On Teaching Math: Part Six

Welcome to A Mother’s Thinking Love: Living Ideas, Lovingly Shared! In my last post, I gave some practical advice for teaching math family-style in the homeschool. Whether you are ready to go all-in or are still testing the waters, I hope you will keep reading this series. My ultimate goal is to give the mother-teacher a sense of freedom in this area and bring delight back to math. In today’s post, I want to continue discussing practicalities. I will introduce some useful tools and discuss how to help children in a family-style model. Join me for: “On Teaching Math: Part Six”!

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LINK TO OTHER POSTS IN THE SERIES

Tools of the Trade

In a previous post, I mentioned that my students, and my daughter in our homeschool, had access to manipulatives and other tools for solving problems. The manipulatives included things like: counters, base-ten blocks, cuisenaire rods, clocks, money, etc. I won’t give a list of “must have” manipulatives here, but you get the idea.

Dry erase boards, markers, and erasers became a nonnegotiable for me early on. With pencil and paper alone, students became very reluctant to try, especially if they experienced defeat with math in the past. One or two wrong moves, and they had erased holes in their papers. With erasers spent and pencil leads broken, students felt embarrassed. With a dry erase board or manipulatives, students could easily make changes. They were more willing to try because they could do so without visible signs of struggle all over their desks. Again, these are not requirements but ideas.

Moving On

I said this before, but I wanted to repeat it here. Children will generally be motivated to find the simplest way to solve a problem as quickly as possible. The won’t want to use manipulatives or draw models for one second longer than is necessary. Now, I recommend keeping all those options open, as they can be used for math discovery or as helps in the future, but children seem to naturally use what they need as they need it.

Every once in awhile, however, I encountered a student who needed a little prompting. Let’s say this particular student was drawing a model based on a word problem involving ten groups of ten. The student made the choice to draw ten boxes and draw ten, very disorganized, dots in each. Next, the student proceeded to count the dots one-by-one. I would come alongside the student and strike up a conversation. I might say, “I wonder if there’s a more simple model to draw.” At this time, the student may bring up that he could write the number ten in each box instead of drawing the ten dots. I would commend the idea and direct the student to try it. If the student did not come up with an idea, then I would suggest one. 

Sometimes I had a student who would finish every problem very quickly. This student may even use the standard algorithm to solve a multiplication word problem on day one. Well, I decided early on that I would strive to make each student’s time in my math class worthwhile. I knew my students were people. I didn’t want to waste their valuable time. For the student who wrote “10×10=100” to solve the word problem, I might challenge him to solve “100×100”. Then, I might ask him to look at 10×10 and 100×100 and see if he “noticed” anything. I will come back to this idea of “noticing” in math later, but it is a very powerful tool. Sometimes the student would notice something but sometimes not. Either way, at least he learned that noticing was part of math.

 Let’s say the word problem involved two groups of five. This same student immediately wrote down “2×5=10”. Noticing that he quickly, and correctly, solved the problem, I would say something like: “I see you used 2×5 to solve this problem and it equals ten. I wonder what 5×2 equals? I would leave the student to solve the problem. Excitedly, he would call me over and tell me that it equaled ten too! I might say, “Wow! I wonder why that is?” and leave him to discover the associative property of multiplication on his own.

Moving Along as a Group

Charlotte Mason said it best when she proposed that students take what they are ready for from each lesson. I found this to be true in math as well. Of course, a teacher could push students to memorize rules, facts, and steps. Those students might be able to replicate them on a test, but that isn’t learning. This is where communal, family-style math benefits all.

As I’ve said, probably multiple times at this point, “concrete, representational, abstract” isn’t a linear track. The strongest lessons touch all of them. In a classroom, or family, with students at various points of understanding, all of those will likely be represented in a single lesson. The mother-teacher in the homeschool can gently illuminate some points, where necessary, but shouldn’t talk too much. I remember Charlotte Mason writing that a young girl once told her mother something like, “I think I could understand, if only you didn’t talk so much.” This wasn’t a comment directed at math, but I think it applies. The fewer the words of a mother-teacher, the more poignant they will be. The fewer things we fill our children’ s minds with, the more space they have to contemplate and chew on their discoveries.

A mother-teacher may, gently and without much expectation, point on something like, “You drew two groups of five. Did you know you can also write two groups of five like this: “2×5=10”? Briefly explain how the standard algorithm represents the model, and let that be it. This could be necessary in the case where the oldest student in the group has not encountered the standard algorithm for multiplication yet. You can use precise math vocabulary but don’t labor over it. I will talk more about math vocabulary in a future post.

Records

Although I rarely handed out worksheets, I did find math journals quite useful. I will discuss those shortly. As a side note, I want to add how I gave grades since they were required from me as a classroom math teacher, just in case any homeschool families are required to do the same. I would give three question quizzes and five to ten question tests. If a student struggled with those short assessments, a 20+ question assessment wouldn’t help. I also gave credit for participation and math journals. 

Math journals were informal for me. Teachers Pay Teachers offered a plethora of cutesy journal cut and paste options, but I did not find these to be useful. Math journals were primarily for students to memorialize their own learning and discoveries. Each student’s math journal might look different. Occasionally, all would add a class discovery to their journals. Looking back, I can see this was another form of narration that I, unknowingly, employed. Knowing what I know now, I can also see why they were so valuable.

I hope you have enjoyed “On Teaching Math: Part Six”! If you have read through this series so far, thank you so much for joining me. Do you have any thoughts or questions? Requests for future posts? Leave them all in the comments below.