I've always believed we can learn so much from observing what and how children learn in other cultures. I've received permission from Jenny, a representative of Singapore Math, to post a few of the questions from the Grade 6B placement Test and discuss them. This post will focus on Question 13:
Last month David and Mary saved some money in a ratio of 3:5. This month they saved an additional $154 together, and David now has three times as much money as he had last month while Mary has two times as much money as she had last month. How much money did they save last month?
Jenny is the curriculum advisor for the US based distributor of Singapore Math, has an intimate knowledge of the materials and how they they are implemented. From looking at the 6A Placement Test, which included some algebra, I had assumed that children would solve this using a variable x as follows:
Dave Mary
Originally 3x 5x
Additional ___ ___
Afterward 9x 10x
Therefore the additional savings would be 6x and 5x for Dave and Mary respectively.
Thus, 11x = 154 and x = 14. Originally, Dave and Mary saved 8x or (8)(14) = $112.
Straightforward, right?
Well, the children in Singapore are taught a visualization for complicated ratio problems like this which, in my opinion, powerfully lays the foundation for algebra. Instead of the variable x that I used, children are shown how to represent the given ratio using unit bars and solving for the value of a unit:
David and Mary saved money in a ratio of 3 : 5
|----|----|----] D
|----|----|----|----|----] M
After this month, David now has three times as much, and Mary now has two times as much.
|----|----|----]----|----|----|----|----|----| D
|----|----|----|----|----]----|----|----|----|----| M
The total additional amount is $154.
From the diagrams, you can see that there is an additional 11 units. Therefore:
11 units = $154
1 unit = $14
Last month they saved 8 units together.
8 units = $14 x 8 = $112
I had to adjust to this when first reading it. I had mistakenly assumed that the children were introduced to traditional variables earlier on and would be encouraged to use them. But then I began to realize what was different here. Many of our children (including secondary students) struggle with complicated ratio problems (even uncomplicated ones!). I needed to imagine seeing the unit bar construct through the eyes of a young child. The idea of using a visualization of a unit bar for a 3:5 ratio doesn't seem to be that significant at first blush, but now I think it is. Instead of representing the original quantities as 3x and 5x, children can see these quantities in a tangible way. More significantly, they can draw the effect of multiplying Dave's savings by three and Mary's by two. I needed to step back here to appreciate this. Jenny explained that children do not use unit bars for all ratio problems, just for the most complicated ones. So what are we saying here? You mean, it's not enough to just buy Singapore Math materials and give it to kids in our classrooms? Teachers need to be trained in how to implement them successfully? You mean there's no easy short-cut here? You think!
I want to personally thank Jenny for her graciousness in replying so thoroughly to my naive questions. I will have more to say about this but I await your thoughts...
Saturday, August 18, 2007
Singapore Math - Part II - It isn't just the materials!
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13 comments:
minor correction - the line above the final depctition of the unit bar should read "...Mary has twice as much>" (not three times)
feel free to delete this comment!
I too assumed that an algebraic approach with variables would be taken (although I don't know why I assumed this as I know next to nothing about the Singapore program). Now that I think about it, isn't the use of the bars just another way to approach it algebraically? Each bar represents an "x" and the like terms aren't yet combined? so...
|---|---|---| = x + x + x = 3x
I like the idea of the unit bars. The visualization is an important piece that supports student understanding.
I'm doing some work with percents and ratios in my pre-algebra class this year. I'll have to investigate this more (unless reading the post counts as full teacher training!).
Jackie's right, this is still algebra, just not symbolic algebra. But I like the idea of using a more tangible metric for the computation.
tony, jackie--
I felt like both of you did at first. The bar was equivalent to any other symbol or variable. But the metric allows students to work with a concrete representation. Many have tried to find physical representations (manipulatives) for variables and for solving equations. Algebra tiles, balance scales, you name it. But in practice, students rarely make the transfer from the manipulative to paper-and-pencil work. Using the bars to represent the quantities in a ratio is a bit different and I can't evaluate its effectiveness without seeing it in practice. It seems like a great idea in theory.
Jenny added several more comments about the bars that I will share later and a link to learn more.
Dave,
I like that the bars are a written manipulative that the kids can hopefully connect to the formal algebra.
I didn't mean to say that the effect of the represenation on the students was equivalent, but that it was a nice way to show the set up of the algebra (lead in to algebraic thinking).
Pretty neat!
Gives new meaning to the term 'Abstract Algebra'
TC
I just wanted to point out that i said this was a "kind of algebra." There are different types and some work better than others.
For example, the Greeks used primarily rhetorical algebra, which produces interesting results but is REALLY bad for manipulation. However, some of their greatest successes came through forays into geometric algebra, like the the Singapore problem you are discussing. So it doesn't surprise me at all that some students would do better with bars instead of x's.
tony, jackie--
I understood your points about bars being another representation of algebra. My main focus in all of this is that, as a nation, we seem reluctant to acknowledge the possibility that other cultures may have something to offer and, heaven forbid, even do some things better than we do! I absolutely believe that each culture, both present and past, offers something that could be beneficial to our children. Imagine if we gathered some of the best ideas - an eclectic approach - and investigated the possibilities of incorporating them into our educational systems. Imagine, just imagine...
Jenny has made it clear that the fraction bar is just a small piece of what is done in that program. From reading other descriptions of Singapore Math, it seems that it is characterized as the cynosure. By the way, there are are many other blogs that have been discussing this program for some time and they have far more to offer than my brief synopsis. I wanted to use a couple of the placement questions as a jumping off point for a discussion of effective ways of teaching ratios and problem-solving to middle schoolers and beyond. Certainly, we as math people would use algebra more likely than most other methods, but I'm willing to bet that some of us visualize ratios our own unique ways.
My other point in these posts about Singapore Math is that educational institutions and curriculum leaders love to look for the quick fix: "Look, Singapore students are doing so much better than anyone else. let's purchase their materials immediately." How about, "Let's find out what they are doing differently and see what makes sense for our children." Certainly it makes sense to pilot it on a limited basis, then analyze some results before buying into it outright. I simply so not know enough to form a conclusion. On the surface it appears very promising. Beyond the issue of the curriculum and the worksheets however is the culture in which these materials are implemented. There's an extraordinary quote near the end of a recent post in the blog,
Math in Singapore (http://mathinsingapore.blogspot.com) that I read the other day that speaks volumes about cultural differences and priorities. You may not agree but here goes (my loose paraphrase I'm doing from memory):
In Singapore the education of children is seen as an investment; in our country, it's seen as an expense.
Dave-
I feel that my purely semantic arguments have left you feeling that I disagree with you, when in fact I agree with your last comment to the letter.
Tony--
Thanks! I sometimes feel my views are misconstrued, probably because of my lack of eloquence. It is reassuring to know there are others of similar mind who really do give a d--- about the education of our children and are tired of politicizing them. I sometimes forget why I keep writing on this blog. Then I read a few insightful comments from kindred spirits and I remember...
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