On Teaching Math: Part Eight

Welcome to A Mother’s Thinking Love: Living Ideas, Lovingly Shared! In my last post, I focused primarily on the use of math journals. We walked through some philosophy behind their use and some practical ideas for implementation. I hope to strike the same philosophical-practical balance in today’s post about one of my favorite math ideas ever. Join me for: “On Teaching Math: Part Eight”!

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LINK TO THE REST OF THE POSTS IN THE SERIES

A Living Math Idea

I have written before that I approached teaching math, in both 3rd and 4th grades, like a one-room schoolhouse. This was really necessary due to the wide range of readiness among my students. In my one-room schoolhouse approach, although I didn’t call it that at the time, I did not have a curriculum. This led to me necessarily coming up with ideas on my own. Now I had never heard about this idea of a “living education” at the time, but I was instinctively drawn towards creating one. Of course those early attempts were quite feeble, but a few things stuck. In this post, I’m going to share one of those sticky ideas.

The Humble Number Line

Since I didn’t have a math curriculum, I spent a lot of time with the 3rd grade math standards for my state. They were clear as mud most of the time. How they could take simple 3rd grade math concepts and make them incomprehensible to the average adult deserves some kind of award. It’s truly remarkable. But through the fog, I noticed that fractions on a number line were important.

Fractions rank up there with word problems in striking fear into the hearts of both math teachers and students. I once read an account where the writer claimed that the introduction to fractions lesson was the root of math anxiety in most students. After seeing those standards, I could believe it. I remember in my Algebra ll math class in high school, our teacher had to go back and teach us elementary fraction concepts because our class lacked understanding as a group. I was determined to find a new way for my students. Although I didn’t understand how the standards wanted me to teach fractions with a number line, I would find a way.

The Same Old, Same Old

If you’ve read the previous posts in this series, you will find that these lessons look the same as the rest. I would give a word problem involving a fraction like one-half. There would be no mention of the word “fractions” or “halves”. The word problem might be something like, “Sally and Samantha have one cookie left. They both want to eat it. What can they do?” It would be intentionally vague but almost every student would say they should split it. I might ask for clarification about how they should split it. I might also ask students how we could represent how much they would each get. Most students could say, “They each get half.” 

I might draw few models of this with a cookie, split in two unequal parts. In a model, one of the parts would be significantly bigger than the other. Students would protest, saying, “No! One part is bigger than the other!” They would tell me that each part needed to be the same size. Then, I would briefly elaborate.

I might say something like this, “You all have discovered something, and I want to show you another way to write your discovery. 

½

The number on the bottom represents the two equal parts you told me about. The number on the top represents how many of the parts each girl gets.

Each girl gets ½.”

I would use the terms “numerator” and “denominator”, but I wouldn’t focus on them here. You can see the last post in this series for a discussion on how I taught math vocabulary. 

Introducing: The Living Number Line

 Before class, I would put a line on the wall. It was big. It took up the majority of the wall, but it was important and would stay up for the rest of the school year. This number line started with marks for “0”, “1”, and “2”. We would add more to it later, as we made more discoveries. I would then prompt students to do some thinking.

By the third grade students had seen a number line, but they usually had not thought much about them. This was new, but they were always capable of thinking and noticing! I would ask them, “I wonder where ½ belongs on our number line,” I would usually do some thinking and questioning outloud. 

“Well, ½ represented half a cookie. Is it greater than one? Is it less than zero?”

Students would think. I would even permit them to raise hands and give ideas. It did not take long, however, for students to decide that since ½ was half of one cookie, ½ belonged in the middle between zero and one.

And we left it there. We didn’t add more to the number line. We only added what was relevant to the day’s lesson. This could have been added to a math journal, but I don’t think I ever required it, as the gigantic example was on the wall for all to see.

Not Just Fractions

I’m not sure when it happened. All I know is that it was completely organic. We were doing some word problem involving money and the idea of “half a dollar” came up. That’s when I had a “Eureka!” moment myself. We had been adding fractions to our number line, but it was still pretty barren. Suddenly I said to the class, “Wait a minute. Didn’t you already discover something that means ‘half’?” They would remember, “Yes! It was a fraction. ½. It’s on the number line.” I said, “I wonder if we can add half a dollar to the number line too. If we can, I wonder where it would go.” After some thinking, they would conclude that half a dollar belonged with the ½ fraction. So, I wrote “0.50” and “50 cents” on sticky notes and put them above ½ on the number line. 

That’s the day that the “Living Number Line” was born. Over time, we continued to add fractions and decimals to it. We even ended up with percents. We added equivalent fractions, mixed numbers, and improper fractions. At times, we even added pictorial representations. This may seem like a lot, but, again, the number line was huge. We just stacked our additions in neat columns up the wall.

No Systems

I did not do this systematically. I introduced fractions with the idea of “one half”, but I did not take a systematic approach to the others. I found, or came up with word problems, and we went from there. Sometimes, a student would come to me with an idea for a new addition to the number line. I would always make time for this and encouraged other students to come up with ideas too. Again, I don’t think math requires such a rigidly systematic approach like we have been led to believe, especially when we’re dealing with the basics.

Although this idea was born in a classroom, it can work in a homeschool! If you have fewer, or even just one, students, you may need to do a bit more “wondering” outloud to get the ball rolling, but you can absolutely do this. Once students begin to think about math, you can’t stop them. In fact, they may even begin thinking about math even outside of their math lesson time!

The Thrill of Discovery

One of the things I learned through this Living Number Line experiment is that students are motivated by discovery. Sure, I could have created lesson plans and “taught” them these concepts, but the discovery would have been lost. Yes, this approach took a bit more patience on my part, but I found it to be worth it. I could have just told students these facts, but at what cost?

Thank you for joining me for “On Teaching Math: Part Eight”! How is math going in your homeschool? Do you have any questions or requests for future blog posts? Leave them in the comments below.