MathNotations

Note: As usual, the comments section contains insightful contributions from Denise, mathercize et al. I expounded briefly on a ratio approach and provided a link to some other bar diagram solutions. By the way, I've been using the misnomer, 'fraction bars' instead of the correct phrase bar diagram. I have a lot to learn here...

Lots of traffic from people viewing this placement test. The following is Question #11 from this test and I think it can lead to fruitful discussion of methods and strategies for middle school and secondary students. Older students will usually use an algebraic approach, prealgebra students might use 'Guess-Test-Revise' with a calculator. However, Singapore students apparently use a 'fraction bar' model for many of these, which I see as a ratio approach. How would you solve it? How would you present this to your students? What methods would you expect your students to use? What percent of your students would feel they don't even know where to begin and give up quickly? Would group work help here?

My first inclination was to set up an equation in one variable, however, because it was on the Singapore test, I tried a ratio approach which enabled me to solve it mentally. Have fun! By the way, to personalize it for your students, you may want to change the names to two students in your class and place it in the context of spending money at the mall!

Also, consider that if one were to use 'Guess-Test', it would make sense to start with a number divisible by 48ドル (since 240ドル is a multiple of 48,ドル an intuitive student with strong quantitative skills might easily guess the answer!).

The Question:

Peter and Paul each had an equal amount of money. Each day Peter spent 36ドル and Paul spent 48ドル. When Paul used up all his money, Peter still had 240ドル left. How much money did each of them have at first?

Posted by Dave Marain at 9:07 AM 7 comments

Labels: algebra, placement test, ratios, Singapore Math

I deeply appreciate zac's update on Singapore Math he recently published over at squareCircleZ. There's an excellent link there to a site that debunks many of the myths regarding the program (actually a commercial site but very informative). I was aware of most of this from other sites, but some information was new to me. I strongly commend it to your attention. I also asked zac if I could reprint my comment and his excellent reply. Fascinating stuff here...

In the end, regardless of whether or not students in Singapore are primed for these assessments (as in 'teaching to the test'), the bottom line is that the level of problem-solving that is assessed is higher than their grade-level counterparts here in the US. I never apologized for teaching to the AP Calculus Exam for the past 33 years. It has always been a high-quality challenging exam and became less predictable over the past 15 years with the Reform Calculus movement. Teaching to the test simply meant that I covered the syllabus and used released AP questions in addition to other resources to challenge my students. Shame on me! Of course I always added my own touches to the course like we all do.

Now for my comments and zac's reply:

1. zac–
   Thank you for debunking some myths (excellent site) and providing first-hand information. It doesn’t surprise me that there are many inaccuracies in reports I’ve heard about and read. It’s interesting to see that there isn’t a ‘one size fits all’ approach to the materials, however, the comments about ‘essence’ were the most telling. The ‘form’ of individual materials may change but the essential philosophy, not so much…

   It also seems that the supplementary workbooks in the program are significant and, at some point, I will need to order some of these materials to become more knowledgeable about the program.

   I have had the pleasure of teaching and providing SAT instruction for Asian students for the past 30 years (simply the demographic in my area), so I have come to know a great deal about their culture, after-school tuition programs, and their math curriculum. They found my comments about the superior performance of Singapore students interesting. Some characteristics they used to describe the tiny nation included ‘very clean, ‘very strict discipline in the schools’ and affluent. One student commented, “You don’t really believe that every student there can do all of these problems, do you!”

   My blog, however, focused on an actual ratio problem from the 6B Placement test which was really a 7th ‘grade’ pre-test from what I gathered. There was rich discussion about the heuristic of using fraction bars to represent units but, in the end, the quality and difficulty of the problem came through over all of the other conceptions and misconceptions about the program. That’s why the focus of my blog is problem-solving rather than debating overall philosophies which I generally consider futile.

   The calculator vs. non-calculator issue was mentioned in my posts and comments but it was not my focus. It’s not a secret that students with a solid foundation in arithmetic can regain computational proficiency if forced to. They didn’t enjoy it, and for the geometry questions, it was the most time-consuming part of the problem, but they did it - end of story there.

   Again, zac — the proof is in the materials and the level of complexity of the problem-solving. Many teacher and students were taken aback by how complicated some of these questions were. I commented that, if Singapore students, were exposed to these kinds of questions frequently over time, they wouldn’t find them so unusual or formidable. That made sense to them (of course I was only speculating that this was the case since I didn’t have many samples of problem sets).

   Your final comment about the irony of two nations whose assessment philosophies are somehow morphing into the other’s is fascinating but not a shock to me. I’ve read for many years that Asian nations have been watching American education closely and have been interested in fostering more creativity in their students and less rigidity. The problem-solving curriculum adopted by Singapore math in the 90’s is a reflection of some of this. But there’s a key point here that is often missed. We’ve had a problem-solving curriculum in this country for many years now, BUT THE PROBLEMS ARE NOT AS CHALLENGING! Philosophies don’t equate to performance. It’s all about the QUALITY of the materials and instruction as well as the overarching philosophy (the ‘essence’) - always has been, always will be. it isn’t just that students get to advanced topics earlier in some other countries. I’m more interested in the kinds of questions they are expected to solve. Do you see this as significant or do you believe it is the overall educational philosophy in Singapore that distinguishes it?

   I plan on posting another article on Singapore Math, referring to your latest post and including some of your links. i might even repeat some of your comments and mine if that’s ok with you. Of course, I will give you the attribution and link readers directly to this article. Thanks again…

   I am indebted to you for providing genuine and provocative information directly from the source. Thanks!

2. zac said,

   September 2, 2007 at 1:38 pm

   It isn’t just that students get to advanced topics earlier in some other countries. I’m more interested in the kinds of questions they are expected to solve. Do you see this as significant or do you believe it is the overall educational philosophy in Singapore that distinguishes it?

   The “educational philosophy in Singapore” (at the primary and secondary level) is very much focused on external standardized tests.

   I was very surprised when I got here to learn that the ‘O-Levels’ (end of grade 10) and ‘A-Levels’ (end of Junior College, ie 2 years of pre-university after O-levels) are administered from Britain! (The examiners are the University of Cambridge Local Examinations Syndicate.)

   At primary level (grades 1 to 6), the students have the joy of the PSLE (a Singapore-based series of 2-hour examinations in English, mother tongue, mathematics and science). [See also Bilingualism in Politics.]

   So to answer your question - if those questions are going to ‘come out’ in the examination, they will be drilled like mad in class. [One thing that never ceases to amaze me is the inherent ability of Singaporeans for rote learning. Read out a list of 20 words and they can happily recite them back to you. This is after years of testing in the Singapore system…] So I guess the ‘quality’ of the questions is a result of the examination writers’ enthusiasm for such questions.

   You may also be interested to poke around the Singapore Ministry of Education site.

   I might even repeat some of your comments and mine if that’s ok with you.

   Fine with me :-)

Posted by Dave Marain at 8:59 PM 2 comments

Labels: Singapore Math

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ドル 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ドル.
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ドル.

From the diagrams, you can see that there is an additional 11 units. Therefore:

11 units = 154ドル
1 unit = 14ドル

Last month they saved 8 units together.

8 units = 14ドル x 8 = 112ドル

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...

Posted by Dave Marain at 6:18 AM 13 comments

Labels: Singapore Math

[Note: As promised, I have now posted a lengthy comment to this post, which only scratches the surface. I welcome your comments.]

Although Singapore Math is no secret to many homeschooling parents and districts around the country, I thought that the sample placement test for Grade 6B, available for free downloading, might lead to some fruitful discussion. Pls try all 14 questions before commenting. I did and I will have much to say later. There are many other placement tests at the same site so have fun! Go to the Singapore Math home page for background if needed.

Posted by Dave Marain at 9:48 PM 5 comments

Labels: placement test, Singapore Math