A couple of recent threads have got me thinking about mainstream US textbooks a couple online systems (ALEKS and Khan).

Problem 1. Has anyone else noticed that different books/systems have different methods of doing the same operations? And that every method requires a lot of instruction to teach the same simple idea?

Typical example. A Khan question asked, "You have a block that is 1 1/6 units long. What do you have to do to it to get a block that is 2/3 of a unit long?" DD twigged immediately that this was a subtraction problem. But she couldn't just type in "cut off half a unit." She had to go through a complex process that involved using the software to carve the original block into pieces of a certain length that could be removed. I swear, stuff like this just creates confusion. I see what Khan was trying to do, but IMO, the method has the exact opposite of the intended result and muddies the concept by forcing the kid to focus on WHERE TO CLICK TO CUT THE BLOODY BLOCK!!

Problem 2. Also, has anyone noticed that textbooks/systems don't present things in an orderly manner in which each chapter/section builds on the one before it and leads into the next one?

DD did the first two sections of the Khan algebra course. Section 1 had her solve for x, do basic decimal addition, plus exponents and square roots. When questions in a given topic came up again, they were exactly the same as the originals, except with different numbers. This was a constant in ALEKS last year with DS14.

Similarly, Chapter 2 of her new Algebra book has the following concepts: rational numbers, the distributive property, "theoretical and experimental probability," and more properties of numbers. Not sure why distributive gets its own section, while nine others are squished into one section. Chapter 1 discusses basic expressions with variables, but has scatter plots and mean/median/mode tossed in. It's a mashup of concepts --- and this book isn't even the worst I've seen. Like Khan/ALEKS, all the questions are basically the same. Online or print, the level of difficulty doesn't really increase until they throw something out of left field.

I compared with Brown and AoPS basic Algebra. Both start on a subject, move stepwise to the next one, and continue in the same pattern. The exercises are focused, and get harder as you go. Unlike the mainstream books, there are no brightly colored pictures on every other page, nor are the margins full of distracting, irrelevant notes.

I've been studying algebra and other math books for a couple years now, and I've concluded that it's no wonder so many students are mathematically incompetent. How could they be anything else, given the material?

#2 is because "spiraling" is a hot concept in math right now. I don't have a particular opinion on it, but the ideas is that they study several concepts, them move up a level and student the same concepts again. They build on them, but it's like circling around a mountain to get to the top, not going straight up a set of steps. This didn't seem to bother my DD at all though I admit, it drove me a bit batty.

#1 I think you sorta have two complaints here. There ARE a lot of different ways to get to the same end, and different textbook companies pick different ones. Different kids really do learn better from some methods vs others. As a very visual learner, I've always found visual methods, diagrams, etc extremely helpful to me. I haven't seen that particular Khan Academy problem, but that sort of methods could make good sense and really help some kids. Our kids aren't necessarily those kids because they "get it" so quickly through pure logic.

You get some applause from me there--I actually laughed.

Maybe these texts should incorporate a lot of red and the letter "M" given their content light approach to *waves hands* math.

There are textbooks that do things differently, and a really neat digital one coming out early next year. Sometimes bells and whistles actually provide a function and engagement, which is easy enough to disdain when kids like many of ours don't need the carrots.

Overall, I've seen some real schism factors:
- philosophy (universal design, inquiry, top down, etc.?)
- priorities
- pedagogical methodology (spiral and such)
- quality of implementation
- sensitivity to teacher talent

Like looking at Everday Math, the philosophy might sound reasonable ( http://everydaymath.uchicago.edu/about/understanding-em/em-philosophy/ ), but if it takes a nose-dive from there, then everyone condemns the philosophy. "Encourage use and sharing of multiple strategies" is not the same thing as "force everyone to guess all the multiple strategies the teacher or developer had in mind and train in the use of all of them."


Val,
Do you have some framework concepts like these that you've found useful in looking at math books?

I approach any math textbook published after 1970 with great caution. Today a colleague dropped a copy of Dolciani's Pre-Alegbra (1970 ed.) in my campus mailbox, and I just got an e-mail that her Modern Algebra: Structure and Method, Book I (1962 ed.) is waiting for me when I get back to campus on Monday. I love that the mathematicians I work with like to give presents to English profs. like myself who appreciate a good math textbook.

Anecdotally, I think EM adoption is in decline although mostly due
to the push for Common Core alignment. At least here in the Seattle area it was almost universally in use across both public and private schools up until maybe 3-5 years ago and that has been
completely reversed in a series of recent curriculum decisions.

"Calculus" by Michael Spivak was always highly recommended (and is available on Amazon) as a coherent and mathematical introduction to Calculus.

I think that the problem is there are a lot of substandard textbooks and online programs out there. I don't have a whole lot of experience but so far what I have seen haven't been terribly bad. Having said that, our district develops a ton of curriculum to be used with published textbooks. In elementary school, the math curriculum is heavily subsidized by our district's own curriculum so that less than half of the materials came from the textbooks. Two years ago DS used a relatively older (Pre-2000) version of McDougal Littel for "Pre-Algebra" titled Passport to Algebra and Geometry. I think that the newer versions of that McDougal Littel text is less GT oriented. The ordering of the units seem sensible enough. DS is currently using Holt McDougal for Geometry (Common Core version) but that is to be used with our district developed curriculum as well. This particular text was chosen after a lengthy piloting process involving three or four different textbook companies so the hope is that a good choice was made but who knows. I have glanced briefly through the Geometry text and it looks okay but then I like pictures and colors.

As for multiple approaches, I am actually heavily in favor of exposing students to them but not forcing them to use every approach. The best math teachers regularly present multiple ways of the solving the same problems. One way that I survived too easy math courses was because the teachers always challenged me to come up with a different method of solving the problems than what was just presented.

Here's what I look for. I'd be very interested in learning about how anyone else approaches this task. In summary, I ask myself, "Does this book have ADHD?"

1. First, I look for distracting content. Distracting content includes irrelevant color photos, factoids in the margins, notes in the margins, "hints" and "tips" in the margins, and other margin content ("Go to...; "What you'll learn...And why," etc.). And colors. Lots of bright colors.

Examples. DD's (school) Algebra[/i]-1/dp/0131339966/ref=sr_...=Prentice+hall+algebra+1] Algebra 1 book has 12 irrelevant color photos on pages 2-20. They include a girl climbing a rock, a guy who worked in a shoe factory in the 19th century (colorized to give him rosy cheeks), and a dude mowing his lawn. There's a "Real-World Connection" in the margin under the lawn mower photo. It tells us about the amount spent by Americans annually on lawn care. It's completely irrelevant, but it does make a good distraction for students who are bored with learning about what variables are.

There are also drawings of dollar bills and what appears to be a park (with a fountain!).

Alternatively, DD's (home) Algebra 1 book (Richard Brown) has only 2 pictures on pages 1-20. Both are related to the word problems next to them. There is absolutely no content in the margins --- because the margins aren't wide enough for it.

We have an old copy of Mary Dolciani's Algebra 1 book. Again, no content in the margins. Pictures are minimal and restricted to content between chapters, such as mini-biographies of mathematicians. The Brown book has these, too (Brown is a new edition of Dolciani's books).

Richard Rusczyk's Introductory algebra book has no photos and no content in the margins, but there are drawings accompanying problems.

2. I look for actual text. Many modern textbooks skimp on explanatory text. All that stuff in the margins seems to be (IMO) an attempt to put prose into a sound-byte kind of format. Dolciani and Brown do the best job with the text, followed closely by Rusczyk. The text in the school's book is risible and limited to short paragraphs scattered throughout a section.

3. I look at the problems. Do they start easy and get harder? How many sections have a set of computational problems followed by a set of word problems? How often are you asked to write a proof from scratch? The problems in the school's book are mostly pretty same-y (forget proofs). They use what I see as tricks to make them "harder": Simplify 2xyz + 4xy - 18zyx + 2xy. Basic variable expression problems in the other books are just...harder because they're longer or have nested parentheses or whatever. No one is trying to trick you in the other three books.

4. Does the book toss out random concepts that are above the student's pay grade at that point and/or are off-topic? Or does it proceed in logical order from one idea to the next? Dolciani, Brown, and Rusczyk generally move in a logical way and go into depth. The other book tosses out random ideas and treats them superficially.

Examples. The school's book tosses out matrices on page 59 in a section on adding rational numbers. It provides exactly one sentence of information about them, unless you count the sentence that says, "The plural of matrix is matrices (pronounced MAY-truh-seez)." I don't count that sentence.

We return to MAY-truh-seez on page 79 and then meet them again on pages 394-5. Brown and Dolciani discuss them in depth in subsequent books. Rusczyk has a single problem about them on page 154 and a discussion of them on the next two pages. IMO, it's nice information but is maybe too much for someone who's supposed to be learning about solving systems of equations for the first time. But this is very different from the school's book's florid case of ADHD.

5. I look at layout. Is it easy to find my way? Do I know where I am? Brown wins hands down here. Sections always start on a new page and are clearly marked. The beginning and end of each set of problems is easy to see. Etc. Dolciani is second, with the school's book and Rusczyk coming in way below them. The school's book is just too full of distractions and is therefore beyond hope in this regard, whereas the Rusczyk book just needs (serious) help from usability guy and a typesetting/reformatting chick.

Okay, that was really detailed.

Thanks for taking the time to post this Val it is very helpful information.

In fairness, I think that the Rusczyk book expects someone to have worked through a rigorous pre-algebra class before encountering that book. My DD will start their Algebra I class which uses this book (first half) later in October following on for the pre-Algebra class. I suspect (I don't have the book in front of me right now) that the matrices bit is in the latter half of the book and is it is advised that more Number Theory and Counting & Probability (introductory) are completed first prior to tackling the second half of the book which is their Algebra II class.

I am impressed with their pre-Algebra classes and their ability to hold my DD's 9 year ild attention and keep her engaged. I will report back on how the Algebra I class worked out in late Feb of next year.

I may have see if I can dig out an old Dolciani too...

Yes, I have noticed those things. I have never found an online math program that I like. But there are some texts that are amazing. Primary Mathematics ("Singapore math") is an example. They did a fabulous job with many things, and the order of presentation is one of them.

This is all great information. We're using AoPS books, and are well satisfied, but it's good to know which other books are the good one's especially some old one's I didn't really know about.

Thanks Val, really appreciate you taking the time. All good detail. Textbooks are likely right around the corner for DS. So far, he hasn't had any except an Algebra and a Geometry book he bought himself at a used book store.

Reading it here, I'm thinking, oh supporting pictures aren't so bad. But the Algebra book he picked up (can't remember the title) has a ton of full-color fluff in it; so, I get that idea and it is distracting and wasteful and doesn't help engagement that I can see.

I guess that also trumpets the state of things. I had actually imagined it was something more pedagogical on progressions and which topics lead at what levels to what topics. But if cutting through noise is significant, then it's hard to even have a functional dependency discussion.

Personally, I'd be looking for scaled problem solving, clear overviews with theory and related material. I'd most like to see a flowchart to see the how/why/what behind topics and sequence. But that would be for me, DS has his own instincts and seeks out material of all forms; I'm not certain I'll ever have much curricular input on what he does.

I really have no advice to offer up on this thread - I've honestly always just gone along with whatever text our children's school throws at them for math classes, but I had to lol about the ADHD comment, Val, because my older dd's school has a new math curriculum… not EM but of the "spiraling" variety… and Oh Good Grief.. that is the very first thing when I saw the new mathbook - it looks like the book has ADHD!

Actually, it looks like it was authored by the folks who publish the "Yoga/Car Maintenance/Everything-under-the-Sun for Dummies" genre. It's filled with outlined boxes of random methodologies, little hints flying in from the side, crazy fonts of all different sizes and shapes, none of the chapters are more than two small topics long, then the next chapter is about something else entirely unrelated to the previous chapter, and they purposely throw in at least one problem that has nothing to do with the current chapter in each end-of-topic set of questions… just to be sure the material is "reinforced" and the kids don't forget something they might have learned three chapters ago. I got a headache just looking at it, and I can't even imagine how my dd (who has both vision issues and a challenge with staying focused in the presence of the smallest of distractions) can even begin to not want to just toss the book off a cliff

Thanks for allowing me to have my little rant over my dd's totally silly math book

polarbear

Among the many treasures DD9 picked out for herself at the library sale was an above-level (8th grade is my guess) earth sciences textbook. She flipped it open to a random page, and as I looked over the disjointed mass of colors, images, and text boxes, I was prompted to say, "It looks like somebody barfed all over the page." Then I pointed out the absurdity of the text box, in a text box, in a text box, in a text box.

I declared the result to be unreadable, but DD assured me that she could read it with ease, as she understood the patterns underlying all the barf.

(returning the title to its regular upright position.)

Hmm, that is another consideration the modern construct of busy pages, txt msging and such. Or maybe it just takes a natural instinct in forensic toxicology.

A few years back I complained because the "planner" our school was asking our children to buy was like this. It was crazy, they want to teach our kids to USE the planner but it's impossible to find the page you want. And the page is so busy it's hard to figure out where to write, and then you only get a tiny amount of space. The thing was 3X bigger than it needs to be. It had all this extraneous stuff about schools and even ads. So the kids they are most desperate to get to use it can't even function with it because it's simply too busy.

Thankfully someone listened because the planner this year is a lot smaller and more streamlines.

Well, I've seen a lot of different math textbooks in the pre-algebra through calculus sequence over the past six years or so.

I have to say that I have nothing much to add to Val's detailed and (IMO) accurate assessment on the previous page. Just one thing, perhaps-- that I suspect that the distracting content is intended to make such textbooks less threatening/intimidating to those children who have math phobia. Unfortunately, they also teach very little in the way of mathematics and numeracy, but they do not frighten students with walls of text or pages filled with derivations.

{sigh}

I do think that some things about modern pedagogy in mathematics COULD be very good. It is mystifying to me why it isn't a better state of affairs there, honestly. Because what was done in 1970 wasn't terribly effective for a huge segment of students who had learning challenges or differences, and the pedagogy exists (in spades) to reach that segment now. Technology alone makes some things possible that used to be excruciatingly slow and laborious to implement. I have no idea why we've thrown the baby out along with the bathwater here, however.

Most of the educational theorists seem to focus on equity. So there's a lot of emphasis on how to reach non traditionally successful students. That's then filtered through the textbook companies who have incentives to make thicker and more flashy books which are then selling to school districts with checklists for what the books need to meet.

Ds11 uses Browns Structure and Method Algebra book 1. It is jammed packed.
To cover this in one school year your dc will have study the text on their own to as to not miss things.

We also have AoPS Introduction to Algebra at home.

http://www.zooktutoring.com/the-best-algebra-book-in-the-world/

Yes, it's very strange. Stranger still is the idea that if any of these books are trying to reach that segment of students, they're probably making things a lot harder for them. That mishmash of random ideas I mentioned is primarily responsible for that problem. And there's more; the school's book also uses terms with exponents casually in the absence of any instruction about them. That kind of thing must be crazy frustrating for students (who probably blame themselves for not understanding any of it).

The pre-algebra book foisted on my second son last year was even worse than the one I described. Chapter 2 (!) had students solving thorny inequalities with variables. The book was full of out-of-place stuff that required a huge amount of background knowledge that hadn't been taught yet. The students were just given recipes to follow.


Yes, the problem definitely isn't restricted to math books. It's pretty much everywhere. Multiply-nested text boxes, indeed.

6th grade I had to go make a planner for my son that worked better than the stupid one they made...now her uses his calendar on his phone in 9th grade. I said I was going to market my better planner.

OK... I'll put away my 60's Dolciani Structure Method books for another day But, I dispute that anything we might know about pedagogy has actually made it into most modern books. Plenty of old books are better than anything I've seen today for both high and average ability students.

The best book I've seen for either very young or math phobic folks is Harold Jacobs' "Elementary Algebra". It feels like it is written by a smarter version of your goofy 8th grade math teacher. It has clever engaging examples. It has relevant picture, cartoons, and diagrams widely interspersed. It is written by an actual mathematician. It is recommended by some of the previous Epsilon Camp staff. It is more rigorous than most current texts but somewhat lighter than AoPS, Foerester, or Dolciani... and it has a copyright of 1979.

My copy of Jacobs' Geometry is from '74. The '86 edition is also fabulous. The 2003 3rd edition is more discovery and light on proofs for some peoples tastes.

The Jacobs' books are designed to be both rigorous and approachable. There doesn't have to be a tradeoff. Regardless of student ability, I would pick these books from the 70's over any research based, committee designed, standard public school text commonly in use today.

Man, I know that Jacobs text of which you speak. I LOVE that book. I learned geometry from the 1974 imprint. Love-love-love.

{sigh}

What mystifies me is that a few decades' worth of research should have made for a landscape filled with examples like it-- and it hasn't for the reasons that Val illustrates so amply (both in her original post and in the one above).

Spiraling is a complete disaster in the hands of a publishing industry that doesn't actually pay Subject-Expert AUTHORS to write textbooks. I'm convinced that the people assembling such a dog's breakfast genuinely fail to appreciate that they haven't taught the students some of the things they're presenting out of order.

Nutty as Christmas fruitcake, but there it is.

I found this article interesting, A Brief History of American K-12 Mathematics Education in the 20th Century by David Klein . He discusses the role colleges of education have played in promoting progressivism in mathematics education. It's one explanation of why math textbooks are so bad.

If you haven't read this NYT article about the Japanese method, and it's relationship to historical US pedagogical methods, you might find it worth the time:

http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?_r=0

My daughter's honors calculus course is using some of the Matsuyama-Takahashi-Lampert methods discussed in that article.

I'm thrilled for her-- it's exactly what she needs to really light up with a subject. That Socratically-oriented approach is so powerful.

I'm just not sure how a textbook fits into that framework.

Eta my favorite quotes from that article:


I've frequently been APPALLED at the level of preparedness in the subject that my DD's teachers have themselves confessed to her (or me)-- while lovely and caring people by and large, there is something very wrong with a high school geometry instructor who "doesn't really understand calculus."



Indeed...



YES, YES, YES-- this is the kind of dynamic classroom that isn't exactly "flipped" (though I suspect that it is the engine that drives success in flipped classrooms by keeping students actively engaged) and is highly successful for pretty much ANY discipline, and even moreso in STEM where students must master their own internal set of concepts in order to build a framework of real understanding and not just memorization.



I'm wondering if anyone has read Elizabeth Green's book, which was the basis of that NYT Magazine write up?

Yes, exactly. The Brown/Dolciani books "spiral," but in a very different way than the current circus of books. Brown/Dolciani stay focused on one theme at a time. They present information, provide problems of increasing difficulty...and then, at the end of each section, they give 15-ish mixed review problems that are there to remind you about the stuff you learned last week.

I ask myself how the spiral idea originated. Maybe someone out there was thinking about mixed review exercises and decided that if 10 milligrams of spiraling worked well to cure forgetfulness, then, boy howdy! --- maybe we should try 10 full grams of the stuff! Dunno.

I tried a web search, but only found this kind of stuff:




In other words, our curriculum has a florid case of ADHD. It can't sit still and it can't focus.


Some people even say it seems "spaced out."

That article is titled "Why Do Americans Stink at Math?", but I have not seen evidence that this is the case when accounting for demographics. East Asians do well on international math comparisons, but children of East Asian parents in the U.S. also do well in math.

Joanne Jacobs http://www.joannejacobs.com/2014/09/japanese-learn-math-in-cram-schools/ linked to a blog post by a Japanese writer Big Doubts on the NY Times Article: "Why Do Americans Stink at Math?" who pointed out that a large fraction of Japanese children attend afterschool (juku) programs, so to the extent that the Japanese are good at math, we don't know if the schooling or the afterschooling should get the credit. And if the afterschooling used the same teaching methods as the schooling, parents would be less likely to pay for it.

I have added the Spivak and Jacobs books to my wishlist to keep track of for future reference.

This whole subject is why I still have my old textbooks, and why I buy old textbooks at yard sales and library sales -- I figure they're only getting worse, so we can always go back and refer to the old ones if the new ones are incomprehensible. And, indeed, we do refer to the Algebra and Geometry ones when DS's e-school materials leave us scratching our heads.

Maybe, but how many high-performing Asian (or other) kids in the US are going to Kumon/the Mathnasium/other tutoring services (or getting parental help)? At least in the American case, we can be reasonably sure that a lot of those kids are getting their knowledge from sources outside the schools.

I have a couple dozen random math books in my house (my analysis stuck to four Algebra 1 texts). I've also looked closely at every book my kids have brought home over the years, and I can say without reservation that today's mainstream US math books suck eggs.

I don't know about Japanese math books because I can't read Japanese. But my eldest went to a French school that used the French national curriculum. The math books he used were superior to the American books (I still have them). The Irish mathematics curriculum is also high-quality. Certainly, at least some students in those countries get extra help, but that fact doesn't diminish the quality of the math education in those places.

I would consider Math Mammoth to be an elementary program similar in style and tone to the Dolciani books. The author is not a mathematician, but her background is sufficient to do a better job than most American elementary textbook authors.

We mostly used Singapore for elementary, but I really think I like Math Mammoth better for teaching the Why's directly to the student. The key is to read all of the blue boxes and only do a few key problems in each small section then focus on puzzles and word problems. Otherwise it's overkill for gifted kiddos.

I use it also to tutor kids who are being taught with sub-standard texts at local public school. http://www.mathmammoth.com/books.php

Do others have experience or viewpoint to share regarding TERC investigations as a math pedagogy, such as the math process shown in this youtube video:

Our district used TERC Investigations for 12 years and it was a huge reason I decided to homeschool my younger son.

I have extensively reviewed the fourth grade TERC books alongside the fourth grade Primary Mathematics (Singapore) and Saxon books. TERC had *far* less practice and the practice it had was with numbers that were "easy." So, fraction problems stuck with halves, thirds, quarters, sixths, eighths, and twelfths whereas the Singapore book had kids dealing with more complicated fractions, like 15ths or 17ths (I'm making these up, but it was like this). TERC had something like five longish division problems in the whole book for the kids to do whereas in Saxon it was up around 100 and Singapore was somewhere in between.

The TERC books I reviewed were an older edition. My understanding is that they came out with something to address concerns about the lack of practice later on.

TERC came to my attention when my younger son was going to be entering kindergarten. When we looked at a well regarded private school for him, they very proudly announced that they used the Investigations program which in kindergarten would be focusing on the number 6. The number 6 for the whole year? Apparently yes. This was my kid who understood place value into the thousands among other things. I knew that wasn't going to work, so I looked at the local public school, thinking that whatever they were using had to be better than that, and that's when I found out that literally *all* of the schools in the area, public and private, were using TERC.

(Sorry for the rant.)

Our district (public) uses TERC. However, I'm not sure my opinions would be helpful, because our district supplements heavily with locally-derived materials.

From what we have seen, I agree that it is light on practice and 'drill' in the early years of the program, but our schools were aware of this and I feel the (supplemented) curriculum offered here did a good job addressing this (too much for my kids, but they are outliers). There is also too much spiraling for our tastes, but again, my kids do not seem to be typical math students. There is also a fair amount of explaining and writing required; this was irritating at times but not an issue for my kids; they are comfortable writing and often enjoyed the opportunity to think more deeply. There was a fair emphasis on manipulatives, IIRC, but not sure if this was inherent to the program or our district; specific teachers used their discretion and my kids were not forced to use these if they didn't need/want to.

My biggest criticism is that there was no opportunity for acceleration or differentiated work, but again, I suspect this was district-specific, not TERC-specific. They do have challenge problems, which were often assigned to be thought about and completed over several days; my kids usually completed them during the class period in which they were assigned, if that tells you anything.

The general consensus here seems to be positive; our district consistently has some of the highest math scores in the state on standardized testing, including the new common core tests (take that as you wish). However, and this is a big however, the teachers did not use the materials exclusively, or even most of the time, depending on the unit, grade, teacher, etc. Also, our district has a well-developed program for math help, remedial and otherwise- it is tough to fall through the cracks.