I find myself struggling with the technology question, how much is too much. Or, more realistically, how much is the right amount.
I am a stickler for working out a problem yourself, but there are some problems I like to assign that are too time consuming, or the student does not have adequate knowledge, to be done without technology. It seems that once I show how to use the technology, the student uses it for everything.
My dilemma; is this a problem? When I was learning mathematics 30 years ago there was no technology and so we could not do really difficult problems. We were not able to “see” what was going on and had to trust more in formulas. Some classes I teach today are nothing like the classes I took because technology has changed them. Students today get to experience the mathematics in ways I never dreamed of and that gives them a better experience and a better understanding.
I have to be honest, though, I am distressed when a calculus student has to use a calculator to draw a simple y = x2 graph or a simple sine or cosine curve. I want this student to have a full calculus experience and when he struggles with basic skills he cannot get to the meat of the course and have the full experience.
The same exact thing is happening in our developmental courses. Their basic skills are multiplication facts, division, and fractions. These students struggle so much with these basic skills that it is difficult for them to learn algebra. They have to put so much energy into figuring out something that should be second nature, that there is no energy left to understand the next level. I sometimes find myself telling that student, “Just use your calculator.” But this is only a temporary fix.
So then I begin to wonder, what is the important thing? I know the most important thing is to understand the problem at hand and to be able to figure out a strategy to solve it. Is the actual solving of the problem important? Technology can do those tedious calculations if I only know what calculations to ask it to do. So thus is my dilemma.
This might sound like a lame and lazy answer, but you ask, “what is the important thing?” Maybe the better question is “What ARE the important things?”
I say that partly because I am really poor at math and always wished there were more right answers. Mine were never right.
I also say that because looking for only one solution keeps us in a narrower road and less likely to discover other routes to the destination.
The advances in technology that changes the way people do math is not unlike the technology that takes people places. Not too long ago no one would have dreamed of driving a car two blocks to get a quart of milk (beer) and some bread. Now no one walks that far!
I wonder what your questions would look like if the technology part was taken out. What is the goal of the lesson? If it is a process, and that IS the most important thing as you see it, then do that thing. If simply getting an answer is the goal, let the calculators rule the day.
I will sit and add up bill payments on my phone rather than write them down and do the math on paper. My immediate goal is to see just how much money I don’t have. I get that. There is a time and place for everything. Not an everything for every time and place….
Todd,
I really don’t believe you want there to be more right answers. When the engineers build the bridge, I want them to have all come to the same right answer before I drive my car over it.
The beauty of mathematics is that there are many roads to that one correct solution, some better than others.
I did not make it clear that I was not talking about computation alone in my post, but about the computation that is done to solve a problem. In my calculus class it is the calculating of the double and triple integrals the student has set up to calculate a volume or a center of mass. A lot of work went into setting up the integrals correctly, my question was, is it then necessary, or useful, for the student to do the computational work. At times like these, I would rather the student tackle the next problem rather than spend the time on the computation. Generally, the student is not content leaving the computation undone. There is benefit in the student doing the computation, but there is more benefit in the student tackling the next problem. There’s just not time for both in a class period.
You are right about right answers. Surgery comes to mind. Buildings and cars and refrigerators. This things are great to have when the right answers have been found to make them work.
As a high school English teacher I think I had a similar experience to yours with spelling and “spell check” features. We would do vocabulary and we would write sentences and write out definitions. Just like we all have done. But when it came to writing essays and stories, sometimes I was looking for other things in the writing and would be paying less attention to an individual word.
And there was just not enough time in the day to get all the essays written with 100 percent correct spelling!
Tricky, what skill to focus on as the class progresses. Time is difficult.