We are seeing critical reviews of school math texts by EdReports and much talk about evaluation of OERs on LINCS. But there are many math websites and there are only occasional mentions of a few scattered ones on Math & Numeracy, and these mentions almost never say anything good or bad or detailed about the content on the site or its relevance to teaching some math topic. With so many math websites you can waste a lot of time trying to cull through them to find something that you might find useful for your class. It would be very helpful to have a growing list of some math websites with detailed annotations and critical reviews about the content by more than one teacher. One commentator might say they really liked the treatment of Topic A and give their reasons why, but another person may object and point out faults, missed opportunities and missed connections and say they prefer another website better for Topic A.
I would have liked to consult such a list last October 30 2014 when Brooke Istas mentioned the Get The Math series produced for WNET thirteen.org and said she was going to try using in class the Math in Basketball (MiB) video. On Nov. 18 she reported back briefly about her use of it, but that left me a bit puzzled and skeptical. So I watched the video of MiB, checked the challenges, and read carefully all the associated Teacher Guide explanations. And I came away irritated and dismayed. I thought I saw misleading info and false statements about the physics and graphs. Moreover, only the vertical component of velocity was mentioned - the horizontal was ignored. Where was a reasonable discussion of speed in this quadratic setting where it is very simple?
I have waited long, hoping others would try MiB and report back to Math & Numeracy with their own detailed reports - perhaps very different from my negative comments. I hope some of you readers will do that. And maybe I will later write up a detailed report on my observations.
Comments
Ladnor and others, there are so very few engaging and realistic applications of quadratics for students to engage in. The MiB video is one of those resources that I felt was forcing itself into the math rather than drawing students from the math into a real application. Anyone who has ever taken a free through (or tried to hit a baseball over a fence) knows that the action is hardly a simple one and is only duplicated most every time by those very few masters who have perfected these very complex motions. As such, most of us can not learn quadratics and then go out there and apply our learning in a meaningful way as depicted in the many basketball and baseball examples that are on the Internet and in books. I have never seen a professional player pull out a slide rule, do some quick calculations and then swing that bat to ensure if that baseball will clear the fence. Likewise, the basketball player in the video had to admit that he does not use the mathematical knowledge he is sharing when he shoots, but it is related. All of these contrived experiences sell our students on the fact that quadratics is simply some mathematical trivial pursuit topic that someone will quiz us on somewhere.
I engaged students with this lack of application years ago and we struggled for a good number of years to come up with something engaging, related to the learning needed and safe. Initially, students and I decided that we would form teams and construct catapults that would be able to project water balloons at the other teams. Each team had to stand within one hoola-hoop and was not allowed to move while others fired. Unfortunately, such a proposal was deemed a safety hazard. We modified the experience to build siege engines in class that had to be within a size range that fits on the student desks and it had to be sturdy enough to launch pingpong balls at least 30 times. In teams of 3, the students would launch their machines from three different initial heights and record the data of how long the ball was in the air, and how far the ball traveled before it hit the ground. Lacking any high tech precise tools there could be much subjectivity in time and distance readings, so we decided to do 10 flights at each initial height and students would take the average (which average to take became a great discussion later). This was all done BEFORE quadratics was introduced. I then posed the question of who's machine launched the fastest and who's launches traveled the highest based on the data we had. Of course the students now wanted to know how fast and high their machine went and how that compared to others. It was amazing to see how quickly ALL students dove into this crazy formula I shared (the quadratic of course). Since our data tables had labels that were in fact the variables in the formula, substitution was quite easy for all. Gravity did take some time to get a grasp on because many of the students had very limited exposure to formal study of gravity. Within a couple class periods (one for show/demonstration, one for peer collaboration on finding the individual values of each student) everyone had the velocities figured out (we did not follow strict physics as air resistance, stresses and fatigue of equipment, cleanness of release and data recording and so many other real errors were ignored). Now students wanted to know the height so the introduction of completing the square and what that formula and process looked like was very well received. In all, students started sharing with peers their experience and I would start hearing with every new class, "Is this the class we get to launch ping pong balls?". Not only did they love it, the ability for students to replicate the process and accurately determine which form or the quadratic was useful given any set of data convinced me that having real experiences that students can do produces much more proficiency than the contrived examples that are out there.
From a physics point, you were correct to be dismayed in the given video. In high school and adult ed settings, every resources I have seen simplifies the physics to facilitate the math learning. There is no horizontal velocity mentioned often nor is there friction, air resistance, or any discussions of how one might control those elements in a testing setting. Sadly, mathematics is taught in such isolation that both science and math people get frustrated. Science can constantly point to how inaccurate the mathematical examples and context used in construction is not real and the math people get frustrated with a lack of applications that help students concentrate on "just the math" elements. Science and Math, like so many other disciplines that have strong connections, should be taught together. This takes time and coordination and collaboration. Our teaching environment often seems short on all three of those elements today.
I agree with Mr. Latham that in the case of Math in Basketball, the player himself admits that he is not USING math, it is the students who are to use the math to ANALYZE the path of the ball when thrown up for a basket. So this is an example of a contrived "use" of algebra. And it may be true that most "applications" of quadratic functions that he has seen in math materials for adult education are indeed contrived.
But there are many authentic applications of quadratic functions and you could find lots of them in the book Elementary Mathematical Models by Dan Kalman. That was the text that I used for my freshman (non-math major) math class at UNC-Chapel Hill for some years. Most of his examples are in the form of sequences of data that satisfy simple difference equations, and linear and quadratic sequences occur frequently. Whenever you have a sequence L which is linear so that L(n) = A*n + B then the sequence S of partial sums S(n) = L(0)+L(1)+...+L(n-1) = A*(n-1)*n/2 + B*n is quadratic. So for example if you are making some gadget in a factory and you make B of them during the start up 0 year, and each year thereafter you make A more than the year before, then by year n you will have produced a total of S(n) gadgets.
I think the most interesting case where quadratic functions appeared was in the results of the experiments of Galileo in the early 1500's. That's where we first saw that distance travelled was quadratic in time if the motive force (acceleration) was constant, and that meant also that speed was linear in time. This was a revolution, the beginning of experimental science, of empiricism. So the context of balls rolling down inclines, dropped balls, balls thrown upward, etc. is I think perfectly appropriate for discussions of speed, distance travelled in time, and quadratic functions. But MiB doesn't come close.
Ladnor, and others,
Our Connecticut math teaching and professional development colleague, Connie Rivera, suggested New York State's CollectEdNY website math collection at http://www.collectedny.org/category/math/ . CollectEdNY project staff and content specialists review or describe adult basic skills math websites there. Although there are only 16 reviews so far, some already have several practitioner comments. Perhaps our colleagues in New York who are involved with this web site could tell us more about its purpose(s) and its plans. Does this website come close to what you are looking for Ladnor? What do others think of this collection?
Are there other state collections of reviewed math or numeracy websites, including instructional video websites, that anyone here thinks fit what Ladnor is looking for? If so, please tell us about them. An important part of integrating technology and learning, and aligning local curricula with CCR standards, is having high quality, practitioner-reviewed, adult learner-appropriate websites in each content area. Are we there yet?
David J. Rosen
Technology and Learning CoP Moderator
djrosen123@gmail.com
Many of the lessons on the www.oercommons.org site are reviewed - and some of the sources for the lessons therein were set up so that commenting and reviewing are part of the process. This is nifty because some of the lessons have been revised and adapted through time to make things clearer.
I know my math teacher associates in the #MTBoS ("Math twitter blog-o-shpere") explore and share lessons and sites all the time. "Three-Act" lessons per Dan Myer are discussed ( see http://wmh3acts.weebly.com/3-act-math.html )