I think Cynthia Reynold’s article (“Have you finished your homework, Mom?”) in the latest issue of Maclean’s magazine makes some valid points and raises some important questions about math education in Canada.

Reynolds writes of parents Anna and Ross Stokke—both professors of math at the university of Winnipeg—who are having trouble figuring out how to answer the questions in their daughters’ elementary and middle school math textbooks (note: as pointed out in the comments, the professors were misquoted by Maclean’s, and merely meant to say that the new methods are confusing to students and parents). How can this be? Did math suddenly change? Are addition, subtraction, multiplication, division, fractions, algebra, and geometry so different now that we have to use paper strips, colored blocks, and other diagrams to explain them? The new methods of teaching math are unnecessarily confusing, and the problem has no end in sight, as it’s part of a vicious cycle.

We’ll get in to the details of this vicious cycle in a moment, but first I want to highlight what I think are three major problems (in addition to the many smaller ones that also exist) with math education in Canada (and perhaps elsewhere), which are all touched upon in Reynold’s article. I’m going to approach these problems by analyzing what I think are two common misconceptions about learning math, and then lastly by expanding upon Reynold’s conclusion.

Misconception # 1:* Rote learning is evil – you must never drill students or assign repetitive homework.*

This just isn’t true. We’ve shifted so far to the conceptual side of math education, we’ve forgotten to give students any time to practice just calculating things. We need to find moderation between abstract conceptual examples and concrete practice and manipulation of numbers and formulas and don’t get me wrong—I think it’s extremely important to offer visuals, manipulatives, and alternative explanations of concepts such as addition, subtraction, multiplication, division, and fractions. While these visuals can be helpful for developing an understanding of how and why something in mathematics works the way it does, we still need to practice using the new concepts, formulas, and methods. And often, the best way to practice something is to to do a short drill.

Misconception #2: *Calculators should never be allowed in math class.*

Every so often I am shocked to see students struggling over multiplying seven digit numbers by hand. I’m not shocked because they can’t do it quickly. I’m shocked that teachers make them do it. When was the last time you had to multiply something like this:

77894.28

x 6410.339

¯¯¯¯¯¯¯¯

… by hand?

We have calculators, computers, and Google to do that for us. Should we know how to estimate? Absolutely! We can reason that 77894.28 is close to 80000 and that 6410.339 is close to 6000, bringing us an estimated product of (80000 x 6000=) 480000000, which is off by nearly 20 million! But with calculations with big numbers like this, that’s only about a 4% error, which isn’t so bad for estimation of large products.

When you’re shopping, baking/cooking, or travelling, you’re a lot more likely to need skills of estimation. “*How much would 6 of those tomatoes cost? If I want to double this recipe, what will 1 1/3 cup turn into? If I need to make it to Calgary from Kelowna by 5:00pm, how much time should I allow myself?” *These are things that come in handy. Multiplying 7-digit numbers by hand is not very handy, and frustrates students more than it aids in understanding.

For calculations that require accuracy down to the last decimal point, we can turn to our calculators, at least once some level of understanding has been reached. In my experience, students aren’t being given enough time to play around with their calculators to actually understand how numbers work. Without calculators, they get through a small amount of questions either in work periods at school or at home, become frustrated with the tedious calculations done by hand, and quit. If we allow them to use calculators the majority of the time, we allow them to practice a lot more examples. In my experience, I’d say you can finish up to five or six times as many questions (for some types of homework) if you simply have a calculator or a browser open to Google or Wolfram Alpha for tough calculations.

Once students build more confidence with calculators, they become more comfortable with numbers, and they’ll start to trust their mental math and powers of estimation more than ever before.

Of course, there are plenty of cases where students should not be allowed to use calculators, such as when they’re learning the process of long division, in certain parts of trigonometry, logarithms, geometry, and basic algebra, to name a few. Many types of these calculations can be done entirely without calculators, if the questions are designed accordingly, with the tools of special triangles, sine and cosine graphs, logarithm and exponent laws, and a few geometry formulas, and that’s kind of the point of some questions in the first place. The Greeks solved them without calculators, and we can too. And once you learn a few tricks, some questions become much quicker without a calculator than with it. These are important mental math skills that need to be developed.

But in addition to these skills, now that we have calculators, we should be teaching kids how to use them as well. People use calculators every single day for a wide variety of purposes. A calculator is an essential tool for all types of fields including engineering, experimental science (chemistry, physics, astronomy), business operations, accounting (and anything related to finances for that matter). Do you trust your banker to do calculations by hand? Do bankers even do calculations? Not really. It’s more like they hire people to write the programs for computers to do it for them. Does this ever come up in high school math class?

Einstein once said that *‘A pencil and I is smarter than I.’* and I think if he were alive today he’d be okay with making the logical extension to an updated 21st century version of the quote: *“A calculator and I is smarter than I.’* We need to teach students how to think both with and without their calculators, and increasingly in the digital age, more with than without.

**The Domino Effect**

Finally (and wow, I didn’t know I had so much to write about calculators!), to build on the final couple of paragraphs of Reynold’s article, I’d like to add that as an experienced tutor and new teacher, I’ve seen the mentioned domino effect in action so many times, in so many different forms. Students never learn multiplication or division properly in grades 3-7, and as a result don’t learn basic algebra properly in grades 7-9. Then they can’t do Physics, Calculus, or Chemistry in grades 11 and 12. I think this is a massively overlooked problem.

Like the article says, math professors are complaining that first-year students are increasingly unprepared each year for the rigors of algebra they’ll be seeing in first-year calculus class. I think this stems directly from not learning the times tables properly in the primary and intermediate grades. Once these students reach first-year university, they decide math isn’t for them, and major in something that requires the least amount of math possible — probably something like history, philosophy, geography, English, or something else in faculty of arts. As the article suggests, many of these students will go on to become teachers, probably teaching in the primary and elementary grades.

The Reynolds article concludes that the domino effect of misunderstanding is leading to a visual cycle of mathematical inadequacy. When students don’t understand basic operations, they struggle, finding a way to make it through math class each year, and then drop it from their timetables soon as they can, only to be rudely confronted by it ten years later, when they’ve become teachers, perhaps standing in front of a group of grade 7′s attempting to explain how equations work. Hundreds of new teachers are graduated every year in BC, and many of them are woefully ill-prepared to teach much math beyond grade 8 or 9, and yet they do it anyway. It’s probably not their fault—they were simply thrust into the role and were just happy to have a job, but now they have to figure out a way to teach math to some uninterested teenagers. This ”never-ending cycle of innumeracy” is real, and it’s certainly alive and well in the Lower Mainland of British Columbia, gradually lowering our mathematical adequacy with each passing year.

**Summary and (non-scientific) Conclusions:**

**1)** Rote learning is not completely evil, and should not be abandoned completely. A few short drills every other class can help reinforce concepts learned in earlier classes, especially when learning times tables and algebra.

**2)** Calculators should be used wisely, with increasing use in later grades, especially in the applied sciences such as physics and chemistry.

**3)** The third problem is much bigger and harder to fix. Teachers who hated math as students, who haven’t taken a math class in years, and who trudged their way through maybe one ‘How to Teach Math Class’ course during their training to be a teacher, are teaching your kids math in their elementary years. With the added confusion of new visual methods, the kids are inadvertently getting confused as well, and are likely to develop a distaste for math by the time they reach high school.

Thank you for your excellent post about this article. Before commenting, I feel the need to point out an error that was in the printed version of the article. Ross and I do not find it difficult to figure out our daughters’ homework. Many of the new methods for teaching math are confusing for children and parents, and I told the Maclean’s writer this. However, I did not tell her that these methods were over our heads and was quite angry that Maclean’s misquoted us in this way. (Somehow, we felt that this was an insult to us professionally and were very unhappy with Maclean’s.) Maclean’s has promised to print a letter from me, which corrects this, in the next issue.

I agree with everything you have written here. The pendulum has swung so far in the other direction in math education that teachers are discouraged from drilling kids at all and this often sets them up for future failure. I want kids to understand the concepts behind the math they are doing but this does not need to come at the expense of mastering concepts and skills. And no one ever mastered anything without a sufficient amount of practice – the same principle holds in mathematics.

I agree with you on the calculator issue. The key with technology is to use it wisely. In the classroom, teachers need to be careful that calculators aren’t replacing thinking entirely – especially when students are first learning concepts. I will point out, though, that the number of students I have in my first year math classes that are completely calculator dependent is shocking! Many need to reach for the calculator to do simple math calculations they should be able to do in their heads. This tells me that something is wrong.

The issue with teacher preparation in math is huge and is, in my opinion, one of the most important factors in this equation. There are some excellent teachers out there but there are far too many teachers who are uncomfortable with math and math anxious and this can transfer to their students. This is the fault of a flawed system that allows for this to happen and does not provide adequate training / support for teachers in math.

Thanks for pointing this out, and I’m glad to hear that they’ll be printing your letter. Sorry if I implied that you weren’t able to figure it out. Obviously, finding something confusing and not being able to figure it out are two different things entirely, and I think you have every right to be upset with being misquoted in the article. I’ve added a note to my post regarding that point.

I mostly teach and tutor Math 10-12, Physics 11 and 12, and Calculus. Occasionally, I have worked with grade 8 students who are learning algebra by using red and yellow tiles in Math Makes Sense. Perhaps I’m just out of touch and/or used to doing algebra in a more traditional sense, but I’ll be the first to admit that these methods make very little sense to me (i.e. they confuse me). I have yet to meet a student who seemed to prefer the tiles, or who seemed to benefit from their use.

I’ve also seen the phenomenon you’ve mentioned regarding calculators, where grade 12 students are confronted with something like 1-2, and they reach to their calculator to confirm that the answer is -1. I agree that this is a huge problem.

Thanks for commenting. Looking forward to reading your letter!

The following are the suggestions I have for making improvements to the numeracy of students in the K to 12-school system currently in Ontario:

Establish SAT testing for all students leaving Grade 12. Simply adopt the US system of testing to implement right away. At the present time, there is no sorting being done by the secondary school system and students are leaving that system with over inflated grades and getting into programs at the post secondary level they cannot handle. I would also include both the comprehension and writing parts of the test to cover it all, namely, “reading, writing and arithmetic”. If someone writes the SATs, I have a very good idea how that student will do in university and college level courses. Teaching to an exam forces everyone to at least learn the need to know.

Eliminate the distinction between college and academic courses in the secondary school program in mathematics. Pushing someone into the college courses too early only sends and reinforces the message that those students cannot handle mathematics. Obviously the academic courses are under performing based on the failure rates of students taking introductory level mathematics courses at the post secondary level at university, so they will need some sort of overhaul also. What does the student need to know to progress to the next level is clearly not being achieved by many secondary school students.

Following along the thoughts of the above paragraph, there has to be something wrong with the way mathematics is being taught in the secondary school system. There certainly is with the way much mathematics is being taught in the universities. In 1968, at a university in southwestern Ontario, a bunch of applied mathematicians got together to try to come up with a standard way of teaching their applied courses. We all decided that teaching the theory first was not always the way to proceed. Instead showing the student the usefulness of the technique they were about to learn with one example after another was sometimes the best way to proceed. When the technique was understood, then if the theory backing the technique was helpful, it was also taught, but mostly it was decided just to summarize it. In a survey of all mathematics students in the faculty, most students did not care about how the result was derived and cared much more on how to use it. This is a formula I have used mostly teaching aviation students the many ground school courses they have to take throughout their career.

Make the taking of mathematics compulsory every year of both primary and secondary school and do not let a student progress from one year to the next unless they actually pass the preceding year.

There needs to a recognition of the fact that many secondary school teachers in mathematics completed only a general BA in mathematics, or took mathematics only as minor. Hence, require all teachers of mathematics in secondary school to undertake a professional upgrading course in the subject at least every three years, a summer course or a night course during the school year and pay for the course. If they cannot pass the course, then freeze their salary increment until they do.

When SAT testing is unsatisfactory, establish as an additional transition for students leaving Grade 12 in Ontario that they spend at least one year at a community college taking upgrade courses in mathematics if they want to proceed to a university or college in a technical program. Do not let students return to high school for a 5th year if they can actually graduate. This is basically the system that is employed with the cegeps in the Province of Quebec.

If education is really a continuum, at the present time it is not, then do something about the level of university teaching in mathematics. This is definitely part of the problem. Just because someone has a PhD does not mean they know how to teach what they know. In fact, some of the worst teachers I ever had tried to teach me mathematics at the university level. A short instructional techniques course is all most would need and for foreign instructors with a poor command of English or French, some basic instruction in improving the spoken language., especially annunciation. I have gotten a lot of comments about this latter point from students in the past.

I would be careful asking too many mathimaticians both teaching at the secondary level and especially at the post secondary level what is wrong with the mathematics skills of students leaving the current secondary school system. There will always be those small percentage of students that will excel in mathematics in spite of the system who will go onto the post secondary level and want to study mathematics for the sake of itself and end up teaching the subject somewhere. What we are and should be talking about is not turning everyone into a mathematician, including those who will be teaching mathematics at the primary and secondary level, instead we should be talking about teaching the necessary need to know skills so that both student and teacher know how to use and successfully instruct mathematics. Mathematics is nothing more than a tool for the vast majority to help them better understand and deal with the world we all live in.

Thanks Paul for the detailed comment. I agree with a lot of your points.

I think the debate about theory/derivation of formula vs. application/usefulness of formula is an interesting one, and can go either way depending on the class, the level of the students, and the type of formula. For myself, if I’m learning the Quadratic Formula in grade 11 math, it’s neat to see where it comes from, but I’d rather see examples of how it works, and how to use it to calculate roots of quadratic functions. Once I know that the formula was created to solve for the x-intercepts, I’m good to go and should start practicing. However, if I’m learning a new physics equation, say Work = Force x Distance, I’d like to see the derivation, showing me an older formula that I already know (Force = Mass x Acceleration) is used to find work, meaning that Work is actually equal to Mass x Acceleration x Distance. A lot of students seem to think that formulas are just magical things that were discovered one day, and showing them some theory/derivation certainly can’t hurt in some cases.

Of course, these would just be my preferences, and everyone is different.

Secondly, I think your idea of having teachers write professional upgrading tests every few years is a great idea. It brings a measure of accountability, which the government and public should like, and (most) teachers should embrace it too. If teachers are truly passionate educators they should be learners themselves, and if they want what’s best for their students, they’d recognize that learning new techniques and writing a test every few years is not only good modelling, but valuable experience. You said secondary teachers should attend courses but I think all K-12 teachers could benefit from some math upgrading every now and then.

If you’re teaching math 6 or 7, for example, you should know a variety of techniques for demonstrating and solving questions with/in order of operations, fractions and decimals, geometry and algebra, to name a few. And you should be able to demonstrate that you’re capable of writing a test using the techniques you’re supposed to teach For some elementary school teachers, they might not have taken a math class since their first year of university, which could have been five, ten, or thirty years ago. As teachers, we should be reminded of what it’s like to learn (or re-learn) something new every so often. It makes teaching easier.

Lastly, I’d like to add myself as another person who can attest to the ‘annunciation of foreign instructors’ point you brought up. It’s definitely common in BC, in K-12 and post-secondary.