Wednesday, March 12, 2014

Return of the Math Wars

The math wars are heating up in Canada.

First, some background: 

The math wars are nothing new. Some could argue they are timeless. Others might say they started in the late 90s when the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics which "called for more emphasis on conceptual understanding and problem solving informed by a constructivist understanding of how children learn."

If I had to distill the math wars down to a simple idea, I would probably say that constructivist math calls for an increase emphasis on understanding while simultaneously calling for a decrease emphasis on direct instruction of facts and algorithms. The math wars get heated when critics come to see these changes to mean an elimination of basic skills and precise answers. 

Let's run through some frequently asked questions from critics of constructivist math:

Math hasn't changed and neither have kids, so why are we changing how we teach math?

The argument isn't necessarily that math or children have changed, but that our understanding for how children learn math has evolved. Decades ago, when we embraced behaviourism and psychometric tests, education moved from an art to a science that said knowledge is acquired by internalization from reinforcement. Behavioural mathematics is about drill and reinforcement and might be summarized as all about teaching mathematics at students.

In his book The Glass Wall: Why Mathematics Can Seem Difficult, Frank Smith writes:
The constructivist stance is that mathematical understanding is not something that can be explained to children, nor is it a property of objects or other aspects of the physical world. Instead, children must "reinvent" mathematics, in situations analogous to those in which relevant aspects of mathematics were invented or discovered in the first place. They must construct mathematics for themselves, using the same mental tools and attitudes they employ to construct understanding of the language they hear around them. 
Maybe math and children haven't changed, but our understanding for how children learn math is more sophisticated than generations ago. Jean Piaget was an epistemologist who studied the nature and origins of knowledge, and his 60 years of research tells us that children learn mathematics by constructing them from the inside, with the artful (and scientific) guidance of a teacher and their peers. Constructivist math is less about teaching math at students and more about math learned by students.

Behaviourism and Piaget's Constructivism are both scientific theories that
have been verified all over the world. An interesting phenomenon in a scientific
revolution is that while the new theory makes the old one obsolete,
the old theory remains true within a limited scope. While Piaget's theory can explain
 everything behaviourism can explain, behaviourism cannot explain children's
acquisition of knowledge in a broader, deeper sense. Piaget's constructivism
goes beyond the primitive theory of behaviourism by encompassing it.
Memorization is important and it is a very real product of learning -- but memorization is not the primary purpose. Memorization is something that happens because children learn and understand mathematics first. Winning is an important part of sports, but we don't teach kids how to win -- we teach them how to play. Like winning, memorization has its place, but we need to keep them in their place. Like winning, memorization becomes ubiquitous because it is a feature of learning and understanding.

For more on what works better than traditional math instruction, check out Alfie Kohn's article on math.

In math there is one right answer. Doesn't this fuzzy math just confuse kids and convince them to hate it?

First of all, let's not pretend that traditional math instruction didn't confuse and turn a lot of students off of math. (Full disclosure: I grew up with traditional math instruction that included memorizing my times-tables and Mad Minutes! and I learned that I was terrible at math and hated it.) We have to be careful that our knee-jerk reaction to change isn't an act of Nostesia: a hallucinogenic mixture of 50% nostalgia and 50% amnesia that distorts rational thinking.

When my friend Dave Martin tells people that he is a high school math teacher with his Masters in Mathematics, people look at him like he's left-over mashed potatoes -- most people can't imagine why Dave would put himself through such needless torture. We have generations of adults who have graduated from traditional math instruction who break into a cold sweat when confronted with long division. For too many students, the extent of their enthusiasm for math climaxes when they are told they only have to do the odd questions. (Listen to Dave Martin debate math on CBC here.)

As for right answers, there is only one right answer if we limit ourselves to asking questions that have only one right answer, such as 4 + 3. Some of the most provocative questions that hook students' curiosity are questions that have no one right answer, such as how much does it cost to redesign your bedroom.

One of my favourite elementary math questions comes from Constance Kamii's book Young Children
Constance Kamii's three books on
Young Children Reinvent
Arithmetic are must-reads.
Reinvent Arithmetic
Grandpa said he grew up in a house where there were 12 feet and one tail. Who could have lived with grandpa?
I like this kind of question because there are of course right answers, but there isn't one right answer. I also like it because it allows the children to construct mathematics out of the necessity of their reality. Constructivist teachers create questions, projects and games that give children the opportunity to invent arithmetic out of their reality.

It might defy common sense, but teaching children algorithms and tricks before they've had a chance to construct them for themselves actually sabotages and confuses children. Kamii writes:
It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development.
For generations, math students have asked out of frustration, "when will I ever use this?" To be clear, I'm not suggesting that everything in mathematics should be reduced to real-life application -- the significance of mathematics should not merely rest on its practical value; and yet, I would like to hear students say they use math to solve problems and understand the world rather than just to complete the odd questions from the textbook or worksheet.

I still remember being taught how to divide fractions in junior high. I was told to flip the second fraction and then multiply. It was a trick that enabled me to get high scores on the tests.

Here's the problem...

To this day, I have absolutely no idea why I flip the second fraction and multiply. I have no idea what the mathematical reasoning is. I can get the right answer on the test, but there is nothing mathematical about my (lack of) understanding for dividing fractions. If we want to confuse and turn students off of math, I can think of no better strategy than to make math a ventriloquist act where children are merely told the most efficient ways of getting the right answer. This is mindless math mimicry.

Graphics like this from Alberta's
 Wildrose Party
 is an effort to turn
 pedagogy into cheap political points.
The temptation to teach children carrying and borrowing as soon as possible comes from the need for efficiency, but this reminds me of what Martin Luther King Jr. said:
The function of education, therefore, is to teach one to think intensively and to think critically. But education which stops with efficiency may prove the greatest menace to society. The most dangerous criminal may be the man gifted with reason, but with no morals.
If a teacher is provided the appropriate professional development and they understand the theory behind Jean Piaget's constructivism, then this "new" math actually reflects the very essence of how people learned arithmetic before we had all these tricks and algorithms - essentially making this "new" math a very "old" math.

Canada's rankings on international tests like PISA are dropping. Doesn't this mean we should go back to basics and traditional math teaching.

PISA's rankings on their own are useless. If we focus too hard at the competitive rankings, and reduce the point of school to "test scores are low, make them go up", we risk ignoring the real lessons of PISA.
Corporate Reformers in the United
States use infographics like this to
encourage people to focus on
meaningless competitive rankings
while ignoring the real lessons
of PISA.

The real lessons from PISA are found from researching how each nation achieved their results and then assessing their methods via ethical criteria that is independent of their results. Things go very wrong when we allow education policy to be driven by circular logic: define effective nations as those who raise test scores, then use test score gains to determine effective nations. (Things go equally wrong when standardized tests move from a means to measuring education to the purpose of education.)

Since 2009, Alberta has dropped from 9th to 10th place in world rankings. Jonathan Teghtmeyer writes
A 2 per cent reduction in our raw score on math over a period of three years led to ministerial handwringing, parents initiating petitions, newspaper columnists launching crusades and CEOs descending from on high to chastise teachers.
It's important to also note that the children who wrote the 2012 PISA test had the old traditional math curriculum for their first 7 years of school and only 3 years with new curriculum. To be clear, this doesn't prove that "old" or "new" math is responsible for the change in PISA scores -- there are too many other variables, including other in-school factors, out of school factors and unexplained variations. When I take my umbrella to work it rains, but that doesn't mean my umbrella caused the rain. Too many people confuse causation and correlation in an attempt to draw convenient conclusions that they simply can't prove. No one can prove that the change in PISA scores were because of teacher instruction.

PISA's 2012 rankings show Finland has been replaced at the top with a handful of Asian countries (and cities). By idolizing the rankings, people might drop Finland like a hot-potato to chase after Asian countries who achieve their high scores with very different priorities and questionable means.

PISA envy can lead us to aspire to be more like top-ranking East Asian education systems even though East Asian education systems are desperate to reform their schools to look more like ours. Yong Zhao writes:
While the East Asian systems may enjoy being at the top of international tests, they are not happy at all with the outcomes of their education. They have recognized the damages of their education for a long time and have taken actions to reform their systems. Recently, the Chinese government again issued orders to lesson student academic burden by reducing standardized tests and written homework in primary schools. The Singaporeans have been reforming its curriculum and examination systems. The Koreans are working on implementing a “free semester” for the secondary students. Eastern Asian parents are willing and working hard to spend their life’s savings finding spots outside these “best” education systems. Thus international schools, schools that follow the less successful Western education model, have been in high demand and continue to grow in East Asia. Tens of thousands of Chinese and Korean parents send their children to study in Australia, the U.K., Canada, and the U.S. It is no exaggeration to say that that the majority of the parents in China would send their children to an American school instead of keeping them in the “best performing” Chinese system, if they had the choice.
If we change how we teach math, doesn't this mean our children will get a fundamentally different education than we got?


If we want school to improve, then we have to allow it to change.

The nature of society's first reaction to changes
to school is resistance. It takes time for us to give
up our vested interest in our old ways of thinking.
People who argue that school doesn't need to improve (or should just go back to basics) are no different than a commissioner of the patent and trademark office resigning because everything that can be invented has been invented. If we are not careful, blind self-justification can mislead us to believe that the here and now is as good as it gets. Wishing tomorrow to be just like yesterday won't make today a better place.

Don't get me wrong. Change for the sake of change is no better than tradition for the sake of tradition. But let's keep in mind that too many of us merely endured math or flat out hated it -- I think it's safe to say that not enough of us loved it.

And we aren't going to get more children to love math (or school in general) by pretending that school already doesn't have enough lectures, direct instruction, worksheets, textbooks and memorization.


  1. Good summary, and of course, constructivism is necessary. Teaching as if the brain constructs knowledge, as if meaning, understanding and internal motivation is essential for learning as well as love of mathematics. Also, all these either-or fights are silly fights in the sandbox. Mathematics education necessarily includes learning math facts; AND they will be learned better in the context of total brain involvement.
    HEY YOU GUYS: watch out for either-or-ism.

  2. I do believe children have changed. Evolution is constant. They process information differently. Perhaps their emotional layer is the least evolved, but children have definitely changed.