Context should add something to the problem, whether it be intrigue, interest, or a way for students to pull from their intuition, and prior knowledge. It is the absence of reaching these measures, that makes me characterize this problem as pseudocontext. No student is going to read about the daffodil logo, and then feel compelled in anyway to prove the leaves to be at congruent angles.
This particular problem became more interesting after I took away the context. That could become another measure of how we judge context v. pseudocontext. Is the problem more interesting after the context has been stripped away? If so, then the context was actually pseudocontext.
You are designing a logo to sell daffodils. Use the information given. Determine whether the measure of angle EBA is equal to the measure of angle DBC.
- Scan an example of pseudocontext.
- Email it to firstname.lastname@example.org
- List the textbook title, edition, and publisher.
- Give me your interpretation of the term “pseudocontext.”
- Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.
CalcDaveOctober 9, 2010 - 8:51 am -
While I agree that this specific context seems poor, I do think it’s another example of a time in someone’s life where someone would say, “I never thought I’d need that stuff from geometry, but THIS is that time!”
It seems artificial to high school students to try to imagine themselves as a graphic designer and put this kind of context in there, but I am sure graphic designers work with angles all the time and this is the sort of thing they might deal with often.
Now, is a proof like this important to them? Probably not. But, I would imagine that constructing congruent angles and looking for visual symmetry is.
WhitneyOctober 9, 2010 - 10:15 am -
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?
Elizabeth SOctober 9, 2010 - 12:39 pm -
As someone who managed an internationally well-known family of brands for many years, may I be the first to say, “AAAAAAAAAAAAAAAAAAAAAAAAAGH!”
Elizabeth SOctober 9, 2010 - 12:41 pm -
PS — And to think, I actually believe the McDougal-Littell Geometry book is pretty good!
mmmsoapOctober 9, 2010 - 3:36 pm -
@Brian Miller…..If you actually cared about the logo of the daffodil, wouldn’t you just *measure* the angles? This is throwing weird math at a problem that doesn’t need any. Sure, it might be interesting to know that these angles are *always* equal, but with the “context” given, you don’t care about all angles, just the ones in your logo.
Elizabeth SOctober 9, 2010 - 3:46 pm -
@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.
BrianOctober 9, 2010 - 4:06 pm -
Actual execution of this:
“You are designing a logo to sell daffodils. How should these three petals be arranged to be most appealing to the consumer?”
“You are designing a logo to sell daffodils. Research has shown that symmetrical logos entice consumers to buy more goods. How can you ensure the symmetry of the figure?”
Probable solution to the second scenario: use a computer program. Done. Time to go work on the next project.
NathanOctober 10, 2010 - 4:13 am -
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.
Dan MeyerOctober 10, 2010 - 6:57 am -
I hope to avoid turning these features into a mob so I appreciate CalcDave’s generosity to the problem. Graphic designers do need to know their angles pretty cold. Graphic designers do deploy symmetry in their work.
This problem barely nods to those concerns, though.
I think we could make this work:
“A company that specializes in floral arrangements needs 500 individual logos for each of the flowers they work with. They will supply you with a petal from each flower. You need to turn that into three petals, rotating one 40Â° to the left and the other 40Â° to the right. They need symmetry of every kind (including the kind mentioned above). We have created a script that will do that for every single petal, but it taxes the computer a great deal, and we need to know in advance that the symmetry will work. Can you prove to your bosses that this will work?”
With one logo, you can just trial-and-error the thing. You can measure those angles to make sure they’re equivalent. A teacher could say, “Yeah, sure, they look equal, but can you prove it,” and I’d be lying if I said I hadn’t played that game. Bulk is better. With bulk, you can’t measure each one individually. With bulk, you have a great incentive to get it right academically so that in practice you aren’t wasting processor cycles or ink or paper or whatever.
Pam RichmondOctober 10, 2010 - 1:25 pm -
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?
ComputerGeekOctober 10, 2010 - 3:48 pm -
Dan, I appreciate your attempt to find some justification for the use of proof here, but I’m pretty skeptical. “Saving computer cycles”: but today computer cycles are super-cheap, and human time is super-expensive. From a financial point of view, simple trial-and-error seems likely to be much more effective.
Then again, I always hated geometry class: I felt like the proofs were utterly pointless exercises. (And I say this as someone who majored in math in college, so in general I love proofs — just not my stupid high-school geometry class.)
BrianMOctober 10, 2010 - 5:31 pm -
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.
Maria DroujkovaOctober 11, 2010 - 7:15 am -
What frustrates me no end… There are gazillions excellent geometry problems in graphic design. Anyone who has made any symmetric logo met such problems. However, designing a good problem, even if you are familiar with the area, takes 10-20 times as long as making up something like this.
This is the house that Jack built.
Students are driven through hundreds of “quick exercises.”
Teachers are required to drive students through at that pace.
Curricula have to provide exercises for that pace.
Curriculum publishers set up content writing practices matching the pace, including writer salaries and automated problem generation.
Students, teachers, and curriculum makers need to spend TIME on their problems. Like, ten times more time.
Matt EOctober 11, 2010 - 7:28 am -
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.
Maria DroujkovaOctober 11, 2010 - 8:16 am -
Why not prove things that aren’t obvious?
Dan MeyerOctober 11, 2010 - 9:08 am -
This is right on. Pseudocontext offends my work ethic more than it does my delicate mathematical sensibilities. I spent hours filming one problem this weekend. One. I’ll spend a few more hours editing that video. These people are making billions and they’re putting out junk.
Spend ten times the time. Make a single problem that’s ten times as good as the status quo. Drill your students on ten times fewer problems.
Maria DroujkovaOctober 11, 2010 - 9:50 am -
Let’s talk about billions.
We spent almost two hours this Thursday playing with “star formulas” – basically, one problem about star polygons, with its “relatives” (related problems) that came up. This is a math club; we will also spend a few hours writing the story and dealing with photos, and maybe later some time putting stories together into some form of a site or a book. The math club involves very little money exchange: members contribute $15 per six meetings, which we spend on snacks and polyhedra modeling straws and such.
How can we make this work sustainable economically? Will it always be very indirect, such as increases in for-fee consulting invitations?
Bill BradleyOctober 11, 2010 - 11:38 am -
@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.
Matt EOctober 11, 2010 - 4:11 pm -
@Maria: We get around to that, of course. I would do more of it sooner if I didn’t have to be so concerned with what my textbook is doing, and when.
KarimOctober 11, 2010 - 4:33 pm -
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?
Bill BradleyOctober 11, 2010 - 6:21 pm -
Not linear so much as consistent. There are recurring themes in both science and mathematics and to not not reflect back on “where have we seen this before” or set up a logical sequence removes a great deal of effectiveness. Neither Science nor Mathematics were discovered in a random fashion, yet we teach them as if they were (or even could be).
BowenOctober 12, 2010 - 6:12 pm -
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 DroujkovaOctober 13, 2010 - 3:34 am -
@Bill, @Bowen: it is possible to achieve rather large flexibility in the order of topics. There are several learning designs doing this that I like. Here are two examples.
Learning based on problem-solving can be rather open, within each level. Each problem is based on several concepts, which tend to be presented more independently from one another, because learning is not organized around them. As a student works on a problem, she fills in whatever concepts are needed to solve it. As a result, a single problem may take hours or days to solve, unlike exercises. This requires access to knowledgeable mentors.
Learning based on “big ideas” and projects is indexed by overarching ideas that come up in daily work, such as the idea of transformation or taxonomy. Individual activities are indexed by big ideas and levels within the ideas, and may or may not have dependencies. You can visualize this as a dense network of activities, with mapped MULTIPLE paths among them, each connected to big idea “hubs.” This requires a good expert system to navigate, which are usually computer-based. However, an expert teacher can help students navigate without a computer, individually or in small groups.
Ed GeometryOctober 13, 2010 - 6:38 am -
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 DroujkovaOctober 13, 2010 - 7:46 am -
You put it perfectly: “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.”
There are some good examples in Lockhart’s book. There are other collections, such as The Art of Problem Solving accumulates in various forms. However, this brings us back to the issue of quality and time.
It is significantly harder, and takes significantly longer, to find these GOOD (let alone “perfect”) examples than random problems easy enough for novices.
Who is going to spend ten times more time?
Who is going to pay for it?
BreedeenOctober 13, 2010 - 7:50 am -
@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.
BreedeenOctober 13, 2010 - 7:51 am -
oops. That should have been @ Ed Geometry.
Ed GeometryOctober 13, 2010 - 8:03 am -
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 DroujkovaOctober 13, 2010 - 8:09 am -
Breedeen writes: “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 love how GeoGebra clarifies this particular issue. I can fully acknowledge that a shape looks “close enough” to a square to be a passable representation of one. However, what I require of students is that the shape stays square when I drag points and lines around, and otherwise change the construction within its degrees of freedom. If there is too much freedom, it won’t be a square anymore! Obviously so!
Computer construction provides a good language to discuss the issue.
Ed GeometryOctober 13, 2010 - 8:32 am -
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”
CalcDaveOctober 13, 2010 - 8:38 am -
This is a “puzzler” to prove your point: http://www.computermagic.gr/Fun/StrangeMaths/FromWhereComesThisHole/DefaultUK.aspx
Bill BradleyOctober 13, 2010 - 1:32 pm -
I didn’t say it wasn’t possible, but it certainly isn’t being done by the major textbook publishers!
DougOctober 14, 2010 - 6:02 pm -
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