Homeschooling Corner: Math Philosophy

Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.–Gottfried Leibniz

Fibonacci Spiral

You may have noticed that I talk a lot more about math than reading or writing. This is not because I dislike the language arts, but because they are, once learned, not very complicated. A child must learn to decode symbols, associate them with sounds, and then write them–tricky in the beginning, but most children should have the basics down by the age of 6 or 7. For the next several years, the child’s most important task is simply practice. If a child has a book they love to read, then they are already most of the way there and will probably only need some regular instruction on spelling and punctuation.

Math, by contrast, is always advancing. For every new operation or technique a child masters, there is another waiting to be learned.

I don’t hold with the idea that mathematical concepts must be taught in a particular order or at particular ages–I introduced negative numbers back in preschool, they’ve learned about simple logarithms in elementary, and they seem none the worse for the unusual order.

Count on Math gives the logic behind Particular Order:

Developmental sequence is fundamental to children’s ability to build conceptual understanding. … The chapters in this book present math in a developmental sequence that provides children a natural transition from one concept to the next, preventing gaps in their understanding. …

When children are allowed to explore many objects, they begin to recognize similarities and differences of objects. When children can determine similarities and differences, they can classify objects. When children can classify objects, they can see similarities and difference well enough to recognize patterns. When children can recognize, copy, extend and create patterns, they can arrange sets in a one-to-one relationship. …

This developmental sequence provides a conceptual framework that serves as a springboard to developing higher level math skills.

This logic is complete bollocks. (Count on Math is otherwise a fine book if you’re looking for activities to do with small children.)

Humans are good at learning. It’s what we do. Any child raised in a normal environment (and if you’re reading this, I assume you care about your children and aren’t neglecting them) has plenty of objects around every day that they can interact with, observe, sort, classify, etc. You don’t have to dedicate a week to teaching your kid how to tell “similar” and “different” in objects before you dedicate a week to “classifying.” Hand them some toys or acorns or rocks or random stuff lying around the house and they can do that themselves.

Can you imagine an adult who, because their parent or preschool skipped straight from”determining similarities and differences” to “making patterns,” was left bereft and innumerate, unable to understand fractions? If the human mind were really so fragile, the vast majority of people would know nothing and our entire civilization would not exist.

More important than any particular order is introducing mathematical concepts in a friendly, enjoyable way, when the child is ready to understand them.

For example, I tried to teach binary notation this week, but that went completely over the kids’ heads. They just thought I was making a pattern with numbers. So I stopped and switched to a lesson about Fibonacci numbers and Pascal’s triangle.

Then we went back to practicing addition and subtraction with regrouping, because that’s tricky. It’s boring, it’s not fun, and it’s not intuitive until you’ve really got base-ten down solid (base 10, despite what you may think, is not “obvious” or intuitive. Not all languages even use base 10. The Maya used base 20; the Babylonians used base 60. There are Aborigines who used base 5 or even 3; in Nigeria you’ll find base 12.) Learning is always a balance between the fun stuff (look what you can do with exponents!) and the boring stuff (let’s practice our times tables.) The boring stuff lets you do the fun stuff, but they’re both ultimately necessary.


What else we’ve been up to:

Fractions, Decimals, and Percents, by David A. Adler. A brightly-colored, well-written introduction to parts of numbers and how fractions, decimals and percents are really just different ways of saying the same thing.

It’s a short book–28 pages with not much text per page–and intended for young children, probably in the 8 to 10 yrs old range.

I picked up Code Your Own Games: 20 Games to Create with Scratch just because I wanted to see what there was outside the DK Workbooks (which have been good so far, no complaints there.) So far it seems pretty similar, but the layout is more compact. Beginners might feel less intimidated by DK’s larger layouts with more white space, but this seems good for a kid who is past that stage. It has more projects than the shorter DK Workbooks but they’re still pretty simple.

I also happened across a Singapore Math Workbook, which seems fine. Sample problem:

Emily and Jasmine had the same number of stamps. After Emily gave Jasmine 42 stamps, Jasmine had twice as many stamps as Emily. How many did Jasmine have at the end?

At a movie, 1/4 of the people in the theater were men, 5/8 were women, and the rest were children. If there were 100 more women than children, what was the total number of people in the theater?

Our recorders arrived, so now we can play music.

Finished reading The Secret Garden, planted seeds, collected and identified rocks. Nature walk: collected fall leaves and pressed flowers. Caught bugs and observed squirrels for Ranger Rick nature workbook. Read about space and worked with cuisenaire rods. Etc.



10 thoughts on “Homeschooling Corner: Math Philosophy

  1. Love. I have a question: I am not a child but I was wondering why they have the book coding 20 games for children? I making small strides into learning coding, and honestly the Boolean logic already has my mind and knots. Would not this child book be good for adults too?


    • It’s probably way too simple for an adult.

      Scratch is a system where the code is already written in chunks and then you can arrange the chunks how you want to create your projects. It’s good for little kids who aren’t good at typing yet and would probably accidentally misspell a lot of code if they were manually entering all of the lines. I doubt it’s what an adult would want.

      There are some good comments here on teaching/learning to code that are probably closer to what you’re looking for.

      Good luck!

      Liked by 1 person

  2. My personal experience with stupid mandatory order for math instruction was in kindergarten, when my mother told them I could do multiplication up to ten by ten, and their response was that I couldn’t because I wrote some of my numbers backwards. My mother was unable to convince them that handwriting was a separate thing from conceptual understanding of math…

    On the other end of the spectrum, I was later in a “progressive” school that had no regular formal math instruction until pre-algebra. We were “allowed” to do as much math as we liked, but the teachers were often innumerate, and one even taught me and my friend a completely incorrect and useless method of long division, which my mom her best to remediate, but unteaching is harder than teaching the first time… (I’d previously learned the correct method, but it’s one of those things that requires practice.) Anyhow, advanced algebra is really hard if you have to learn long division for the first time with polynomials…

    One thing that I think is missing in a lot of math pedagogy is the idea that there are really several different things that aren’t necessarily contingent on each other, the way that a child can read without having the fine motor control to write, or can learn lots of vocabulary while still being completely illiterate. (Granted, some people are still clueless there. So it goes.)

    Re: Singapore math, if you want to go that direction, it’s worth acquiring a teaching manual and reading it thoroughly. Not that it’s going to be harmful without it or anything, just that some things make more sense with it…


    • My experience with early elementary math instruction is that it… kind of isn’t. And even if you tell the teachers, “Hey, this kid can already do X, Y, and Z,” they’re still going to do XYZ for the next 3 years because the other kids can’t do XYZ yet.

      Reading curriculum seems very flexible. Kid at level A gets A Level books to read, and a kid at level G gets G Level books. Kids can write at whatever level they write at. But kindergarten teachers just don’t have multiplication worksheets and they’re not going to stop and teach multiplication to the kids who are ready for it.

      Pure free-form is not going to cut it. Someone has to actually explain things like “here’s how place value works,” “here’s how to divide.” But concepts like “If we want to add the same number multiple times, we call that multiplication,” can be understood pretty young, even if the kids haven’t memorized their multiplication tables. “Division is just undoing multiplication” can be understood before they really master long division.

      I got really frustrated with the school when they didn’t actually teach the younger elementary kids how to write down addition and subtraction problems, but were instead having them add multiple two-digit numbers in their heads by, like, making tens and 100s. There’s nothing wrong with making tens and hundreds, but they need to know how to write down the problem!

      Thanks for the advice. :)


      • So, the thing with the school I went to in kindergarten and early elementary, is that all grades had math and reading at the same time and students often moved to different rooms to accommodate different levels (there were a mix of professor’s kids, military brats, and factory and farm kids) so it wouldn’t have disrupted anything to send me to another room for math the way it was set up, and I could write the numbers legibly, just not quite correctly. (When I was in first grade at this school, I was in a third grade reader by the end of the year, with another girl my age plus a boy from the third grade. The following year, the school got a new principal, who insisted we all be in the reader for our grade level, not our reading level. If you want to train a child to be bitter and cynical, I can give some pointers… I’m kind of inclined to avoid public school for my kids just to avoid at least some of the “flavor of the week” pedagogy… (If a homeschool parent tends toward the “flavor of the week”, that’s just an issue for them and their kid(s).)


  3. […] This is part of the reason I have written on the cocktail, since it is only in the process of mixing that alcohol turns from something to be enjoyed for itself into something to be enjoyed for its effect: the worst cocktails are those which hide the alcohol most effectively, because they are drinks designed for and by people who drink to get drunk. Alcohol is most naturally suited not for intoxication, but for the cultivation of discernment. Consider the aphorism of Don Colacho that “the man of culture is obligated to be intolerant”: what does this mean if not that an elevated spirit is necessitated by his nature to discern and discriminate higher from lower? Not merely to discriminate, but to be discriminating in response as well, to avoid or even eliminate the lower and to glory in and to glorify the higher? Such a sense of discernment at the most basic level is a fundamental part of development as children, as evolutionistx has pointed out: […]


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