On Teaching Math: Part Three

Welcome to A Mother’s Thinking Love: Living Ideas, Lovingly Shared! In my last post I outlined how I took modern math textbooks and flipped them upside down, so they would be, in the end, right-side up. If you are totally confused, go back and read “On Teaching Math: Part Two!” In today’s post, I want to connect my ideas directly to the homeschool. I also want to discuss my ideas for “family style math”. Join me for: “On Teaching Math: Part Three”!

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Concrete, Representational, Abstract

Before I begin discussing today’s topic, I want to briefly address this idea of “concrete, representational, abstract” continuum that I mentioned in the last post. When given a word problem, students will naturally use one of those three methods to solve it.

• Concrete – Students may choose to use some type of physical manipulative to solve the problem. In this case, students may not write any numbers at all, except maybe their answers.
• Representational – Sometimes called “pictorial”, students may choose to draw a type of model to solve the problem. In the case of “Sally has two baskets and each basket contains five books”, students may draw two boxes with five in each. The five may be represented by dots or the numeral five.
• Abstract – Students may choose to write “2×5=10” to solve the word problem.

One misconception that I attempted to address in the last post, is that students have to begin at “concrete”, then go to “representational”, and finally end up in “abstract”. It is viewed as a linear, one-way street. I didn’t require this. I let students choose what they needed and move on when they were ready.

When is it Time to Move On?

Generally, most students do not want to be in the world of “concrete” or “representational” longer than is necessary. They will move along at their own pace, but they will move along. Occasionally I might push a student a bit, but I usually didn’t need to. In the discussion portion of the lesson, outlined in “On Teaching Math: Part Two”, you can see how students are introduced to other ways to solve. This may not happen as organically in the homeschool, so I will give suggestions on how to encourage it in a later post. The goal is to use the simplest method to solve the problem. This will eventually be the standard algorithm.

Family-Style Math

Like I mentioned above, I think the process of learning math is not as linear as textbooks make us believe. Just because math concepts are introduced in step-by-step increments in a textbook doesn’t mean that students learn in that way. In my experience, I saw that students usually took leaps forward. After mulling a concept over for a few weeks, and feeling stuck, they may jump ahead in one class period. This is why I think family-style math can, with some consideration, be done well.

In the previous post, I gave this word problem:

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

Many, or most, elementary aged students with a strong number sense can solve this problem. After a bit of time to work, there could be a communal math discussion time, as I outlined in “On Teaching Math: Part Two”. If your children are many years apart in age, you might consider a Forms approach and do two math sessions a day. For example, K-2 students could be grouped together and 3-5 students could be grouped together.

This may sound far-fetched, but let me share this from my classroom experience. At the beginning of each school year, students were required to take diagnostic assessments. At the end of these assessments, I received reports with each student’s math grade level score. Every year, I had students who entered my classroom with the math readiness equivalent of kindergarten. In the same class, I had students with the math readiness equivalents of grades 5 or 6. In order to be a good teacher to all my students, I had to operate like a one-room-school-house instead of a 3rd grade math classroom. This is why I think family-style math can not only work but be a rousing success.

Don’t Panic

I know this seems overwhelming and too good to be true. You’re thinking “there’s no way we could enjoy math as a family”, and by using modern textbooks and lesson plans, you would be right. But I think there is some freedom to be found here. I’m going to outline that freedom in the next post and even suggest a resource! Would you like to experience more freedom in your family’s math lessons? Feel free to share in the comments below!