Comments

There's a typo on this Singapore math page, did you notice? 'B's weight' should be 'C's weight'.

-- CarolynJohnston - 09 Jun 2005

Oh my gosh!

Nope, didn't notice.

-- CatherineJohnson - 09 Jun 2005

Using the Singapore Math books is always funny, because I don't think they're translations; I think this is the English they're written in.

I've forgotten the stats on Singapore now, but I think everyone speaks English (don't quote me!), but English isn't anyone's first language. (They have zillions of languages, and IIRC Chinese is number 1. I must check . . . )

So the wording on the problems is often a little other-worldly.

-- CatherineJohnson - 09 Jun 2005

I came into post that there was a typo, but you already found it. Great bar graph! I appreciate the scan!

-- KatherineProuty - 09 Jun 2005

300 bar models!

I have some catching up to do!

-- CarolynJohnston - 10 Jun 2005

The bar model approach to solving this problem would make a constructivist proud. It is a good way of seeing what is going on, although I don't know how many students would construct this approach. If there is a big benefit to constructing your own solution, then is the benefit all lost if the student who figured it out "teaches" it to another student? So much for constructivism. It's neither necessary or sufficient. As for bar models, they have their place, but should never be used to avoid or delay the transition to defining variables and equations, and then solving those equations.

-- KtmGuest - 10 Jun 2005

As a follow up to my previous comment, bar models can get much more difficult as the problem becomes more complex. This is not necessarily true if you learn how to define variables and equations. It is often very easy to find at least one equation. In the above problem, it is A+B+C=102. Once you believe/trust the idea of solving a problem by finding equal number of equations for your variables, it is extremely liberating. This will hopefully progress to where the student understands the classes of problems to solve where m equals n, m is less than n, and m is greater than n, where m is the number of variables and n is the number of equations.

-- KtmGuest - 10 Jun 2005

The bar model approach to solving this problem would make a constructivist proud.

Ouch! ;)

(Actually, there are some constructivist math tricks that I really like. I'll steal cool ideas shamelessly, I'm afraid).

I agree with you wholeheartedly that they should never be used to delay or replace the introduction of formal algebra. But I think bar models really do provide extra leverage for setting up a problem. They are really just a slight systematization of the 'first draw a diagram' school of math problem solving.

-- CarolynJohnston - 10 Jun 2005

I've got to get Sybilla Beckmann's article uploaded pretty quickly here.

The point of bar models is to advance the introduction of algebra, not delay it.

In the bar models a question mark is used in place of a variable. A question mark, IMO, is no less abstract or symbolic than a letter.

5th grade students in Singapore solve two-variable algebra problems in 5th grade using bar models.

Only our most advanced kids are solving two-variable algebra problems in 8th grade.

I'll do a front-page post about this at some point, but I've been snooping around, and I don't think any other country uses bar models. The Chinese do not; I'm fairly certain the Japanese do not.

Singapore students have far outstripped Japanese students in just 20 years, after the introduction of the PRIMARY MATHEMATICS curriculum.

I'm also intrigued by the fact that the first girl to win the Math Olympiads is from Singapore, and grew up using these books.

I suspect that the bar models increase 'spatial-visualization' ability. I don't know that, but I've seen mathematicians on the NY Math Forum say that scale drawing increases one's ability to do geometry, and bar models are scale drawings in the sense that you determine equal units. The examples are always drawn proportionally as well.

There's a lot going on with these bar models, I think.

-- CatherineJohnson - 10 Jun 2005

I didn't mean to imply that bar methods are bad or wrong, unless you become infatuated with them over progressing to more powerful techniques. I guess the main point I was making was that anything like this is fine if the kids are taught the technique and you move right along. The fallacy of constructivism is that it is necessary or sufficient. The fallacy of constructivism is that it's the only way. The fallacy of constructivism is that in a group setting, all kids will construct something of value. The problem of constructivism is that it wastes a lot of time and that afterwards, they usually don't teach the kids a proper way of solution and then require lots of practice. Understanding the concept is enough. I'm not sure what concept that is, however.

I'm a pragmatist and a big fan of teaching kids any method that works. The more tools in their mathematical toolbox,the better off they are - of course, with lots of practice. However, I would get a little worried if teachers tried to force everything through a bar model approach. (At what grade level they attack certain problems is another question.) If you have two equations and two unknowns, the students first have to see what type of equations they are. If they are linear, then it is a simple matter to graph the equations (on paper) and see where the two curves intersect. This is the solution and the point where the two equations are true or satisfied. This understanding can be applied to the multiple solutions (perhaps) of the intersections of nonlinear curves or finding the roots of an equation. Bar models are fine, but they are not the only or best tool in the toolbox.

-- KtmGuest - 10 Jun 2005

I think they may be the best!

I don't know, of course, and we should take this discussion 'up front' so others can join.

What intrigues me about Singapore Math's bar models is two things:

• Singapore's curriculum may be the single most effective curricular reform ever undertaken (more on that later) and the introduction of bar modeling, taught at a very early age, may be one of the essential reasons why
• these bar models have often radically changed my own perceptions of math as I've gone about re-teaching myself
  

Interesting question about graphing & lines, though . . . as far as I can tell, the Singapore curriculum introduces this idea much later. (I could be wrong; I haven't read the books cover to cover, as I plan to.)

I'd love to know the history of Singaporean bar models: who invented them; why; did they field test them extensively; etc.

One thing I do know is that when Singapore set out to reform its math curriculum some 20 years ago, they partnered real mathematicians with real math teachers.

My husband did something like that in CA when he was working on the history-social-science frameworks, and he said it was incredible. The history teachers were starved for content themselves, and starved for colleagues.

It worked.

I'm assuming, though I don't know, that the bar models emerged from the partnership between mathematicians and math teachers . . .

-- CatherineJohnson - 10 Jun 2005

KtmGuest and Catherine raise some really good questions here.

I haven't had enough experience with bar models to know whether they are the best tool in the toolbox. At this point, Catherine has worked with them much more than I have. They seem to be especially good for visual learners (Catherine seems to be one, and so is my son, judging from the way he's taken to bar models when I've used them).

Kids do seem to get more out of concrete diagrams and mathematical manipulatives when they are older, so maybe we shouldn't simply throw out the simple models completely when the kids are sophisticated enough to use more complex tools, but rather work with all the tools together for a while (see CalStateStudyOnManipulatives).

I think your point is not to delay the introduction to more abtract concepts and tools. But the use of these diagrams in Singapore Math does seem to advance, not delay, the introduction of algebra in that curriculum.

-- CarolynJohnston - 10 Jun 2005