KTM User Pages
05 Oct 2006 - 00:25
UPDATE 10-4-2006: Be sure to see the bar model problem Barry left in the Comments. I'm going to have Christopher do it now (it's a much simpler version of the problem below, which is exactly what I was about to write myself.)
My newfound appreciation for the Work Backwards genre is fading fast. Work this backwards, why don't you?
Keri bought a new pack of graph paper for math class. When she got to class and opened it up, three people asked her for some. She gave one quarter of the pack to Christopher. Alyson got one fourth of what was left. Then Mark took one third of the remainder. That left Keri with 36 sheets. How many sheets were in the pack of graph paper? source:
I was finally able to work this backwards using bar models. Thank God for bar models. LESSON: ALL OF YOU SHOULD BE TEACHING YOUR CHILDREN HOW TO CONSTRUCT BAR MODELS. YOU MUST TEACH YOUR CHILDREN HOW TO CONSTRUCT BAR MODELS BECAUSE YOUR CHILDREN WILL BE REQUIRED TO SPEND YEARS OF THEIR YOUNG LIVES SOLVING ALGEBRA PROBLEMS BEFORE ANYONE TEACHES THEM ANY ALGEBRA. SOLVING ALGEBRA WITHOUT KNOWING ANY ALGEBRA WILL BE CALLED CRITICAL THINKING. OR, POSSIBLY, LOGICAL REASONING.* DEFENSIVE BAR MODELING! WE'VE DISCUSSED THIS BEFORE!
OK, seriously, I was finally able to solve this using bar models. I still haven't managed to solve it using conventional algebra, though that is almost certainly because I'm making a careless error. But I don't see, offhand, or even onhand, how to do this problem via a Work Backwards approach without algebra. Obviously it's possible, since I just did it with bar models. But I don't see it.
I'm going to have to add the bar model book to Christopher's stash.
* This problem appears on a page entitled "Critical Thinking Skills" in Skill Builders, as one might expect.
-- CatherineJohnson - 05 Oct 2006 Back to main page.
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John spent 3/5 of his allowance on some books. He had $10 left. How much money did he start with? Working backwards won't help much with this one. But guess and check will. So will bar modeling. -- BarryGarelick - 05 Oct 2006
HEY! Barry, THANK YOU! I was just sitting here thinking I need to make up a SIMPLE work backwards bar model problem involving a fractional part so I can show it to Christopher AND YOU JUST DID IT FOR ME! -- CatherineJohnson - 05 Oct 2006
Barry - I got the Zaccaro book on problem solving (it's pretty good, I think) and he says that the bar model problem you used in your article is one of THE biggest sources of confusion and error for kids doing algebra problems without algebra. He doesn't put it that way, of course ("algebra without algebra"). That confirmed my own sense that my entire Math Brain expanded when I saw that particular model. (I'll check to make sure they're the same. Must go have Christopher do your problem first.) -- CatherineJohnson - 05 Oct 2006
Just had Christopher do it — wonderful. Thanks! (I had to talk him through it, but that was fine. It was great, actually.) -- CatherineJohnson - 05 Oct 2006
Catherine, Have you seen the algebra problems in the Thinking Blocks multiplication program? Some of those are very challenging to do without any formal knowledge of algebra. With bar models or blocks, it's a snap. The direct link is a long one: http://www.thinkingblocks.com/ThinkingBlocks_MD/TB_MD_Main.html Click on the cube next to advanced problems. I should have problems similar to the one Barry shared ready by January. The questions are dynamically generated and no two should ever be exactly alike. Colleen(publisher) -- ColleenKing - 05 Oct 2006
Hi Colleen! I think someone left a link to your thinking blocks the other day - am I right? Thanks! -- CatherineJohnson - 05 Oct 2006
How did you think of the "thinking blocks"? -- CatherineJohnson - 05 Oct 2006
SOLVING ALGEBRA WITHOUT KNOWING ANY ALGEBRA WILL BE CALLED CRITICAL THINKING. OR, POSSIBLY, LOGICAL REASONING.* DEFENSIVE BAR MODELING! At least bar modeling is useful. What I'm worried about is all of the other c*** that is used to judge whether my son can think critically or not. As a supplement to EM this year (5th grade), the teacher is giving out brain teasers to the kids. The most recent one required them to figure out how to take a number from one to eight and combine it 8 times to make 1000. Another problem told a story about how a snail was climbing out of a hole 30 feet deep. Each day it climbed up three feet and each night it slid back two. On which day does the snail get out of the hole. The trick is "which day", not "how many days". This supplements 5th grade EM, where the teacher is reviewing partial sums addition and how to write numbers like 268,123 in word form. What is it - week 5? In English, for writing, the teacher is using peer and teacher review of their work. The final (peer and teacher corrected) draft of a book report he brought home today still had mistakes in punctuation, grammar, and common sense. My wife was livid. Process over thinking. My son thinks that all he has to do is to follow a step-by-step formula. My email to the head of school explaining my "concerns" about Everyday Math, and offering my services in reviewing new math curricula was politely, but clearly brushed off. He may agree with me on some points, but they will deal with it themselves, thank you very much. As a side issue, I've gotten myself pulled in as a mentor or advisor (baby-sitter) to our school's new after-school robotics program. This is the First LEGO League NanoQuest? for 2006/2007. Their web site is full of the usual fuzzy real-world, hands-on (fun, fun, fun) talk. It could work out OK, but only if we advisors make sure the kids are taught something, rather having them teach themselves and construct their own knowledge. Has anyone been through the process and have any thoughts? This year, the kids are supposed to do (along with the robotics) a report (invention/development) on the use of nano-technology. Right. They can even present the report as a play - with costumes. Right. -- SteveH - 05 Oct 2006
Pre-algebra, multi-variable problems usually have one of the equations (if you were to do it with algebra) with one equation and one unknown. the trick is to find that equation or formula and solve it first. Then you use that answer to find another equation in one unknown and solve it. It may not be a backwards approach, but it all depends on finding that key equation. Most sets of equations (problems) cannot be solved this way. Do the students know the difference? With algebra, it doesn't matter. All you have to do is start writing down equations. Get an equal number of unknowns and equations and you are just about done. I have mixed feelings about whether problems like this are appropriate in the pre-algebra world. If the kids are ready to tackle this problem, then they are ready to start learning the basics of algebra. However, the real problem, as always, is their unwillingness or inability to teach the kids how to solve the problem using anything other than vague generalities like drawing a picture, looking for a pattern, and guess and check. You would think they would LOVE bar models. By the way, when I solved the problem with a bar model, I made a mistake. When I did it with algebra, I didn't. "Keri bought a new pack of graph paper for math class. When she got to class and opened it up, three people asked her for some. She gave one quarter of the pack to Christopher. Alyson got one fourth of what was left. Then Mark took one third of the remainder. That left Keri with 36 sheets. How many sheets were in the pack of graph paper?" The key is to see that for the final remainder, Mark gets one-third and Keri gets two-thirds. This means that Mark has 1/2 of what Keri has, or 18 sheets. With bar models, this relationship jumps right out at you. With algebra, this doesn't jump out at you, but it doesn't matter. N = the total number of sheets K = the number of sheets for Keri C = the number of sheets for Christopher A = the number of sheets for Alyson M = the number of sheets for Mark 1) N = K + C + A + M 2) K = 36 3) C = N/4 4) A = (N - C)/4 5) M = (N - C - A)/3 Five equations in five unknowns. Substitute so that equation 1 is defined with just N and solve for N. The problem with my algebraic approach is that you do not see the relationship between Keri and Mark. If the final remainder is (N-C-A), then K = 2/3(N-C-A) = 36. The final remainder (N-C-A) is then 54 sheets and M = 18. This is faster than using algebra if you can "see" this relationship. If not, you are lost. The advantage of algebra is that it works for any problem, and if you get good at defining unknowns and equations, is really pretty simple and mechanical. This means that you do not have to worry about finding the key trick to the problem. Math is not about having some sort of Zen knowledge or math brains. It's about knowledge and skills that make it possible to solve all sorts of problems without guess and check or some magical sort of thinking. The goal is to make the process mechanical so that you don't have to think (too much). I'll call it no-thinking math. Just start writing down equations and see where it takes you. Let the equations lead you to the answer. This is true for bar models. Just start drawing the bars and see where it takes you. The relationship between Mark and Keri will jump right out at you. You would think that the fuzzies would love bar models - algebra without knowing algebra. -- SteveH - 05 Oct 2006
I worked it forward. Keri bought a new pack of graph paper for math class. She started with 1 pack. She gave one quarter of the pack to Christopher. 1 - 1/4 = 3/4 pack remaining. Alyson got one fourth of what was left. 3/4 * (1 - 1/4) = 3/4 * 3/4 = 9/16 pack remaining. Then Mark took one third of the remainder. 9/16 * (1 - 1/3) = 9/16 * 2/3 = 18/48 = 3/8 pack remaining. (When I did this in my head I said 2/3 of 9/16 is 6/16, and left it unreduced.) That left Keri with 36 sheets. 3/8 pack = 36 sheets. How many sheets were in the pack of graph paper? 36 sheets / 3/8 pack = 12 * 8 = 96 sheets. (I did (36 / 6) * 16 when I did it, because I had not reduced the 6/16.) This is closer to algebra than working backwards, but is still not algebra. I think working backwards with bar models is a fine way to check after you've solved the problem, but won't teach you much algebra. -- GoogleMaster - 05 Oct 2006
"I think working backwards with bar models is a fine way to check after you've solved the problem, but won't teach you much algebra" The question is whether I teach my son "Defensive Bar Modeling"(or some other technique), or do I jump right into algebra (at home, of course). There is a real problem with these pre-algebra problems and the morons who think there is some magical math thinking or conceptual understanding that has to be developed. At least bar models give kids some tangible place to start. Bar models are much more specific than the vague admonition to "draw a picture" of the problem. Bar models could also be used as a lead-in to help kids define algebraic equations. The goal should be pictures or models that help define variables and equations. There are many other drawings that can help this process other than bar models. Bar models are not an end in themselves or a substitute for algebra, but can be invaluable in helping kids get through the fuzzy pre-algebra years. Ironically, schools are probably having students solve these problems before the kids have mastered the fractional manipulation shown in GoogleMaster?'s solution. Students will be confused at both ends of the problem. -- SteveH - 05 Oct 2006
I once ran across a guy who was arguing that schools should be teaching in maths problem-solving skills and communication methods rather than algebra. Which, since algebra is the most generally useful problem-solving method for maths problems I know, and rather essential to communicating mathematical/physics/economics/etc formulae with other mathematicis/physicists/economists/etc, left me rather gobsmacked. I asked him if he also thought schools should be teaching communication skills rather than reading and writing. -- TracyW - 05 Oct 2006
I have mixed feelings about whether problems like this are appropriate in the pre-algebra world. I'm almost ready to say they're dangerous, because when you haven't been taught the notation, you are tempted to make it up as you go along. That leads to writing things that aren't true. Mark took one third of the remainder. That left Keri with 36 sheets. So 36 sheets is 2/3 of "the remainder" (Mark got the 3rd 1/3) Now let's assume that I'm clever enough (or have done enough problems like this) to understand that I need to figure out what "the whole remainder" is. Okay, what's the first thing I've been taught? *Write down what you know!* But since I don't have algebraic notation (and I've never been taught dimensional analysis), I write:
-- OldGrouch - 06 Oct 2006
Old Grouch, That is exactly the way that I visualized it, and probably the only way I could get a pre-algebra kid through the problem. Defensive Bar Models are essential to survival in the pre-algebra "higher order thinking" world. There are probably other ways of drawing, but at least with a hint link -- use a bar model -- my kid has some sense as to how to begin. SteveH?, FWIW, my 5th grader is also doing the brain teasers as well. The teachers must have gone to the same conference. We have multiplying mice -- it could have been a nice fibonacci introduction, but they gave them numbers that wouldn't work. It took a long time to painstakingly draw out generations of mice. Now it is Friday and I just discovered the Problem of the Week is still on the kitchen table. Now, despite the hard work, she'll get no credit. We had a major project due this week and the started another -- 5th grade is project year I guess. It's not going well. -- LynnGuelzow - 06 Oct 2006
"I'm almost ready to say they're dangerous, ..." At the very least, students will think that math requires some mysterious thinking process, when in fact, it can be quite mechanical. Algebra (naming variables and solving equations) should be started much earlier. It's really very easy, especially for simple problems, but many treat it as some monumental leap of conceptual understanding. For example: Mary has 25 marbles and gives some to Frank. Afterwards, she has 10 left. How many did she give to Frank? N = the number she gave to Frank; what we want to know or solve for. 25 - N = 10 Solve for N. Learning the basics about algebra is much easier using problems like this. If you put off algebra too long, then you are learning the rules while trying to do Keri-type problems. Bar models really aren't a great solution, except as a help to visualize the equations. Even bar models get difficult when the numbers aren't nice. For example, make the following changes to the Keri problem: 1) Mark gets 1/6th of the total number of sheets (not remainder) 2) Keri gets 38 sheets. With algebra, this change doesn't make it any more difficult. The process is exactly the same. However, the bar model gets difficult. Maybe someone can do this with a bar model, but I can't figure it out. (Actually, I can, but it's not easy.) Bar models work when the bars (fractions) line up nicely. With quarters, sixteenths, and sixths, you have to go to 48 divisions of the bar to get what is, in effect, a common denominator or alignment for the bars. They try to find advanced problems that do not require advanced tools (algebra), but these problems are all usually "nice number" or "trick" problems where the solution technique cannot be applied to other problems or worse, other numbers. Imagine a solution technique that only works for some numbers or some problems. How do the students know when it can or can't be applied? This must be extremely frustrating for kids. It makes them think they are just not good in math or do not have "math brains". That's because they will see some kids figure it out or "see" the solution. They won't realize that "seeing" is not a required skill to be good in math. High-level thinking is not the magical ability to "see" things. It's the knowledge of how and why algebra works, and the ability to apply that skill. When I approach a new problem, I don't have to "see" anything. I just start writing down equations and let them show me the way. Of course, with all of my content knowledge, skill, and experience, I am able to "see" lots of things ahead of time (like linear independence or the shape of a curve), but there is nothing magical about it. Bar models are a great help and work in many cases, but they have their limitations. Algebra (naming variables and defining equations) has no such limitations. One technique does it all. That's why algebra is so important. -- SteveH - 06 Oct 2006
"...my 5th grader is also doing the brain teasers as well." My son's math teacher told us parents (and agreed with me) that EM is a mile wide and an inch deep, and that there is not enough practice. I was optimistic. But what do we have? Brain Teasers. They must be included as an optional supplement in the EM teachers' packets. I have just one word for her. Saxon. The problem is "drill and kill". They can't deal with it. They think it's either not necessary or that there is an easier way. More parents are beginning to complain. They acknowledge that there are proponents and opponents. BUT, I get the polite brush-off when I offer (with other parents) to help them with a curriculum change. Basic assumptions are hard to change, along with their unwillingness to accept any sort of outside influence. -- SteveH - 06 Oct 2006
Old Grouch WOW!!!! -- CatherineJohnson - 06 Oct 2006
What I'm worried about is all of the other c*** that is used to judge whether my son can think critically or not. It really is staggering. -- CatherineJohnson - 06 Oct 2006
They try to find advanced problems that do not require advanced tools (algebra), but these problems are all usually "nice number" or "trick" problems where the solution technique *cannot be applied to other problems or worse, other numbers.* Which raises a question: Exactly what are they trying to teach by assigning these problems now? Perhaps it's to demonstrate (after thrashing around and gnashing of teeth) that some problems become impossibly complicated without algebra. So that when algebra is finally revealed, the kids will all cheer, "Halelujah!" But if that's the idea, why "points off" for wrong answers? And why the "nice numbers?" Obviously, the bar model is a crutch, and it becomes harder to set up as the numbers become "less nice." IMO, it's still useful for bringing home the process of getting from the first to the second step: What's 1/4 of 3/4? -- OldGrouch - 06 Oct 2006
This year, the kids are supposed to do (along with the robotics) a report (invention/development) on the use of nano-technology. Right. They can even present the report as a play - with costumes. Right. I was actually thinking, last night, "I'm not going to make it through another 9 months." -- CatherineJohnson - 06 Oct 2006
That is NOT the kind of thought I normally have about anything at all. I don't believe in thinking that way (I don't say this as a judgment on anyone) and I'm not temperamentally inclined to think that way (due to general bounciness & manic-ness).... And I was thinking, over and over, "I'm not going to make it through another 9 months of this." -- CatherineJohnson - 06 Oct 2006
We've had another Ms. Kahl eruption, and the new principal has imposed collective punishment on the entire school because of the bomb threat....and I lost it enough over Ms. Kahl to fire off a sharply worded email to the principal which I DIDN'T want to be doing this early in the "game"..... That's another thing. Obviously I'll "make it" through another 9 months. That's not the issue. The issue is can I keep it together for another 9 months. -- CatherineJohnson - 06 Oct 2006
I have mixed feelings about whether problems like this are appropriate in the pre-algebra world. If the kids are ready to tackle this problem, then they are ready to start learning the basics of algebra. I agree absolutely. This one is ludicrously hard (I never did solve it using algebra that night! Which reminds me, I was planning to do it again in the daytime while I'm awake.) Ms. K didn't assign this problem, btw. I took it from a workbook. -- CatherineJohnson - 06 Oct 2006
Four weeks into the school year Ms. K has given her first test of material the kids haven't been taught and don't know. -- CatherineJohnson - 06 Oct 2006
Plus she may have given her 9th period class, which is Christopher's class, an extra-hard version of the test because thanks to the bomb threat they didn't get to take their test the day before and thus were, according to Ms. K, "talking about the test," which she regards as a form of cheating. I know this because another mom had what was apparently a fairly extensive exchange of emails with Ms. K last spring on the subject of talking-about-the-test being a form of cheating. So she gave the 9th period kids a test none of them could do. Several of them couldn't finish, apparently. -- CatherineJohnson - 06 Oct 2006
Then I semi-lost it in an email. -- CatherineJohnson - 06 Oct 2006
The school board wants another $9 million for fancy playing fields. -- CatherineJohnson - 06 Oct 2006
And Jimmy's special needs class hasn't gone to any vocational training at all in 4 weeks because THERE'S NO BUS. He's had no education for 4 weeks. There's no bus. Plus there's been an embezzlement scandal on Long Island so the state passed some godforsaken new anti-corruption law so none of the teachers can spend their own money on materials and get reimbursed. So now Andrew's class doesn't have a toaster oven & Jimmy's class doesn't have the promised digital camera (used to take pictures & set up picture schedules - essential) and one of the students broke the microwave so that's gone, too. (These classes really do need microwaves because the kids all have funky eating disorders AND THEY'RE SUPPOSED TO BE RECEIVING VOCATIONAL TRAINING IN HOW TO USE THINGS LIKE MICROWAVE OVENS.) Yesterday I went to Bed & Bath and bought a toaster oven. Then last night I found out about the microwave and the camera, so I'll almost certainly be buying those, too. Plus I spent $70 (I think it was) buying Singapore Math books to send to school so Andrew could have a math curriculum. -- CatherineJohnson - 06 Oct 2006
And the head of special ed, who is retired and receiving a NY state pension, is being paid $700 per diem. We didn't sign our IEP last spring, which means it's in dispute (this is a legal situation nothing like our saying we "haven't resolved" an issue with Ms. K). After we refused to sign the guy stopped responding to email and stopped taking our calls. So then we saw an attorney who gave us the appropriate legal language to use in our threatening letter. As soon as the letter went out our new CSE meeting happened. At the meeting Ed asked the special ed director why he had stopped responding to email. Special ed director said the email system was down all summer. (True, but not OK as an excuse in this case, which involves federally mandated entitlements.) Then Ed said "When I call your office I'm told you're in a meeting. Usually that's what people have their staff say when they don't want to talk to a person." Director of special ed said, "I didn't get the messages." -- CatherineJohnson - 06 Oct 2006
And that's just scratching the surface of the barrel as far as I'm concerned. -- CatherineJohnson - 06 Oct 2006
You would think they would LOVE bar models. I know. Basically they're opposed to anything that they didn't come up with themselves or that might be characterized as "rigorous," "precise," or "elegant." -- CatherineJohnson - 06 Oct 2006
Bar models by the way, and I've mentioned this before, aren't intended as a problem solving method - although Barry's observation that in fact they (often) work beautifully as a problem solving method are correct. The material I've read (this may be Parker & Baldridge??) says that they are intended as an aid to a child trying to discern which operations to use. I've used bar models both ways, and so has Christopher. -- CatherineJohnson - 06 Oct 2006
I don't think anyone would tell you to use bar models routinely for complicated problems - though in this case I was able to solve using a bar model & didn't solve using algebra. (not good, and this wouldn't be the goal of the Singapore Math curriculum. This is a sign that I have a ways to go in algebra.) -- CatherineJohnson - 06 Oct 2006
Will read the rest later! (THANKS SO MUCH EVERYONE - I set up a whole long complicated equation with fractions - "ugly") -- CatherineJohnson - 06 Oct 2006
Did I already say that when I had Christopher do it as a bar model we completely blew it?? When I did it alone as a bar model, thinking it through really carefully, I got it. But it was all wrong as a problem for Christopher, and I've never seen anything like it in PRIMARY MATHEMATICS (though I haven't read the entire curriculum). -- CatherineJohnson - 06 Oct 2006
-- TexasDesert - 06 Oct 2006
-- TexasDesert - 06 Oct 2006
-- TexasDesert - 06 Oct 2006
hey! Tex! someone's stepping on your post! -- CatherineJohnson - 06 Oct 2006
My 4th grader has WEEKLY MATH CHALLENGE. After doing Saxon over the summer, I dislike even more than ever these ”challenge” problems that allow the students to “discover” the answers. They usually are meaningless or frustrating for my daughter. But I should feel better because now I know that . . SOLVING ALGEBRA WITHOUT KNOWING ANY ALGEBRA WILL BE CALLED CRITICAL THINKING. OR, POSSIBLY, LOGICAL REASONING.* Nah, it doesn’t make me feel better. Here’s an example of a WEEKLY MATH CHALLENGE Apple Lake Bike Shop has a total of 32 bicycles and tricycles for rent. The shop owner checked all 74 wheels at the beginning of the season to make sure they ere in good condition. How many bicycles are there, and how many tricycles are there? The teacher confirmed it should be solved using “guess & check”. We try to spend our precious homework time learning other stuff that seems more practical. (Sorry about the multiple empty posts. Not sure what I did.) -- TexasDesert - 06 Oct 2006
YOU CRACK ME UP! -- CatherineJohnson - 06 Oct 2006
Catherine, was your equation something on the order of...
N = number of sheets in the pack = Christopher's sheets + Alyson's sheets + Mark's sheets + Keri's sheets 1 1 3 1 3 3 = -*N + -*-*N + -*-*-*N + 36 4 4 4 3 4 4 60 = --*N + 36 //solve for N 96? The "trick" is, there's only one unknown: the number of sheets in the package. Everything else is expressed as a number or as a fraction of the unknown. -- OldGrouch - 06 Oct 2006
TexasDesert?, how old is your child? You can, of course, solve the Apple Bicycle Shop problem algebraicly -- B+T=32, 2B + 3T = 74 and then solve for T by substituting 32-T=B for all B, but that's probably a lot harder then most elementary kids can handle. I agree, we've had all of these challenge and enrichment problems that are incredibly time consuming and impart very little "insight" to the child. When she reaches a solution thru guess and check, she has no idea how to tackle any similar problem in the future. She doesn't even know how to tell when a problem is similar. -- LynnGuelzow - 06 Oct 2006
Lynn, My daughter is 9 years old. She definitely does not know near enough algebra to be able to solve the bicycle problem. My 14-year old son commented that he thought that type of problem was not unlike some he had seen in his algebra class. From my conversation with her teacher, most if not all the kids need to guess & check. BTW, these weekly challenge problems are extra credit. The children receive gold coins if they complete the problem. So, I’m somewhat torn. On one hand, why bother with these. On the other hand, we want those gold coins! (The teacher has a system whereby the students can earn coins for various actions, and then turn in the coins for various rewards. The ultimate reward, which iirc costs 250 coins, is lunch with the teacher.) Tex -- TexasDesert - 06 Oct 2006
"The teacher confirmed it should be solved using “guess & check”." This is educational malpractice. There is such a thing in math as searching for a solution, but it applies to very specific algorithms, not random guess and check. And, these techniques would never be used for this kind of problem. If teachers would explain how a first guess would lead to a better second guess, and then, if needed, a third or more to close in on the solution, that would be fine, but I have NEVER seen this done. Random. The more I see and hear, the crankier I get. I keep wanting to find someting I've missed. I'm willing to learn and keep my mind open. Nothing has even come close. You would think that criticism by mathemeticians, engineers, and scientists would make them feel uncomfortable, but I don't see it. -- SteveH - 06 Oct 2006
there's a sort of systematic guess-and-check-like trick
that might very well be worth thinking about ...
something like "okay. what if all 32 cycles were bi?...
let's see: 32*2; 64 wheels. too small.
how about 31 bikes & a trike? gee: one more wheel.
30 bikes, 2 trikes ... one more wheel.
oh, and sure. every time i "trade" a 2-wheeler
for a 3-wheeler, i'll get one more wheel.
i want 74 wheels ... so i should trade in ten
of the bicycles from the inital "all bikes" setup:
22 bi; 10 tri" (now of course, check:
2*22+3*10. ok.) this seems like good mathematics to me.
there's probably even a name for it.
exercise: my life savings consists of
300 coins, all quarters and nickels.
there are $50 worth of coins altogether.
how many of each denomination are there? no fair using algebra. -- VlorbikDotCom - 06 Oct 2006
"So, I’m somewhat torn. On one hand, why bother with these. On the other hand, we want those gold coins!" It's like the Brain Teasers with my son (5th grade). They are extra credit and he wants to solve them. It's OK, as long as they don't take a lot of time. I usually like to help him figure out the interesting points or tricks. Since many think the solution of these problems indicate some higher level of intellegence (including some job application tests I've seen), then it's important for my son to get used to them. It reminds me of the books about Dick Feynman. When he was young, he studied all he could about these sorts of problems. He loved being able to answer any one of these trick questions. Of course, the reason he could solve these problems is that he had seen and worked on them before. Other people wouldn't know that. He built his reputation on these tricks. (The fact that he was incredibly smart didn't hurt either.) Most other people at his level could see through his tricks. One big one question I heard before (asked on many job application tests) is to explain why a manhole cover is round. Whether you know the answer or not probably depends more on whether you've heard this before rather than coming up with the answer on your own. However, the person hearing your answer doesn't know which it is. You could put on a big show and then pretend that a light bulb went off in your head before giving the answer. I've met a number of people who go out of their way to do these sorts of things. Many fall for it. I keep meaning to get some of these math (or Mensa) trick books and go through them with my son. I'd like to think that I don't have to stoop to that level. -- SteveH - 06 Oct 2006
"... what if all 32 cycles were bi?... let's see: 32*2; 64 wheels. too small. how about 31 bikes & a trike? gee: one more wheel. 30 bikes, 2 trikes ... one more wheel." I think this is what they are hoping the kids will construct. The problem is that they don't formalize the process and show how it works with other problems. They don't practice it to mastery. In most cases, the kids are too young for this anyways. Any explanation like this will just seem like a special trick. "this seems like good mathematics to me. there's probably even a name for it." There are many techniques for iterating (from an initial guess) to find the solution to a set of linear or nonlinear equations. This would be fine if the goal of the math class was to introduce the idea of smart searching, but I have never seen this done. That's content knowledge. -- SteveH - 06 Oct 2006
Old Grouch BUSTED!! -- CatherineJohnson - 06 Oct 2006
Hi Steve, I really agree with what you've written. If you like brain teasers, great, but I don't think it says much one way or the other about how smart or how knowledgeable you are. The best brain teasers are ones that are kind of unique, so they don't really help you with solving a different problem. I'm also partial to the ones where there's a simple solution if you approach the problem a certain way. (An example is adding all the numbers from 1 to 100. I have no idea whether or not I ever figured that out on my own--it's been too long since I first heard it.) I didn't know the story about Feynman. That's funny! -- SusanJ - 06 Oct 2006
I agree, we've had all of these challenge and enrichment problems that are incredibly time consuming and impart very little "insight" to the child. People so take these things for granted now that one time when I was telling a fellow whose daughters are in college about the difficult problems kids in other countries do, he countered by telling me about the difficult problems his kids had to do. (He wasn't remotely a fuzzy; he was just saying, 'My kids did difficult problems.) Then he said, "The whole family used to spend hours doing them together at the dining room table." I said, "But in Singapore the whole family doesn't sit around doing them. The kid does them." He was actually stunned. It had never occurred to him that child-doing-difficult-math-problem or rigorous-math-curriculum meant CHILD KNOWS HOW TO DO THE PROBLEM HIM/HERSELF BECAUSE HE HAS LEARNED HOW TO DO THE PROBLEM IN SCHOOL & PRACTICED IT IN HOMEWORK. -- CatherineJohnson - 06 Oct 2006
And let me STRESS that this guy was no apologist for Everyday Math and all the rest. Not remotely. In fact, I believe his kids went through school before Everyday Math really took over. He'd simply never seen anything different with all 3 of his kids. "The bad gets normal." -- CatherineJohnson - 06 Oct 2006
"An example is adding all the numbers from 1 to 100." This is the boy Gauss problem; the one (as legend goes) where he impressed his teacher. This is a staple of many fuzzy curricula. I can't remember when I heard or figured out the simple approach. I like to remember that I figured it out. I told my son to think of 100 stairs and divide them half way up. Then flip the upper stairs upside down and fit them neatly on the lower stairs. You have a rectangle 50 wide by 100 high or 5000 if you multiply it out. Actually, you have to flip steps 51-100 upside down and stack it on steps 0-49, with the 50 step in the middle left over, so the sum is really 5050. For even numbers, you multiply the top number by the middle number and then add in the middle number. For odd numbers, subtract the top number first to get an even number, do the calculation, and then add it in at the end. This is another good one to pretend to think about for a while before figuring it out to everyone's amazement. -- SteveH - 06 Oct 2006
"In fact, I believe his kids went through school before Everyday Math really took over." The old ways of traditional math were not necessarily so great. However, the newer methods are worse. I remember my first thought when I heard that our town was using MathLand. "They're going in the wrong direction!" If I had to generalize, I would say that "traditional" math content was better, but the teaching was poor and more hit-or-miss. Schools now seem to get more teachers up-to-speed on the program, but the curriculum is lousy. At least the older curricula allowed people like me, with no help at home, to progress to Calculus in high school. With fuzzy math, this is almost an impossibility. -- SteveH - 06 Oct 2006
Hmmm, I hate to say it, and I don't think it's great for every kid, but my two kids (12 and 15) have a far better number sense and math sense than I ever did at their ages. And, I did well in math -- although I was usually surprised by it, too -- that is, I would do the test, not be at all sure I'd done it right...and then when I'd look back at it a few weeks later, it would all make sense then. My sons, however (both products of Everyday Math -- and before they added back in more drilling in at least the older ones case)seem to see problems in a way that I didn't at their age (really, I didn't begin to see math problems until my 20's, long after I'd stopped doing any school math). FWIW. A lot of the math competitions that Wicklegren (sp?) talks about in his book have problems like the bike and trike wheel problem. As far as I remember the sort of systematic guess and check described above is explicitly taught in Everyday Math; I'm remembering pages in the workbook that show partially started charts that the kids were to finish and then similar problems for them to solve on their own. And to repeat -- it definitely didn't work for every kid in their classes. In the last few years there was a big emphasis in about 4th grade on picking ONE method of doing [adding, subtracting, etc]problems for the kids that were not doing well, something that the kids that were doing okay had already done, even if they didn't quite think of it that way. -- JenL - 07 Oct 2006
" ...but my two kids (12 and 15) have a far better number sense and math sense than I ever did at their ages." It's hard to judge what you are comparing here. The "good 'ol days" had their own problems. And actually, I don't even know what "number sense" is. This idea of "seeing" is just so vague. Either a student can do the problem or not. Either a student knows what he/she is doing or not. There is nothing I have seen in EM so far that indicates it would provide a better understanding of math than Saxon or Singapore. The goal in lower school math is to lead students to a proper course in algebra by 8th grade; one that prepares students (content and skill-wise) for the AP Calculus track in high school. This goal can be clearly defined. Vague goals like number sense, seeing, and higher-order thinking are usually just a cover for a slower pace, less practice, and lower expectations. Our public schools now use EM, followed by CMP, both of which emphasize number sense and conceptual understanding. All you have to do is look at the content and skill gap between their 8th grade algebra lite course (CMP plus a few algebra topics) and the high school honors math track to know that something is wrong. Also, compare EM side-by-side with Singapore or Saxon and you can clearly see the difference. My son (fifth grade) has EM in school and I have him do the Singapore Math workbook at home. There is no comparison. The pace of coverage is faster, there is much more practice, and problems are more complex. " ... before they added back in more drilling..." Gee, I would hate to see the earlier version, since one of the major problems with EM now is its huge lack of practice. Even my son's teacher is surprised. "As far as I remember the sort of systematic guess and check described above is explicitly taught in Everyday Math; I'm remembering pages in the workbook that show partially started charts that the kids were to finish and then similar problems for them to solve on their own." Oh boy. I can't wait; charts for solving (wasting time) equations. I'm afraid that I cannot see any value in even this improvement to random guess and check. This is really the crux of the controversy - number sense, "seeing", conceptual understanding, and higher-order thinking; all vague, undefined terms. Just show me the problems that the students can do and I will judge their mathematical abilities. Just look at the math workbooks side-by-side and compare. EM comes up short even for these vague criteria. -- SteveH - 07 Oct 2006
This is really the crux of the controversy - number sense, "seeing", conceptual understanding, and higher-order thinking; all vague, undefined terms. Just show me the problems that the students can do and I will judge their mathematical abilities. Just look at the math workbooks side-by-side and compare. EM comes up short even for these vague criteria. -- SteveH? - 07 Oct 2006 Or better yet, have your kids take the Singapore Math placement test. http://www.singaporemath.com/Placement_Test_s/86.htm -- NicksMama - 07 Oct 2006
<<The goal in lower school math is to lead students to a proper course in algebra by 8th grade; one that prepares students (content and skill-wise) for the AP Calculus track in high school.>> Well, the older one is doing 11th grade (IB) in his 10th grade year and will do calc next year. The younger one will take algebra in 8th and calc in 12th. I guess perhaps I was being oblique and not wanting to say, they are good at math. They get the concepts and they know when to use them. They weren't tutored at home. They score at the top levels of standardized tests. Maybe this isn't due to Everyday Math, but it certainly didn't hurt them, either. -- JenL - 07 Oct 2006
<<All you have to do is look at the content and skill gap between their 8th grade algebra lite course (CMP plus a few algebra topics) and the high school honors math track to know that something is wrong.>> <<" ... before they added back in more drilling..." Gee, I would hate to see the earlier version, since one of the major problems with EM now is its huge lack of practice. Even my son's teacher is surprised. >> These both sound like problems in your school/district? My understanding is that EM added back in lots more drilling years ago, however it required more materials? I am definitely fuzzy on this, it was explained by a 2nd grade teacher who was copying pages of the one workbook they had, rather than having copies of the workbook for everyone. So it was/is available, but I'm not sure how many people have taken them up on that version. -- JenL - 07 Oct 2006
There seems to be some disparaging of bar models above, which I don't understand. Singapore uses them to illustrate what is going on in the problem: What information do we know, and what don't we know? So something simple like "I have $5 now and had $3 earlier, how much did I make" can be illustrated easily with a bar model. Since Singapore uses bar models or a variant in explaining addition and subtraction, this problem is just an extension of what they've learned, and kids easily see that the solution to the problem is subtraction. They don't have to use bar models to solve each and every problem; SM considers it a "heuristic" to help them understand how to solve such problems. As one progresses through the SM books, the problems become more complicated so you eventually get to the two variable problems, that the fuzzies like to solve using guess and check. The bar model approach allows students to come up with a systematic method. The next step is to translate that approach into algebra. And it does lend itself to that. In other words, Singapore does what another commenter said should be the goal of lower math courses: to prepare students for algebra. Singapore can get them ready for algebra by 7th grade. Bar modeling does have its limitations, very true. But it isn't intended as a device to solve every problem. One can explain how to multiply fractions by using shaded areas of gridded rectangles and using problems like 2/5 x 3/4. Once the student understands you multiply numerators and denominators, then they can do problems like 3/17 x 51/73 which would be quite cumbersome to do using a rectangle diagram. It allows for generalizations to occur in other words. SM uses one type of problem quite often: "Maria spends 3/5 of her allowance on a doll. She has spent $9. How much allowance did she receive?" This problem is presented before students have learned division by fractions, and it's easily solved using a bar diagram. When they get to division by fractions, the problems are stated in quotative form rather than the partitive one above: How many 3/5 foot segments are contained in a board 9 feet long? Illustrating this by bar modeling is confusing and SM doesn't use bar models for this type. They rely on student's knowledge of division by fractions. Using division by fractions to solve both types of problems then starts to get into algebra. Students learn how to set up an equation. Then, as Steve says, the equation takes over, and the student has less to keep track of. As Wayne Bishop has pointed out about SM, by the time students get to the sixth grade, the bar modeling has educated them enough that they're practically begging to use algebra. But we're talking about using SM from Kindergarten on--the books are developed consistently, using the same basic methods from day 1 and embellishing upon same all the way through the program. -- BarryGarelick - 07 Oct 2006
Steve H said: It's like the Brain Teasers with my son (5th grade). They are extra credit and he wants to solve them. It's OK, as long as they don't take a lot of time. I usually like to help him figure out the interesting points or tricks. Since many think the solution of these problems indicate some higher level of intellegence (including some job application tests I've seen), then it's important for my son to get used to them. Hung-Hsi Wu comments on "brain teasers" in his exhaustive and illuminating review of the IMP program, located on his web site at: http://math.berkeley.edu/~wu/IMP2.pdf He says: "A judicious use of mathematical puzzles has its place in a mathematics curriculum as a tool for training mental agility. However, in view of the fact that the IMP curriculum gives a consistent impression of teaching mathematics that is truly basic and relevant, the not-infrequent appearances of such puzzles in POW’s [these are a kind of in-class worksheet...BG] and homework problems without any preamble can only reinforce the popular (and unfortunate) misconception that mathematics is nothing but a bag of cute tricks. It would be far better if each puzzle is prefaced by a disclaimer to the effect that “This is a test of your ingenuity”. Incidentally, I would hesitate to recommend making puzzles part of an examination (e.g., Sally’s party in the IMP Final). An examination should test only whether a student has learned well, not whether she is inspired at the particular moment of exam-taking. To most of us, solving a puzzle does require inspiration." -- BarryGarelick - 07 Oct 2006
"Maybe this isn't due to Everyday Math, but it certainly didn't hurt them, either." I'm honestly glad for you and your kids, but this is not a basis for judging math curricula, especially ones with variable degrees of supplementation. "My understanding is that EM added back in lots more drilling years ago, however it required more materials?" The only reasonable supplementation to EM I have seen did not come from EM. Most smart schools supplement EM on a carefully-defined basis. But, as my wife says, "Why use a math curriculum that you have to supplement?" The reason is that many schools will never adopt a program like Singapore Math or Saxon. -- SteveH - 08 Oct 2006
"The bar model approach allows students to come up with a systematic method. The next step is to translate that approach into algebra. And it does lend itself to that." This is important, but I guess my feeling is that the sooner, the better. I've started to have my son do bar modeling (fifth grade) and he loved the few "nice" problems I gave him. The solutions jumped right out at him. We'll see what happens when I give him ones that are not quite so easy. I have also started to teach him algebra - very simple equations. I'm still trying to have him follow Singapore Math at home (after everything else is done). I'll let you know how it goes. -- SteveH - 08 Oct 2006
Steve, Steve: Yes, the bar models have limitations with non-nice problems, but as I indicated are used to help students see what's going on with fractions, etc. For example, the quotative division type problem cannot easily be represented by a bar model, so the student has to know division by fractions at that point, but if they know that, you can now use algebraic equations to solve both types of division problems. So using simple algebra equations to extend the story problems to more complicated ones is a good tactic. -- BarryGarelick - 08 Oct 2006
OK, I still haven't read the thread carefully (AAAUUGGHH!) I love the Hu paper (read it awhile back). Old Grouch asked: Exactly what are they trying to teach by assigning these problems now? Your question is emblematic of the problem parents have with progressive ed - we all think they're trying to teach "something" and we just don't see what it is. Progressive educators actively oppose the teaching of "something." They reject content; they call facts "mere facts" or "dead facts" etc. Steve's teacher, I think, told him his son knew a lot of "superficial facts." They believe, and this really is a belief system, in teaching process. They believe in "critical thinking skills," and that's what they're going after here. They believe that a child who just uses "rote algebra" to solve a problem has not, in fact, solved the problem, because he hasn't used logical, reasoning, and critical thinking. They believe only in critical thinking and logical reasoning. Period. Which is a huge problem for us seeing as how it turns out there's no such thing as critical thinking and logical reasoning divorced from the content you are reasoning critically or logically about. Experts can't transfer even the most brilliant "critical thinking" from their own fields to fairly closely related fields - a reality that is highly counterintuitive. I've seen it often, though, because I'm married to Ed. The most distinguished historian can write absolute pap about contemporary politics. Back before the election Bob Dallek, whom we knew at UCLA, was writing lots of anti-Bush op eds. This guy is a very well-respected U.S. historian and he was writing things like "George Bush will not be elected because the American people do not reward bad presidents with re-election and George Bush is a bad president, therefore the American people will not re-elect him." It was ludicrous! I know we don't "do" politics at ktm, but I'll exempt myself for a second and say that Ed is second to none when it comes to Bush-loathing (ok, he's second to quite a few people you can find on the web)....and even he would read one of Dallek's op-eds and just shake his head. No historian would ever construct an argument like that when writing history. The cognitive scientists are right. "Logical reasoning" and "critical thinking" aren't abstract intellectual tools that can be applied here, there, and everywhre. They are content bound. -- CatherineJohnson - 08 Oct 2006
Actually, it's worth reading - I just looked at it again. Dallek is making a prediction based on his expert knowledge of U.S. history. But of course historians specifically do not predict the future. They have a saying that historians can't predict the past. Predicting the future is absolutely outside the bounds of expertise. So here is Dallek using critical thinking and logical reasoning to do something he's not expert in & does not have the domain knowledge (in polling, statistics & probability, contemporary political practice, etc.) to do. And it doesn't work. -- CatherineJohnson - 08 Oct 2006
"They reject content; they call facts "mere facts" or "dead facts" etc. Steve's teacher, I think, told him his son knew a lot of "superficial facts." I recall an incident that occured when my son was in 4th grade. He’s now in 9th grade. My son reads voraciously, with special interest in history and geography. He knows a lot of stuff and can usually hold his own against adults in debates. During one conference with his teacher while she was droning on about how he was doing fine, she remarked how he possessed a lot of “general knowledge”. She said it in a somewhat patronizing tone. It was clear she considered it curious and not very valuable to have too much “general knowledge”. We still remember this and joke about our son’s abundant “general knowledge”. -- TexasDesert - 08 Oct 2006
My son reads voraciously, with special interest in history and geography. That right there is an 800 on SAT Verbal. Seriously. -- CatherineJohnson - 08 Oct 2006
"We still remember this and joke about our son’s abundant “general knowledge”. And my wife and I still remember and laugh (now) about our son's first grade teacher. When I told her that he loved geography and that he could find any country on a globe, she said: "Yes, he has a lot of superficial knowledge." Ouch! We were a little naive back then. Then, later that year, my son had to show the student teacher where Kuwait was on the map when they were having a thematic unit on sands from around the world. -- SteveH - 09 Oct 2006
And my wife and I still remember and laugh (now) about our son's first grade teacher. When I told her that he loved geography and that he could find any country on a globe, she said: "Yes, he has a lot of superficial knowledge." Ouch! We were a little naive back then. I have NEVER gotten over this. Ever. -- CatherineJohnson - 09 Oct 2006
my two kids (12 and 15) have a far better number sense and math sense than I ever did at their ages Still haven't read the whole thread (!) but I wanted to chime in on this. I definitely have seen that in the two kids I know who are using Trailblazers. Better number & math sense, definitely. And I do attribute it to Trailblazers, not just to them or their parents. -- CatherineJohnson - 09 Oct 2006
oh for heavens sake here's one of the problems on Christopher's endless impossible test (grade: 74) Karen was given money for her birthday. She spent $15 on two new CD's, and then spent 2/3 of the remaining money on a new radio. After she deposited 3/5 of what was left into the bank, Karen had $30 remaining. How much money did she get for her birthday? Show all work. (6 pts) -- CatherineJohnson - 12 Oct 2006
I realized tonight I was having another case of inflexible knowledge. The reason I couldn't do these problems was that I wasn't extending the commutative property to a more complex problem. I was having the same problem I had back when I didn't see that you could find the total price of an item (P) + 5% sales tax by taking 1.05 x P instead of .05P + P The problem wasn't that I was using lots of different variables. I wasn't seeing the simple way to write the equation. boy after all this time yikes -- CatherineJohnson - 12 Oct 2006
Another child dropped out of Ms. K's class on Tuesday (one child dropped over the summer). The one fantastic piece of news is that the new principal is....I can't quite think of how to say good things about the new principal without carrying on being negative about his predecessor, so I'll just say that I feel a basic compatibility with him. More importantly, there is zero "your child is the only one having a problem" talk. Zero. It's gone. The new principal's attitude seems to be that if our child is having a problem, then that's a problem. There is zero "you're the only parents complaining." The other positive development is that two other parents have written incredibly emails to the principal laying out the issues. One of those parents is an attorney; the other is a lifelong math teacher. The combination of the two makes it impossible not to see that Ms. K needs serious mentoring and guidance. Both emails, too, were quite beautiful. Both were from fathers and the expression of father-love through their precisely chosen words and images was affecting. Anyway, the new principal is on the case. He may be able to improve this situation since no one seems to feel that the problem lies in her knowledge of the mathematics she's teaching. The problem is in the teaching, the tests, and the lack of integrated formative assessment. (sling the lingo) Ed and I pushed the idea of "collect and correct" (a friend came up with that slogan). "Collect and correct" as in collect and correct the homework. She needs to find out, probably every day to start, what the kids actually have absorbed. A quick quiz on just one homework problem would be better than what has been the case until now. Christopher thinks her expanations are better this year. He didn't think her explanations were bad last year, btw. That was never the complaint. The fact that he thinks the explanations are better this year is good. Also, he has said this about brand new material which is very good. It strikes me as possible that a principal could work with her on checking homework and on testing the material that has actually been covered on the level at which it was covered. The math teacher dad was amazing on the subject of the tests. I wish I'd written down what he said. "The tests are so long and the problems have so many multiple, verbal steps beyond anything they've ever done that they can't even think, and they can't finish" - something like that. Christopher managed a 74 on this test, so I've told him we have to go back to reteaching the content every night. Period. The math teacher dad reteaches every single lesson. He has a huge advantage over me, because he knows the state standards inside and out. I need to go pull whatever the New York state website has on the standards for this year and commit them to heart. -- CatherineJohnson - 12 Oct 2006
One more thing. I haven't been able to imagine how the fact that the new principal comes from an "urban" school might affect his work here. I've seen one positive thing, which is that he's used to bomb scares. He sees bomb scares as something middle school kids do. That is a tremendous relief. Our district has developed a tone that, for my money, verges at times on moral panic. We're getting emails from the School Board president warning us that if we don't have family dinner our kids will take drugs — and that's just the beginning of the woe that is going to rain down on our heads. Every bomb threat is treated with utmost seriousness and severity of tone; the feeling one gets is that the offending child is going to be arrested, tried, and hung. Last year the parent messages that went out after bomb threats were so laden with condemnation of the as yet unknown but already despised perpetrator that my only feeling was one of deep sympathy for whatever family was going to get clobbered once its child was Found Out — that and a deep sense of relief that it wasn't my kid. Sympathy, relief, and a secret half-hope they didn't find the kid and didn't destroy a whole family's life. The new principal, coming from a school where apparently the kids could get going on waves of bomb threats, is matter of fact. Much better. -- CatherineJohnson - 12 Oct 2006
I definitely have seen that in the two kids I know who are using Trailblazers. I've seen the exact opposite. One kid seems to have no number sense at all after two years of it. -- SusanS - 12 Oct 2006
I'm with Susan on this. I can't understand what is meant by number sense. All of my parent newsletters tell me that my child is getting lots of number sense and we should use our family time together to further increase it by talking about shopping. Sometimes I wonder if the goal if EM is to create a solid class of checkout clerks, or maybe trophy wives (that can shop), or gamblers. We just spent the last several weeks working on data and charts and graphs, with dice rolling and marble picking, etc. Any day now I expect a problem of the week asking kids to calculate odds at the track. -- LynnGuelzow - 12 Oct 2006
i've never been able to figure out what
"number sense" is supposed to refer to either.
i'm pretty sure i haven't got it, whatever it is.
maybe they should pull my doctorate (in math). -- VlorbikDotCom - 12 Oct 2006
I've spent way too much time at work looking at that test problem about Karen's birthday money. Using algebra was getting way too messy and I couldn't imagine a 7th grader doing it that way. So I tried a bar graph for the first time. You can now count me among the converted. It was so much easier to figure out, but I don't know the solution Ms. Kahl was expecting. If the test was more of the same, I would think kids would be getting extremely frustrated. -- KathyIggy - 12 Oct 2006
I'm with Susan and Lynn on this too(if you couldn't guess). I don't know what number sense is. All I can think of is that it's when you know that 22 times 9 is about 200 rather than 2000. But, does number sense mean that you really don't have to be able to do 22 times 9 in your head? Our school does Brain Teasers as a supplement, but wouldn't better number sense be achieved by spending the time on a book like Arithmetricks? For example, one chapter talks about multiplying by 5 as dividing by 2 and adding a zero at the end. It's easier to divide by 2 than multiply by 5. This would be great - more than one way to solve a problem. They should love it. 22 times 9 would be 22 times 10 - 22, or as I like to do it; 9 times 20 + 9 times 2. Maybe number sense means talking about place value over and over and over and then not using it for anything. Then again, my fifth grade EM son has gotten this spiraled into his head time after time, but he looks at me with blank eyes when I try to talk about how the technique is applied to binary numbers. However, he understood the concept a lot better when I had him PRACTICE converting numbers to and from binary. The fuzzies cannot admit to linkage between practice and understanding because practice and hard work are a filter. Conceptual understanding, number sense, and a calculator are all you need. The fuzzies can't even define what number sense is and they can't really say that what they do promotes number sense. My son has to do multiplication 3 different ways, but I don't think even the teacher can explain mathematically or algebraically how they are equal. They seem more concerned about doing things multiple ways than number sense or conceptual understanding. I told my son that in multiplication, each digit of the first number has to be multiplied by each digit of the second number. I told him that all multiplication techniques have to do this. I told him that it has everything to do with place value, but not to worry about it until algebra. -- SteveH - 12 Oct 2006
Number sense is what they call the stuff that's getting taught in math class that's not actual math. Number sense means they are teaching about math, rather than actual math. -- KDeRosa - 12 Oct 2006
Maybe number sense means talking about place value over and over and over and then not using it for anything. Lol! Only those in the know understand how funny a comment that is. Or is is just a really elevated form of rounding and estimating that you can only achieve by rounding and estimating a lot. Maybe number sense is much like horse sense. -- SusanS - 12 Oct 2006
OMG! I love this thread about number sense! I have never really understood this. You all make me feel so much better. -- TexasDesert - 12 Oct 2006
Or is is just a really elevated form of rounding and estimating that you can only achieve by rounding and estimating a lot. That MUST be it, maybe tied in somehow with the fixation on data collection and graphs. It's having the "sense" to do something with numbers which serve no purpose. Maybe I can add to my daughter's diagnoses. Instead of "sensory integration dysfunction," she probably has NUMBER sensory integration dysfunction too as she hates to round and estimate, asking "why can't I just find the real answer?" -- KathyIggy - 12 Oct 2006
When I hear the phrase "number sense", I think of two things: 1. Math contests for middle and high school students, as in the Wikipedia definition of "number sense". My college roomie and roomie's sister were both big into math contests in high school. Their father led the state teams to national contests. In this context, "number sense" is a 10-minute test where the students do math in their heads -- no scratch paper or calculators allowed. 2. John Paulos's fine book Innumeracy. In this context, "number sense" is the ability to tell whether the answer you got is even close. That is, you should be able to tell whether you're looking for a number in the negative hundreds (-100), or the positive thousandths (+0.001), etc. It is number sense that allows you to be skeptical when you read that 40 million babies were born in the U.S. last year. Does that make sense? The population of the U.S. is almost 300 million. Does it make sense that one out of every 8 U.S. residents had a baby last year? No, that sounds too high. The real number is about 4 million. Number sense alone won't help you get the answer, but it can help you determine whether the answer you got has a chance of being right. Number sense is absolutely required for things like compounding drugs -- was that milligrams or micrograms I was supposed to give you? The problem about Karen's birthday money is most easily solved backward, and since the numbers in this problem are very friendly, you (= an adult) can probably do it in your head. A seventh grader might be expected to use scratch paper, unless he's a TAG/GATE/G&T student on a math team practicing for "number sense" contests. BTW, the number I got, and checked by working forward again, seems to be a ridiculous amount of money to give someone for a birthday present. Then again, I'm known to be cheap. ;-) -- GoogleMaster - 12 Oct 2006
This discussion has moved me to actually pull my copy of the EM Teacher's Reference Manual off of the shelf. Wouldn't you know it, but "number sense" is in the index. EM has a whole section devoted to explaining what number sense means. Are you ready? It is everything and nothing all at the same time. It would be hard to craft a definition that exceeds this in vagueness, yet all encompassing breadth. I hope that is helpful. Here it is with more detail: "It is perhaps the single greatest goal of Everyday Mathematics that students completing the program acquire number sense. People with number sense: "
Every one of those elements of "number sense" develops only after a student has ingrained and automatic calculation skills. You know, the kind you get by doing a few thousand rote arithmetic problems. I wonder what kids should start with? -- DougSundseth - 13 Oct 2006
Several years ago, I had a "debate" with a curriculum person about what I thought were the perils of fuzzy math. Eventually, the conversation turned to the topic of number sense. I told this person the story of a college professor friend of mine who was appalled that the students in her class could not easily convert 700,000/7,000,000 to 1/10 without a calculator. My argument was that students who haven't learned and practiced that skill don't have number sense. Her response was to the effect that if students have "number sense," they would just know how to do that skill. I was arguing that the skill was best learned as a result of direct instruction and practice; she was arguing that the theory of how to do it was sufficient. It was at that point that I realized that there was no point in continuing the discussion with her because the gap between us was a mile wide. I also remember feeling incredibly disheartened, as she had a tremendous amount of influence with respect to math curriculum. -- KarenA - 13 Oct 2006
My argument was that students who haven't learned and practiced that skill don't have number sense. Her response was to the effect that if students have "number sense," they would just know how to do that skill. The true test is to find the mythical person who has this magical number sense who doesn't also have calculation automaticity gained either through lots of practice or sheer genius. Such a person, I assure you, does not exist. -- KDeRosa - 13 Oct 2006
This week my DD in 4th grade has been working on: -- rounding numbers up to millions -- word problems that require estimating social studies facts (e.g., the population of Dallas is about how much more than the population of Indianapolis?) -- comparing addition/subtraction facts derived by using pencil and paper or mental math with results using a calculator She’s been having difficulties with some of these exercises. Just last month her teacher told me that students who are still using their fingers to do single digit addition should not be a concern because it’s more important that they understand the concept of addition. It seems to me that they are forcing “number sense” of pretty big numbers before knowledge (quick recall, memorization) of some very basic number facts. Oh, I get it. Number sense before number facts. This is painful. -- TexasDesert - 13 Oct 2006
"BTW, the number I got, and checked by working forward again, seems to be a ridiculous amount of money to give someone for a birthday present. Then again, I'm known to be cheap. ;-) " My thought was that the answer was: "Too much money". -- SteveH - 13 Oct 2006
what do you mean you can't work that backwards? Here: Keri bought a new pack of graph paper for math class. When she got to class and opened it up, three people asked her for some. She gave one quarter of the pack to Christopher. Alyson got one fourth of what was left. Then Mark took one third of the remainder. That left Keri with 36 sheets. How many sheets were in the pack of graph paper? 1) Keri ended up with 36 sheets. 2) She had 36 sheets after giving 1/3 of the subtotal to Mark, the last recipient. Therefore 36= (subtotal * 2/3) Therefore subtotal = 36 * 3/2 = 54 Therefore she had 54 sheets BEFORE she gave to Mark, but AFTER she gave to everyone else. 3) Alison, the second recipient, got one fourth of what was left. After Alison took her share, we know we need to end up with 54 sheets as above. 4) Same math as above, but with 3/4 instead of 2/3: 54=(new subtotal) * 3/4 (new subtotal) = 54 * 4/3 = 72. Therefore she had 72 sheets BEFORE she gave to alison and Mark, but AFTER she gave to everyone else. 5) Christopher got one fourth of the whole pack. Same as above for Alison: whole pack = 72*4/3 = 96. Therefore the whole pack had 96 sheets. -- KtmGuest - 13 Oct 2006
"Oh, I get it. Number sense before number facts. This is painful." This was precisely my discussion with the curricula (lum?) "expert." My argument was that number sense is derived from (no pun intended) number facts and her argument was that number sense is pulled out of one's rear. Okay, that wasn't her argument, but in my opinion, it might as well have been. That conversation was painful. -- KarenA - 13 Oct 2006
Well, I just lost a big post because my second post overwrote the first? -- SteveH - 13 Oct 2006
"The true test is to find the mythical person who has this magical number sense who doesn't also have calculation automaticity gained either through lots of practice or sheer genius. Such a person, I assure you, does not exist." My thoughts exactly. The problem is that hard work is a filter and they don't like filters. Therefore, number sense must not require hard work. QED (Quasi-Ellipso-Delirium) -- SteveH - 13 Oct 2006
"EM introduces students not only to the tradional mathematics parents expect but also to a richer mathematics curriculum that older family members may not have experienced." I assume that some in the education field know exactly what they are doing by saying this and that the rest buy into it witout using any "critical thinking". I remember one open house for parents where the head of curriculum talked to us about how wonderful MathLand was (a math curriculum so bad that it was dumped by the publisher and you can't find any mention of it on their web site) and that (basically), the people at www.mathematicallycorrect.com don't know what they are talking about. I looked at her and couldn't quite figure out if she really believed this or not. She must, at the very least, realize that it was a matter of opinion and assumptions. She could have just said that people have different opinions, but this is what we are doing. She didn't. She must have known that she was arguing against professional mathematicians, engineers, and scientists. She must have known that seated (in little kids' chairs) in front of her were people with a whole lot more subject knowledge and experience than her. However, she proceeded to lecture us using her first grade teacher voice with seemingly no embarrassment. It's not just that she had her own opinion about education, it's that she didn't see it as opinion. I see it as academic turf. This one-sided view is all they know. If you take that away, they have nothing. As one parent told me once: "They are the experts." I said: "Experts in what? Their own opinion?" -- SteveH - 13 Oct 2006
I continually find in my discussion with the administrative "experts" in education that there is this blindness to admitting that there is more than one side to these types of debates. It's as if they can deny away all contrary opinions. I've been told far too often that "all the research shows such and such." When I ask to see the research as I'm interested in reading it, they usually can't even list any research. I'm likely to be sent to the website of the publisher to find out what research they are referring to. The idea that a parent would pull and read research they cite (but apparently never read) is foreign. I am amazed at the level of arrogance. She must have known that she was arguing against professional mathematicians, engineers, and scientists. She must have known that seated (in little kids' chairs) in front of her were people with a whole lot more subject knowledge and experience than her. Exactly. They know they have doctors and engineers and physicists in front of them. These people are in volunteering during the year. We leave near UT's main headquarters. There are LOTS of scientists and engineers in this town and the administrators know it. But the arrogance persists that these highly accomplished mathematical professionals didn't get a "rich mathematical curriculum" back when they were in school. -- LynnGuelzow - 13 Oct 2006
Steve, was this the post you lost? "Or is it just a really elevated form of rounding and estimating that you can only achieve by rounding and estimating a lot." They don't like "drill and kill" and they don't like single correct answers, so they work on estimation and call it "Number Sense". Less is more. "The true test is to find the mythical person who has this magical number sense who doesn't also have calculation automaticity gained either through lots of practice or sheer genius. Such a person, I assure you, does not exist." My thought exactly. Number sense is a nice way of saying lower expectations. Hard work is a filter and they don't like filters, so obviously, it isn't necessary. QED. (Quasi-Ellipso-Delirium) "That MUST be it, maybe tied in somehow with the fixation on data collection and graphs. It's having the "sense" to do something with numbers which serve no purpose." Last night for (fifth grade) EM, my son has to collect data. He took this calibrated strip of paper that one person had to drop (vertically) and another had to grab between their fingertips. The paper had a nonlinear calibration that was related to how long it took for the person to grab the paper. Cute. This test had to be done 10 times for the left and then the right hands for two people, one of whom is older than 25. Forty data points to be written down on the page. The question was "What is the median time for each hand and each person. Well, I can't tell you how many times they went over "median" last year (they apparently don't know enough yet to calculate AVERAGES!). Who says they don't do drill and kill! You would think that by fifth grade they would have spiraled up to calculating average. They could even use a calculator! Do they talk about experimental control and accuracy? No. Do they talk about outliers? No. Do they talk about faked results? No. (A few times, my son and I missed grabbing the paper completely. We didn't record those values.) They are teaching process, not understanding. Eighty percent play; twenty percent learning. Just because you talk about conceptual understanding doesn't mean that it's happening. His math teacher knows that they need more, so they get brain teasers. Do you know what yesterday's teaser was? A farmer had 20 cows and ducks in the barn. They had a total of 64 legs. How many cows and ducks did he have? If a child can answer this without any background knowledge, what does that tell the teacher - that this problem helped CREATE that ability or simply allowed the student to practice? If a student gets nowhere with the problem, what does that tell the teacher - that they need more practice doing problems they have never seen before and don't have the basic knowledge to figure out? That they don't have a "math brain"? What if the student has seen this kind of problem before and was taught how to do it, but the teacher doesn't know that? What does that tell the teacher? One could argue that struggling with these problems is better than not struggling with them, but it is an extremely inefficient (and neither necessary or sufficient) way to learn a subject. -- SteveH? - 13 Oct 2006 -- DougSundseth - 13 Oct 2006
Wow! Where did it go? Usually, I just cut and paste my comments to the clipboard in case the post gets lost. But once I check to see that it is posted, I don't necessarily save the comments. I saw that it posted, made another, separate comment, posted that, and then the first one was gone and my clipboard was overwritten. -- SteveH - 13 Oct 2006
Someone had the page open for editing when you were posting. You posted your comment and then that person posted the edited comment. The edited comments thread overwrites the previous comments thread. Since your comment wasn't in the edit page (it was opened before your comment was posted), your comment disappeared. Since the page has a full edit history, I found the deleted comment by clicking the "Revision history" link at the bottom of the page. (As you might have guessed, I've had this happen before. Without a major change to the software, I don't know of a way to fix the problem, but fortunatly it's somewhat rare.) -- DougSundseth - 13 Oct 2006