Math Mammoth

Crimson Wife recommends the Math Mammoth books.

Life of Fred

How is it that in all these years I have not encountered the Life of Fred series by Stanley Schmidt until today?

Some arithmetic books omit the sugar—which is like lemonade without any sweetener. They give you a couple of examples followed by a zillion identical problems to do. And they call that a lesson. No wonder students aren’t eager to read those books. At the other extreme are the books that are just pure sugar— imagine a glass of lemonade with so much sugar in it that your spoon floats. The pages are filled with color and happy little pictures to show you how wonderful arithmetic is. The book comes with 1) a teachers’ manual, 2) a computer disc, 3) a test booklet, and 4) a box of manipulatives. And they are so busy entertaining the reader that they don’t teach a lot of math. This second approach is also usually quite expen$ive.

We’ll take the Goldilocks approach: not too sour and not too sweet. We will also include a lot of mathematics. (Check out the Contents on page 10.) How many arithmetic books include both forms of the Goldbach Conjecture? (See chapter 17.) The reader will be ready for algebra after completing this book.

survey: who should choose textbooks?

This is interesting:

When asked who should have the final say on what textbooks are used in the classroom, 34% of Americans say teachers, but 24% say parents should have the final say. Fifteen percent (15%) prefer giving the final say on textbooks to local government. Nine percent (9%) each designate federal and state governments as the final word.

Among those with children in the schools, 28% say teachers should have the final decision on textbooks, but just 21% say that decision should be made by parents.

Sixty-one percent (61%) of all adults say parents, if they don’t approve of the textbooks selected by a school, should be allowed to transfer their child to another school that uses other textbooks. Twenty-seven percent (27%) disagree, and 12% are not sure.

But only 29% say that if all public schools in a district use textbooks that are unacceptable to parents, the district should make arrangements for those students to attend an acceptable private school. Fifty-three percent (53%) oppose that idea. Eighteen percent (18%) are undecided.

60% Say Their Kids Textbooks Place Political Correctness Above Accuracy

Only 28% of people with kids currently in the public schools believe teachers should choose the textbooks.

That is tiny.

I wonder how this question would poll if you asked how many adults think curriculum specialists lacking a degree in the subject matter under question should pick the textbooks .....

kitchen table math

I have just spent 3 hours doing Chris’ new homework assignment. The problem set is drawn from two short lessons on parallelograms appearing in one short chapter of Glencoe Geometry New York. The two lessons cover NINE theorems about parallelograms, not one of which I’m able to prove although I am apparently expected to be able to prove all 9 now that I've seen them tidily numbered and listed in an attractive Glencoe Geometry Theorem Chart enhanced with a red-and-tan color scheme.^* One two-column proof of Theorem 8.4^** (Opp. angles of parallelogram are congruent) and one paragraph proof of Theorem 8.10 (If both pairs of opposite angles are congruent, quadrilateral is a parallelogram) and it's off to the races.

That's not all. Having read nine theorems about parallelograms & 2 proofs, I am now Glencoe-ready to solve homework problems involving PARALLELOGRAMS ON THE COORDINATE PLANE USING THE MIDPOINT FORMULA, etc.

This is a textbook written by math educators.^***

Speaking of math educators, thank God I have the Teacher Wraparound Edition. Unfortunately, what I really need now is the Teacher Solution Manual (ISBN: 0078602041). Too bad I didn't think of that.


^* I feel about tan the way I feel about beige, only more so.
^** And, yes, I did spend time Googling the known universe to find out whether these numbers are official: is Opp. angles of parallelogram are congruent always and everywhere Theorem 8.4? Apparently not.
^*** written by math educators, but sounding suspiciously like contemporary geometry textbooks authored by actual mathematicians...

Bob Dixon on the worst textbook you could possibly imagine

What if Zig Engelmann set out intentionally to write the worst textbook he possibly could? ...If you think about it, Zig should be able to pull this off better than anyone alive. ...Extremely confusing concepts all would be introduced at the same time and/or in close approximation. Stuff would be "taught" and then dropped, or more accurately, "covered" and then dropped. New material being covered would logically require mastery of prerequisite knowledge that most of the students most likely wouldn't have.

[snip]

If Zig were to engage in this little heuristic exercise, the result, I believe, would be a textbook that would sell like crazy and generate a fortune in royalties, and it would take about one-twentieth of the time that it would take to write an instructionally sound textbook.... [T]his ... came to me while looking at my daughter's geometry textbook. The thought hit me that if Zig had tried to write the worst possible geometry textbook in the world, it would end up looking a lot like Emily's geometry text, published b one of the few major educational publishing companies still standing.

[snip]

Within two lessons of a single chapter of a best-selling geometry text, eight major, similar concepts are introduced, with eight more or less similar names or labels. Absolutely no one associated with this textbook ever took a single minute to look at all of this stuff through the eyes of the learner.

It's same ole, same ole. A few kids get A's on the tests, which passes as proof that the book is fine, but there is something wrong with the kids who flunk the test or get C's or D's. Zig could have written this book without having seen it just playing the game of trying to make everything as difficult and confusing as humanly possible. I've looked carefully for spots where Zig could, in fact, make the book worse than it is, but they are few and far between.

Isn't that strange? Really. You'd think that any best-selling textbook would get a few things right, would do a few things that actually take a student perspective into consideration. You'd think that Zig's horrible geometry text would be far worse than a best-selling geometry text. This is both astounding and depressing. Someone is making a fortune off a textbook that could be just slightly better than the worst textbook we could imagine, the worst textbook Zig could write. Get out the thesaurus: overwhelmed, dismayed, incredulous, confounded...

What Would You Get if You Set Out to Write the Worst Textbook You Could Possibly Imagine? A Best Seller
by Bob Dixon
Direct Instruction News
Spring 2008

Of course, I'd love to know what textbook he's talking about.

I was thinking Glencoe Geometry, but it can't be because the lead author on the textbook Dixon's talking about is a mathematician.

I just checked the author page for Glencoe Geometry & all 5 authors are math educators.


kitchen table math

is this ironic?

Here is a page with what seems to me a nice explanation of seductive details (aka page splatter) that is itself filled with seductive details.

hmmm....

Just looked at two of the other pages: one seductive detail per page.

I give up.

remembering key concepts in math with foldables

Amazon reviewer SW says:

As an Instructional Facilitator at my school, we received this guide to help students remember key concepts in math, however our teachers have adapted these ideas to use with science and social studies concepts also. The teachers find this to be a creative way to help the children organize the facts and their ideas, and the kids love to get to do something different than just take notes or complete a study guide.

visual learning

foldables
why lawyers burn out
Independent George re: foldables
your tax dollars at work part 2
my busy day
not your father's formative assessment
remembering key concepts in math with foldables
south of the border
Steve H and palisadesk on foldables
homeschooling convention: no foldables

you may have to hit refresh a couple of times to load these pages:
21st century skills in Singapore
the master plan
horselaughs are heard in Singapore
more horselaughs in Singapore

good money after bad

So the question is....do I want to roll the dice again, and see if I can pick the correct Teacher Wraparound Edition (scroll down) for C's brand-new, groovy, four-color Glencoe Algebra book, New York edition on my second try.

?

No.

Enough with the Teacher Wraparound Editions.

I've got so many Teacher Wraparound Editions at this point I'm in danger of being crushed by avalanching Teacher Wraparound Edition stacks, like that hoarding twin in Harlem in the 1940s.^*

Also, where Teacher Wraparound Editions are concerned, I feel my record has peaked. I'm not going to surpass last year's Wraparound triumph, when Ms. K began sending home problem sets she'd Xeroxed from some other textbook, not the one C. was using in class.

The problems were about symmetry, translations, transformations, rotations and the like, a topic I had never been taught. I had no idea what the particular worksheet was asking C. to do, and neither did he.

As it turned out, this wasn't a problem, because she had copied the homework set from the Workbook for the Phase 3 textbook, and it just so happened I had purchased the Teacher Wraparound Edition for that book when we were thinking the school would display ongoing support and cooperation in the matter of moving C. to Phase 3 for the remainder of the year, then moving him back up to Algebra 1 for 8th grade.

I pulled it out, read the appropriate material, figured out what the assignment was trying to get across, found the answer key, retaught the content to C, had him do the problem set, checked his answers, and had him re-do problems he'd missed.

All of this on a subject I'd never studied, as covered by a textbook my kid's class wasn't using.

I'm never going to top that.


^* Correction: apparently Langley Collyer wasn't crushed by a falling stack of hoarded items, but rather by a booby trap he'd built to keep people out.

Ghosty Men by Franz Lidz

Puzzle

Question. What do the following have in common:

Youth Service
Ballooning
Ladder Safety
Deserts
Buying a Computer
Fundraising
Politics
Communication
Wages
Billiards

Answer. They are the "Applications Highlights" from a chapter in a math textbook; the chapter title is Linear Equations and Functions.

Catherine is right; it's always worse than you think.

(This came up on a list about converting math textbooks to braille. I don't have the book title. It's published by McDougalLittell.)


triplets homework
puzzle

Susan J's brilliant lesson on per cent

Susan's lesson is wonderful.

Read these two lines:

Remember that it doesn't make any difference whether you write a fraction with a slanted division line between the numerator and denominator or with a horizontal division line between the numerator and denominator:

This direction comes exactly where it should, arriving at the precise moment in which a student may be starting to feel confused by the horizontal notation 33% = 33/100.

The use of light violet highlighting is fantastic, too:

Don't forget that in word problems, is almost always means equals and of almost always means times.

This next line is a terrific example, IMO, of the proper way to teach a procedure:

You can probably divide 300 by 100 in your head. But it is useful to remember that an easy way to divide a number by 100 is to move the number's decimal point two places to the left. You should also remember that when a number isn't written with a decimal point, you put the decimal point just to the right of the number.

One of the problems with "traditional" math, which I'm sure plagues a lot of reteaching parents, too, is that teachers & parents fall back on purely procedural teaching whenever a student is struggling with a concept. You can see that your student's working memory is already maxed out just dealing with the new material, and you know there's no point adding even more information to the load. He's not going to absorb it, and he may lose focus on the concept he's trying to master.

I've thought about this a lot, because I do more "straight" procedural teaching than I would like.

I'm constantly looking for an "in" to offer an explanation or make a connection between the new material and something C. (presumably) already knows. But if I were actually writing a curriculum, explanations wouldn't be an "add-on," and I wouldn't be looking for an "in." The explanation would be a seamless part of the sequence of instruction and example. My starter examples would be simple enough to do two things:

• incorporate an explanation within the structure of the example
  

• leave enough working memory free to allow the student to read and understand a simple explanation accompanying the example
  

The Singapore Math books work this way. Rarely do they give a student "too much" at the same time. In Singapore Math, the examples are the explanation to a large degree. The written text is spare, even terse.

That's what Susan has pulled off. She has set up a simple per cent problem that gives her a fraction with a 300 in the numerator and a 100 in the denominator. Both of those numbers -- 300 and 100 -- are already "chunked"; the load on working memory is extremely low. The student can hold them in WM while reading Susan's reminder that moving the decimal point two places is another way of dividing by 100.

In short, having set up the correct teaching example, she can in two sentences express and distinguish between two ideas that trip up many a middle school student:^*

• moving the decimal point two places to the left is the same thing as dividing by 100
  

• moving the decimal point two places to the left is just an easy way of dividing by 100, a shortcut, no more & no less^**
  

This is agile writing.

I would describe my own efforts to teach per cent this summer as clunky.

First of all, I don't write my own problems. I use whatever problem happens to be on the page before me in whatever workbook I'm using.

As a result, C. ends up with a fraction along the lines of 286/100, which he can't divide mentally. When he forgets he can move the decimal point, I remind him; then, as he's laboriously writing out 2.86 (because his handwriting, along with everything else, has also not been learned to fluency), I say, "What are we doing when we move the decimal point?" At this point he's focused on getting the number right and only vaguely registers the move-the-decimal-point explanation, which he's sick of hearing in any case.

It doesn't work (not well, at any rate), because the explanation is an add-on.


writing is all about structure

That's the famous William Goldman slogan about screenplays: structure, structure, structure.

Structure is practically impossible to "see" when we read, and it is the single hardest trick of the trade to pull off.

The answer to Susan's question -- Are math books too verbose? -- is yes, for the same reason the answer to that question is yes with nearly any piece of writing. Remember the UK writing assignment:

[Judith] Koren describes how two British women she knows became effective essayists and speakers. “Each week, they’d had homework exercises like this: While preserving every essential point, reduce a 100-word essay to 50 words, then to 20, then to 10. Reduce 500 words to 50, 1,000 words to 100. Week after week, year after year...."
(appeared in American Enterprise Magazine)

The reason you can keep cutting a piece of writing after you've already cut it down to the bone, the reason it gets better with each cut, is that you are correcting and refining the structure.

In a piece of educational writing about math, the structure is refined and the verbiage trimmed by choosing the correct example or sequence of examples of the concept being taught.


Singapore Math for afterschooling

This reinforces my decision to use the Singapore Math books for formal afterschooling. They are superb.

I don't think you can find an unnecessary word anywhere in the series.^***

^* The article The Effects of Cumulative Practice on Mathematics Problem Solving by Kristin H. Mayfield & Philip N. Chase has a fascinating observation about "stimulus discrimination training" in math practice sets -- will post ASAP.

^** This is so important for students just learning math. Especially when I was relearning arithmetic, I found myself constantly confused over the question of whether a particular procedure was "real" or just a shortcut. (Can't explain better than that at the moment.)

^*** This may be true of the Saxon books, too. However, the Saxon books cover all the standards in all the states, or nearly so, which makes them too unwieldy for the time I have to work with.

Then and now

Myrtle commented that I should cross-post this here (though I'll give you fair warning--going through my old high school algebra book is really fascinating, looking at it from the other end of the telescope, so to speak, and I'm probably going to be on a roll). Here it is. (Oh. By the way, no, I was not a freshman in high school in 1961. I'm not quite that old. We were, however, using a textbook first published in 1961. I just thought I'd clarify that, since one blogger has already make that particular boo-boo.)

I bought a copy of the freshman algebra book we used in high school (Pearson and Allen, Modern Algebra: A Logical Approach, 1961). Here's a problem example dealing with fractions (and before you complain about the quality of the scan, you try scanning a book bound in 1961–they don't make 'em like that anymore):

 

 

 

and here are some of the problems we did:

 

 

 

Look at 13, which says:

Why is it that you cannot find the cost of one ball and one bat in this case although you could find the cost of one uniform and one hat in Exercise 12?

This is excellent. You're not asked to solve the problem here. Instead, you're asked to explain why you cannot solve this problem, but could solve a similar problem. In other words, you're being asked to think analytically about the problem–about the mathematics behind the problem. If "higher-level thinking" meant anything that had any pedagogical usefulness, this is what it would mean.

Also note that the problems are literate–that is, they're written in educated Standard English:

Were he to reverse the amounts, the yield would be only $330 per year.

That speaks less to the math and more to the place of literacy in education and the way educators treated students (then) seriously, rather than condescending to them with "hip" nonsense and pathetically trying to be "relevant."

Now, contrast with these current examples of 8th grade math:

In a couple of paragraphs, explain how you would estimate the square root of 170.

x² + 2y = 10

What do you notice about the expression?

In the first, the student is asked to do no math. Instead, he's asked to talk about it. Note that he's not asked to think about it–just write a narrative about how he would approach the problem. Also note that the embedded problem he doesn't have to solve is to estimate, and not find, the square root of 170. The second problem isn't a problem, just as the first isn't a problem, and it's ambiguous. What is the answer supposed to be? That the expression contains an x and a y? That it's a quadratic equation?

And you wonder why we geezers can make change and these kids can't?

Think Math!

Are you familiar with the Think Math! textbooks?

Apparently, this is a new K-5 series published by Harcourt. My school will be piloting Think Math! next month. (Yes, the exclamation point is part of the name.)

At first glance, it appears that the developers have tried to address many of the objections to NSF texts while maintaining some constructivist components.

Here is some info from their website http://www2.edc.org/thinkmath/index.htm:

. . . developed by Education Development Center, Inc. (EDC) in Newton, MA with support from the National Science Foundation.

Think Math! does not pit skill against problem solving. Rather, it builds computational fluency through plentiful practice in basic skills as students investigate new ideas and solve meaningful problems. Lessons provide glimpses of ideas to come, letting students build familiarity and develop conceptual understanding as they apply, sharpen, and maintain skills they already have.

What makes Think Math! unique (not just another NSF-supported program)? Think Math! provides focused practice, which enhances conceptual understanding as it increases computational fluency. The curriculum allows students to get involved in solving real problems, figuring out what to do without first being told. Instruction is then used to provide good explanations of reliable techniques. The materials, and some teaching techniques recommended in the lesson plans, also reduce the number of words, using visual models to convey information instead. By puzzling out what’s missing, students can “read” the mathematics and figure out what to do without written directions.
Perhaps most unique, the program features embedded professional development for teachers both in understanding the mathematics content at a deeper level and in suggested teaching techniques. Professional development is located in a feature of many lessons titled About the Math, and in thoughtful explanations throughout the teacher’s guide.

There’s much more information on their site that I’m just starting to read. The first question that comes to mind is why adopt a text that has no proven track record when there are others (Singapore, Saxon) that have demonstrated success? This sounds like another experiment with our children being used as guinea pigs. Also, will our school receive NSF funding if we implement this?

Background:
My school piloted Growing with Math last fall and had planned to pilot TERC Investigations this spring. I have been vocal in expressing my concerns, and then last month a few days after I circulated an email to a small group of parents the school informed us that they had decided not to pilot TERC. My email, which included the Inconvenient Truth video and other choice references, was apparently forwarded to many other parents and made it to the BOE and to the school administration. At one point a PTA officer asked me to please email my “readers” to let them know that the TERC pilot had been cancelled.

I would appreciate any comments and advice.

Math Education: An Inconvenient Truth Video

Have you seen this video that presents an introduction to the problems with reform math? It includes specific examples from TERC & Everyday Math. I learned about it from Linda Moran’s Beyond TERC Yahoo group. It’s about 15 minutes long.

http://www.youtube.com/watch?v=Tr1qee-bTZI

I would like to share this with some local parents because I want to inform them of what’s in store for them if our school district decides to adopt TERC (currently being evaluated here). In this era of YouTube appreciation I thought it would grab people’s attention in a positive way. (Although I would have preferred another title.)

Any opinions about this?

Video embedded by Rory:

Japanese math textbooks

Instructivist links to a video of a presentation on constructivist math by David Klein and James Milgram hosted by Where's the Math.

I haven't seen it yet, but apparently Klein mentions a translated Japanese middle school curriculum he likes.

Instructivist thinks Klein is referring to these books:

[14] Mathematics 2: Japanese Grade 11 - Kunihiko Kodaira, Editor - AMS, 1997, 262 pp., Softcover, ISBN-10: 0-8218-0582-7, ISBN-13: 978-0-8218-0582-4, List: US$32, All AMS Members: US$26, MAWRLD/9

[15] Basic Analysis: Japanese Grade 11 - Kunihiko Kodaira, Editor - AMS, 1996, 184 pp., Softcover, ISBN-10: 0-8218-0580-0, ISBN-13: 978-0-8218-0580-0, List: US$26, All AMS Members: US$21, MAWRLD/11

[16] Algebra and Geometry: Japanese Grade 11 - Kunihiko Kodaira, Editor - AMS, 1996, 174 pp., Softcover, ISBN-10: 0-8218-0581-9, ISBN-13: 978-0-8218-0581-7, List: US$26, All AMS Members: US$21, MAWRLD/10

[17] Mathematics 1: Japanese Grade 10 - Kunihiko Kodaira, Editor - AMS, 1996, 247 pp., Softcover, ISBN-10: 0-8218-0583-5, ISBN-13: 978-0-8218-0583-1, List: US$32, All AMS Members: US$26, MAWRLD/8

I have in my own files this list, which may be the same as Instructivist's books:

Resource Materials available from the UCSMP Director's Office

• Soviet Studies in Mathematics Education, vols. 7-8
• Japanese Grade 7 Mathematics
• Japanese Grade 8 Mathematics
• Japanese Grade 9 Mathematics
• Russian Grade 1 Mathematics
• Russian Grade 2 Mathematics
• Russian Grade 3 Mathematics
• Developments in School Mathematics Education around the World, vols. 1-3

About UCSMP's Translation Series

Translations of Japanese Elementary textbooks are sold here: Global Resources. (Klein didn't mention these, I gather. I have no idea whether he's seen them.)

You can look at sample pages here

Myrtle (hi, Myrtle!) found this link: Japan: A Different Model of Mathematics Education (pdf file)

I still can't do these middle school entrance problems. At least, not the first one. Not in two minutes.

sigh

UPDATE: Yes, I can.

In fact, I may even be able to do this problem in 2 minutes.

Saxon Math to the rescue.