Schwan Smith, M., and Stein, M.K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.
This article stresses the importance of thoughtfully selecting/assigning/creating mathematical tasks for students. The tasks presented to students should facilitate the intended learning, and to be sure that happens, teachers need to pay close attention to the level of cognitive demand tasks will potentially place on students. Low demand tasks may help students learn math facts and develop procedural fluency, but tasks with greater complexity and even procedural ambiguity have the potential to engage students in high level reasoning and connection-making.
It would be very interesting to have the "Task Sort" discussing with my own math department.
- Here's an example of a low-level task from my precalculus class:
Rewrite | pi - sqrt(5) | without the absolute value symbol.
- Here's an example of a high-level task from precalculus from one of our problem solving days:
A line tangent to a circle is a line that intersects a circle at one point and is perpendicular to the radius at the point of intersection.
The line 2x-3y=-11 is tangent to a circle at (2, 5).
The line 3x+2y=29 is tangent to the same circle at (7, 4).
Find the standard form equation of the circle.
Tuesday, October 29, 2013
Sunday, October 27, 2013
Making an Ambigram
One of our assignments for class this week was to do some math art, and one of the suggestions was to try making an ambigram. Ambigrams are words or phrases that can be read in more than one direction, or from different points of view. In class, we had looked at several examples by Scott Kim. Having never made an ambigram before, I decided to try what seems to be the simplest type - one that reads forward and backward (upside down).
After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere. Geometry seemed like it could work well. It wasn't too much of a stretch to imagine a capital G that looked something like a y.
Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time: G and y, e and r, o and t, m and e.
I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.
I had the hardest time with e and r. I was tempted to just borrow Scott Kim's solution to this one in his "SuperTeacher" inversion, but I kept trying and came up with something else that worked. My solution to m and e, right in the middle of the ambigram, was to go with a basic lowercase m and turn it 45 degrees. Not fancy, but again, it worked. Once I had each symbol figured out (four of them altogether, since it was two letters for each), I set about drawing a large copy of the ambigram.
To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.
The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?
After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere. Geometry seemed like it could work well. It wasn't too much of a stretch to imagine a capital G that looked something like a y.
Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time: G and y, e and r, o and t, m and e.
I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.
I had the hardest time with e and r. I was tempted to just borrow Scott Kim's solution to this one in his "SuperTeacher" inversion, but I kept trying and came up with something else that worked. My solution to m and e, right in the middle of the ambigram, was to go with a basic lowercase m and turn it 45 degrees. Not fancy, but again, it worked. Once I had each symbol figured out (four of them altogether, since it was two letters for each), I set about drawing a large copy of the ambigram.
To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?
Tuesday, October 15, 2013
Geometry at Artprize
My wife's family is very serious about Artprize. Her parents and sister volunteer several times during the event. My wife and daughter and I usually make two to three, if not four, trips downtown during Artprize. The first day we went, I took our camera along, in hopes of finding some geometrically interesting art. I had hoped to notice some geometry that was very subtle yet deeply tied to the art. Most of what I photographed, though, was pretty obvious in its geometry. Nothing too profound here. And this is only from one visit, so I'm sure there were many more "geometric" entries that I never ran across. But here are a few things that I saw.
In the Gerald R. Ford Museum, one of the entries was Reflection, by Josemiguel Perera. Aptly named, I thought, since it has reflection symmetry across a horizontal. I added these circles to the picture to emphasize the concentric circles in each fan. The little circles are tangent to one another. Of course, then the larger circles with the same centers intersect. I thought it was interesting that connecting the points of intersection on the larger circles created a segment that passed through the point of tangency of the smaller ones. I suppose this would be the case any time two congruent larger circles are concentric with two smaller, congruent, externally tangent circles. That's a fairly specific situation, though.
I also found some of the Artprize signs interesting. They suggest geometry at a pretty basic level, so this was no great discovery on my part - all circles, triangles, and squares. From the right side of the sign, this pair of shapes stood out to me (right end, middle row of figures). It's a quarter circle with an isosceles right triangle attached to the side, with legs equal to the radius of the circle. It's divided up differently than that, though, with a long diagonal across the middle. My drawing from GeoGebra shows three triangles, and all of them have the same area. Each of them has the radius of the sector as its height, and half of the radius as the base (the vertical radius is bisected by the diagonal - the two line segments are diagonals of a parallelogram). Each of the three triangles has an area of (pi*r^2)/4. The light blue portion of the this figure (as seen in the Artprize sign) has the same area as that of a full quarter sector of the circle with this radius. These are somewhat random musings, but it seems there is probably more in this picture that could be uncovered and might be interesting.
At another venue, someone had put together a bunch of square Artprize signs into an interesting design. This got me thinking about squares inscribed inside one another, with vertices tangent to the midpoints of the outer squares' sides. It's related to the special, isosceles right triangles that show up when these midpoints are connected, and square roots of 2 canceling one another out, but there was an interesting relationship in these, too. As more inscribed squares were added, every second square would have its side length halved. In other words, the second inscribed square would have side lengths half as large as the outermost square. The side lengths between were, according to the special right triangle relationship, the outer square's side length times square root 2. I thought it was interesting that every second square had half the side length (and one fourth the area).
There were some other geometric aspects of entries that caught my eye. Silkwaves in the Grand, by Al and Laurie Roberts consisted of flags in the Grand River. The flag poles were connected near their bases, which made it apparent that they were arranged in star formations - pentagons with triangles attached to each side. I wonder if there was a particular aesthetic reason this arrangement was chosen.
At the B.O.B., Through the Iris, by Armin Mersmann was my favorite entry. It was impressive to me because I thought the pictures were photographs, but they were pencil drawings! I usually take for granted the perfect (or near perfect, considering my astigmatism) circular geometry of human eyes. It's pretty amazing, and it was nice to be reminded of it.
In the Gerald R. Ford Museum, one of the entries was Reflection, by Josemiguel Perera. Aptly named, I thought, since it has reflection symmetry across a horizontal. I added these circles to the picture to emphasize the concentric circles in each fan. The little circles are tangent to one another. Of course, then the larger circles with the same centers intersect. I thought it was interesting that connecting the points of intersection on the larger circles created a segment that passed through the point of tangency of the smaller ones. I suppose this would be the case any time two congruent larger circles are concentric with two smaller, congruent, externally tangent circles. That's a fairly specific situation, though. I also found some of the Artprize signs interesting. They suggest geometry at a pretty basic level, so this was no great discovery on my part - all circles, triangles, and squares. From the right side of the sign, this pair of shapes stood out to me (right end, middle row of figures). It's a quarter circle with an isosceles right triangle attached to the side, with legs equal to the radius of the circle. It's divided up differently than that, though, with a long diagonal across the middle. My drawing from GeoGebra shows three triangles, and all of them have the same area. Each of them has the radius of the sector as its height, and half of the radius as the base (the vertical radius is bisected by the diagonal - the two line segments are diagonals of a parallelogram). Each of the three triangles has an area of (pi*r^2)/4. The light blue portion of the this figure (as seen in the Artprize sign) has the same area as that of a full quarter sector of the circle with this radius. These are somewhat random musings, but it seems there is probably more in this picture that could be uncovered and might be interesting.At another venue, someone had put together a bunch of square Artprize signs into an interesting design. This got me thinking about squares inscribed inside one another, with vertices tangent to the midpoints of the outer squares' sides. It's related to the special, isosceles right triangles that show up when these midpoints are connected, and square roots of 2 canceling one another out, but there was an interesting relationship in these, too. As more inscribed squares were added, every second square would have its side length halved. In other words, the second inscribed square would have side lengths half as large as the outermost square. The side lengths between were, according to the special right triangle relationship, the outer square's side length times square root 2. I thought it was interesting that every second square had half the side length (and one fourth the area).There were some other geometric aspects of entries that caught my eye. Silkwaves in the Grand, by Al and Laurie Roberts consisted of flags in the Grand River. The flag poles were connected near their bases, which made it apparent that they were arranged in star formations - pentagons with triangles attached to each side. I wonder if there was a particular aesthetic reason this arrangement was chosen.At the B.O.B., Through the Iris, by Armin Mersmann was my favorite entry. It was impressive to me because I thought the pictures were photographs, but they were pencil drawings! I usually take for granted the perfect (or near perfect, considering my astigmatism) circular geometry of human eyes. It's pretty amazing, and it was nice to be reminded of it.
Most of my geometric noticing was fairly mundane, but it was fun to spend the day being attentive to the subject in my surroundings. Here are a few other Artprize entries that stood out as being more geometric than others:
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| Sacrifice, by Tom Panei |
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| Facing Al Aquaba, by Maurice Jacobsen |
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| Hilo - Sacred Geometry, by Kimberly Toogood |
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| I failed to write down the exhibit name and artist for this one. Nice to see pi in my search for geometry, but unfortunately it was backward in every place it appeared in this exhibit. |
Tuesday, October 8, 2013
Conjecturing
Our in class activities in Modern Geometry as well as several of the assigned readings have heightened my awareness of the absence of conjecturing in the geometry curriculum that I present to my students. In general, my students read or are presented with the postulates and theorems that are important to a topic, and then we work with them. Very rarely are they presented with a relationship that they are asked to uncover on their own.
In their article Geometry and Proof, Battista and Clements (1995) suggest that students own conjectures are an ideal segue to the process of developing proofs. Providing students with an opportunity to discover a geometric relationship, and then leading them to defend their discoveries through class discussion and debate can demonstrate the importance of proof to students before they are asked to engage in it. The process of developing and testing a conjecture before proving it is also more true to real mathematical development. The article left me wanting to completely restructure the way proofs are presented in my geometry course, with ample time for exploration leading into informal proofs, and eventually some practice with formal deductive proofs. A progression like this would be more in line with Van Hiele‘s levels of geometric reasoning as well. Letting conjecture be the impetus for proof would make the learning of proof more intrinsic for students, and it would also make their learning a more genuine mathematical experience.
The value of conjecture in geometry courses stood out to me again in Diemente’s (2000) article about the Euler line and its relationship to the sides of a triangle. The mathematics presented in the article are certainly interesting, but I was most struck by the fact that the content of the article was all prompted by students’ questions and conjectures. Taking the time to explore a students’ question about whether the Euler line can be parallel to a side of a triangle, Diemente and his class covered a wide swath of mathematical ground. Their exploration and proof included midpoints, slopes, systems of linear equations, and even conic sections. What a great example of how valuable students’ curiosity can be! Not only were these and other concepts used, reviewed, and learned, but they were shown in connection with one another, and moreover, with a real purpose! While an exploration like that of Diemente’s honors geometry class might be over the ability levels of my geometry students, it might be a great study for students in my precalculus class.
So now I have a heightened sense of the value of providing students opportunities to make and investigate conjectures, and at the same time a greater awareness of the absence of such opportunities for my students. So now I am looking for places to create these opportunities. There are a couple on the horizon.
Tomorrow we will begin our unit on triangle congruence. Our department has an activity designed to help students discover the “shortcut” congruence theorems, like Side-Angle-Side, etc. It’s called “Triangle in a Bag”. Students can ask for one side length or angle measure of a certain triangle at a time, and the goal is to construct a triangle congruent to the one “in the bag” with as few given pieces as possible. The hope is that after doing this for several triangles, they might start to think critically about what key measurements are necessary to guarantee the same triangle, and develop the shortcut theorems. It’s a nice activity, but at the same time, I’m not sure if there’s much learning in it. A good class discussion might need to follow it, with some time for students to defend their conjectures as to why each set of three measures is enough. Any suggestions out there to make this activity better?
The chapter following this one includes the centers of triangles (incenter, circumcenter, centroid, etc.). For the first time, our department is planning to get students working with this with the aid of GeoGebra. I am looking forward to seeing how students respond, and I hope some of them will get excited about what they’re learning. Depending on how much time is available, it would be great to have students or groups of students share what they find about the relationships between these centers, or just the centers themselves. This might also be a good opportunity for some informal/formal proof writing.
I took my class to the computer lab for the first time this past week just to do some basic GeoGebra constructions and find their way around the program. They liked it, and more than once I ran across students experimenting with lines and figures on the program - being curious. I hope I can find more opportunities to let them be curious about geometry and learn through their curiosity. For that matter, I hope I can do that for all of my math students, and not just my geometry students.
References
Battista, M.T., & Clements, D.H. (1995). Geometry and proof. The Mathematics Teacher, 88(1), 48-54.
Diamente, D. (2000). Algebra in the service of geometry: Can Euler's line be parallel to a side of a triangle? The Mathematics Teacher, 93(5) 428-431.
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