Friday, November 28, 2008

Math problems of the week: 2nd grade Everyday Math vs. Singapore Math

A. From the last page of exercises of volume 1 (of 2 volumes) of Everyday Mathematics Student Math Journal, p. 163

4 children share 12 slices of pizza equally. How many slices does each child get? Draw a picture.

Each child gets ____ slices.

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5 + 6 + 23 = _____
_____ = 3 + 3 + 12
4 + 3 + 17 = _____
____ = 9 + 2 + 9

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Count by 2s.

70, 72, 74, ___, ___, ___, ___, ___, ___,___, ___, ___


B. From the last page of exercises of volume 1 (of 2 volumes) of Singapore Math Primary Mathematics Workbook (2A), p.175

312 boys and 195 girls took part in a swimming test. How many more boys than girls were there?

There were _____ more boys than girls.

----
There are 292 men, 149 women and 68 children on a train. How many people are there on the the train?

There are _____ people on the train.

----
Meihua and her sister saved $502 altogether. Meihua saved $348. How much did her sister save?

Her sister saved $______


C. Extra Credit:

Which problem set do you think the University of Chicago Math Department, as opposed to the University of Chicago Math Project, would prefer to see 2nd graders doing half-way through 2nd grade math?

11 comments:

Karl Fisch said...

Thanks for sharing this. I have some questions. And these truly are questions, not statements disguised as questions.

First, my assumption is that these problems are representative of these two texts, since obviously such a small sample doesn’t do justice to either text. So, here are the questions.

1. The three Singapore problems are basically addition and subtraction problems, albeit with higher numbers and a lot of reading involved. The first EM problem is nominally a division problem, along with a visual representation, albeit with small numbers and pizza. There are then four addition problems (with small numbers, but varying the format), then a combo addition/pattern recognition problem. So, do higher numbers and more reading, even though they only involve addition and subtraction, automatically mean they are better problems than lower numbers that also include division and pattern recognition (a pre-Algebra skill)?

2. Should the metric for second grade math be what the UC Math Department thinks is best? This is something I struggle with. While I certainly want the UC Math Department as part of the conversation, I think we can agree that very few of those second graders will be math majors in college. And that math professors have a certain way of looking at and thinking about mathematics that may not (or may, I don’t know for sure) be best for folks that aren’t destined to be math professors. To use your definition of left-brain and right-brain, is a (presumably what you’re advocating) left-brain approach best for teaching mathematics to second graders, many of whom presumably have a more right-brain approach to learning (again, using your definition of brain hemispheres). And, of course, most of the math texts in the U.S. that are compared unfavorably to the Singapore series have been written by math professors, so I’m not sure that supports the argument.

3. Can you point me to anything that discusses the relationship between the UC Math Project and the UC Math Department? I would hope that there is some collaboration/cross pollination there, but I really don’t know.

Lori said...

I just thought I would offer a little background. As a teacher of Everyday Math I can share that the Singapore problems are solid third grade concepts. Second graders are expected to have mastered basic addition and subtraction facts and add and subtract two-digit numbers.

The "equal sharing" concepts in the EM example are not expected to be mastered at this point but in third grade.

This is the EM spiral concepts where concepts are presented in at least three different ways over a period of two years before mastery is required.

While I like the fact that my third graders have excellent reasoning strategies, that is to say, they can explain why things work rather than just supplying the correct answer, the program has been weak in the basics.

On another level, I know our district has been pleased with the near exact match with our PA State Academic Standards, but I wonder, are we asking enough of our kids?

Karl Fisch said...

@Lori - 3rd grade concepts because they are 3 digits?

When you say "weak in the basics," are you referring to addition/subtraction/multiplication/division? Recall, or understanding? Or both? Just trying to get a handle on this.

Mrs. C said...

Karl, I homeschool two boys in fourth grade math. We use BOTH EM and Singapore. Singapore math has a more lineal approach and kids are supposed to "get it" before they move on. Which is fine.

EM "spirals" and I about drove myself nuts trying to teach it. I thought my first-grader was supposed to have a full grasp of the concept of adding and subtracting fractions, which is NOT TRUE. EM Teachers' manual shows the secret... They have "beginning, developing and secure" levels. So, your first grader would have a "beginning" level in fractions because WOW, you showed him a fraction!! Whoopee.

"Secure" is a much higher level, obviously. So sometimes comparing EM and Singapore is unfair because in the EM your students are NOT supposed to have a good grasp of all the concepts. Which is funny because Lefty shows how Singapore math is actually more difficult. The children not only MASTER the concepts, but at younger ages... but are not necessarily INTRODUCED to certain concepts 'till later.

I use both in my homeschool because our public school uses EM. I want my kids grounded in the Singapore methods. So sometimes I'll do an EM unit and then a Singapore unit. Takes longer to teach that way, but I figure it's about how the children grasp the CONCEPT and not necessarily how far the children are in the curriculum.

For all the criticism of the curriculum METHOD in EM, the worksheets are actually very good.

Hope that helps!

:]

lefty said...

Karl,

You're right--my problems of the week, by construction, can't be representative of the entire curriculum. But I'm hoping that, collectively, as I post more and more of them over time on this site, they can give a more global impression of how the different curricula stack up.

The significance of the number of digits is that, the higher the number, the harder it is to avoid using the standard algorithms. My impression is that Reform programs rarely give problems involving large enough digits or complex enough calculations that students are compelled to use the standard algorithms.

Why compel students to use the standard algorithms? I've come to believe that extensive practice with these is one of the best ways to master place value and prepare for algebra. (Yale mathematician Roger Howe has more to say on this: http://64.233.169.132/search?q=cache:5d6TP63BKMYJ:www.maa.org/pmet/resources/PlaceValue_RV1.pdf+%22roger+howe%22+%22standard+algorithms%22&hl=en&ct=clnk&cd=1&gl=us&client=firefox-a).

I'm not aware of any Reform Math text that was written (or co-written) by a math professor (of course, math education professors are a different story). The U of C math dept--some of whom I know--was not part of the U of C Math Project. (I can't find a link demonstrating this, but cf: http://www.toacorn.com/news/2005/0324/Front_Page/001.html) None of the math professors I know like Reform Math, and many have spoken out against it.

Perhaps not all students can handle the problems that Singapore students use--I don't know. I do know large numbers of kids, in Reform programs around the country, who are under-challenged in math, except when their parents supplement them at home (or in after school math teams). I imagine that some of them are what I'd called "left-brained," but probably not all.

Lori said...

Karl,

Having taught EM in 2nd, 3rd, and 4th grade I did notice that the basic arithmetic is a bit lacking. Students are supposed to master facts, primarily through the use of math games and fact triangles (as opposed to flash cards). So, by basics, I mean all facts and basic computation. Second graders are exposed to some three digit adding and subtracting at the very end of the year. Third grade picks up with three digit addition and subtraction and requires mastery.

Mrs. C - I LOVE that you are incorporating both! I can't imagine homeschooling and teaching EM! I really like how they built in differentiation for learners, but I can't imagine doing all of that with several levels at one time!

Karl Fisch said...

@Mrs. C – Thanks. What’s the specific criticism of the curriculum method in EM? It seems like either the Singapore model of mastery before moving on, or the EM model of spiraling through beginning/developing/secure should work fine, as long as both lead to mastery. Kudos for trying to incorporate both for your kids.

@Lori – Thanks. Why do you think that the skills are lacking if they are “supposed to master facts” just like in Singapore? It seems like math games and fact triangles would get you to the same place as flash cards.

@Lefty – Thanks. Some follow-up thoughts.

I agree that place value seems key, but I guess I’m not convinced that the only way to get there is via repeated use of the algorithm with large numbers. I taught high school math in a district that used fairly traditional elementary texts (we had a “back-to-basics” school board at that time) and many of the students I saw in high school had mastered the algorithms but didn’t really understand the mathematics (including place value). This was true for my basic skills students through my honors level students, and most of them hit a wall with the more advanced level of thinking required in our high school classes. My statistically-irrelevant impression was that they hadn’t spent much time thinking about place value (or math in general), but a lot of time just doing the algorithm. I wanted them to do both.

I wasn’t necessarily talking about just “reform math texts” (not completely clear on what the definition of "reform math text" is), but also the more “traditional” texts that are still used (and that we predominantly used when growing up). Those have been compared unfavorably with Singapore, and I thought (could be wrong) that they mostly had math professors as one of the primary authors.

I wasn’t trying to say that all students can or can’t handle Singapore math, I don’t know enough about it. I was simply pondering whether math professors would necessarily come up with the best way of teaching second graders. (Again, I think they should be a big part of this conversation, but have my doubts that they should have the final word.)

In my very limited research on EM and Singapore today (fit in around eating lots of turkey), it seems like they both are pretty good, with each bringing strengths and weaknesses to the table. EM certainly seems to have some research backing it up, although of course I’m relying on their excerpting of the research (but they do cite and quote). And they certainly don’t seem to be anti-algorithm, they just include other approaches and give kids more time to master the algorithm (that spiral approach again).

Singapore Math seems to have a couple of things going for it (in it’s original implementation) that would help – teachers with an understanding of math and significant training (100 hours a year). I’m wondering if mathematically strong elementary teachers with 100 hours of training a year might do pretty well with EM, Singapore Math, or even Karl and Lefty’s Homebrew Math Curriculum (tm). I’m not teacher bashing here, but I think it’s probably tough to implement any math curriculum if you’re not very comfortable with the math yourself.

Mrs. C said...

Karl,

I have to be honest here. My math is HORRIBLE. I homeschool because of abuse issues in public school with my son "Elf," now age 8. He is autistic. I also homeschool his little brother "Emperor," age 7. I would NOT be my own first choice to teach my children mathematics. But now that I'm doing it myself, I am learning how to do the math with the children. Maybe I was just taught incorrectly and am not that stupid after all. :] Someday, if Elf is able, I should actually prefer a co-op for higher levels.

But in any event, I find a few things difficult in MY implementation of the curriculum:

1. I'm a little fuzzy on exactly what constitutes a "developing" or "secure" level according to the manual. I have all the assessments, etc. and have not seen any mastery "cutoffs" or set level for a passing grade. So what I do is use the unit reviews and make sure my children score at least 80%. I can't imagine having my own children fill out the "how I feel about my math progress" sheets and calling that a school day.

2. It's written for larger groups of children and I can't do the games very well, if at all. We rely more heavily on personal drills and/or computer time for those "fun" parts of the math curriculum.

3. It relies heavily on using a group to solve the problem. Sometimes this works with two children, and sometimes it doesn't.

4. This is for what it's worth... When my older children were in public school, there would be homework that would employ strange methods rather than traditional algorithms (lining up numbers for multidigit multiplication, etc.). My oldest son is gifted but found around seventh grade that there was a large JUMP into higher mathematics, and he felt unprepared for Algebra I.

Our public school, fortunately, believes in drilling the children in addition/subtraction and multiplication. Unfortunately, that also came home as "homework" on top of the 8-hour day the children spend at school. :[

PS We have also used the Horizons curriculum, which is a spiral curriculum. It's expensive, but very colourful. I changed to Singapore after reading several math blogs. The downside is that I think Singapore will take quite some time (measured in months) to teach because it is more advanced.

I also, because I KNOW I have a weakness in this area, am carefully going through Everyday Math at the same time with the children so that we're sure not to have "gaps" if they were compared to public school peers.

OK, done with the novel.

PS. Lori, I really do like the worksheets very much in EM. I wish it were laid out a bit differently, because it is rather awkward for homeschooling the way different things are in different books, etc. I just xerox from the masters way ahead of time to help avoid this problem, but it's still problematic.

Karl Fisch said...

@Mrs. C – Wow, what you’re doing with/for your kids is inspiring. I think learning alongside them is a pretty good model to follow. Thanks for expanding on your issues with EM, I hadn’t thought about all the group emphasis and how problematic that would be for a group of two (or three, including you!), but it sounds like you’re adjusting/compensating well.

lgm said...

@Lefty>>Why compel students to use the standard algorithms? I've come to believe that extensive practice with these is one of the best ways to master place value and prepare for algebra.

Why would a teacher want to take all the time for extensive pencil & paper practice, when the mental math methods lead to rapid understanding of the concepts and accurate computation? What insights is a student getting with taking more time and using the right to left algorithm that he can't get with the mental math methods?

For myself as a small child, I did not develop the insights after all those pages of problems, although I was very quick and could execute the algorithms easily. Fortunately new math came in Grade 5 with a unit on other bases, set theory, and careful study of the distributive property ... this gave me the insights needed to find algebra extremely easy.

lefty said...

Finally a moment to reply! (In the middle of a grant application, due Friday)

Karl, I agree that math teachers shouldn't have the final word on curriculum. Where they are key is in what they have to say about what concepts students need to master in order to proceed on to college level math. My impression is that the Singapore curriculum represents a model collaboration between educators and mathematicians. Actually, ideally I believe there should be three different parties involved: people with classroom teaching experience, math professors, and cognitive scientists specializing in math acquisition.

As far as research supporting EM goes, the thing to look for would be randomized trials at different schools. Otherwise there are too many variables. But such research doesn't seem to exist. Another question to ask is: is any other country using a Reform program like Everyday Math, or might we be in uncharted waters here. And if we are, should we be?

Karl and LGM, As far as the standard algorithms go, obviously the key thing, as with everything else, is that they be taught conceptually, not as an arbitrary rote procedure. Taught properly, I believe that most kids can understand them. Certainly that's what I see at Continental Math League practice, though I'm obviously getting quite a biased sample here. But my 20 second and third graders are catching on so quickly that it's hard for me to believe most of their classmates aren't capable of understanding them, too--if only they were given the opportunity to.

So, yes, the poor preparation of many elementary school teachers in math is a big problem...

LGM, I can believe that mental math is an alternative route to mastering place value. Your post suggests there is seem real research out there supporting the efficacy, and efficiency, of this approach: can you send me some links? In particular, I have to wonder whether a mental math approach works well for all kids. I myself seem to have serious short term memory problems when it comes to keeping multiple calculations in my head. For people like me, might the standard algorithms, which (except for long division) I can't remember ever not understanding, be a better alternative? At Continental Math League practice, we're trying to do a combination of mental math and algorithmic practice. Different kids seem to excel at one or the other.

It strikes me that, beyond whatever conceptual benefits the standard algorithms, properly taught, can bring students, there are the pedagogical benefits of setting kids up for algebra problems: adding terms; adding rational expressions; multiplying and dividing polynomials. Whenever my son gets lost in one of these, we got back to a concrete problem involving numbers and the standard algorithms, and he quickly regains his footing.

Mrs C! Wow, it is amazing all the math you're doing for your kids, and I can believe that a combo of EM and SM is very effective. So sorry about the abuse issues with your autistic son at school--truly appalling. Sounds like he's not only in a much safer place now, but a much better one for learning!

Lori, I'm curious whether your school allows you to supplement EM?