From P.J.:
I don't have a simple definition of an "authentic" problem, but here are two examples from the last couple of weeks. Though quite different from each other, both problems generated intense discussion and a variety of ideas.
I don't have a simple definition of an "authentic" problem, but here are two examples from the last couple of weeks. Though quite different from each other, both problems generated intense discussion and a variety of ideas.
Students entering my Geometry classes on Tuesday were confronted by a string running from the top-north-west corner of the room to the bottom-south-east corner, bearing a pink sign with the words "How long is the string?" Even students who ordinarily sit back were up and about, measuring, arguing, and figuring out a way to determine the length of the string. They extended the Pythagorean theorem to find diagonals of three-dimensional boxes, discussed their formula's validity, and had a great time.
The week before, we started our lesson with a problem I found in a samizdat geometry book from Germany (any reference would be appreciated--my copy of a copy of a copy has no title page or publication markings of any kind). Compute the area of the trapezoid, if A is the center of the circle.
Although totally abstract, this problem was also fruitful. It took students a while to draw in the radii:
Then more questions started: what's the length of EF? Does it matter? Is EAF a right triangle, and how can we be sure? Ultimately, our discussion led us to Garfield's proof of the Pythagorean Theorem, #5 here.
These stories have two morals. First, to grab students' attention, problems must be challenging and doable. But (second), those problems don't need to be "real-world" or "applications:" kids can get excited about figuring things out, even when what they're figuring out is "just" a math problem.
What do you all think? What are your favorite problems? What made them valuable? Share links in the comments or email them to us directly.
I think of an authentic problem as one that is doable as you say, but has no known "starting point on how to solve (or answer) it. Tomorrow, in the first lecture for my middle school teacher content course, I plan to pose the problem, "On average, how many students here at the University do you think get a hair cut on a semester weekday?" I hope to not only get the students talking to each other (class of 70), but also hope to touch on six topics I plan to cover during the semester. I will let you know how it goes. Kathleen
ReplyDeleteI like the idea that engaging problems are ones where you can present the setup, and there are natural questions people will ask upon seeing it. For instance, the string prompted the question by having a sign. But even without the sign, after the opening salvo of "Why is there a string here? Why does it go from here to there?", if you asked the room "What questions can we ask about this string", one of the first ones is going to be "How long is it?"
ReplyDeleteSimilarly, the circle+trapezoid only have a few things going on. I like the fact that one of the followup questions to "How long is EF" is "Does it matter?" To me, that's part of what makes an engaging problem fun. There is more than one path to follow, and not all of the facts you can deduce are useful to your ultimate goal.
Conversely, I think a question has some troubles if you have to add in "Assume for the moment that you care about X; how would you derive it?" If "X" isn't some factor that jumps out of the setup, then you've got a serious motivation problem. To measure the string motivation problem, imagine instead that after somebody asked "How long is it?", you went on with some other lesson. You just know that at the end of class, people are going to say "Well, okay, but really - how long is the string?" Then you have won!
Bacon
Seems to me that part of what makes a problem authentic depends upon "to whom is this question being asked?" If someone already knows the Pythagorean theorem and knows that it applies to these questions, than neither is particularly a problem, authentic or otherwise. It's not unlike asking whether 37 - 19 is 'authentic': it is to a child who hasn't been taught multi-digit subtraction. For one who has, it's likely a mere exercise.
ReplyDeleteI would suggest that "real world" and "authentic" aren't isomorphic, either. See, for example, Jo Boaler's comments on 'pseudo-context' in WHAT'S MATH GOT TO DO WITH IT? or any number of blog posts on pseudo-context by Dan Meyer.
This week the Berkshire Eagle, our local newspaper, reported about a math club at a local elementary school. They gave an example of a typical math problem posed at the after-school club, which I used yesterday with my GED students:
ReplyDeleteA single piece of wire is folded to form a square with an area of 225 sq.cm. The wire is then straightened, cut into pieces of equal length, and those pieces are fastened together to form the frame of a cube. If no wire is wasted, find the VOLUME of the cube, in cubic centimeters.
We did area, perimeter, and volume some time ago, and this problem was a good way to review the concepts with them while challenging them at the same time. They struggled with it, but as a class were able to solve it. What was exciting about it was how engaged they were in trying to solve it.
The length of the string problem sounds great also. I would love to hang a string diagonally across the classroom, but since my GED class is in a county house of correction, that would not be possible! Any suggestions other than pretending that a string was hung across the room?