Gifted Issues Discussion homepage Forums Learning Environments Approaches to math pedagogy, and textbooks

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...