Tuesday, December 02, 2008

More hands-on equations

We don't do much that looks like "school" in this family, but every once in a while there's something that comes up that is so quintessentially academic and visual that I just have to take pictures. Hands-On Equations is certainly one of those pursuits. I can't seem to resist snapping photos of Fiona at work with this program. These are the photos I should put in a Christmas newsletter for skeptical relatives who could never understand unschooling.

Borenson describes Hands-On Equations as "Piagetian learning," in that children are learning by doing, without direct teaching of principles and theory, but absorbing the concepts all the same. It is pretty neat stuff. Fiona has easily learned the simple basics of what are considered "legal moves" and the whole thing plays out like a game for her.

The program is divided into three levels, with Level 2 introducing negative x and Level 3 introducing negative integers. We're working through Level 1 fairly systematically to ensure that the rules of play are well understood and are already almost done. We've had a couple of short sessions and a couple of longer ones. It is working beautifully for Fiona. She now often "sees" her way to the solution a couple of steps before the end. For instance she'll see that 2x + 4 = 10 means x=3 just by looking at the initial set-up of the problem with the manipulatives. I'm amazed.

We're almost ready to start Lesson 6. At this stage the expectation is that students will start working without the manipulatives, instead drawing their symbols pictorially on the page. I'm not sure if we'll do this step or not. I'm not sure a 5-year-old is as ready for this as a 10-year-old. Plus she loves the game pieces and that's much of the allure of the program. Although she writes pretty well for a 5-year-old, I also think that sketching out the problem visually would be a lot of pencil-work for her and might detract from the fun of the program. I'm not in a hurry to get her solving algebra on paper, so we might just skip this expectation.

Fiona has set up her equation and done the
preliminary simplification. She started with

4x + 1 - x = x + 13

and has simplified it to

2x + 1 = 13

It's time for the next move.
She removes 1 from both sides, leaving

2x = 12

That's easy.
x = 6!

It's always fun to find the solution!


  1. Oh dear, I have Maths toy envy again! Last time it ended with R410 spent on a base 10 set for someone who turned out to understand base 10 already, demonstrated it in short shrift, and then proceeded to build things with the pieces (and lose them round the house). It must be fun to be justify the expense, something that is trickier with a schooled kid. At least I have 2 more kids....

    Apart from the want-to-get-your-hands-on touchable pieces, I liked the idea of the verbal questions book: do you have that? We do nearly all our Maths in the car, and I probably am doing pre-Algebraic concepts in the way I ask the questions (or the way he does, and his mean mommy turns them back to him with "well, what do *you* think?".

  2. There are a few verbal questions in the kit we got, but not the full 250. I honestly think Fiona is too young to be conceptualizing them as algebraic problems and solving them à la Hands-On Equations. That's a very abstract step, conceptualizing that "how many cards there are in a pack will be represented by x." I'd give it another couple of years at least before she's ready to make those connections. I could probably rote-teach her the process of defining the unknown in a word problem as equal to x and then building an equation from that, but I'd honestly rather she discover the connection between algebra and real-life problems herself on her own time. I've always found mathematical learning more powerful if its connection to real life is discovery-based.

  3. Wow, that is amazing Maths for a 5yr old- very impressive.

  4. Yes agreed. Our car Maths includes some stuff I'd say was pre-algebraic, as in one example on the website, basically saying 5 plus what equals 8, or the even more practical, how many more do we need to get 8? Where it is addition, subtraction or algebra I think is a fuzzy line, because the same problem can be approached from so many angles.

    He seems to get concepts discussed in this very cerebral and non hands-on environment. Although I always think it would be better to have manipulatives, somehow we always seem to chat Maths in the car or at best at supper, without even so much as a duplo block for support. The problem with the verbal environment is that I find it hard to hold back and not give away Maths "secrets" that I know he should figure out for himself. That was why I bought the base 10 set, because I wasn't sure he had mastered the base 10 concepts for himself, and wasn't just going on what I said.

    We are now messing with multiplication. It started a while ago with doubling and halving recipes, then he started working out what 2 or 3 or 4 times something was, and after a bit of that I introduced some formal vocab "times" and "multiply" but the ideas are the same. I still don't know if he has figured out that multiplication is commutative, or whether he rote learnt it from something I told him. He has turned a few non-Maths-designed toys into multiplication toys (some magnetic rods we have, he was arranging in groups), but he likes to ask me "what is 4 times 3", which results in him being told to do the sum, but me being available to mentally "mark" it as it were.

    Part of the problem with manipulatives is that he really loves big numbers, which one really can't manipulate physically. A sum like "1 billion plus 1 billion" or "what is ten 100 billions?" pretty much has to be cerebral. :) The closest we've come to a manipulative for his big numbers is a number card I made at his request, with columns going up to a trillion. Actually big numbers is a fun "grown-up" way to learn basic Maths. 1 billion plus 1 billion sounds so much better than 18 plus 6, but of course is a much easier sum in reality. :)

  5. I do love the look of this and I am so tempted... However, my kids aren't as much into manipulatives as I expected so not sure whether I should spend the money.

  6. I can't believe this is "baby" Fiona, doing algebra.

  7. That sounds like a great way to learn math! We don't have much in the way of manipulatives so I'm going to check into this :) Thanks for sharing.


This blog is moving to archive-only status. Please consider posting comments instead at the active version of the blog at nurturedbylove.ca/blog

Note: only a member of this blog may post a comment.