New Math equals trouble, education expert says

The CBC just ran an article on the problems in our current math system which was terribly one-sided and an example of the worst kind of fear-mongering journalism. They are quoting an article by Michael Zwaagstra, an "educational expert" writing on behalf of the Frontier Centre for Public Policy.

First, let’s examine the article written by Zwaagstra.

A solid understanding of mathematics, also known as numeracy, is an important component of a well-rounded education. The ability to perform basic mathematical computations is a requirement of many entry-level jobs. In addition, careers in fields such as engineering, medicine, finance and all of the sciences require a solid background in higher-level university mathematics, including calculus, statistics and linear algebra.

The first thing to point out here is that the basic mathematical computations … for entry level jobs are much different than the higher-level university level mathematics needed for engineering, medicine, finance, and the sciences.

I have to agree with Zwaagstra that a solid understanding of mathematics is an important component of a well-rounded education, but his assertion that mathematics equals numeracy is definitely false, as I have had pointed out to me on a regular basis. There are many mathematicians, engineers, doctors, economics, and scientistis who struggle with basic computational math, but are fully capable of doing higher level mathematics, and this has been true for a long time; far longer than the new math has been used in schools.

Because math is such an important skill, schools have an obligation to ensure that students learn key math concepts. Unfortunately, schools are largely failing in this regard. First-year post-secondary students are increasingly unprepared for university-level mathematics, and this has led to a proliferation of remedial math courses at universities across Canada. Many parents choose to enroll their children in special tutoring sessions with organizations such as Kumon and the Sylvan Learning Centre to fill in the gaps left by the public school system. Unfortunately, many cannot afford extra tutoring, and this creates a two-tiered system that unfairly penalizes children whose parents cannot pay for extra math lessons.

Now Zwaagstra points out that remedial math courses are on the rise in universities, but he doesn’t mention a couple of key facts. First, under the old system of mathematics instruction, around 50% of students failed first year math courses, which were often included in programs as a tool with which to weed people out of university. Could it be that this issue has always been around, and universities are simply now doing something about the problem? What about the increase in students seeking a university education? Could these two issues be connected? Zwaagstra has assumed a correlation between the number of remedial math courses, and the effectiveness of k-12 math education, without actually finding research which supports his conclusion.

Further, he talks about parents enrolling their kids in after school tutoring programs without discussing the reasons why parents are doing this? Are parents increasingly enrolling their kids for extra tutoring because they are dissatisfied with their kids current educational attainment? Or do they have other reasons for paying for these tutoring services? We don’t know, and Zwaagstra doesn’t provide us with any evidence for the reasons for parents to choose tutoring, he just cherry-picks this fact because it seems to support his argument.

Although there is solid evidence supporting the traditional approaches to teaching math that involve mastering standard algorithms, practising skills to mastery and introducing concepts in incremental steps, most provincial math curricula and textbooks employ a different approach. Constructivism, which encourages students to come up with their own understanding of the subject at hand, is the basis for this new approach to teaching math. As a result, there is very little direct instruction of important mathematics algorithms or rigorous practising and memorization of basic math facts.

There is also solid evidence showing that the longer that people are out of school, the less likely they are to use the algorithms they use in school, but the more successful they are at solving mathematical problems they encounter, as Keith Devlin points out in his book, The Math Instinct. In other words, traditional school math seems to be a hindrance to people being able to actually solve real world mathematical problems. It’s worth pointing out that Devlin’s research is reasonably old, and most of the participants in the research learned mathematics in the traditional method. Is it even worth pointing out that Zwaagstra doesn’t actually include any of the solid evidence in his paper, and the footnote here (see the original article) leads to a definition of the word algorithm?

Our students deserve better. Pupils who are not taught math properly are being unfairly denied the opportunity to enter careers in many desirable fields. The public school system has an obligation to ensure that every child has the opportunity to learn the mathematics required for university-level mathematics courses.

It’s pretty important to note that the new math is not being taught evenly, and that when teachers are given proper training in how to use the new math materials, their students’ understanding improves. To say that the problems in our math education system are entirely due to the introduction of the new math curriculum, is pretty irresponsible, given that any number of other factors could be contributing to the problem. Further, many schools use the International Baccalaureate program, which itself relies on the "new math" with a focus on students understanding mathematics and being able to communicate their understanding and these students are highly sought after by universities. If the new math was so destructive, wouldn’t we see these students being turned away by universities in the sciences?

Zwaagstra then goes on to bash the results of the PISA examinations, citing an article (claiming it is research) written that suggests that Finnish students are not as good at math as the PISA results would claim, and that by extension, neither are Canadian students.

There is a strong consensus [emphasis mine] among math professors that the math skills of these students are much weaker than they were two or three decades ago.

Zwaagstra links to two articles (neither of which is a research study) that state that some professors have found a drop in numeracy skills (again, these are associated with mathematical ability, but are not equivalent), and the other of which makes no mention of math skills at all. In this case, Zwaagstra is completely misrepresenting the articles themselves. He then points to two professors who have done research on the computational abilities of graduates and noticed a decline, but he does not clarify whether or not this is correlated with a decline in their ability to do university level mathematics.

Zwaagstra continues by bemoaning the lack of standards and emphasis on accurate calculations by the National Council of Mathematics Teachers (NCTM) and the Western and Northern Canadian Protocol (WNCP). Clearly the research these two organizations have done for decades is not sufficient for Zwaagstra, especially considering Zwaagstra’s credentials (Hint: He’s never been a math teacher, nor has he any credentialed expertise in mathematics education Update: Apparently, Zwaagstra spent 7 years as a middle school math teacher, so I’m retracting at least this part of my response).

However, there is a big difference between demonstrating a conceptual understanding of mathematics and actually being able to solve equations accurately and efficiently. Just as most people would be very uncomfortable giving a driver’s licence to someone who merely demonstrates a conceptual understanding of how to drive a car, we should be concerned about a math curriculum that fails to emphasize the importance of mastering basic math skills.

To extend Zwaagstra’s analogy, we should similarly be afraid of giving the keys to someone who has no real world experience driving. If someone has spent all of their time in a flight simulator, but never actually driven a car, should they be allowed to do so? Does an emphasis on the mechanics of driving a car (or the mechanics of mathematics) turn someone into who is capable of driving a car (or able to use mathematics)?

Zwaagstra’s solution to improving math education is to move "back to basics" which is as unoriginal an idea as I’ve heard, and it is arrogant of Zwaagstra to assume that this approach hasn’t been tried before. Perhaps Zwaagstra could instead address the issue of elementary school teachers often lacking support and training in how to teach math? Zwaagstra points out (correctly) that having mastered one computation, students are then better able to learn another computation, but this leaves students learning a series of computations, and not spending any time actually using them.

JUMP math is mentioned in Zwaagstra’s article as an antidote to the problem, but he doesn’t talk about the issue of the associated training, or the lack of diverse assessment used in the JUMP math system. I think that the training manuals which go along with the JUMP math curriculum, for example, actually address the misconceptions of the people teaching the math (mostly elementary school teachers) rather than itself being a significantly better system. As one educator has told me, JUMP math is pretty useless without the training materials for teachers.

Just as someone who does not practise the piano will never learn to play well, someone who does not practise basic math skills will never become fluent in math.

Similarly, someone who has not had time to play with a piano, to improvise, and to perform music for others will never develop an appreciation for the instrument. Zwaagstra is suggesting that we should discard the extra parts of math education, like problem solving, and focus on computations, which is the musical equivalent of only learning scales, and never getting to perform music.

No one would stand for that in music education, so why should we accept it in math education?

 

Update: Here’s another good rebuttal to Zwaagstra’s article.

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19 Comments

  • It is frustrating to read about the “push back” against efforts to make maths ed an authentic experience for learners. Great response David- so much more research backs a concept laden pedagogy than a procedural approach in terms of disposition to maths as well as achievement. Why are people defending “long division”? How is it relevant to life beyond a classroom and what understanding does it promote? The debate does raise the important issue of educating parents about 21 cent maths.

  • I agree with you that Zwaagstra mistakenly equates mathematics with computation. If you were to extend his argument to literature it would be like saying that students must spend more time practicing vocabulary in order to read essays, novels and poetry, rather than having students read essays, novels and poetry in order to develop their vocabulary in a meaningful context.

    I was also frustrated by the emphasis Zwaagstra put on the argument that students are ill-prepared for university math when, pedagogically, university programs lag behind elementary and secondary schools. University professors – who are mathematicians first and teachers second – have developed their courses based on students who have learned how to do math by rote. Now that more and more students are coming to their courses with a more conceptual than mechanical understanding, it is not surprising that they are seeing different results. Is there any discussion about whether the university programs should change?

  • I appreciate the time and effort you took to point out the holes in the article. With every program or idea there is always someone, or a group, who has to pick it apart, and often does so with a lack of authority on the subject, or research to prove it.
    Thank you.

  • One of the problems that I have with these types of arguments against this ‘new’ approach to teaching math…

    They assume that all teachers have adopted this approach and this is the reason for the decline in some measure of numeracy and/or mathematics.
    Just because the curriculum has changed does not mean there has been a change in curriculum. Many of us may still use a traditional approach. I know of some teachers who use a resource created in the 80s. That would be a few curriculums ago (I know, I know, probably should be curricula but that always sounds pretentious to me). Maybe this decline that these authors write about is due to not adopting this ‘new’ approach. Maybe it is b/c we have held on to a traditional approach that no longer works (if it ever did) for today’s students.

    By the way, it ain’t new. I recently saw a video clip of Marilyn Burns talking about the difference between math then and now. It appeared to be from the mid-eighties. Sadly, 20+ years later she could still say the same thing. The picture of what math looks like today that she painted back then really wasn’t ever realized.

  • Not a bad rebuttal, but when it comes to the competing analogies of giving someone a driver’s license I have to side with Zwaagstra. It is better to give someone a license who has extensive practice in a simulator than someone who merely has a concept of what real driving is like. Realistically, I believe the “new math” may be helping more students suceed in the lower grades but is not as effective for the minority who pursue higher science degrees. New math helps the average person do basic calculation long after schooling is over, but old math is the way you form physicists and engineers – people who are already gifted enough to grasp the concepts without making “concepts” the focus.

  • David Wees wrote:

    Wouldn’t it be better to have both the practice with the skills, and the experience of being a driver? Does it have to be a one or the other situation?

    As for the assertion that physicists and engineers are formed using only the more traditional methods, I have some physicist and engineering friends who might disagree with you on that…

  • Thanks for the great rebuttal David.

    I did want to comment on one point as some one who was involved in the development of the “new math.” Is it really “new math?”

    I can remember my mathematics teachers (at least the ones that I think did a great job) emphasizing a need for conceptual understanding of mathematics. I remember them stressing the mathematics process (communication, connections, mental mathematics and estimation, problem solving, reasoning and visualization). I did not include technology in this list as it was not a big emphasis at the time as calculators, computers, etc. were “new” and I can even remember the debate in one class as to whether they would catch on. Seems they did.

    When we polarize “new” and “old” math in the fashion that it seems to be in the media I am forced to think the proponents of the “old math” are more in favour of computational skills than conceptual understanding and I do not remember it that way when I was in school. I remember both stressed.

    What I do see as different in the 21st century is the following:
    * What we know about mathematics pedagogy
    * A desire to have all students succeed (I did well in math but many (perhaps most) of my peers did not)
    * The skills students need now are different than when I graduated
    * Technology
    * Parental influence
    * The general education level of the Canadian population
    * Speed of communication and prevalence of information (Think of the ease of obtaining a positive or negative research study on anything)

    I expect that debate on mathematics will continue long after I am gone but I don’t think the idea of a good mathematics students being able to do the math and know the math will. To me these are linked and timeless.

  • David Wees wrote:

    I think the focus on computational skills and/or conceptual skills probably depends a lot on who taught you. I definitely remember teachers who focused on rote memorization, and did not spend ANY time on understanding the computations we were doing, and I could imagine that for some children, that might be their entire experience of math.

    That being said, I also had some good teachers, who as you say, focused on understanding and being able to communicate what we were doing. I remember these people as part of my inspiration for becoming a mathematics teacher!

    So you are right, there is less difference between the "new" math and old math than we are led to believe in this article…

  • This is great food for thought, thanks for your work David! I have just started a class on Mathematical Processes as part of my education program. One of the first things our teacher talked about was a "brittle" understanding mathematics. We were told that that just robotic knowledge of algorithms, without the comprehention, was "brittle", and would be less helpful when faced with a problem than a comprehensive understanding would be. We also discussed the difference between a problem (a question where the student does not know how to find the solution) versus an excise (a question where the student knows how to apply an algorithm to get the answer).

    I think Zwaagstra would be correct in his saying that students need algorithims to solve exercises, but when faced with a problem that doesn’t have a classic algorithimic solution, students need the comprehension gained by exploration to puzzle through. Do you agree? I was sad to see that when this article was persented in the Vancouver Sun, there were opinions offered other than Zwaagstra’s, but only at the very end of the article. Also, the article was headlined: ‘Canada’s public schools doing a poor job teaching mathematics: study’. So someone perusing the headlines would read that and assume that the study is credible and correct. I appreciate the effort you’ve made, David, to educate readers.

  • They’re another neocon group of “experts” like the Fraser Institute whose essential job is to cast doubt on the expertise and skill of anyone working in the public sector. Stephen Harper is a big fan.

    It’s trashy of the CBC and predictable of the Vancouver Sun to feature this material as “a study.” It obviously had a precast conclusion and worked back to any “evidence”, such as it was, from there.

  • I was really surprised by the articles on this topic that the CBC has been running with over the last two days. Today there was an article quoting a Maths Professor who tears down the MathEd programme at his own University. The reporter doesn’t appear to have made the effort to contact anyone in the Faculty of Education for a response. Brutal.

    As a musician, the clincher for me is in your final argument. I would never start someone only on scales and music theory, without giving them at least a taste of what the goal is – the beauty of music and song. I can’t imagine how incredibly dry and boring that study would be. The goals of the new Maths curriculum is the same – to produce good mathematicians: people who solve problems, recognize patterns, and examine cause and effect relationships.

  • Kayla Williams wrote:

    David Wees, I am a student at the University of South Alabama and I am currently taking EDM 310. The article you posted very much surprised me, and it is disappointing that Zwaagstra would make so many accusations against both teachers and the public school system without having sound research to support his arguments.

    I completely agree with you when it comes to the skills required to be successful in mathematics. It is both about understanding the subject, problem solving, and computational skills. As a student, it is important to me to be well rounded and I believe this applies to every subject. In order for students to have mastered mathematics, they must know how to solve problems and compute different equations, but also have an understanding of how to apply it in their lives and careers.

  • Have we actually agreed on what “new math” even consists of, or are we just taking a few things from the article and saying that is everything in new math? If we really want to have a quality discussion about this we would have to layout everything that new and old math is. It’s too difficult having a discussion otherwise.

    p.s. Long division is absolutely useless. No one actually sits there and does long division. That’s one of the reasons we made computers. I can’t believe anyone would use that as an argument concerning anything, but especially using the proficiency of that skill to determine the math education level of a person.

  • David Wees wrote:

    I’m in agreement with you Dave, on the long division issue. It is only useful to learn as a process if you will be using polynomial division later, but that happens MUCH later in the learning typically, and so the skills often do not transfer smoothly, and is itself a skill which can be done using a computer.

    Many skills/algorithms can be done on a computer. One question is, which of these is vital to the understanding of further concepts by students?

    As for the definition of "new" math, it is mostly about how different skills are taught, at least it seems that way according to Zwaagstra, rather than a whole new set of curriculum. As Richard points out, much of this math is not really new at all.

  • That’s what I thought was meant about what new math was. It’s not actually brand new math it’s just the way that it’s taught. If you want to talk about new math then let’s talk about what mathematicians are doing nowadays. So the person/people that made the phrase “new math” popular should be flogged.

    So can we stop using that phrase? All it does is invite non-educators to complain about something that educators aren’t even really talking about. I hope you know what I mean there. I would bet if we sat down with a group of parents and told them we were going to teach their sons or daughters new math they might be a little ticked off or put back for whatever reason, even though they don’t really know the details. But, instead, if we told them that we were going to use new teaching skills and practices that have been proven to increase children’s understanding of math, then they would probably feel comfortable about that and supportive. And the later is really what new math is.

    Every company/business in the world comes up with better strategies and innovations, but if education does it people ask why we would change something that’s been working for years. Sure, the automobile worked great when they put it out the first time, but I’m really glad they did something new throughout the years. How many people are complaining about that?

  • David Wees wrote:

    Yeah, I don’t use the phrase new math myself, except it seemed appropriate in the context of rebutting Zwaagstra’s article.

  • As I’m not a teacher I don’t share the same vocabulary some of you are using. I’ll keep it very simple. As a father of a child in grade 5, I am absolutely frustrated with the “new” way of teaching addition, subtraction, multiplication and division. Instead of teaching kids to add / subtract / multiply one column at a time from right to left, they are teaching what I presume to be the “conceptual” approach some here have made reference to. My child doesn’t quite get it and I can’t help her because I have no idea what the hell it is. It seems to involve many more steps and calculations than the way I was taught in the 70’s. As a result, I teach my child to get the answer the way I know how to do it. She learns this way easily enough and finds it easier than what her teacher is trying to teach. She recently told me – as she explained that she could not understand the teacher’s way of doing division – that all of the other kids in her class are also learning from their parents the “old” (traditional) way.

    This new format can’t be properly taught if the parents don’t understand it. Even with the best of teachers, if the parents can’t help their child with their homework and ensure that they understand what it is they’re supposed to be learning in school, they aren’t going to learn it. There has been no communication to me about this new system other than vague assertions from teachers that it is somehow “better”. I’m not buying into it; and furthermore, I am aware of several articles indicating that it is failing kids – something that becomes evident when they get to post secondary school. I feel as though I’m being defrauded by the school system which I’m paying to operate and provide a service.

  • David Wees wrote:

    Hi Dan,

    This will always be a problem with curriculum changes in any subject area – it leaves parents struggling to help their children. Some schools are addressing this by having parent nights on a regular basis where they talk about the curriculum changes and help parents understand the reasons behind them, and understand some of the techniques being used. I’d recommend talking to your school to see if they are open to this type of discussion.

    One of the problems with the "old" way of teaching that many teachers use is that it can leave students unprepared for some of the more advanced concepts in math, particularly starting with algebra, trigonometry, and pre-calculus. The reason why is that these areas of mathematics require some conceptual understanding in order to be able to do well, and students who have memorized the standard algorithms without understanding why they work often struggle with these later topics. If your child learns the reasons behind why these algorithms work, then they are far better off later on in their mathematics career.

    Also, as is true with any initiative, the skill teachers have at teaching the new curriculum can vary greatly. It may be that your daughter’s teacher needs some more training in how to teach the to the new curriculum so that your daughter does not struggle so much.

    David

  • As a mere parent, I am not a math expert like David Wees, but I do have a very good perspective on how this new math isn’t working.  Our own experience with our 2 school age daughters led us to enrolling 1 of them into a math tutoring centre by the time she was in Grade 5 because she could barely understand 2+3 or 2X3.  After a mere 7 months in the learning centre, she was multiplying and dividing numbers in triplicate (eg. 769X423 and 768/342), and was also UNDERSTANDING and multiplying fractions galore.  Her confidence soared.  Her marks improved.  Her demeanour changed incredibly, and math, once again, became her favourite subject.  Did we enrol her to become a star math student?  No.  We enrolled her so that she would receive the most basic of the math fundamentals which were being denied to her at her own school. And her success at this was, surprise, due to BASIC math drills and repetition.  3 years on she can still perform these math functions at a high level due to the instruction she received at the learning centre previously, and she is still loving it.   It saddened me to see that 4 years of schooling had failed to provide her with even the most basic understanding of math, because her teachers were using the Discovery Based Math concepts in the classroom, as dictated BY our provincial curriculum. As for our other “math brained” daughter, she was equally confused by the Discovery Based Math methodology she was receiving.  Luckily she had an excellent Grade 5 teacher who provided her with an “old fashioned” textbook which she used for extra work at home. That, coupled with regular multiplication, long division and fraction drills in the classroom, prepared her for middle school.  There is no doubt in my mind that she wouldn’t have been nearly as successful in math if it wasn’t for the intervention of this one particular teacher. 

     

    More and more of us parents are completely frustrated with our kids’ (lack of) math instruction in our public schools.  The overwhelming majority of parents, David, that I have spoken to, have their kids in tutoring centres and in the JUMP Math programs…because they WORK and they are not getting the same type of instruction in our schools. We are not putting them there because we want them to become stars in the classrooms.  We just want them to understand math, in it’s most basic form.

     

    I have also read Zwaagstra’s article. Wees isn’t being entirely accurate in his blog about this article.  I don’t think any math educator is advocating a return to the “Kill and Drill” technique that was in the old ways of teaching, but our kids still need to learn the basic fundamental building blocks of mathematics.  What Zwaagstra illustrates in his article are the differences in the methodologies when it comes to solving math problems.  This newer discovery technique employs a greater number of steps, and it is completely confusing to small children when trying to solve a problem.  These multitude of steps in trying to solve these problems completely overwhelm the brain for wee kids.  These new techniques would be better employed with older students AFTER they have already mastered their most basic algorithms. 

    Another problem is with the textooks and learning materials which are being used in our classrooms.  When parents point out inaccuracies and errors in their kids’ textbooks to teachers in our schools, they are being ignored. And even after this has been done, some parents have been told, “Oh well. It’s part of the curriculum.”

     

    Is this what we should accept for our kids?  I don’t think so.  I say this.  Ensure that basic fundamentals are still part of the math curriculum.  I have been told it still is, but actions speak otherwise.  Why is it that over 40% of the kids in my daughter’s class were receiving extra tutoring just so they could understand basic math?  Why do so many schools now use either JUMP Math, or other math programs, as a SUPPLEMENT to the curriculum?  This should all be PART of the curriculum.  Understanding multiplication tables and long division should be mandatory before the kids reach middle school.  There should also be a very firm understanding of fractions.  Currently this is not being done.  Furthermore, ensure textbooks and learning materials support these outcomes, and get some much needed training for our teachers, because I can guarantee you, their math skills are probably no better than yours or mine. 

     

    Goto http://www.wisemath.org and see the hundreds of signatures of other concerned math professionals, parents, grandparents and other citizens who feel very strongly that this new Discovery Based Math methodology is failing our kids.  They’ve already changed the school math curriculum BACK this year, to including basic understanding of algorithms, in the province of Manitoba. Nova Scotia is looking at changing their policies.  Ontario and British Columbia has heard the frustration and concern over this issue as well.  We’re fed up with having kids counting on their fingers in Grade 8.  So let’s go back and ensure we can give them the basic fundamentals that they deserve before employing these other concepts later on. 

     

    Sincerely,

    Tara Houle

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