Entries from ElementaryMath

Multiple Approaches to a Computational Procedure: Flexibility Rooted in Conceptual Understanding

Although proofs and explanations should be rigorous, mathematics is not rigid.... Dowker (1992)asked 44 professional mathematicians to estimate mentally the results of products and quotients of 10 multiplication and division problems involving whole numbers and decimals. The most striking result of her investigation "was the number and variety of specific estimation strategies used by the mathematicians." "The mathematicians tended to use strategies involving the understanding of arithmetical properties and relationships" and "rarely the strategy of 'Proceeding algorithmically.'"

"To solve a problem in multiple ways" is also an attitude of Chinese teachers. For all four topics, they discussed alternative as well as standard approaches. For the topic of subtraction, they described at least three ways of regrouping, including the regrouping of subtrahends. [they also talked about which approach worked best in which situation] For the topic of multidigit multiplication, they mentioned at least two explanations of the algorithm. One teacher showed six ways of lining up the partial products. For the division with fractions topic the Chinese teachers demonstrated at least four ways to prove the standard algorithm and three alternative methods of computation.

For all the arithmetic topics, the Chinese teachers indicated that although a standard algorithm may be used in all cases, it may not be the best method for every case. Applying an algorithm flexibly allows one to get the best solution for a given case. For example, the Chinese teachers pointed out that there are several ways to compute 1 3/4 dividedby 1/2. Using decimals, the distributive law, or other mathematical ideas, all the alternatives were faster and easier than the standard algorithm. Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations--the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics.

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ExtendedResponse 08 Nov 2005 - 22:52 CatherineJohnson


My sister-in-law, a fantastic teacher in central Illinois, says the Big New thing in math is extended response. She's going to fill me in when she finds out what it is.

In the meantime, I found this page of released extended response items on the ISAT.

my extended response to extended response

OK, my initial reaction to extended response is: I'm against it.

Actually, make that mixed. My initial response is mixed.

Here's one of 2 released 2004 extended response gr5 items:

A company makes a wall calendar each year. The company sells ad space
around the calendar to local businesses. The cost of ad space is based on
the number of square units each ad contains. The company charges $40.00
for Ad Space D. Using this information:

Draw an Ad Space that costs exactly $60 in the gridded space on page 10 of
the answer document.


And here's the illustration:




I like this problem, although wiser heads here at ktm may give me reasons why I shouldn't, in which case I'll revise my opinion.

I like it because it's visual & spatial as well as 'numerical' (if that's the right word), and because I've found Christopher to be very challenged by any problem that asks him to combine numerical thinking or problem-solving with spatial 'thinking' or problem solving. And of course I love the Singapore bar models, and this problem reminds me of them.

I also like it because it has 2 steps: you have to figure out how much each square costs & then you have to figure out how many squares $60 would buy.

I like the open-endedness of this particular problem, too. A child could simply count the number of squares in Ad Space D (40) and then divide 40 dollars by 40 squares to get $1/square. Or he or she could notice that Ad Space D is a standard multiplication array, and multiply 4 by 10 to get 40. I'm sure a lot of kids would start out counting & then notice, mid-stream, that they could have arrived at their answer more efficiently by multiplying instead. Which is good. A little Math Object Lesson buried inside a story problem.

I like that!

Last but not least, I kind of like the fact that each square turns out to cost exactly one dollar. I don't know why. It reminds me of a genre of problems in Russian Math, in which you go through all kinds of elaborate, painstaking calculations only to end up with an answer of ONE. Or maybe TWO. Or, when things get really fancy, ONE HALF.

Interestingly, I'm finding, as I work my way through RUSSIAN MATH, that I'm becoming quite attached to the number one. Every time it crops up as an answer I think: I should have seen that coming. An answer of one always seems like a flag, a sign that there was an easier, more elegant way to do whatever it was I was doing.....but I missed it.

Russian Math has all kinds of 'surprise answers,' and I think a surprise answer in the middle of an ISAT could be slightly.....fun?

An answer of one is like a little joke.

What I don't like...

...is the injunction to Explain in words how you got your answer and why you took the steps you did to solve the problem.

That is a terrible, terrible idea for a test.

It's a good thing to do on homework once in awhile, or in the classroom. RUSSIAN MATH asks students to write out explanations, although it doesn't ask students to explain how they did a problem. It asks them to restate the definitions & explanations given in the lesson.

Items like these can't possible be graded well on tests. They are far too time-consuming, and graders will end up scoring on length or number of explanations given. When you have items like these teachers are going to end up devoting all kinds of class time to writing extended responses, as Susan H says is already happening. We're looking at a massive waste of teachers' and students' time.

Last but not least, I'd bet the ranch you learn nothing from the verbal explanation that you didn't already learn by looking at the student's work.

Being able to produce a fluent, intelligible verbal explanation of a mathematical solution is almost certainly important for math teachers.

It's not important for the rest of us.

I really don't like this one

The number of fifth-grade students going to the museum is greater than 30
but less than 50. Each student will have a partner on the bus. At the
museum, each tour group will have exactly 6 students.

How many students are going to the museum?

Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.

Unless 5th graders in Illinois are doing a lot of prime factor problems, I don't see any reason to include an item like this one on a timed assessment.

First of all, no one should have to be doing discovery ON A TEST.

And second, this problem has two answers (36 & 42, right?), but the wording implies that it has just one answer, and that one answer is findable.

I am DISCOVERING the fact that I don't think red herrings belong in math classes. Certainly not in elementary school math classes.

What is the point? You are teaching children to distrust the English language at the precise moment they're learning grammar & composition. An unreliable narrator in a work of fiction can be a terrific device.

But an unreliable questioner in an examination is just wrong.

I'm against it.

update: I forgot 48!

sigh

(thank you, Dan K)

extended response in 8th grade

Here's the 2004 released 8th grade item:

Peter sold pumpkins from his farm. He sold jumbo pumpkins for $9.00
each, and he sold regular pumpkins for $4.00 each. Peter sold 80 pumpkins
and collected $395.00.

How many jumbo pumpkins and regular pumpkins did he sell?

Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.


The problem is fine, assuming these kids have actually been taught some algebra.

If they haven't, this is a discovery problem on a timed assessment, and I'm against it.

So, assuming they've learned how to set up & solve equations with unknowns, the problem is good. IMO.

The demand that the student explain each step in words is not.

Russian Math rocks

Instead of writing about Russian Math, I should be downstairs (at the kitchen table!) actually doing some Russian Math.

So I think I'll sign off.

But tomorrow I'll give some examples of what a proper extended response item should be.

A proper extended response item should be a RUSSIAN MATH EXTENDED RESPONSE ITEM.

update: scoring rubric for extended response

'Student Friendly' Mathematics Scoring Rubric

Assuming I'm reading this correctly (I feel a little distrustful), students must get all computations correct in order to earn the highest possible score of 4. They can earn a score of 3 with minor mistakes in computation, which I feel is fair, though others may disagree.

What I reject absolutely is the explanation section:

• I write what I did and why I did it.
• If I use a drawing, I can explain all of it in writing.

This is wrong. I don't believe a 4 should depend upon being able to supply an explanation in any case. But here you have a child who can explain why he or she did what she did in a drawing, which is no mean feat (and I'm in a position to know) and even that isn't enough.

Pace Anne, you'll notice that it's not OK for a child to explain what he/she has done by offering a mathematical demonstration, as the teachers in Liping Ma's book do. Anne's right about that; it struck me, too. Over and over again, when Liping Ma asks a Chinese teacher why he/she teaches an idea a certain way, the teacher responds by writing out a proof-like mathematical demonstration. That's what makes the book incredibly difficult (and incredibly valuable) to read for most of us; the teachers don't translate math into words, and neither does Ma.

For Chinese teachers, math is math.


This drops you to a 3:

• I write mostly about what I did.
• I write a little about why I did it.
• If I use a drawing, I can explain most of it in writing.

A couple of years ago the head of our school board sent out an email explaining the adoption of TRAILBLAZERS that included this line (from memory): In recent years math has become language-based.

I think that would come as a surprise to actual mathematicians.


extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

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WallStreetJournalSingaporeMath 12 Sep 2005 - 19:32 CatherineJohnson


I'm teaching my little Singapore Math class again this fall, in the Main Street School after-school program. Last year I had one blinding success, a boy who took to the Singapore bar models like a fish to water and decided, apparently as a direct result, that he liked math and wanted to do well in it. He was a Phase 3 kid, now boosted to Phase 4!

So I'm looking foward to it.

(The other kids all did great, too; I don't mean to draw negative comparisons. They just didn't experience major life epiphanies as a result of drawing bar models.)

I was revising my course writeup today, and had to go hunting for the WALL STREET JOURNAL article on Singapore math, which I apparently had neglected to post anywhere on the site. So here's the link.


Excerpts:

Singapore's curriculum was developed over the past few decades by math experts hired by the Ministry of Education, who continually interviewed math teachers to find out what works and where kids need help. The elementary textbooks cover only one-third of the topics typically found in U.S. textbooks, but the material is taught far more thoroughly. While rote learning plays a part, kids in Singapore also learn to use visual tools to understand abstract concepts.

Singapore math texts, for example, ask kids to draw bars and other diagrams to visualize problems -- a technique called "bar modeling." When this strategy is applied consistently over a number of years, children tend to be better able to break down complex problems and do rapid calculations in their head.

[snip]

The National Council of Teachers of Mathematics in the U.S. suggests that it might not be possible to copy what Singapore's done simply by importing its books. The success of its math program may have roots in Singapore's highly disciplined culture, where the entire community -- particularly parents -- expects kids to buckle down and work hard, argues the NCTM.

There's little doubt, though, that math teaching in America needs to be overhauled. Tuesday, Boston College will release a four-year global study that is expected to show the math gap with Asia remains. The college's last study, the 1999 Trends in International Mathematics and Science Study (TIMSS), ranked eighth-graders in Singapore the best in math, while U.S. kids came in 19th, just behind Latvia. American kids also fall further behind the longer they're in school; as fourth-graders, American kids ranked 7th on the 1995 study.

That decline has already had an impact on U.S. universities.

today's horror factoids:

• Among U.S. freshmen who plan to major in science or engineering, one in

five requires remedial math courses

• Enrollment by U.S. citizens or permanent residents in graduate science and engineering programs, meantime,

dropped 10 percent between 1994 and 2001. Enrollment of foreign students grew 35 percent.


another link to the WSJ article: As math skills slip, U.S. schools seek answers from Asia


key words: decline in U.S. engineering math and science enrollment

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EverydayMathLongDivision 13 Sep 2005 - 15:06 CatherineJohnson


Thanks to NYC HOLD I have a graphic of Everyday Math's substitute division algorithm. TRAILBLAZERS teaches the same approach, which it calls 'forgiving division.'

...instead of teaching long division, students are taught to divide numbers using the partial products method, a technique where children guess how many times a number goes into another and keep subtracting the guesses until they come up with the answer (see box). This method works, but it takes more time and doesn't allow the student to divide past the decimal point.

[snip]

Isaacs and others defend the alternative algorithms by explaining that they teach students how math works. The partial product method of division, for example, is a lot more transparent to students than the long division method.

I'm sure he's wrong about this. I found partial product division quite confusing myself when I used it.

otoh, I think partial product division might work as a teaching tool when used on simple demonstration problems. (I tried it on a complicated division problem and got completely lost mid-stream.) I might use a problem like 16 divided by 2 to show that division is repeated subtraction, analogous to multiplication being repeated addition.

I haven't tried it with any children just learning long division, but if I ever get a chance to, I'll take notes.

the honeymoon

Some parents like the program as well. "It's sort of incredible," said Susan Pottinger, whose son Theo attends kindergarten at P.S. 261 in the Cobble Hill section of Brooklyn. "For him it's great fun. He's fascinated by numbers. He sees patterns everywhere," she said. "He'll put shoes away and alternate shoes with sneakers and say, 'See I'm making a pattern with my shoes.' "

We parents (well, some of us) spend those early elementary school years in a wonderland. Then the you-know-what hits the fan in 5th grade.

source:
Weighing the Factors Does the City's Standardized Math Curriculum Measure Up? By Amy Sara Clark

update

Lone Ranger supplies this link to lattice multiplication, the method Everyday Math teaches children when they cover multiplication. Carolyn points out that lattice multiplication is distinctly opaque; it obscures rather than reveals the fact that multiplication depends on the distributive property.

Here's another link to lattice multiplication at Math Forum Carolyn posted awhile back.


why long division?

Milgram & Klein links:

• Why Long Division? speech by James Milgram, Stanford University
  
• The Role of Long Division in the K-12 Curriculum
  by David Klein, Department of Mathematics California State University, Northridge and R. James Milgram, Department of Mathematics, Stanford University
  

Everyday Math's alternative division algorithm
forgiving division
forgiving division, part 2
try this with forgiving division
who says long division is hard?
advice from Canada
Everyday Math division algorithm fighting innumeracy at CO
conceptual understanding vs numbers

keywords: Columbiajournalismstudent EverdayMatharticle

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KumonMathInDetroit 17 Nov 2005 - 13:28 CatherineJohnson



fyi:
KUMON math program
KUMON reading program


I've had an amazing email from an engineering professor who learned of Kitchen Table Math while she was in China (!)

(Apparently, not being listed on Google isn't a problem in China.)

She also sent me a copy of her paper on Kumon supplementation in Detroit schools (the results were incredible), and I'm waiting to see whether it's OK to post. In the meantime, she says it's fine to post her email:

I'm sure you must have come across Kumon mathematics? I'm a professor of engineering at Oakland University, and so mathematics is obviously very important to me. As a consequence, to make up for the problems with the American school system I've had my own daughters in the Kumon program for about ten years each--between the ages of three and thirteen. Their math skills are far better as a result. I was so impressed with the ideas behind Kumon (it is an outstanding supplement that provides the additional practice missing from K-12 math), that I started a program using the Kumon method in a local inner urban school district, Pontiac. The results are described in the attached paper.

Kumon provides the easiest, smartest way I've ever seen for a Mom to help her kids with math. I couldn't recommend it more highly.

One last thought. I've taught in China as well as the US. The US is definitely way ahead on the "creativity" side. But we are so far behind in math that it is ridiculous--and it is potentially crippling for our source of engineers and other professionals. There are many aspects involved in good engineering, for example, where a good math background is critical. Most of the engineering professors where I work now (Oakland University), are foreign born. Although I greatly respect my foreign-born colleagues, it's really an indictment of the American system that we can so rarely grow our own any more.

Thanks for your blooki, which I have bookmarked and will be following!

Kumon for children with severe disabilities, too?

And, in a follow-up:

Actually, the woman who ran one of the Kumon centers I brought my children to originally got into Kumon because she saw how much it was helping a profoundly mentally disabled child who she was working with. So I suspect it may be surprisingly beneficial for Andrew. I couldn't have done the outreach in my local inner-urban outreach without the incredible help I got from Doreen Lawrence, the Vice President of Research for Kumon, North America. Her phone number is 248-755-2587, and her email is dlawrence@kumon.com. Doreen is a wonderful person who is deeply oriented towards helping children. I'm sure she'd be glad to answer any questions you might have about Kumon (she knows EVERYTHING about the program).

You can feel free to post anything from my letter that might help. I just apologize for the poor writing. I just got back from China and am still jet-lagged.

Over the next week or two I'll read through your website more carefully and get a better feel for what's going on (I just found out about your website while I was in China, but scarcely had any time available while I was there). I've a lot of thoughts and background information related to what you're doing, and have some interesting and relevent experience with national policy setters in academia on this topic, but am a little bogged down now working on a book, research papers, experiments, and grant proposals. You know, the usual academic stuff! So I will try posting some once I feel I understand more fully what you are doing and how you are doing it.

Thank you ever so much for providing a forum for something that is so important to our children!

Her name is Barbara Oakley & she has had an amazing life (e.g., she met her husband at the South Pole.....)

Plus--and I MUST post this--she's started a page of things she finds funny, which, thus far, has one link to a pdf file of what looks to be a PowerPoint presentation: Yours is a Very Bad Hotel.

All you World Traveling Kitchen Table Math denizens will relate.

it's getting clearer now

Back when Carolyn and I started Kitchen Table Math, my one question was: Why?

Why exactly, in the middle of my life, am I spending 18 hours a day WRITING A MATH BLOG? Excuse me, a MATH BLOOKI.

This was my husband's question as well.

I'm just coming off a newyorktimesbestseller, the goal nonfiction writers spend their careers aspiring to reach.....shouldn't I be Following Up with another book? (I will follow up with another book; Temple and I are working up steam. But still. Kitchen Table Math is a detour.)

So what was I thinking?

Somehow, it seemed like I was supposed to be writing a math blooki.

That reason turns out to be, in large part, the people who write comments and set up pages and create dimensional dominoes and, now, send me an email out of the blue telling me I need to take Andrew to Kumon.

That is exactly what I need to do. I need to take Andrew to Kumon.

Andrew is my little locked-in boy; he's bright--so bright, it's there, you can see it--and I don't know how to reach him.

The folks at Kumon may not know how to reach him, either, but it's obvious to me I'm supposed to give it a shot. If they don't know, something there will give me a new idea. It's a lead.

I wasn't going to figure this out on my own.

I was telling my neighbor about this today, complaining that I can't think of these things myself. I have to have complete strangers tell me: take your severely autistic son to Kumon Math.

My neighbor said, 'You can never think what you're supposed to do about your own life.'

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OakleyPapersOnline 19 Sep 2005 - 17:20 CatherineJohnson


Chris Adams found all of Barbara Oakley's research papers posted at her web site (something I probably could have done if I hadn't gotten sidelined by the humor page.....)

This is why it's a bad idea for me to try to learn math from textbooks with pictures of diving penguins.

Thank you, Chris!

update

Oh, boy.

I'm gonna be reading all of her stuff.

Check out this title: IT TAKES TWO TO TANGO: HOW ‘GOOD’ STUDENTS ENABLE PROBLEMATIC BEHAVIOR IN TEAMS

This paper was written to describe a successful program developed to forestall non-cooperative behavior in team-related activities, and to provide an explicit guide for students on how to handle such problematic behavior if it does arise. The program involves creating self-awareness of the deleterious effects of typical, seemingly ‘nice’ behavior in a dysfunctional team situation. Indeed, it has proven to be a revelation to many students to find that their ethical, industrious, and well-meaning responses to non-cooperative behavior can often enable such unacceptable behavior to continue and even escalate.

I myself have Personally Experienced the deleterious effects of seemingly nice behavior in a Dysfunctional Team Situation, and I've never had the first clue how to deal with it.

Mostly I just fume and glare and fire off furiously angry body language in all directions, & end up looking like a lunatic.

I once did this on cable TV, trying to speak my piece at a school board discussion of TRAILBLAZERS.

update update

OK, this paper is not going to solve my looks-like-a-lunatic-at-school-board-meetings problem.

It's about dealing with Hitchhikers & Couch Potatoes.

More t/k.....

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MathLessonsPage 21 Sep 2005 - 15:48 CatherineJohnson


I've started to get the Math Lessons page pulled together. I'm sure I've forgotten posts that should be indexed there, so if you know of any, let me know. (Any lessons you especially like from other people's sites, like MathandText, for instance, should also be added.)

There's a link to 'Math Lessons' on the sidebar.

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TeachnologyFreeWorksheets 21 Sep 2005 - 20:07 CatherineJohnson


Teachnology seems like a useful site.

Here are free online word problem worksheets.

And here are lots of free math worksheets.

I like this addition and subtract equations worksheet.

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BestMadMinutesBook 22 Sep 2005 - 04:06 CatherineJohnson


I keep forgetting to ask.

I'm teaching the Singapore Math after-school class again, and I don't want to use Saxon's 5-minute sheets.

I need a 1-minute sheet (or online source).

Thanks--

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OnlineMathResources 22 Sep 2005 - 22:30 CatherineJohnson


I came across all kinds of interesting-looking math web sites last night while looking for:

• integers worksheets
• downloadable number line worksheets

I didn't find either of the things I wanted (and almost spent $29.95 to join some teacher site linked to by FunBrain just to be able to printout their number line sheet...).

But I found all of these:

• AAA Math (resources listed by grade thru gr8)
  also has a potentially interesting page called World Education Levels. Unfortunately, I can't tell what 'world education levels' are without spending a lot more time on the site than I want to spend. LOTS of online quizzes that are corrected by the site, and they seem to be selling a software program on arithmetic.

• Computer Lab: Math, Numbers, and Counting HUGE survey of 163 different online math sites for kids. Terrific collection.
  Item number 163: Disaster Math for Kids
  From FEMA! Interactive word problems graded by the FEMA site.

• the aforementioned FunBrain Math Baseball is a classic.

• FunBrain's teacher site, the page that almost sold me a $30 sheet of number lines. Has articles on behaviormanagement in the classroom that look good.

• Harcourt School Publishers' number line express Blecch. But maybe little kids would enjoy it. There's a talking lion railroad engineer.

• MacTutor History of Mathematics

• Math Cats how-to for teachers Definitely worth looking at.

• math clip art! possibly for autistic kids (I was on a major clip art tear a few years ago, when Andrew was in his PECS genius phase...)

• Maths Dictionary (British?)

• Mathsurf teacher's site word problems from Pearson Scott Foresman. If you're looking for story problems with multiple answers, this is the spot. Possibly (probably?) a good site to visit for problems your child may encounter in constructivist math courses -- worthwhile problems, as far as I can tell on cursory inspection.

• Mathsurf telling time worksheet (to print)

• NCTM Illuminations Looks decent. Lots of interactive 'tools': affine recurrence plotter and--Yay! nets!! I've been looking for nets!

• National Library of Virtual Manipulatives for Interactives Mathematics I've visited this site several times before. It's pretty good, as I recall. (If I'm remembering correctly, the pan balance problems are kind of cute & potentially useful.)

• Room 108 Looks decent. You can create online Mad Minute pages (must be answered & graded online)

• odd & even numbers possibly good for autistic kids? this site speaks the directions, although I don't think the directions are also written out in words. But any time an autistic child can hear the same words spoken by the same recorded voice it's a good thing, I believe. Site is simple and graphically compelling. Has a HUGE cursor (also great for autistic kids.)

• Primary Games good for autism? I have a feeling this might work with Andrew at some point in the near future. Very simple, has ONE moving image--'Squigly,' a little worm inside one of 10 apples who pops out of his apple and then disappears back inside every couple of seconds. The child has to tell which apple Squigly is in (first, third, fifth, and so on). The only bad part is that there's a lot of advertising crud at the top and the bottom of the page.

• Primary Games 'Diamond Mines' knock-off I spent a couple of years as a Diamond Mine addict.

• Primary Games fishy counting game good for autism? terrific. Very, very simple counting game (as nice as the counting game they used to have on the Barney web site....

• Primary Games Tetris bubbles Great! I've been meaning to post a TIME MAGAZINE article saying girls improve their spatial-visualization skills when they play Tetris. This is, I think, a somewhat slower version of a Tetris game. (Slower is always good for me....) Stupid music, though.

• Primary Games time clock Terrific! Very simple & cute. You have to be able to use a mouse (Andrew & Jimmy both have huge MOUSE difficulties, unfortunately.)

eureka

I will never, ever speak ill of the NCTM again.

They have FREE NUMBER LINES, 8 to a page!

Unfortunately, all 8 number lines start at 0 and contain only positive numbers....

update

I take it back.

I will carry on saying bad things about the NCTM.

They do not appear to have posted a single number line on their web site that includes negative numbers as well as positive numbers and 0.

keywords: online interactive math resources tools nets manipulatives

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WickelgrenOnYoungChildrenAndMath 17 Sep 2006 - 01:14 CatherineJohnson


back story:

My neighbor, the statistician, showed me her copy of Math Coach: A Parent's Guide to Helping Children Succeed in Math quite awhile back, before either of our kids had had any trouble in math class. I ordered a copy just because I order lots of copies of books I'd like to read but then don't.

So the book was sitting there on my shelf when Christopher came home with his 39 on the Unit 6 test & I subsequently failed to teach him fractions using SRA Math. I needed help.

It was the right book at the right time. A page-turner.

Most of what I believed to be true of math ed & math achievement, I discovered, was wrong. Severely wrong. I had been operating on the basis of sheer ignorance, naivete, and boneheaded cliche.

This is the observation that probably shocked me the most. It appears in Wickelgren's chapter on finding a school for your child:

There are schools with even less structure than Eastside. Take the Sudbury Valley School, a private K-12 school in a Boston suburb. This school gives each child complete freedom to choose how they spend their time at school. There are no classes except those specifically requested by a group of students. Children learn largely on their own, reading books, talking to each other and to teachers or outside experts, solving problems, playing games and sports, practicing musical instruments, doing arts and crafts, and anything else that can be done on the school grounds.

While you can read at length about the school's strengths on its web site, one of its biggest potential benefits is that every child can proceed at his or her own pace, in math and in other subjects as well.

There are also potential drawbacks. Since young children are not generally highly motivated to learn math, they may choose not to study much of it.