he had been teaching for 6 years.

Meyer suggests that the real problems besetting high school mathematics education is the lack of real-world sensibility in the work. He figures that questions where all of the relevant information is provided, or where the abstractions are beyond everyday experience are alienating for students. His solution is use digital technology to create problems that can be modeled in the classroom, and that will provide meaningful experiences for his students.

I can go along with this to a point. The first fact that Meyer avoids is that mathematics requires basic skills and that these skills require practice. It isn’t sexy. It isn’t fun for everybody. But without mastery of basic technique, math becomes significantly more difficult than it needs to be.

Second, the purpose of stereotyped problems is that they help students  learn general procedures.

pool problem

In this case we have a type of problem that is difficult for most students the first time they see one, but becomes easier with only a little repetition. OK for some students, once is enough; for others, 10 times or more might be necessary.

(For those who forget such things, the first hose fills 1/12 pools per hour and the second fills 1/10 pools per hour. Add them and you’ll know how many pools per hour they fill together. I’ll leave the rest to you.)

So now we have a matter for teacher choice. Some will teach only the textbook stereotypical problems. Some will teach stereotypes first, then maybe do a “real world” problem with incomplete and imperfect data. Others may start with the real world to motivate the development of technique necessary to solve the real world problem.

Meyers suggests skipping the stereotypical problems and jumping into the real world and staying there. If this is really what he does (and I doubt it) then I have a hard time believing that it’s successful.

Real world problems are interesting in lots of ways, but they are time consuming. The business of learning the technique has to take place somewhere.

Practice is important; but it isn’t everything. Real world engagement is important; but it isn’t everything either. What we require is balance. But how to balance?

The short answer is that there is no short answer. Teachers are professionals for a reason. Judgments have to be made about the students in the room before us. The reality is that not every student is the same as the others. Not every class or community or teacher is the same as the others. Often I will have two classes for the same course during the same term and find myself teaching the classes very differently from one another.

One size fits all solutions make for great TED talks. But they make for pretty sketchy classroom practice.