On Teaching Math: Part Two

Welcome to A Mother’s Thinking Love: Living Ideas, Lovingly Shared! In my last post in this series, I discussed why I think many math curriculums are set-up “exactly and precisely backwards.” I want to take that a step further, then I want to present a different way. Join me for: “On Teaching Math: Part Two”!

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Asked & Answered

In my last post, I shared this quote from Paul Lockhart:

“Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time – there is nothing left for the student to do.”

Following our multiplication example: presenting a standard algorithm up front, telling students the answers, asking them to memorize the facts, and then requiring them to apply this to a word problem takes the discovery out of math. A few years ago, I was reading a biography about Archimedes with my daughter, and I was reminded of the importance of discovery. First, you have to ask a question. Then you have to take time to think about it, try solutions, fail, be inspired. Eventually, like Archimedes, you too can have a “Eureka!” moment.

As adults, I think we take the four basic operations for granted. We’ve known about addition, subtraction, multiplication, and division for a long time. We weren’t inspired by them in elementary school, and we take no delight in them now. But children delight in the simple things. The whole world is new to them. They, if given time and space, are constantly discovering something that’s new to them. Our modern methods for teaching math remove every opportunity for those moments.

Righting the Ship

I don’t want to suggest that I have all the answers. I also don’t want to imply that I was, or am currently, a perfect math teacher. Please, let me humbly suggest a different way to approach this often dreaded subject. I think it’s worth a try. Besides, how much worse can math lessons get, truly? Also, like I said in the last post, don’t dump your math curriculum just yet. With a few adjustments, you may be able to get some use out of it still. I want to illustrate how I approached the idea of multiplication by highlighting an example from my third grade math classroom. Once you have the idea, I think you will be able to apply it to other math concepts.

The Math Lesson

Let’s say, for the sake of continuity, I was introducing the idea of multiplication to a class of third graders. I would START with the word problem. Yes, the dreaded word problem that strikes fear into the heart of teachers and students alike. You could likely find a more organic way to go about this at home, but classrooms are artificial by nature.

Sally has two baskets. She put five books into each basket. How many books does Sally have in all?

After telling students to get to work answering the question, I would do nothing. I would wait. I would watch. Each student had access to a basket of manipulatives and a small dry erase board, marker, and eraser. They knew the routine for those items and were free to use them at their discretion.

Now, at the beginning of the school year, this was hard. Students couldn’t believe they had to actually think in math class instead of just memorize. But I smiled, and I waited. I told them I was very patient and could wait as long as was necessary. Eventually, someone started to work and others did the same. If any students seemed truly stuck, I might direct them to try to imagine the baskets and books in their minds. I would remind them they had some tools in their baskets to help, if they needed them. With this added encouragement, a student rarely needed additional help from me to get started.

Don’t Skip This Step

Next came the discussion, or communal learning, aspect of the class. I know these terms have a bad reputation in a lot of classical circles these days, but, from my observations, I have seen their value. As long as students had guidelines for these times, I didn’t have any real issues. Students were excited to share and listen. After learning about Charlotte Mason and narration, I saw why these times made such a difference.

I didn’t want to embarrass anyone, so I tried to choose volunteers that I knew arrived at the correct answer. I would also try to ask a wide variety of students to share. Meaning, I tried to choose students who had employed different methods to solve the problem. Each student would be asked to explain to me how he/she solved the problem. The methods fell along the concrete, representational, abstract continuum.

If a student used manipulatives to solve the problem, I reenacted step by step his/her process on the overhead projector. If a student drew a model, I drew step by step his/her process on the board. I did the same for a student who used the standard algorithm. A sometimes comical benefit from this process was that students learned to more clearly articulate themselves. I reenacted or wrote EXACTLY what they told me, as I best understood it anyway. This led to some funny outcomes at times, and indirectly taught students that communicating clearly was important.

The Never-Ending Math Wars

I soon learned that there was always some war in the world of education. While I’m sure some of the battles were necessary, I didn’t have much time to engage in them. I spent my time learning about math, honing my craft in teaching it, and being present with students in my classroom. There is common misconception that I want to address here because I still hear it in the homeschool world.

I regularly hear that there are two opposing ways to teach math: memorize the facts and become proficient in the basics versus conceptual understanding and reasoning. I heard this when I began teaching math over a decade ago, and I hear it today. Here’s the thing, proficiency in the basics and understanding math are not opposed. In fact, I think they are the best of friends.

An Example from a Different Subject

For the sake of illustration, I want to use the teaching of history as an example. In the neoclassical world, young students are often taught to memorize timelines. The students do enjoy memorizing through song it seems, and this is done with an end in mind. In the early years, students memorize, so in the upper years they can organize the stories from history that they learn. The idea is that the memorized dates will be a “peg” that students can hang their future, real history, learning on. As Rachel Woodham outlines in “Charlemagne and the Case Against History Timelines”, however, this does not really work. Children remember stories, not dry as dust bits of information.

I think this is the approach that modern math textbooks take. We give students dry as dust math facts to memorize, in the hope that, in the upper years, they will remember them when it’s time to do real math. Here’s the problem with that: students hate math in the upper years and still don’t remember those basic facts. Facts absent context are tricky to remember. They don’t have much to hold onto or connect with in the brain. Facts attached to context and meaning, however, have a purpose. They are connected. They can hold on in the mind.

What Are We Fighting Over, Again?

This is why I used the “concrete, representational, abstract” continuum to teach math. I will outline this approach more in my next post, but I want to clear up a tragic, common misconception. When I began using this approach, I didn’t require students to use particular strategies. I wanted them to experiment, build, and try things on their own. This was not the case in other math classrooms, as I later found. In many districts, students were expected to master every strategy for every operation.

However, this is not necessary. Students don’t need to memorize all strategies. They merely need to employ strategies until they can make sense of the standard algorithms. This is what happens when there aren’t any principles or ideas given to the math teacher. This is what happens when those in charge don’t understand the “why”. Math is again reduced to a lifeless system to be memorized.

On Our Way

I feel like we are finally making some progress. I think you can pull out some ideas from this post and apply them to your homeschool tomorrow without totally starting from scratch. If you still aren’t there yet, or even if you are but want to learn more, hop on over to the next post in the series. Have you been inspired to change your approach to math in anyway? Share about it in the comments below!