On Teaching Math: Part Nine

Welcome to A Mother’s Thinking Love: Living Ideas, Lovingly Shared! In my last post, I discussed my favorite ever math teaching discovery: The Living Number Line. It followed me from third grade, to fourth grade, and then to our homeschool. In today’s post, I want to discuss an issue that I highlighted previously: The Math Wars. Should we teach math for reasoning or math facts? How do we pick a side? Do we even need to pick a side? Join me for: “One Teaching Math: Part Nine”!

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LINK TO THE REST OF THE POSTS IN THE SERIES

Fighters, To Your Corners

When I first began teaching third grade math, as a very new teacher, I heard a lot of debate over the new math standards. I discussed my lack of participation in this debate in a previous post. Essentially, I was too overwhelmed to engage in the fight. I decided, instead, to learn about math and how to teach it. Then, I worked with my students, kept written records, and pressed on. A battle raged on around me, but I was barely aware of it.

Students Must Memorize the Facts

I remember my third grade math experience quite clearly. It was made up of a lot of timed multiplication tests. I had a good memory and passed them all with flying colors. The very next year, however, in the fourth grade, I struggled intensely with math.  Although I had memorized my basic facts, I didn’t understand what to do with them. Once a straight “A” student, I was now struggling to keep my head above water.

When I began teaching third grade math, I understood the need for students to memorize their math facts. Students came to me without having addition doubles facts or addition facts within ten memorized. This made the process of teaching even third grade math standards quite a bit more challenging. This led me to consider ways in which I could help my students memorize these addition facts and, after, their third grade multiplication facts.

Reasoning About the Facts

Like I stated above, there was a math war going on. One side believed that you should teach math only for conceptual understanding. The other side believed that elementary students should focus on memorizing their basic facts and that the conceptual understanding should be saved for later. As I have outlined more thoroughly in previous posts, I taught math through word problems. This meant, apparently, that I was on the “conceptual” side. But, I also knew that students needed to know their basic math facts. So, I guess I was also on the “facts” side.

It didn’t take long for me to decide that I just wasn’t going to pick a side. And, actually, I wasn’t even going to engage in this war. I was simply going to teach the students sitting in front of me each day. I was going to teach them to reason about math through word problems. Then, I was going to help them memorize their basic facts. 

One Question Remains

But, one question still remained: HOW was I going to teach students to memorize their basic facts? I already knew that timed tests did not guarantee lasting knowledge. I experienced this first hand myself, as a student. Also, I saw the results of addition timed tests as students came to me. The facts just did not stay with them.

In the beginning of this series, you may remember that, although I didn’t have a math curriculum, I did have a math mentor. She gave me books, ideas, and taught me how to think about teaching math. Thankfully, she also had some ideas about memorizing math facts. She told me about the idea of “Number Talks”. In these Number Talks, students would not be memorizing facts and taking timed tests. However, they would develop fact fluency by strengthening their mental math abilities. In fifteen minutes day, students participated in Number Talks starting with addition and worked all the way up through division.

I won’t outline Number Talks here, I may do so in a future post, but I will share the books I use:

Number Talks: Whole Number Computation

Number Talks: Fractions, Decimals, and Percentages

These aren’t the only Number Talks resources out there, but I stuck with them in our homeschool, as they were already familiar to me.

More on Memorization

As I stated previously, although they help with mental math fluency, Number Talks do not ask students to study facts with the goal of memorizing them. Since I did see the importance of basic math facts, I wanted to add another way to help students memorize them. When I began studying multiplication tables, something dawned on me: there aren’t actually a lot of multiplication facts students need to memorize.

Think about it. Without much effort, students learn the facts of: 0, 1, and 10. With slightly more effort, they learn 2s and 5s. This leaves 3s, 4s, 6s, 7s, 8s, and 9s. That may seem like a lot, but there aren’t many facts within those that are hard to memorize.

Consider the 8s facts:

0x8

1×8

2×8

3×8

4×8

5×8

6×8

7×8

8×8

9×8

10×8

x0, x1, x2, x5, and x10 are part of the simple group.

This leaves only:

x3, x4, x6, x7, x8, x9

If you do this for the other facts in the challenging group, and account for the Associate Property of Multiplication, you are left with a very small list of facts students need to memorize. This can be accomplished through skip counting songs. Also, students can make, and continually review, flashcards. Or, you can simply let them use a multiplication table to solve these problems. This leads to constant review, and, through the table, students notice helpful patterns.

The Power of Noticing Patterns

Again, I always tried to let students discover math for themselves. I would not just say “The answer to every x0 fact is 0.” Instead, I would, after students worked with multiplication word problems for awhile and were familiar with the standard algorithm, write this on the board:

0x0=0

0x1=0

0x2=0

0x3=0

0x4=0

0x5=0

0x6=0

0x7=0

0x8=0

0x9=0

0x10=0

I would then ask, “What do you notice?” After giving sometime to think, I would take volunteers to share. Of course, students would notice that the answer to every x0 fact was 0. Then I wonder say, “I wonder why that is?” Usually, a student would write a word problem to show an example. It might go something like this:

“If Sally has 0 baskets of 8 books, she doesn’t have any books.”

Or

“If Sally has 8 baskets of 0 books, she doesn’t have any books.”

Another student may draw a model as an example.

After these explanations, I would direct students to make an entry in their journals.

Staying Out of the Cross-Fire

I am aware that The Math Wars still rage on. I’ve been accused of teaching “new math”. I’ve been accused of focusing too much on math facts. I choose to opt out of the debate. Instead, I remember myself as a young, scared, struggling math student. I remember what it felt like to be confused. I remember how it felt to want to be a good math student, and the crushing disappointment I experienced when it was out of my reach. 

That’s one of the beautys of homeschooling. We, too, can opt out of the war. In doing so, we can see our children as born persons and meet them where they are. We can show them that, although it involves work and thinking, math is within their grasp. They can become good math students. Have you, too, felt in the middle of The Math War? Did you feel like you had to choose a side? Comment below!