Mathing…

So I dusted off my old javascript/css/DOM reference and created a quadratic grid in html and javascript in about 4 hours. This happened because my last attempt at quadratic factoring visualization had limited interactivity as a learning tool and also happened to be visually disappointing. I am much happier about the end product but it got depressing when I started considering how students would be using it in the classroom and what students were expected to learn from it, especially because it took hours upon hours to create them.

Sam’s original post seemed more about reminding students that we teachers give factorable quadratics to students because we want to save them time while they are learning new material. To prove his point, he might toss up a graphic/table/chart on a projector and take 10-30 seconds to make a point and then just move on hoping students are aware that more difficult (laborious could be a better word) problems exist, but given the time constraints students face, the teacher makes certain concessions when teaching material.

I see a few ways a teacher can make/teach the above point.

1. A teacher can, during the process of teaching factoring, remind kids they are getting easily factorable questions for homework and that it is important they know all the different methods for factoring and solving quadratics so that they can handle even the most difficult among them. Maybe the teacher will show a few examples and then toss up a graphic as an informal proof of the teacher’s claim that very few quadratics are factorable over integers when they have integer coefficients. (Time: 1-2 min)
2. A teacher can create a Quadratic Factoring Worksheet (PDF) and have students investigate individually or as a group. To save even more time, a teacher can assign each student 2-5 quadratics to solve (depending on class size, 50 students ->2 each, 20 students ->5 each) and then have students fill out a table on the board corresponding to their worksheet. They write their results and the results of their classmates and discover for themselves what the answer is. Students also get the benefit of additional practice factoring. (Time: Varies, but 5-30 min)
3. A teacher can ask the question and have students figure out for themselves using calculators, Excel, or even Google Spreadsheets like Eric did (Google Spreadsheet) in a response to Sam’s post. This may end up turning into a lesson that’s more about learning the technologies than the point at hand. To shortcut the problem, students will probably have to learn what discriminants are and how they can be used. We already teach what it means to have a discriminant that is negative, zero, and positive, but we don’t teach what it means when a discriminant is negative, a perfect square, and any other positive number that is not a perfect square. The latter is more relevant to our question. What happens when students can’t figure it out? How much scaffolding/support do we give them? Do we setup the entire excel sheet with all the formulas ready and have them just enter the number? Do we give step-by-step directions that help the students setup the formulas to figure that out? Is completing the assignment proof of anything? (Time:Dependent on student familiarity with the technologies. My guess is a minimum of 30 minutes)
4. A teacher can also created a sandboxed tool, like the quadratic grid mentioned above, to guide students in exploration. The statistics that goes along with the grid has very little meaning and is probably not appropriate for the high school level, even if students are curious. The visuals are interesting, each line corresponds to a factor. First just x, then (x+1) forms the 45 degree downward line, next is formed by (x+2), (x+3) and so on… Reducing the boxsize for large grids produces some interesting patterns. (Time: 2-10 min)

Do any of the above promote/nurture the kind of natural curiosity that breeds good questions and good hypotheses and learning beyond the standards? Is there room in schools for this kind of investigation? Is this too far from, what seems like, our schools’ goal of “just learn to factor” and learn it well for testing (in what soon could be our only measure of effectiveness and accountability)? Am I just way off the standard and what is essential to students’ future success?

It’s only been 48 hours since I started investigating the above question and thought about the different ways to engage students in that question. In that time, I’ve learned (or relearned) more about Processing, Excel, Google Spreadsheets, JavaScript, DOM, CSS, HTML, quadratics than I ever knew possible. I’m very satisfied at the time I spent just thinking, learning, and working on the above. It’s almost 3AM. I don’t want to give an impression that I do this all the time, but sometimes I can’t help myself when an idea excites me. If only I could translate all my enthusiasm and learning into workable and engaging lessons. It might turn out that I was way off course and it was all for naught, in which case it might be a good idea to temper my eagerness and not light the other end of the candle.

So which of the above has good mileage in the classroom? Is any of the above worth the time in the classroom? How much time is too much time spent on lesson planning? Will student’s confuse not factorable over integers as not factorable at all? Does the risk outweigh any possible benefit form this exploration?

UPDATE: Screencaps from Quadratic Grid