Yesterday I was teaching some linear programming to my seniors. I had to graph the constraints and all I had was a whiteboard and a marker. I have to draw the coordinate axis, make tick marks and sort of wing the constraint graph. This is my usual technique and as usual the resulting graph was less than optimal. Lots of room for artistic interpretation of the resulting picture. I have a laptop and a projector in the room so with all this nifty technology I should be able to find a better solution for this task. I know there is software out there that will solve LP problems; enter the objective function, the constraints and stand back. Magic will occur. I do not want to go quite that far, I just want pretty graphs. The first option was Geometer’s Sketch Pad. It did not like x = 32. It would draw it, it just would not do anything useful with it like finding intersections. I then tried Geogebra, a free app I have very little experience with. Our Sketch Pad teachers gave me poor reviews for it but I figured for the price it deserves a try. It took a few minutes to figure out the people who wrote the tutorials were geniuses (or idiots) who live in a cave and never deal with people who do not know the software. So I decide to just tinker. Five minutes later I am drawing functions (and x = 32), and finding intersections. It will even do inequalities and the appropriate shading of the half-plane. Just what I wanted. But now comes the big question, do I want to show this to the kids.

What it boils down to is this – is finding the intersection of two lines, which they supposedly know how to do, a critical skill that needs to be constantly reinforced or can I just get on with solving LP problems? These are very simple LPs with only two unknowns so doing them by hand is not complicated but it does take time. The flip side is do I want to teach math and software? In the real world (whatever that is) the math and software is the obvious route. In the very contrived world of the math classroom being beat over the head with hand-based mathematics is the traditional winner. Years ago it would have been a no-brainer, beat them over the head until they cry for mercy. Maybe I am getting soft in my old age but the software just seems to just make more sense. Learning software is a 21^{st} century skill, finding the intersection of two lines by hand seems not so much. If the software was not free or multi-platformed I would not be so tempted.

Nuts on it. My drawings are confusing even to me. I am going with the software. I want to teach senior level math, not 8^{th} grade algebra.

There is sort of a weird evolution going on with math and technology. It started with the introduction of the affordable graphing calculator (I still remember the great graphing calculator wars of the 1980s), was further confused with the TI-89 symbolic manipulator and now there is WolframAlpha for free in the internet. How much of the hand stuff do we hang on to? When I was in high school if I wanted a square root I whipped out a slide rule. If I wanted more accuracy I had an algorithm that worked sort of like long division on steroids. Now I suspect there is not a math teacher in the US that expects square roots be found with either method. Is there a point in this evolution where we throw out the baby with the bath water? In the workplace almost nothing mathematical is done by hand, there is not time. In the classroom all sorts of archaic methods are used to solve problems. As a math teacher I sort of pick and choose but I know that my picking and choosing is different for one of the other math teachers in the building. She loves old school math, I like solving problems with anything I can get my hands on. If I want the roots of a polynomial I grab a laptop while she starts looking at factors and alternating signs. Interpreting Shakespeare is a piece of cake compared to the math and technology evolution debate. Too bad I am in the middle of the hard one.

October 15, 2014 at 7:02 pm |

So I think it boils down to 2 things: 1) how authentic do you want the exercise to be and 2) what will they remember and be able to replicate in a few years. Studies show that learning tasks that are more authentic produce better learning outcomes (just ask any senior CS capstone student producing a program for a “real” customer). Second, if they ever have to find the intersection of two lines “in real life” which approach would be better?

I use this argument for why my children had to learn their multiplication tables – there are too many situations when they’ll need to use their head (and not their cell phone) to calculate something – and it makes the remaining task at hand so much quicker–think about trying to solve algebraic equations without knowing your multiplication tables.

In your situation I can see pros and cons for both approaches – but in the end what do you want them to remember – solving for line intersections or the LP?