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Idea Bank for Approaches to Math Instruction

(Share your ideas, too!)

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(including math resource books and
links sprinkled through the discussions)
Best practices in math

> I am part of a task force to study best practices in math, nationally
> and in Illinois. The grade level that I am interested in is 4th grade.

My best practices include allowing the students to find out what makes sense and then explain and defend their own thinking. For example, this year we put a number of fraction addition equations on the board. They had to figure out which ones were reasonable and, using any method (manipulatives, numbers, explanations), justify their thinking. I feel it is very important to ask students to explain their thinking. I use a strategy called "think, pair, share" a lot whereby they initially think about what they did or what they believe to be true about a problem and then get in a pair and share with one another. They are expected to challenge what does not make sense in their partners' thinking.

I expect them to be open to learning from one another and tell them that very clearly. They start out with what they know and then, in fourth grade, should be able to clearly explain what they learned as a result of listening and watching others' ideas about a mathematical subject (including mine). I expect them to use mathematical language. What I came to realize is that many times math class resembles the scientific process. You start out with a hypothesis, look at the evidence and then draw a conclusion.

My role as teacher is to help them make connections, seek out what makes sense, and be part of what is available to them to learn with. I also present problems a lot and consider it my responsiblity to think, based on the immediate assessment of what goes on day to day, what is the next right step for my class. I keep the standards structured into my annual plan but also plan my daily and short term lessons to meet the needs of the students. Where are they weak? What understandings need to go to a deeper level? What calculations need work? What does this lesson tie into?

Ramos, on math board

improving math instruction

> I teach grade 4 and have concentrated heavily on revising my
> language arts program this year. It is time to refocus my energy
> on math instruction. Please, offer your favorite methods and programs.
> What are you using? Skill and drill? peer learning? How do you instruct
> students in critical thinking?
> Kelley

Today as an example:

We did Daily Math Review---a mixture of many math skills I hand out every morning and we go over it at the beginning of math time.

Then we did a mad minute for multiplication: 0,1,2.

Then we did a textbook lesson on rounding. I did a diagram of a roller coaster, and showed how getting to the middle (5) sends you flying down to the next side. We practiced some exercises whole class, then I let groups read and solve some word problems involving rounding.
"You have 5 minutes---go!"

We went over the answers, then I assigned homework.

So, to answer your question, I do many things in the space of a day.
I also have a game day on Friday, where I introduce and play a variety of math games.

Karen/PA, on math board

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I teach using cognitively guided instruction. It was developed at the Wisconsin Center for Education Research, University of Wisconsin. Chidren are taught mathematical thinking. They just recently published a book:
Children's Mathematics: Cognitively Guided Instruction (by Thomas P. Carpenter, ed.).

Dee, on math board

When will we ever use this?

> Thankfully, for the first time, I am going to be given the
> chance to put my mathematics degree to use. However, even
> though I should be dealing with students who want to take
> these classes and want to get ahead in the world. I am
> still wondering how to answer the "When will I ever have to
> use this... but I'm not going to be a ..." question. The
> first part is simple, I know an answer to most of those.
> It's the second part that bothers me. What do you say to a
> student like that?
> John

regarding "... But I'm NOT going to be..."

Well, I think 2 strong ideas come to mind when kids bring this up. But it would take a little work on your behalf.

1. Kids usually think concretely and want examples they can relate to (that's why they said "I won't use this," as if they know exactly what they will do forever). I would find some pretty easy but important examples of famous people who started doing something different in their life and completely changed careers. This will stick in their minds that what they think they may want to do as a career or profession might not always be what they end up doing. We change as we grow.

2. I would investigate and talk about how math is used in fields that they probably don't even know is used. There is also a book called When Are We Ever Gonna Have To Use This? by Hal Saunders (Dale Seymour Publications, currently out of stock.) It isn't the best book, but it does chart all the areas of math from like grades 3-12 and how they are used in different fields. Comes with a chart and is fascinating enough so that they will not ever WANT to ask the question again. Mission accomplished :-)

Posted by "me," on math board

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I hear this all too frequently - in the middle school years more than in the high school. My stock answer is that if they are asking WHERE this skill is used, I can tell them that. But if they want to know if THEY are ever going to use this, that I can't answer. No one knows where they will end up. Careers change. Interests change. But learning the early material paves the way for later interests. If they decide to become an engineer (instead of a writer) in college, they will at least have the basic, prerequisite material needed to change majors. They are too young to make irreversible decisions. I tell them that they need to learn everything they can, so that they can make a better choice later.

I also remind them that solving problems in math makes it easier to make non-math decisions in life in a more logical fashion. I use the example that they are driving down a major local road. Suddenly, there is an accident a short distance ahead of them. Should they wait it out and creep past it? Is it better to get off of that main road and take a parallel road? Is there a way to exit, and go back to a different cross street? These types of decisions are made everyday - and lightning quick. We don't realize all of the processes we go through to think this through - we just do it. The ability to process all of the parameters quickly comes from being a good problem-solver in math. The neural pathways are set up by doing something over and over (like riding a bicycle) until they become second nature, with no conscious thought. Hope this helps.

BTW, worse than this is when one of your honor students comes to you and tells you that his father told him that one of the math logic puzzles you assigned for homework (from the textbook) was useless - because he will never have to do puzzles in his life. This actually happened to me. And it was so sad.

DSF/NJ (Donna), on math board

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When will we ever use this?

> I have a difficult bunch of geometry kids this year. I am getting
> more than the usual "When will I ever use this?" I am running out
> of reasons to convince them why we do deductive proofs. If anyone
> has any suggestions I would really appreciate it. After 19 years of
> teaching you would think I could answer that question but sometimes
> I just don't know what to say any more.
> Joleen R. Christianson

Depending on the ages of the students, I remind them that they really can't know at this time what they will need to know and use later in life. I can show them WHERE these things are used, but a guarantee that any individual in the class will use them???? not possible. Also what is wrong with a well-rounded education? Education for education's sake?

I also like to explain the theory of why we never forget how to ride a bike (although it probably doesn't hold as much truth for many of today's kids). I learned in college psychology, that every time you learn a new skill, a "path" is cut into your brain tissue that connects the different neurons needed to perform this skill. Since most children ride their bikes excessively, that "channel" gets cut deeply into the brain. When they get the first car, and stop riding the bike, the brain slowly starts to fill in that channel. But because it is SO deep, it never fully fills in. So, when in our forties we decide to exercise and pull out the old bike, we wobble for a few minutes until the brain finds that old channel and slices through the new tissue quickly and "it all comes back to us." The more you do something, the more you own it.

Why deductive reasoning? We are setting up those neural pathways in the brain that allow us to make coherent, logical, NON-MATHEMATICAL decisions later in life. We practice the math problems so that we can be cool under pressure later in life. When driving on a busy road and there is an accident suddenly in front of us, we quickly consider our options - do I wait it out until I can go around or turn off here and take a parallel road? Do I get off and go back an exit to find an alternate? Decisions such as these are made everyday in our lives - and need to be made quickly. People who are good problem solvers can quickly sort out the options and make the most reasonable choice. Does it always work? No, but it increases your odds. BTW, no matter what line in the grocery checkout I choose, it will be the one that comes to a grinding halt. One of Murphy's laws. Math doesn't help me there.

DSF/NJ (Donna), on math board


Math Philosophy?

There appear to be two camps in math (like reading - phonics vs. whole language): hands-on, math writing vs. paper pencil skill and drill. I've tried both. Here is what I'm thinking:

Last year we departmentalized. We had a lot of really low math kids. Fifth grade math has always been a toughie. State test scores always drop big time from 4th to 5th. The thing is, for the first time ever in our district (it's not too commonplace other places either), we had 100% of our kids pass the state test. Go Figure! It was like a miracle. So I pick the brain of the math teacher. What did you do?

She used the math textbook (Harcourt), the overhead, math workbooks (are you seeing pencil paper skill & drill here) and that's it! Now, granted, the kids complained about math all year. It was too hard, there was too much homework, etc., etc., etc.... Occasionally she threw in a hands-on activity for fun. So what's up with this Marilyn Burns (who I follow faithfully)?

So, I'm thinking balance. Pencil, paper, skill and drill because you don't throw out the baby with the bath water. Hands-on to make it fun, bust the boredom and to reach the Tk kids. And writing for the Language Arts folks like myself. (After reading about math writing I was mad that I never got to write about math in school - figured that's why I shut down).

Just thinking...What do y'all think?

Mae in Texas, on math board

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Mae, I wished we worked together

I'm a high school math teacher and I totally agree with your philosophy. I think there is more to math than drill, although it does have its place. Nothing aggravates me more than when we're doing a really involved application and they can do the hard parts but get stuck in the quadratic formula, something they should know.

Also, I think making math meaningful and interesting is key. Students should know how powerful and important and beautiful this subject truly is. When we spend 3 weeks on factoring, it's no wonder kids hate math. Writing, manipulatives, technology, hands on activities are all things I love to use.

My kids rave that they've never had a class like mine because we do so much. But, you know what I think really makes the difference? The fact that I have a tremendous amount of enthusiasm and I really love math. I see other teachers do activities but the kids don't have the same response. My kids get a kick out of me because I get so into it and am so interested in what I do. Again, I guess it's the teacher that makes the difference, not the gimmicks.

Kathy, on math board

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Math is my toughest subject. I'm not mathematically inclined. Fifth grade math scares, high school math would send me into orbit! You sound like a wonderful math teacher. I wish I had a teacher like you! I shut down in 4th grade after division. It really didn't help when the teacher yelled at me and let me know how stupid I was! So now I teach 4th grade. At the end of the year, I always ask students what they learned, and math is always on top of the list! Go figure!

An older, wiser teacher once told me that bad math students make great math teachers because they understand the kid that doesn't get it. I hold onto that. I also don't leave a kid in the dust. I'm determined that we will all get it together or die trying. Fractions drive me nuts! They are hard for me to explain. I've been known to cry after a day's battle with fractions!

I like math writing because that is something I can understand. I figure there are students who need math writing the way I do. I like hands-on because it makes math fun. I'm not sure it really develops the concept (it might just be me though), but it takes out some of the fear of math.

The one thing I have found to be effective is working on the overhead / whiteboard. It's like a magic trick I can't explain. It works though! I have to use a good textbook with a good sequence to keep myself focused. That is why I liked it when I used Saxon.

Now that I have been teaching awhile, I feel much smarter in math. It occurs to me that kids who are having trouble in math need to teach the concept to someone else. That responsibility sheds a whole new light right through the frustration block.

Mae in Texas, on math board

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super article that addresses this very subject

A super article that addresses this very subject is from the AFT American Educator magazine. Click the link for a .pdf file of the original article.
Highly recommended!

"Basic Skills Versus Conceptual Understanding," by H. Wu.

Harbinger, on math board

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Harbinger, I read the article and learned from it. I have 21 years of experience and find myself becoming more and more like those "old" teachers I used to hate when I first taught. You know the type, the ones who teach memorization and use workbook pages. I came from a gifted classroom which was very much conceptually based to a small first grade using ABeka materials. I can't believe how much better my kids are at math using the traditional method. Sure we play games and explore tangrams and geoboards but the basics come first. Now I'm the teacher down the hall who won't change with the times. Aghhhh!

Pam/1/GA, on math board

Math Research

> However, there doesn't seem to be as much shared about MATH
> RESEARCH. Is there any? I would welcome any thought
> provoking math research.

You raise a very interesting point. I went to the site of the National Council of Teachers of Mathematics ( ) to see what research they might share, and their featured link was to a report issued at the end of September by the Glenn Commission. NCTM highly endorses this report and its recommendations. It is called: Before It's Too Late: A Report to the Nation from The National Commission on Mathematics and Science Teaching for the 21st Century .
To go straight to the Executive Summary of the report, go to:

In reading the Executive Summary, I notice that it has a lot to say about teacher education and nothing to say about at- home support. It does follow logically that if students are being encouraged to do at-home reading each day, and parents are encouraged to read aloud to young children etc., comparable attention should be placed on at-home math, too. (See more below on Family Math.)

The NCTM site also provides links to its numerous journals, some of which focus on mathematics research. Reading the online summaries of some of the research from the Journal for Research in Mathematics Education, it appears to me that these articles are written more for the population of math researchers rather than for teachers who would like research overviews without all the technicalities of specific studies. A better bet might be "Teaching Children Mathematics." I quote from the NCTM site:

"Teaching Children Mathematics is an official journal of the National Council of Teachers of Mathematics. It is a forum for the exchange of ideas and a source of activities and pedagogical strategies for mathematics education pre-K - 6. It presents new developments in curriculum, instruction, learning, and teacher education; interprets the results of research; and in general provides information on any aspect of the broad spectrum of mathematics education appropriate for preservice and in-service teachers."

NCTM produces other journals for middle school and high school math teachers. A subscription to the NCTM journal of your choice is included with your membership in NCTM, and I strongly encourage anyone interested in mathematics education to join the NCTM! Go to their site for info.

Over the years, I have gotten a lot of use out of the outstanding book,
Family Math, by Jean Kerr Stenmark, Virginia Thompson and Ruth Cossey, which summarizes activities developed during a three-year, thoroughly-tested program to promote more at-home involvement in mathematics. The Family Math program was funded by the Fund for the Improvement of Postsecondary Education of the U.S. Dept. of Education.

There is another book by the same lead author (Jean Kerr Stenmark) and others, Family Math for Young Children: Comparing.

Wendy P. of Math Cats

great N.C. site for math instructional strategy

Strategies for Instruction in Mathematics K-5

This is a superb site with very readable, do-able math lessons and games--a lot that will help you with the drill and repetition that they will need, in a fun way. Scroll down to your grade level and the year is divided into 4 quarters plus a section of blackline masters.

Darcy, on primary elementary board

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If you have trouble getting to that great N.C. math instruction site, just type in

Click on instructional resources.
Under Assessment resources you will click on Strategies for Math Instruction.

Good luck! It is worth the trouble to get there!

Cindy, on primary elementary board

Teacher-trainer looking for good math methods books...

> Do you have any ideas on interesting and thorough methodology books?

I purchased the book, Activity Math: Using Manipulatives in the Classroom, Grades 4 - 6 by Bloomer and Carlson. It has all the units by chapter that you would teach to this grade level. It focuses on introducing the concept at a concrete level with manipulatives--some which can be xeroxed at the back of the book in the resources section. It's an excellent book.

[Also available: Activity Math: Using Manipulatives in the Classroom, Grades K - 3 ]

Jody, on math board

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I have taught the math for elementary teachers course for almost 20 years and have used a variety of texts. My favorite is Musser, Burger, and Peterson. It has a wonderful problem solving chapter to start the book, and then continues the theme throughout. It is readable and challenging in the best ways; it provides good historical contexts as well as talking about open questions. It makes a real effort to share some of the excitement of mathematics while it works to ensure a strong understanding of the foundations of arithmetic. It does not talk down to the students. It is now in the 6th edition.

Lee Sanders, Miami University Hamilton, on MathTalk discussion list

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You might also want to examine "A Problem-Solving Approach to Mathematics for Elementary School Teachers," 8th Edition, by Rick Billstein, Shlomo Libeskind, and Johnny Lott, ISBN 0321156803, July 2003. Some good supporting materials are available. It will push your community college students, but their efforts will be rewarded. [If the 8th edition is not available for sale in the USA, you can get the 7th edition from Addison Wesley Longman, ISBN 020134730X, 2001.]

Ron Ward, Western Washington University, Bellingham, WA, on MathTalk discussion list

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I am a novice. I am teaching a course for elementary teachers using Helping Children Learn Mathematics (seventh edicition) by Reys, Lindquist, Lambdin, Smith and Suydam. I find all of the Math Links for each chapter very rich. Here is the Student Companion website:


Since math is not the strength of the non-traditional students that I teach, I am using mostly hands-on activities to involve the students in the beauty of the connections they need to see in mathematics. The students I teach need to walk into their first teaching experience without fear of tackling math. This text gives many, many references through electronic links and connections to elementary literature. I am enjoying the opportunity to share mathematics with future teachers.

Linda Hall, Fairleigh Dickinson University, on MathTalk discussion list

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I'm not a college teacher, but I have used the John Van de Walle book, Teaching Elementary and Middle School Mathematics for professional development. Teachers have responded in a positive manner to this text. It is a standards based text that helps teachers understand math and give suggestions how to help students in their classroom.

Craig Morgan, Haddon Township, NJ, on MathTalk discussion list

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Although I am a big fan of the Van de Walle text, and have used it for many years in all editions in my METHODS classes for K-8 teachers, I thought the initial question concerned possible texts for math CONTENT classes. I would question its appropriateness for those, unless you are really attempting some kind of integrated content/methods class and, perhaps, supplementing or augmenting the content portion from some other resources. Also, since the initial question had to do with community college students, I think it would be better for them to focus on mathematical concepts and procedures -- getting a good content foundation -- and save math pedagogy for their university work.

Ron Ward/ Western Washington University/ Bellingham, WA, on MathTalk discussion list

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At The University of Texas at Austin, we have switched to Bassarear, Tom (2001), Mathematics for Elementary School Teachers, Second Edition. Houghton Mifflin Company, Boston, MA. Text and explorations manual in all of our mathematics content courses for elementary education majors. The students seem to find the text readable and the explorations manual has many good activities. When I supplement, I have found myself using a preliminary copy of Masingila and Lester published by Prentice Hall. Good luck! Jennifer, U. Texas at Austin, on MathTalk discussion list


Mathophobic teacher in need of help

> I am hoping all you wonderfully skilled math teachers can
> help. I am a first year teacher in 4th grade. I love it
> so far. My problem is in math. Always has been. I truly
> want my students to have a good understanding of the
> concepts they are learning but it is difficult when I am
> having trouble teaching it to them. Sure, I can follow the
> manuals with all of the correct answers but I know I am not
> presenting the material clearly...

> Does anyone have any ideas for me or a book I can read to
> improve my own understanding of elementary math? That way
> I can teach it more clearly. Thanks, I am really upset and
> need to correct this right away.
> tt

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A book that I highly recommend is About Teaching Mathematics by Marilyn Burns.... an excellent book that has almost become a "bible" to me in teaching math. She is a great advocate of helping children to use methods and strategies that "make sense" to them in solving problems. She also has a large number of math lessons in the book that are generally in the primary-middle school spectrum.

In addition, check out her web site,, for her one-week workshops in the summer, which I found very helpful. Good luck!

Debbie B, on math board

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Check out the Marcy Cook books ... all hands on and a lot of fun!!

Jam, on math board

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I especially recommend Math Starters! 5- to 10-Minute Activities That Make Kids Think, Grades 6 - 12

Gisele, on math board

Mountain Math

> What exactly is mountain math? Who prints/publishes this
> program? Where can I get information on it?
> Mrs. B, on math board

Hi Mrs. B.,

By now you probably have found the Mountain Math/ Language website... but just in case here is the link below. I have used Mountain Math in my class for 5 years and I do like it. Make no mistake: it is a bulletin board not a complete curriculum. It is a review program that can supplement whatever program you are currently using. MM/L is not a total program, and the authors are quick to tell you that. It is to help in reviewing and aid in the retention of previously taught concepts.

I enjoy using it because it provides great vocabulary development for my kids in the Math/Language area. It also provides a kind of "spiraling" (for a better word) curriculum review. Each grade level kit comes with a list of objectives and all of the bulletin board pieces on very colorful card stock kind of paper. All you have to do is cut it apart and put it up. You can use all or as much as you want. I find with younger kids it can be used as a morning calendar activity. However, in the older grades it could be used as a type of Daily Oral______(language/math) activity. Or even centers, if you prefer. Even if you haven't taught a specific concept yet or the kids have not had enough practice to gain mastery on a concept, just the fact that you have the board up leads them to ask questions or gives you the opportunity to share the "language" of the particular concept that you will, in the future, teach to them. For all it is worth, it is a great way to have daily review of learned concepts, without a worksheet--although the upper grade level kits, I understand, have such things!

Mountain Math/Language

Laura, on math board

Liping Ma

I am reading Liping Ma's book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States , and am greatly interested in discussing what she is saying with other elementary math teachers. I especially would like to talk about using her understandings and insights with our Standards to make some professional development for our school. Any ideas out there about how to bring the two together?

Ramos, on math board

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I've been reading her book, too, and think its really interesting and insightful. I think she's correct when she says that Chinese teachers have a deeper understanding of mathematics than most American teachers. Case in point: there was a message posted on t-net from a kindergarten teacher who asked how to figure out what a child's grade would be if he missed 7 words out of 27. The thing that struck me was that although I'm sure the teacher is a kind, caring person and is probably very good with young children, anyone with a college degree (let alone a teaching certificate) should know how to figure percentages. I've run into teachers who can't add fractions or perform basic calculations without a calculator, so when Liping Ma says American teachers need to have a better understanding of mathematics, I can believe it.

SGE, on math board

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I'm glad to hear someone else is finding her book to ring true. What impresses me is the connectedness of the way they see things and the thorough understanding. I was hooked in the first chapter on division with regrouping when the connection was made between composing a 10 in addition when there are too many ones and decomposing a ten in subtraction when there are not enough ones to subtract. I have been practicing this concept on my seven year old daughter, who I realize from reading the book, has a procedural not a conceptual understanding of much of addition and subtraction as she is about to graduate first grade. I also support completely the social nature of learning math, the talking, discussing, and challenging that needs to go on instead of the endless math work.

Ramos, on math board

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Hi, I don't know if people are still reading this, but I found it important to have the same skills (that Ma describes for teaching elementary math) for teaching college mathematics. The same skill of diagnosing a student's subtle misunderstanding, and explaining the nature of subtle differences in math operations (as in the multiplication / neglecting place and zeroes example) is relevant at all levels of mathematics, and probably other disciplines.

I cast "the ability to diagnose misunderstanding" as one of the teacher's most important skills. What emphasis is placed on this skill in teacher training and education programs? How do you test it?

Owen Ozier, on math board

[The Liping Ma book has also gotten rave reviews from readers at - W. Petti of Math Cats] top

Multiage Math 1-2

> I am looking for ideas to successfully teach first and second
> grade math students in a multiage setting.
> Gina

I suggest using Kathy Richardson's series, Developing Number Concepts. It is an excellent program for a multiage classrooms, because it is very hands- on, and allows for students of varying ability levels. Younger students work with the same materials as the older students, but because of their various levels of understanding, they work in different ways.

Developing Number Concepts, Book 1: Counting, Comparing, and Pattern
Developing Number Concepts, Book 2: Addition and Subtraction
Developing Number Concepts, Book 3: Place Value, Multiplication and Division
Planning Guide for Developing Number Concepts

Kris, Japan, on math board


> Does anyone know where I can get the book Math-a-pedia?
> It's published by Addison-Wesley. Math-a-pedia is an
> encyclopedia of math terms for elementary students.
> I saw a copy of it at a conference yesterday.
> Teri W., Gr.1, on math board

It is at

Math-A-Pedia: Primary
Math-A-Pedia: Intermediate

(Author is David Brummett.)

Lauri/ms math, on math board

Singapore Math

> Is anyone using the Singapore Math textbooks? If so,
> what do you think of them? I think they're terrific...

> Tell me more about them, since I know you are interested in
> Liping Ma, I am curious to hear about what you have found.
> Ramos

Ramos, you can order the Singapore Math textbooks (as well as the science program used in Singapore) at I've been using the math textbooks for two years now and, if you'll excuse an excess of parental pride, my son has scored in the 99 percentile on the Iowa Test of Basic Skills these past two years (1st and 2nd grade), which was well above the school average. The math textbooks are terrific and present each topic in a logical, sequential way, without jumping around from topic to topic like so many of our American math programs do. They emphasize strong basic computational skills, as well as conceptual skills. As you can guess, I'm totally sold on them and next year, my son's whole school will be using Singapore math.

The Singapore math website also offers enrichment materials in English, science, and math, as well as two math software programs. You can also order a copy of the math section of the leaving certificate exam taken by 6th grade children in Singapore. The leaving certificate exam was a real eye-opener for me because it demands math skills that far exceed the skills of most of our 6th grade students.

SGE (Suzanna), on math board

Open-ended questions?

> I am looking for a resource that can help me write open-
> ended questions that are similar in structure to those used
> in testing.
> I am also looking for suggestions, strategies, etc. for
> teaching students how to respond to such questions. Our
> students are clueless. We have modeled example after
> example, and the kids still give incomplete answers.
> Debbie/NJ, 6/02/00



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