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?

The Value of N

I've been thinking a lot lately about the way math is visually presented to young students. Looking at equations and the meaning of equal signs, as written about in this recent post, I also began to wonder about the use of answer blanks, such as can be found in questions like this one:

5-3=___

There are reasons to write it like this, most obviously to give the student a place to put the "answer" or, better put, to complete the equation. However, following the equal sign with a blank might also be the cause of some confusion when students reach introductory algebra. They may have learned to associate the = sign, and/or the blank with "this is where the answer goes" rather than understand that this is an equation in which the value of each side must balance.

Consider the following way of representing the same question:

5-3= n

which could be followed with:

n = ___

or simply the words, "What does n equal?" or perhaps better still, "What is the value of n?".
(I chose the letter "n" because it can stand for the word "number", but any letter would work as well)

While there is still that problem of the equal sign followed by the blank, the way the first part is represented manages to help convey some information that is missing in the first example, such as "what are we looking for?" and "how can we represent the unknown number that will balance the equation?".

Maybe a picture would help get the idea across better (please forgive my crude drawings!):


Of course, if you have a balance and unit weights handy, you could always use those to help solve the problem.

I wonder if presenting simple arithmetic with a variable rather than a blank from the outset would help students better understand the concept of equation and equality better, and also predispose them to accept variable notation when it becomes more crucial in algebra.

If you choose to use this idea with your students, I'd be very grateful if you would let me know how it goes.

What is 4?

I'm currently reading The Glass Wall: Why Mathematics Can Seem Difficult by Frank Smith.

Early on, he describes how number is not the same as quantity. He uses 4 as an initial example, but then moves on to use a large number, somewhere over 7 hundred million, to demonstrate that the number is valid whether you have an associated quantity of something that it represents or not. Number is number. However we develop number sense, he argues, it occurs separate from natural language which tends to be ambiguous where math, buy its nature is not ambiguous (at least not to those who understand it!).

So, what in fact is 4? How do we truly understand the concept of 4 (or any other number)? Smith argues that a number can only be put into context when it is compared with other numbers. He sees mathematics as a separate existence than the rest of the world.

I'm not quite sure if this rings true for me or not, but it is an interesting thought to explore.

This also made me wonder what certain young mathematically inclined students might think about it, which led to the following idea for math enrichment.

Lesson Plan Idea: What is 4?

Students are challenged to brainstorm how they would explain the concept of 4 to people who had no numeracy (aliens, young children, etc.). They then move together into groups and share their ideas. The group chooses several to share with the class. One person (either the teacher or another student) plays the role of the learner while the students attempt to explain the concept. The learner should do their best to avoid using any previous mathematical knowledge and base their "understanding" purely on the information given by the students.

Class discussion should include:

•  counting--number/object connection; meaning of each number name
•  quantity--did they use concrete items (manipulatives) to demonstrate their ideas, and how successful might this be in getting the idea of number across without ambiguity (were the shape of items, colour, function or other characteristics confused with sense of number)
•  use of geometry and/or other drawings or models
•  other ways of relating the concept
•  would their system of explanation work for very large numbers, fractions, decimals, negative integers, zero, etc., and if not, how could they adapt or change it so that it will
•  was this difficult, and if so, why do the students think it was

The activity is open-ended and intended to provide deeper insights into the complexity of seemingly simple mathematical concepts.

Other related lessons might include:

• working with different number systems
•  working with different bases 
• writing computer programs to solve very basic mathematical problems (using a machine-based language or something without built-in mathematical algorithms that the students can access)
• a study of the historical use of "zero" and what it really means in mathematics (hint: it has a more complex meaning than simply "nothing")

For more math activities, see the Lemonade Math Page