This issue is one that has been coming up lots on message boards, blogs and within the discussion community at the SelfDesign program we’re part of. What do we do with young children who have quickly and efficiently mastered the primary mathematics curriculum? The obvious answer is to move ahead into the secondary mathematics curriculum. But it’s not quite that simple, for two main reasons.

First, the presentation of the material, when intended for a teen audience, often becomes dry and dense all of a sudden. Our beloved Singapore Math curriculum is a case in point. Look at the difference in presentation between the end of Primary and the beginning of the Secondary NMC books:

Goodbye friendly cartoon kids. Goodbye white space. Goodbye clever self-checking ‘secret code’ exercises. From a book which covers two problems in a two-page spread, we move to a book with over 70 short problems on a single page. My unschooled 8- to 10-year-old kids, who think of math as a nifty brain game occasionally aided by some fun pencil-and-paper exercises, have not been ready to make the transition illustrated above. It just looked too nasty and grown up, even though they were ready for the mathematics.

The other major issue is that the vast majority of homeschool curriculum choices out there are US-based, and follow the standard US practice of discarding almost all the interesting mathematical threads for two years in order to delve deeply into one particular thread, that of algebra. Variety and the discovery of inter-relationships between different areas of mathematics has always been what has kept math interesting to my kids. To set aside probability, number theory, geometry, statistics, series, patterns, algorithmic logic, topology and all that in order to focus exclusively on algebra seems such a misguided approach, particularly for fairly young children for whom math has typically been interest-led, capricious and playful.

I wish I could say that we’d found the solution. We have found a few non-solutions, and maybe a mix of partial solutions. First, the things that didn’t work:

  1. Singapore’s secondary programs. Though Erin did eventually get through Book 1 and part of 2, these were too dry and killed her interest in math for about four years.
  2. Teaching Textbooks was mathematically shallow and too slow-paced. The computer-based presentation was fairly friendly, but my computer-loving kids preferred their math not to be dressed up as computer entertainment (just like they prefer to just eat their broccoli, rather than having it sneaked into something else) and opted to use just the textbook. We all detested the algebra-only mono-diet as well as the excessive repetition and review.
  3. Life of Fred was a refreshing change. The humour and quirky narrative approach grabbed Noah’s attention for a while. But again it was that algebra-only mono-diet that tired him out. And after a while the narrative thread began to feel like broccoli being disguised as something else.

What I’ve decided so far for certain is that for young pre-teens a straight algebra course is not the best way to go. That pretty much eliminates most US-based math curricula. And that’s not a bad thing, though it makes for pretty slim pickings. In terms of English-language offerings it leaves Canadian school curricula, the Singaporean selections, and a few eclectic American offerings. (There may be some British or Australian or New Zealand stuff out there, but I haven’t stumbled upon anything yet that has been readily available or impressive enough to warrant ordering sight-unseen from overseas.)

Eclectic is really where we’re at right now. Over the past year and a bit, as Sophie closed in on the end of the Singapore Primary Math sequence, we’ve done a lot of eclectic grazing. I wish we’d done more, and I intend to keep grazing with her for a while. By the time I take Fiona through this transition maybe I’ll have it down!

Here’s what we have used, or have on hand, or are planning to use for our eclectic grazing.

  1. Calculus by and for Young People. Sophie delved into this at age 8 and it was great fun.
  2. Theoni Pappas’ children’s books. Delightful exploration of mathematical topics through stories, explanations and demonstrations.
  3. Hands-On Equations by Henry Borenson. Wish I’d had this for the older kids. Fiona is loving it.
  4. Alge-Tiles manipulatives and resource binder. Great fun for factoring quadratics and exploring negative unknowns and integers.
  5. The Man Who Counted by Malba Tahan. A story about a fictional mathematician full of brain-teasers woven into the story.
  6. The Number Devil by Hans Enzensberger. A very fun fictional story which romps through some of the basic concepts of number theory in creative ways.
  7. Art of Problem Solving introductory books. These are US-based but unique in that various topics like number theory, probability, geometry and algebra can be taught in parallel. I can’t see us working systematically through the whole shebang, but we’ll likely keep at least two or three of these around as resources. We just received our first AoPS book in the mail today and we’ll see how it pans out as time goes on.
  8. MathPower textbooks. Overall these are the best systematic texts I’ve seen for my kids, and I found them right under our nose at the local public school. They’re Canadian and cover a variety of topics at each level, but are friendlier and more varied that their Singaporean counterparts. They’re visually busy though, with some classroom-oriented garbage (“discuss with a classmate and formulate a hypothesis, then present it to your class…”) but they feel less intimidating than the book above. And mathematically robust? To a degree. They are conceptual and expect a fair bit from students, especially in the sidebar challenge projects and exercises.
The math watershed

20 thoughts on “The math watershed

  • January 13, 2009 at 3:28 pm

    Are you kidding Miranda – this is wonderful!! You have just saved a whole bunch of us a whole bunch of time and money.

    Thank you!

  • January 13, 2009 at 4:56 pm

    Have you ever checked out ? I’ve used it and it is great. They offer courses from 3rd grade through graduate school math. They’ve just added Chemistry courses. My kids have done well with it. Also, you can sign up for a free trial for each child and if you don’t sign up after the free trial, they will send you a gift certificate for a free month. We were able to really explore the different courses in that month.

  • January 13, 2009 at 6:33 pm

    Thanks for this, Miranda. DS (6) will likely be done elementary math ahead of time unless we really spread it out and I’m looking for things to do afterwards or even concurrently to increase the depth/breadth of his interest in math.

  • January 13, 2009 at 7:38 pm

    Yes Chari, we did a trial of Aleks years ago … when Erin finished Singapore Primary. It didn’t work for us. First there was the fact that my kids don’t like to sit at the computer to do math, and also that American business of separating algebra out as a separate course which we find tedious and constraining.

  • January 13, 2009 at 11:24 pm

    Someone recommended the CIMT site to me before but I honestly can’t figure out what’s good about it besides that it’s free. Maybe I’m missing something. It’s fairly clear, and there’s a lot of it, but it doesn’t seem at all mathematically challenging or likely to inspire mathematical creativity and excitement. What do you like about it?

  • January 14, 2009 at 9:23 am

    Ah, yes, eclectic math learning is what we do, too. My eldest stagnated for a bit with his math work and we tried to find different things to keep with his interests. He’s enjoying the Life of Fred series, but has also stalled there a couple of times. He likes the humour, but having read through the book completely before actually doing the work he’s not enjoying it as much the second time through.

    My middle ds learns quite differently from his older brother, who intuitively ‘gets’ math. We’re using a fractions book based on using cuisinaire rods right now and he’s doing quite well. He’s a very hands-on learner!

    My youngest varies in what he’s wanting to work on. We play math-related games and he likes to do paperwork, but only when he’s really interested. Right now he’s going through an interest in math phase which is great. I actually think he may be ready for the same fractions book my middle ds is working on, but we’ll wait a bit on it.

    We do a fair amount of out loud math work as it applies to life in general and they all respond well to it.

    I’ve been tutoring a 10yo boy for math for a short time and found it interesting to see that his grade 5 math book was covering the same topics Singapore Primary 4 did. Not all math programs are created equal!

    I think we may just check our library to see what else they have in the maths. We’ve had some cool books before! Of course I can’t remember which ones they were though!

    I don’t think any one curriculum is going to be tailored to fit all the needs of all the kids anyway. They may not even meet all the needs of a single kid of mine. That’s okay. As long as their needs are met for even a short time with a certain program and they are enjoying it, I try not to stress and go with it.

  • January 14, 2009 at 2:59 pm

    i think i suggested cimt before. the problem is i don’t use it, but lots of maths parents are using it with their kids. it doesn’t work in a straight line, as the primary stuff is apparently far more vigourous than the next level, and that the sheets seen on the website, they use for approx 10 – 25% of the work [according to who i am speaking to!] the rest is experimental. and i guess this is the bit that makes it exciting] the families using it have maths gifted kids, but instead of racing through do far more lateral work. and they feel that at the end of the primary section is equivalent to a gcse level knowledge [16] if the programme is followed as planned, rather than as worksheet based. iyswim.
    i think there is a cimt yahoo group, where some of the uk users suggest how to get the best out of it?
    we are still in singapore primary land so have a way to go,
    the other ‘english’ being used frequently over here is galore park books.
    but we also do maths reading books and experiments to go off the path and investigate

  • January 14, 2009 at 3:04 pm

    Thanks Helen, that helps explain a lot.

  • January 15, 2009 at 7:28 am

    Thanks for addressing this math issue in your blog!….glad to see we are not the only ones grasping as to what to do after primary math…….

  • January 15, 2009 at 7:45 am

    I called McGraw-Hill Ryerson to get more information about the Mathpower books. They will sell you the student texts, but not the teacher texts with the answers, etc. This does not help math challenged moms!

  • January 15, 2009 at 9:16 am

    Susanna, the student texts have the answers too, in the back. That’s one of the things the kids really like about them. They don’t have step-wise solutions, but they do have the answers. Not sure if that makes any difference to you.

  • January 16, 2009 at 5:01 am

    Have you ever investigated Stanford’s EPGY online math course? It is terribly expensive, but I do know a lot of people who think it is wonderful. It is, of course, American, which does mean separation of courses. Talking of alternate English-language curricula, there’s also South Africa. No idea how good. I do know people whom I could ask, if you are interested.

  • January 16, 2009 at 9:53 am

    Beccy, I’ve never considered EPGY because for us it really has three strikes against it — it’s computer-based, it has separation of algebra, and it’s a very expensive gamble, way beyond what I’d consider spending. It used to be fairly popular here in N. America at the primary level but has been largely replaced by Singapore Primary — at about a tenth the price.

  • January 18, 2009 at 7:45 pm

    I find CIMT useful as one resource among others. It has plentiful white space, clear explanations, and multiple strands. It therefore makes a nice extension for the mathmatically advanced very young. I never use all of anything, as most of the time there are too many problem sets. I did have one who liked math problems as a sort of bedtime soothing activity and that kid went through hundreds of problems. I also like Life of Fred, Harold Jacobs, and Art of Problem Solving for those kids. We’ve used EPGY but if one is going to spend the money I think another time I might try out EIMACS instead. I have to include the fact that mine have a daddy who is a physicist, so we range fast and far with the math and science stuff. Sorry to take so much time to get back to you.

  • January 20, 2009 at 5:59 pm

    Have you looked at the Challenge Math materials? Very quirky for a workbook – similar in layout to Singapore but at a higher level (higher than Singapore 6 – not sure where you left off).
    We like the Algebra, Challenge Math and 10 things Scientists Need to Know.

    Challenge math touches on calculus, graphing equations, solving simultanous equations, trig etc. The sample page on the website doesn’t show the depth of the program.
    Cartoonish characters do the explaining, Einstein pops in with hints and the repetition is low with questions expanding on the concepts rather than repeating them. They are designed for gifted kids and I would say they meet the needs of that audience well.


  • January 28, 2009 at 2:49 pm

    I remember really liking “Uncle Petros and Goldbach’s Conjecture” – it’s more of a reading book than a working through maths book – but it made me excited and inspired about maths! (I guess I read this when I was an older teenager).

    Amazon UK:

    Amazon US:

    It has pretty good reviews!

    I don’t remember so much about what the story was about, but I remember how enthusiastic it made me feel about maths – in a similar wayy to James Watson’s “The Double Helix” (reading around the blatantly sexist bits) making me want to dash out and become a research scientist! 🙂

  • October 4, 2009 at 2:34 pm

    Wow, thank you so much for posting this! I'm a bit earlier on this journey — I have a 7yo girl and a 5yo boy who are racing through math as fast as I can find appropriate materials for them — but it's been an interesting journey so far, and I'm glad to find somone discussing stuff a bit further down the road. It's hard to find resources for teaching an active 5yo about fractions and multiplication; I'd love to read anything more you've written about math, if you have any such writings. 🙂 Thanks!

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