Oh my. I am interested in graphic design, and this problem makes me want to go to sleep. It’s a shame, too. Programming and graphic design are fields that high school students can actually go into while still in high school. There is probably something that can be done here.

The thing that really gets me about pseudocontext is that it completely obscures some of the awesome beauty in mathematics. I mean, think about the things that excite mathematicians about their work. Are students really that different?

PS — And to think, I actually believe the McDougal-Littell Geometry book is pretty good!

@mmmsoap – If you actually cared about the logo, wouldn’t you put it into the hands of a trained graphic designer?

Every observation that gets made leads to the inevitable, forehead-smacking conclusion that pseudocontext is just plain counterproductive. If not evil.

I hate this book – this is a perfect example of what Michael Serra means when he talks about textbooks convincing students that math is simply writing a boring 5 or 6 line proof to show that two angles that _look_ to be exactly the same size actually are. This book is filled with similar pointless exercises in formalism.

Does anyone know of a good math text series, one that is readable, interesting, age-appropriate, and more ‘real’ in its applications? I haven’t found one yet. Sometimes I think of wanting to create a text (I love to write materials), but the challenge overwhelms me. Any recommendations?

I think this problem is savable as well. Dan and CalcDave got us in the right direction. Granted processing poower is cheap, but having the students realize that cutting redundant checks for symmetry increases processor speed could be a great lesson for the techie types.

It also doesn’t really matter if someone would actually use proofs to check this problem… what matters is if the context makes the problem interesting and worth solving in the students eyes.

I think there is a buy-in here, if the problem is presented as Dan did. One issue could be that saving processor power will not be understood conceptually by some of the class, but the percentage of which I am not sure. It may be the same as the percentage who won’t care anyway.

As a previous engineer, I can attest to the fact that if you went to your boss and said that something we were programming was not necessary, and you showed him/her why, you would be the type of employee gets raises. Telling students something like that would increase the buy-in as well.

I feel like trying “motivate” proof with “real-world” context at this early stage (high-school freshmen/sophomores) is an exercise in futility. Either we need to just say up front, “Look, one thing we’ll be learning is how to break down the process that’s happening in our heads when we say ‘That’s obvious’ into small logical deductive pieces. It is not easy to do, but it will serve us well in the future,” or we need to go the more radical Paul-Lockhart route of ditching the rigid two-column method for plain-old English sentences, and lighten up on the “rigor.” Right now I do the former, and I kind of wish I could do the latter. But we’re currently using this very Geometry textbook, so I’m a little limited in my options.

@Dan If you haven’t read A Textbook example of what’s wrong with education it clears up a lot.
What amazes me is that some publishers tout the worst aspects of their texts as positives! The series that we are (unfortunately) using for 9th grade science is compartmentalized and “designed to be used in any order.” Which is to say that there is no context, anticipation, reflection, or overall theme. The fact that it’s also often factually in error is just icing.

I agree that we should spend far more time planning, but also that we should use technology to divide that effort. I’ve been pushing for several years to get common planning time (by departments) at my district so that we can more effectively share and divide the labor.

@Bill:

I’m curious about your comment that content “designed to be used in any order” necessarily suggests a lack of context or coherence. Are you thinking of math (or science?) as being fairly linear?

Amen to Bill’s comment. I don’t understand how textbooks, especially math textbooks, could say that the topics can be taught in any order. The ability for Chapter 3 to build on concepts from Chapter 2 is critical to students’ understanding of the whole of mathematics, instead of thinking of it as a sequence of tiny topics with no connection.

But there are definitely texts, even successful ones, that do this very thing. No wonder kids don’t remember what they learned a month ago — older concepts fail to stay relevant or useful in that environment.

Dan, you say “These people are making billions and they’re putting out junk”. Who is this calling out, and where can I get a share of these billions? ;)

Thanks for the series of these, it’s fun and sad!

“Maria Droujkova
Matt,

Why not prove things that aren’t obvious?”

As a Geometry teacher currently working through the foundations of logic and elementary proofs right now, I am often vexed by this question- I spend much time cautioning my students that “Although we can all tell this is obviously true, we need to slow our minds down and explain every step”. Indeed, we do end up proving much that isn’t obvious, but we end up needing more sophisticated parts put together in subtler and more sophisticated ways to get there. How do you prove things that aren’t obvious at the start, when the need is greatest for developing the motivation for learning proofs. I suspect the answer is in using the perfect set of specific examples- ones that strike that delicate balance between not obvious and possible for the novice geometer- but which ones work for this?

@Maria: I’m going to piggy-back on your comment by saying that many times beginning Geometry students think that things are obvious because they make assumptions about a problem/situation. They sometimes see the obvious when it isn’t there. I’m sure we have all had students who strongly, firmly believe in their hearts/guts that a shape *must* be a square because it “looks like one”, or other less face-palm inducing, but still problematic, misconceptions. I think that part of the slowing down that you mention is getting students to realize the assumptions they are making and to question them.

The problem I see with getting students to prove only things that are obvious is that this doesn’t challenge their misconceptions. If all you do in Geo is prove things that you already know are true, then you won’t see the point of doing it.

@Breedeen:
Well, we spend a good bit of time parsing what is legal to assume (from a diagram or from the given) and what is not. I have not problem taking students to task with falsely grounded assumptions in this respect, seeing the “obvious when it isn’t there”.

I’m more referring to those examples when the concepts of proof are first introduced- when a student can intuitively apply the transitive property of congruence, say, and needs to understand the importance of slowing down and proving every step along the way. It’s a challenge to convey the utility of proofs when the answer is legitimately “obvious”.

But I completely get your point.

And to your last point, we sure do end up proving stuff we don’t know to be true, but it comes later after skills have further developed.

Maria,
Good point- with a properly constructed square in GeoGebra, the properties of ‘squareness’ are preserved. As you know, though, it takes a little familiarity with the program to properly construct a durable square.

I would also note that this is a powerful entry into the need for formalism around labeling diagrams and requisite symbols. I often stress that the formalism saves a poor artist like myself, as a quickly sketched square is not really a square until I label all angles as right and all sides congruent. Internalizing this helps my students avoid the intuitive potholes of “well it looks like a square” or “that angle looks right” or “those triangles look congruent”

@Maria
I didn’t say it wasn’t possible, but it certainly isn’t being done by the major textbook publishers!