Here We Go Again...

Every couple of years we see a huge public outcry about how we need to "go back to basics". Sometimes this is in response to test scores being "below average". Here I will (once again) weigh in on this.

Because most people have spent years in classrooms, they feel like they are qualified to weigh in on educational policy. They appeal to politicians who are not likely to have a background in education or psychology. However, the fact that Canadians are concerned about and value education is something that we can definitely be proud about.

Several news agencies have recently published articles calling for a "back to basics" approach to mathematical education. But what exactly are "the basics"? Are we speaking of numeracy, or simply a fluency with basic number facts?

Many opinion pieces cite methods such as Dewey's constructivism used in the current approaches. What is actually being taught in teacher education is inquiry learning, which shares some features with constructivism, but is not entirely the same thing.

An inquiry model does not require a student to "construct" their understanding of a concept and then leave it at that. The basic lesson has three parts: a minds-on section in which a problem that builds upon prior knowledge is introduced and students are asked to think about how they would go about solving it; an action section in which students collaborate and share their ideas, applying them to a new problem or problem set that extends the concept, with the same concept; and then a consolidation phase in which students share their work. Various approaches used by the students and introduced by the teacher as needed are compared and evaluated for clarity, consistency and efficiency. This phase is where the students consolidate their learning. Students are often asked to complete a new problem or problems using the concept as an "exit ticket" to show their understanding. The teacher uses these to determine the next steps needed for the class, as well as individual students, in order to further their learning.

In subsequent lessons, students are also asked to apply their mathematical understanding in various hands-on ways, which might include projects built in maker spaces, coding, or geometric art.

Taking a lesson to look at mistakes every now and then is also common. Students are asked to look at a teacher-chosen problem and solution, and demonstrate why the reasoning used is not correct. The ideas are that in learning from mistakes, students realize that making mistakes along the way is part of the process, and it also encourages them to work on their own mathematical reasoning skills and means of communicating their mathematical thinking.

What is missing from this approach? Memorization of an algorithm and repeated practice. Memorization of an algorithm provided by the teacher, with detailed steps on how to complete the algorithm, is what many adults equate to math instruction. It is what is familiar to them, since many learned it this way. However, simply knowing the times tables and how to do long division alone do not make a person numerate, any more than knowing the alphabet and phonetic sounds makes someone literate. Maybe you can sound out a simple word, but to gain meaning from the text requires comprehension skills. This is also true of math.


It is true that memorization of times tables helps with the quick completion of worksheets in higher grades. Computation abilities are still important. Even though we have tools everywhere that can complete this with greater speed and efficiency than people can, being able to process these smaller steps with ease and fluency frees up working memory needed to manipulate more complex problems. However, we do have computational tools (calculators, electronic devices, computers), so placing our priorities on those computational skills alone is not beneficial and does a disservice to our students. We need students who are able to apply those concepts, program the computers, choose a strategy, solve problems, make connections, find patterns and apply and extend those patterns, plan and strategize. We need to prioritize higher-order thinking skills that allow us to move beyond basic computation. Students need to develop a sense of number, quantity, additive and multiplicative reasoning, proportional reasoning, patterning, balance, spatial reasoning, estimation skills and so on.

To remain stuck at memorization of number facts and algorithms alone is simply not enough.

Practice is one area that in my opinion could use more balance. We have gone from reams of worksheets, usually all of a single problem type that does not require reflective thought, to the use of 1-3 problems in a day to illustrate a concept. Somewhere in the middle is a place where students have a chance to work on problems that reinforce a concept while being required think critically and strategize, not only with the algorithm of the day, or by matching a pre-determined vocabulary list with a given operation, but in visualizing and manipulating the information given until they make sense of it, then applying an appropriate strategy and computation for solving it. Students need to also be encouraged to search for and find the answers to the age old question, "(when) will we ever use this?". If they don't see a purpose in it, how can we expect them to find the motivation to struggle through a problem or concept? The purpose must be clear.

Another recent push in education is the concept of developing a Growth Mindset, as described by the work of Carol Dweck, and elaborated upon by Jo Boaler. The ideas here are that students need to be open to learning, and accept that there will be some struggle when they are truly learning, but that they are capable of working through this struggle to gain competency. This is especially important in math, since there are many myths that abound about people having a "math brain" or not having one, which is simply not how brains work. While we'd never shrug off being illiterate, common phrases and ideas such as "I'm not a math person" and "you must be so smart to understand math" show how our society reflects an idea that numeracy is out of reach for many people. If students are to learn math, they need to first believe that they can learn it, and the adults around them need to also believe they can.

The Latest False Dichotomy in Education

Years ago when I was a student teacher, there was a battle of sorts being waged in the language arts curriculum: the phonics vs. whole-word approaches to teaching reading. On one hand, students were asked to sound-out all words they did not know, and on the other, students were asked to memorize thousands of words. Reality: fluent readers use a variety of techniques including but not limited to the ones espoused by such methods. When some people tried to call this approach "whole language" it created confusion as many missed the difference between "whole word" where individual words are memorized out of context with 'whole language" in which context, along with phonics and sight words is an important component.

Many moons later, we see a similar battle being waged in the teaching of mathematics. 

There are the traditionalists who value "sage on the stage" and "drill and kill" methods in which students memorize algorithms and focus on answer-driven tests. Students become walking calculators, and weaker students are often left without the conceptual understanding to allow them to know when and how to apply these algorithms to solve everyday problems. Mnemonics such as "Yours is not to reason why, just invert and multiply", "FOIL" (which only works for up to two terms), and what I've only recently discovered, the "butterfly method" for multiplying fractions are examples of how conceptual understanding is replaced with memory tricks in order to gain a correct answer on a standardized test.

Then there are the constructivists who believe that students must create their own knowledge set through experimentation. They offer an overwhelming range of options for students to explore, but often neglect the final stages of consolidation and review, as well as time for practice with additional problems. Critics argue that since it took centuries to develop the fields of mathematics, expecting students to "reinvent the wheel" is a waste of time for everyone. Such teaching can also be time-consuming, and students who are struggling can become overwhelmed and confused with the large variety of methods to solve a given problem.

Again, just as in the reading example, the polar extremes reflect a false dichotomy when it comes to learning.

Since there seems to be a reluctance for educators, policy makers and the general public to consult the literature, examine what we know about cognitive development and read the studies, there becomes a tendency to grasp onto the methods one is familiar with and hold these as sacred. In many cases in North America, this means that the traditionalist methods are held in higher regard than the constructivist methods. Looking to other countries that tend to do well in mathematics, there are some interesting cultural differences that appear in the approach to teaching and evaluation. One example can be seen in this video with Phil Daro http://vimeo.com/30924981.

In the middle, is student inquiry (again, the name is often used as a substitute for pure constructivism, which causes confusion) in which students are encouraged to try out problems using whatever means they can, discuss the various methods that worked and didn't, share and yes, memorize the methods that work consistently, and connect these methods and patterns to problems they encounter in everyday life. There is structure to the lesson, but there is also a place for students to work with problems on their own terms, experiment and make connections with prior learning. Lessons are scaffolded so that they build on concepts already mastered. Consolidation happens with the whole class and is reviewed again at the start of the next lesson. Students still memorize times tables and formulas, but they also understand where these come from and what is happening with them. They can use a matrix to show multiplication and can tell you why ax + by + c= 0 is a different way of showing y=mx +b, and how various different values of "m" will change the slope of a line when graphed. They can relate this to situations in their everyday life. They know what to do when confronted with 

(2x + y)(3x +2y -z)

 because they have learned the underlying pattern of how this works, rather than just a convenient but limited mnemonic. The understand that BEDMAS is an mnemonic shortcut that helps them use the distributive property, and that the multiplication/division are interchangeable as are addition/subtraction.

Without context, understanding of the underlying pattern, and sufficient understanding to apply the concept widely, a math student's abilities are no more useful than a calculator, and are likely much slower at that. We need people who can not only calculate, but choose the appropriate algorithm and problem-solve in a variety of situations. We need people who understand how to program the algorithms in the first place. Math is not a religion to be taken on faith; it is a science that stands up to scrutiny. We would do well to remember this as we approach the subject in the classroom.

Students learn by doing and thinking, by struggling through problems. When this is connected to their everyday experiences, it becomes meaningful. If we can recognize this in other subject areas, then why not in math?

Working Backwards

Looking through many resources in preparation for teaching math in the school system, there seems to be a very common pattern that is rarely broken: introduce algorithm, apply it to increasingly difficult pre-determined problems, review, then tack on a "real life" or "word" problem to add application as almost an afterthought. Some resources even skip this last step, and few indeed involve proofs, aside from a couple that show how to derive the quadratic equation.

In mathematics, we have a tendency to ask students to accept algorithms without question or debate. We essentially eliminate critical thinking from our teaching.

This is not the way teachers are trained in my province, yet many resources that are used in our classrooms still follow this sequence, and many teachers drift toward this in their practice. The emphasis remains on the lower levels of Bloom's Taxonomy at the expense of activities that promote higher-order thinking. In my review of the literature, it appears that teachers are most likely to work in this direction for two reasons: this is the way they were taught, and their comfort level with the curriculum is low.

When I say their comfort level is low, I do not mean that they do not necessarily hold a deep conceptual understanding of the topic, but that for various reasons (most often relating to allotted classroom time), they feel the need to get the basics covered as quickly as possible, and for many, teaching algorithms is how they view "the basics" when it comes to math.

But what if we were to reverse this direction, and start with the applied problem?

Critics say that this leaves students high and dry, with the need to reinvent conceptual knowledge that took mathematical superstars many years to develop. They say it leads to confusion when the approach they might try is not the most efficient method.

However, no one is saying that we withhold the algorithms from the students, only that we let them think about the problems that lead to them in order to foster a sense of pattern and deeper conceptual understanding of the processes involved in applying mathematical thinking.

All the memorized algorithms in the world are useless if students never learn when or how to use them outside of math class or standardized testing.

Sure, it takes a little more time for students to think through the "why" of a problem, but feeding them algorithms to memorize and apply does students a disservice. Computers can work through algorithms, and they do it faster and more accurately than people. What we need are people who can reason mathematically, and this requires that we provide a space for applied problem solving and reflection.

It is my gut feeling based on what I've seen with the students I've worked with that students who develop applied problem solving skills aka mathematical reasoning skills, begin to make deeper connections quicker with later topics. In this way, the time invested at the outset may offset the time needed to cover later related topics.

For teachers to abandon the chalk-and-talk and promote these skills will take a leap of faith. It is much more comfortable to stay with the known, particularly when there is a perceived crunch in terms of curricular content demands and the allotted classroom time to cover it. Students may resist since they are used to being given the entire topic at the outset. People tend to resist change.

How many times have we heard the question from our students, "When will we ever use this?". Students, particularly those for whom math does not come easily, need to understand this in order to invest their time and effort accordingly. Resources like this one: 101 uses for a quadratic equation and this one: "Why study math?" can be good places to start. Starting with a relevant real-life problem is also an effective way to connect theory with application.

I issue the following challenge to all teachers who read this blog: choose one topic this year to present this way and see how it goes. Start with a real-life problem, challenge students either individually or in groups to devise a way to tackle it, and share results using Bansho or another similar method in which the different approaches can be grouped in a meaningful way. Discuss which ones work and which don't and why. Finish with a review of those ways that work best. Follow up with some practice problems.

Did you or your students resist? What challenges did you face? Did different students participate than usual for your class? How might you use this to best encourage mathematical reasoning in your students?