Adventures in Homeschool: Our Math Additions (and Divisions)

When I was a kid, I was a star memorizer. At an early age I could memorize sections of prose, poetry, and all kinds of facts. When I started working in mathematics, we did more memorizing. I was good at that part, memorizing math facts. Trouble was, I didn’t really understand them as deeply as I needed.

Now that I found myself in charge of my son’s math education, I started to branch out from the lesson plans and exercises I had purchased and tried out. Since we were working together, and he was curious, I decided to experiment a bit. First of all, I wanted to help him with a different approach to math than I had experienced. I had struggled a lot with math, up until university. Next, I wanted to be sure our approach to math was appropriate for his learning. Instead of being guided by my own experiences, I would let them inform us, but try to get my direction from kiddo and his interests.

Looking at my own struggles with mathematics required a bit of self reflection. Over the years I had thought a lot about why I had struggled, and for me, it was often because critical information was withheld. Someone, somewhere, had decided that parts of the topics I was learning weren’t age appropriate, so I would learn a certain set of facts at a certain grade level. Later on, more information would be added, and the facts and approaches I had learned were challenged or upended completely. Those transitions were hard for me. I also had trouble because I needed to understand the topic more deeply. When I would ask questions about math topics, particularly with regards to their application in real life, I didn’t get satisfying answers.

Most of the time, the explanations from the teachers didn’t leave me feeling like I fully understood the topic. In fact, I wasn’t able to delve deeply enough in math concepts to truly own them until I took math classes in university. I memorized and applied, but I didn’t fully understand. It wasn’t until I had a math tutor in Grade 12 that I found someone who could fill in those blanks for me. When I went to university, that changed. Whenever I asked math professors “why” questions, they usually appreciated it and explained further, or pointed me to publications with alternative explanations, or trips to the library. I was also inspired by how some of my math and computer science professors taught math to their own kids. They focused on simple, intuitive application that was relevant to kiddos, and they didn’t gatekeep and withhold information. Once things began to click for my math brain in university, I vowed that my kids would get help on the “why” questions. If it was from me, then so be it.

To brainstorm, I talked to programmer friends and asked about their math learning experiences. Many of them just understood math naturally, and didn’t struggle with any of it. Many had mixed experiences, and some had experiences that sounded like mine. They didn’t like how rules changed and wished they had been given a fuller picture earlier. Furthermore, when any of us were curious about topics beyond our grade level or outside of the curriculum, that was usually discouraged. “You’ll learn that later.”

All of us found that programming helped out math skills since it was a real world application. The math facts come alive, and you understand a lot of the “whys”. For example, an abstract concept like a variable in algebra (a letter symbol in an equation) is part of what you always do when you program: you create variables and do things with them. High volume automation of math provides visualization and patterns that you can’t replicate with paper and pencil, and that provides more insight. Even better, small errors can have huge implications in a program, so your attention to detail changes.

My question for my programmer friends was: “What did you miss out on in your early math education that you think I should provide kiddo exposure to?” Their answers provided this list:

  • Zero-based counting
  • Negative numbers
  • Working with fractions earlier
  • Multiplication basics alongside arithmetic
  • Understanding decimals
  • Using variables
  • Thinking about functions
  • Sequencing and loops
  • Abstract data types: lists, arrays
  • Data operations: stacks (LIFO and FIFO, evaluation, etc.)

Some of these are quite simple and actionable. For example, starting to count with 0 instead of 1 is trivial, and should always be done, in my not so humble opinion. When you start looking at place values, or even counting beyond 10, it is extremely helpful to have started at 0. Negative numbers aren’t that easy to grasp, but you can work with them, especially with number lines. Fractions, multiplication, decimals and variables can be worked into early math exercises, but it takes a bit of thought.

About half the list relate to computer programming, so I wasn’t sure how to handle those. Kiddo wants to learn how to code, but I would really need to gauge his interest and follow his energy there. However, they can easily be broken down into concepts he can understand at 5-6 years old.

That led me to thinking I needed to help him understand math concepts at a simple, intuitive level first. If he could relate a math concept to everyday life, and to things that mean something to him, we could build on that. From there, he would need to learn and apply the concepts, and through repeated exposure, he would memorize rules and facts by doing something with them.

I went in two directions with this: utilizing informal math concepts on every day life at home, and a formal math learning structure.

First of all, I found Betty Choi’s blog, and she provided a lot of ideas on how to incorporate math into everyday activities with our kiddos. I got a lot of direction from her post: How to Teach Basic Math for Free Anytime and Everywhere and printed out the PDF and taped it to our fridge. (If you’re teaching math to young kids, download this PDF on teaching math.)

She has wonderful ideas about counting actions (claps, steps, etc.), adding and subtracting toys, multiplying with groups of objects, dividing up food (this is great for fractions too) and applying fractions from recipes. These were great, and we were able to do a lot with them to help make these concepts intuitive. For example, a kiddo who understands how to divide up chocolate chips equally will have an easier time learning division formally later on.

Food division is a lot of fun because you can really focus on the basics of division, with a payoff of a tasty treat. To start, I gathered 10 chocolate chips and asked kiddo to divide them equally so we could share. Kiddos often have an innate sense of fairness, so he carefully arranged them in two rows. At first he made a pattern and tried to make a symmetrical pattern with the other group. That was fine, but he was worried that one person might have more than the other. We played with the patterns by re-arranging them into shapes that were easier to subitize. Five is a wonderful number for this because you can create unique patterns. Next, I asked him to count each group, just to be sure. Once he was satisfied we had divided the chocolate chips equally, we ate them. Nice!

After we had worked on dividing up food objects into equal parts, either for two or three of us, I decided to see how he would manage a remainder. Instead of 10 grapes, I gave him 11 grapes, and asked him to divide them into five equal parts. He started in by making pairs, but then he ended up with an extra. I left him to twist in the wind a bit, to see how he would manage it. He tried adding it to one group, but one group had three grapes while the others had two. He kept rearranging, trying, going back to his original pairs, and puzzling over the extra. Finally, he set it to the side and asked me for help. “I can get 5 pairs, but there is an extra that doesn’t fit, so I just put it to the side.” I praised his thinking because he had just discovered remainders. “This is exactly what you do when you divide numbers and end up with some numbers left over. In division, this is called a remainder!”

Kiddo wondered what use a remainder might be. Using food as a guide, I suggested that if he was the one divvying up the chocolate chips, he could secretly eat the remainder. This is one of the perks when you cook food for others. You can surreptitiously snack as you prepare. This appealed to him greatly. It turns out, stolen edible division remainders are an effective math learning tool.

I also discovered a helpful homeschooler blog from the Natural Math community, Duct Tape Rocket. I got some helpful ideas on games to get kiddos thinking and working on simple programming concepts. After reading some of the posts on the blog, kiddo and I started playing around in the Scratch Jr. program.

We also discovered CodeSpark Academy, a fun app where kids learn programming basics in a simple game format. CodeSpark not only helps kiddos learn about sequencing, but all kinds of concepts including loops and working with stacks. Kiddo would get rewarded for his algorithms, but he got more points when he used constructs like loops. When he was disappointed he wasn’t getting full stars for a task, we had a chat and worked through the concept. He was motivated to achieve more in the game, so he started to use loops in his own work and was rewarded for it. We also spent a lot of time together figuring out how to do LIFO and FIFO tasks in the game.

Next, I formalized what we should cover based on this list from Natural Math:

  • Subitizing: The ability to instantly recognize quantities without counting
  • Counting: Addition and subtraction are based on sequences, dealing with objects one-by-one
  • Unitizing: Multiplication and division are based on equal groups or units
  • Exponentiating: Self-similar structures, such as fractals

Patterns are a clear underpinning shared with all of these concepts, so getting kiddo to work more formally with identifying and creating patterns was a natural place for us to work on. Identifying patterns and groups and performing operations on them can be done with just about anything, anywhere, especially if you are out in nature. Subitizing is also a patterns activity. Thankfully, I had a lot of helpful exercises from my kindergarten no-prep lesson plans, so we were able to move forward with pattern and subitizing exercises for a while.

Exponentiating work involved watching videos and looking for repeating patterns in nature, such as in the veins of a leaf, or in the crystal formations of ice. Unitizing was more difficult. I needed to figure out activities that were appropriate for a 5 year old, so I started to research further.

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