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In response to a question in one of the comments regarding the spirited discussion at Eduwonk's site regarding Everyday Math, here is the link to what students should know by what grade. Table 2 (page 20) of the National Mathematics Advisory Panel's final report, contains the panel's recommended benchmarks for the critical foundations of math skills/concepts that students should master in order to be prepared for algebra in the 8th grade
Fluency With Whole Numbers
1) By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.
2) By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.
Fluency With Fractions
1) By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percent, and with the addition and subtraction of fractions and decimals.
3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.
4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.
5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.
6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.
Geometry and Measurement
1) By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).
2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three dimensional shapes and solve problems involving surface area and volume.
3) By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.
UPDATE: On page xvi of the NMP's report is the following:
"A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided"
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19 comments:
Thanks so much. I just posted a comment before noticing your post. Let me repost part of that here:
I guess I'm curious both what the minimum pace should be, and what the argument might be for setting the pace even faster.
With my 6 year old, I feel like we kind of see-saw back and forth (using Singapore Math), because he is able to pick it up very quickly, but he will then start "perseverating" on math all the time, which gets to be a little overwhelming. So, I'd also be curious about pro and con arguments for accelerating as fast as a kid can go (though I also want a better sense of the minimum pace that should be set for "typical" kids)
My opinion is that the NMAP report sets a good standard for "typical" kids. It can be implemented in all sorts of ways, but if you are using Singapore Math, then there should be no issue at all.
My son loved to do math when he was in Kindergarten. I would leave out worksheets and he would do them. He didn't get weird about it. I'm sure it wouldn't have taken much on my part to have him doing eighth grade algebra by fifth grade. Many kids are capable of so much more in K-6.
My view is that education is not a race, but I don't want it to crawl along when my son is such a sponge. A slow pace can be much worse than a fast pace. Since he is not home-schooled, the problem was to find the right school or balance. I didn't find it with the Everyday Math at the K-8 private school he attended. I had to work just to keep him at a proper grade level. (along with paying the school a whole lot of money) He didn't like having to do both school math and home math. You could say that I didn't try to accelerate him more for practical reasons.
When we decided to bring him back to our public school in sixth grade (they used Everyday Math too), I wanted to avoid that at all costs. Our public school had evolved over the years and they were willing to work with us. I think this was only the case because he was getting to the upper grades where a certain level of tracking happens. I also think that some pragmatism has tempered their understanding of differentiated instruction.
So we went through the whole sixth grade Everyday Math books in the summer and he took a test that allowed him to skip directly to pre-algebra, which uses a traditional textbook. He is now in seventh grade and taking algebra. Unfortunately, this screwed up his schedule, so the only reasonable solution (for us) was to tell them that I would teach him the material and give him the tests. So, when the rest of his classmates are in math class, he is in another room by himself going over the algebra unit I tell him to cover. With geometry next year, we will have the same problem and then have to deal with any issues with taking Algebra II as a freshman in high school.
I apologize for rambling on, but I just wanted to explain that for us, many of the pros and cons have to do with practical issues. Although I think there is plenty of time in high school and college to get all of the math he could ever want, I feel that I could be doing more to tap or test his full potential. Perhaps the biggest issue is to make sure he is challenged. There are times when he gets very frustrated because he can't figure something out immediately. I want him to learn how to work hard and deal with this. Then, he should be able to handle anything on his own.
I apologize for rambling on
well, I for one appreciate it--it's not the sort of information I can find elsewhere!
What you are saying makes sense. I have to say, while the math my son does for first grade at his public school (Investigations) is absurdly easy for him, it kind of seems fine for now. As a kid w/Asperger's, he does really benefit from being in a co-teaching classroom--it's hard to explain, but the teachers encourage the other kids to use math as a way of socializing with him, which is not something we can do for him at home.
What you described is reassuring to me--we can continue to supplement w/Singapore to challenge him, but it doesn't sound like he is falling behind or anything, so our on again, off again pace is probably right for now.
Also, just to be clear, I wouldn't describe his math obsessiveness as weird, exactly, or certainly not unpleasantly weird. I feel incredibly lucky to get to work with him and see a little bit of the beauty of math through his eyes. But sometimes it's good to have a break and spend time talking about all the other things he loves to talk about (science, history, etc.)
Well now I *am* rambling on, so sorry about that, but I really appreciate your help--thanks!
Laura,
Investigations is even worse than EM. If he likes Singapore and is doing well with it, please continue by all means.
The only "disadvantage" is he will be learnng math. Good luck!
Can someone explain to this person why students must absolutely gain fluency with the long division algorithm for polynomials?
She seems to "need a clue" :D
http://www.edexcellence.net/fordhamfellows/blog/index.php/2008/09/long-division-sucks/
Thanks--we definitely will continue w/Singapore Math. I'm thinking about ordering one of the writing workbooks, too.
I've heard that Investigations is worse than EM. We were given a copy of the textbook to take home, because they want us to help w/homework using the Investigations methods.
It's a shame--it does seem to have some good stuff in it (similar to what I've seen in Singapore, though obviously at a much slower pace), but I don't understand why, for instance, four options for solving an addition problem are presented as equal choices when they seem more like stages of understanding (from concrete to abstract).
I also was annoyed that one "option" was to solve 3+4 by adding one to 3+3, but there was no attempt to explain how the student in the example knew that 3+3=6 in the first place. It almost seems coy to me.
Can someone explain to this person why students must absolutely gain fluency with the long division algorithm for polynomials? She seems to "need a clue"
Ordinarily I would do so, but that particular entry was written by Catherine Cullen, the person who held court at Eduwonk's site, discussed in another post on KTM. It generated 66 comments. Many people tried to explain to her why EM was a poor program, why Singapore Math was superior, but she clung to her version of reality. Given this track record, it would be a waste to try to explain anything more to her.
I've heard that Investigations is worse than EM. We were given a copy of the textbook to take home, because they want us to help w/homework using the Investigations methods. It's a shame--it does seem to have some good stuff in it (similar to what I've seen in Singapore, though obviously at a much slower pace), but I don't understand why, for instance, four options for solving an addition problem are presented as equal choices when they seem more like stages of understanding (from concrete to abstract).
Investigations is nothing like Singapore. You might want to look at the website for the folks in Prince William Co., VA who are fighting that program in their schools. It's at:
http://www.pwcteachmathright.com/
Investigations is nothing like Singapore.
Sorry--I expressed myself poorly. I didn't mean to imply that the two curricula were at all comparable. I just meant that there are techniques represented in the Investigations textbook I flipped through that seemed potentially helpful for learning arithmetic, if only they had actually been taught (let alone systematically), which they weren't.
Maybe the 3+4=3+3+1 thing isn't from Singapore--I remember reading about that somewhere (maybe Wayne Wicklegren?) and my son found it helpful (after some practice, he moved from counting on his fingers to using those kinds of techniques as he was picking up math facts).
I just thought it was odd to show these increasingly abstract ways of performing addition and then to represent them merely as "choices," as if adding 3+3+1 rather than counting on your fingers was just a preference, rather than something you can do if you've already learned 3+3=6.
Perhaps I'm still missing something though--I'll take a look at that site.
Barry,
Thanks!
It's just a shame that CC's students lost the opportunity to learn higher level mathematics because she didn't take the time to understand the algorithm well enough, or its usefulness in their mathematical preparation!
A real shame!
"Any approach that continually revisits topics year after year without closure is to be avoided."
Of course they are referring to EM. In the Eduwonk thread, a link was given to a scope and sequence chart for EM that has now magically disappeared. (Can anyone find that chart somewhere else?) There was talk about how EM ensures mastery, but when the discussion got into details about how this works, the details were vague and inconsistent. EM talks about "secure" dates that could be years after the first introduction of the material.
For multiplication facts, all I saw in the chart was “Practice multiplication/division facts”, which is marked as secure in grade 4, unit 3. How do you secure “practice”?
Perhaps they call this closure, but I never saw it in any of my son's classes, and the EM supporters on the Eduwonk site couldn't explain how this worked.
Regarding the strategy of adding 3 and 4 by comparing it to 3 + 3:
More than one curriculum teaches 1st graders basic addition facts by starting with doubles (such as 3 + 3) and then progressing to "doubles plus one."
There are two issues, short and long term. In the long term you need students to recall that 3 + 4 is 7 without using any strategy (i.e. not figuring it out, just knowing it). Short term you want to them to handle all the arithmetic that comes their way and to begin using some quantitative reasoning rather than finger-counting; recognizing a sum as one more than a sum you know is an example of quantitative reasoning. Singapore does this kind of thing all the time -- it is one of their strengths. Later on they teach adding something like 7 + 5 by completing to 10 (7 wants 3 to make 10, take that from 5 leaving 2, you get 12 ...) This is where Singapore's number bond diagrams are very useful. Eventually you expect students (via drill) to know that 7+5 is 12 without a strategy -- just recall. In the meantime you have helped them do all the arithmetic that comes their way before they have mastered recall, plus you have once again made them familiar with using quantitative strategies. Down the line there will be more, one hopes, once basic addition facts are well memorized.
I don't quite follow Laura when she brings this up -- did she see this in Singapore at home or Investigations or EM at school? At any rate, taking more than one look at simple things is not the sign of a bad curriculum.
At any rate, taking more than one look at simple things is not the sign of a bad curriculum.
True, but there are other things that lead many people to conclude that Investigations is not a good program.
Okay, one more time, as clearly as I can manage it.
1.The Investigations textbook had one page which depicted 4 different strategies for adding 3+4.
2. One of these strategies was a picture of a little girl with a bubble that said: "I know 3+3=6. So 3+4 must be 7!"
3. My objection was not to the strategy--in fact my whole point was, "hey, that in itself is a useful strategy--if only they would actually explain the technique for kids who haven't already yet been taught that 3+3=6."
When, Barry Garelick replied that Investigations is nothing like Singapore, that confused me, so to try to make sense of what he said, I wondered if I was misremembering that "doubles plus one" technique being used in Singapore.
So far I still don't understand what I said that was confusing or wrong. I wasn't suggesting that Investigations is a good curriculum, nor was I suggesting that "doubles plus one" isn't a good technique. In fact, I was saying the opposite on both counts.
I was just surprised that they obviously are aware of good techniques like "doubles plus one," yet they don't seem to want to teach the technique consistently and explicitly. I'm wondering what the ideological reason is for that.
sorry for the double post, I forgot to address this point:
taking more than one look at simple things is not the sign of a bad curriculum.
Of course not. But treating counting on your fingers vs. using "doubles plus one" as a matter of mere personal preference is disingenuous. It's extremely unlikely that a 6 year old is going to "prefer" that technique if they don't already know that 3+3=6. Nor are most kids likely to apply that technique to other problems if they are only taught--"hey, if 3+3=6, then 3+4 must be 7!" rather than being taught and reminded that they can use the same technique for other problems. My own son, who, as I mentioned, picks these things up pretty quickly, needed to be reminded of the technique several times before he started applying it himself. And that was after he had learned a number of "doubles" facts and after I had taught him the general rule to the point that he understood it well.
Sorry to confuse you. Investigations may have some good techniques now and then, such as doubling 3 and adding 1, but it does not provide the mechanism for ensuring that students know what 3 + 3 is in the first place. Which is what you surmised, so your instincts are correct.
When Singapore provides examples of different ways of solving the same problem, it does so in the context of what has been learned and mastered previously--which is done in a sequential and logical form. Investigations does not do this. That's what I meant by "Singapore is nothing like Investigations"
Much has been written about Investigations. Start with Wilfried Schmid's essay at:
http://www.nychold.com/forum01-schmid.html
okay, thank you! This:
but it does not provide the mechanism for ensuring that students know what 3 + 3 is in the first place
and this:
When Singapore provides examples of different ways of solving the same problem, it does so in the context of what has been learned and mastered previously--which is done in a sequential and logical form.
are exactly what I was trying to articulate. I don't want to misrepresent the arguments against constructivist math curricula, so I want to make sure I'm not missing the point. I participate in an on-line discussion forum where (on one thread) I go back and forth with parents who are on the fence about these issues, so I'm trying to find ways to communicate convincingly about why Singapore is a better approach.
John Hoven and I wrote an article about Singapore Math which was in Educational Leadership magazine last year. It might be helpful to you in your discussions with others. It can be found at:
http://www.nychold.com/art-hoven-el-0711.pdf
Great, I'll take a look at that and the other article you mentioned. Thanks!
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