In my famous Circle Of Power post, I described a strategy I used to teach some trigonometry that involved students doing a lot of measuring by hand. They had a ruler, a protractor, and a circle physically printed on some physical paper.
What if they had used a Geogebra applet instead? I could have whipped up a “drag the angle and measure the x- and y-coordinates” applet in 5 minutes, and saved the students 20 minutes each. The data in class would have been accurate to three decimal places instead of (generously) one. By a lot of objective measures, the computer would have been better.
And yet…
When do you have students do work by hand?
CalcDave
December 7, 2011 at 8:16 am
That’s a tough one because I’ve always been a “thought experiment” kind of guy and messy labs or even just rulers got in the way of precision and speed. There were also always too many dang variables to account for!
I realize, though, that not everyone is like that. Abstract math and concepts don’t appeal to everyone (thus the infamous “When will we ever use this?” questions). So, it is good to mix them in when you can.
Riley Lark
December 7, 2011 at 8:44 am
Also, isn’t there brain research* that implies that doing stuff with your hands helps you remember the things you were thinking while you were doing it? If a student can make more neural pathways around the same subject, the whole concept is strengthened?
*Please notice that I am not citing any brain research here.
Lori Johnson Morse
December 7, 2011 at 8:27 pm
Yes, Riley, there is research. See “Better Learning Through Handwriting” http://www.uis.no/news/article29782-50.html
and “Handwriting Boosts the Brain” http://online.wsj.com/article/SB10001424052748704631504575531932754922518.html
I can also testify to the power of writing things out for my students. There is no replacement for writing things out especially at the early stages of new concepts.
Riley Lark
December 8, 2011 at 7:22 am
Thanks for these links!
I can’t agree more about the power of writing things out. But do you think writing them into a computer can match writing them on a piece of paper? What about constructing a virtual model vs. a physical model?
Martin Graham
December 7, 2011 at 1:48 pm
I’ve found I get very little mileage out of virtual demonstrations of new concepts. Especially when the concept has a spatial meaning (ie, sine and cosine as x and y values). Spatial concepts need spatial experience it seems. I like to pull in the virtual stuff later if I can.
Riley Lark
December 8, 2011 at 7:20 am
Interesting observation. I always started with the virtual stuff as soon as I could, but I did have a high threshold for “fluency” in the virtual medium. I spent a LONG time teaching the students to program in Geogebra and letting them experiment with it before dumping them in a simulation. Still, maybe it would be better to try harder to find physical demonstrations early on.
Eva
December 7, 2011 at 4:45 pm
I’m with Riley on this – doing stuff by hand is going to imprint itself more easily in the brain. If you need more practise later, then go for the computer screen. Don’t we all get quite enough computer time already?
Ok, that last question is more a personal frustration than a rational argument for better learning…
For lots of people though, a hands-on approach is very different to a computerscreen-approach.
Riley Lark
December 8, 2011 at 7:18 am
And better for your eyes! Don’t miss the research Lori cited above!
Julia Tsygan
December 8, 2011 at 2:06 am
I don’t agree that it is the using of the hands which is the key difference between geogebra and measuring/checking manually. Rather, the difference is that geogebra has an element of mystery/magic, it somehow manages to occlude the down-to-earth-ness of trigonometric ratios, or whatever other concepts and rules we use it for, because it does its calculations in secret, inside the black box. I think after a geogebra demonstration students are often left feeling a little… dissatisfied. At least I do. After all, why are we putting so much faith in a piece of software?
Because of this, I’m switching now to using geogebra to summarize and reinforce what students have learned through other means.
When I introduce logarithmic laws on Monday, and have students discover them and test them (and maybe even prove them), the way they will do it to begin with is by simply “plugging in numbers”, then generalize to variables. I would argue that even though this approach is not hands-on the way using rulers on triangles is, the main principle is the same. Have students test, try, play with the rules using the tools that they understand completely: whether those tools are rulers or plugging in numbers simply depends on the situation.
Another, related, thought: in my experience, when students work with ratios, log laws, or whatever general rules we like them to discover, that generality can be made convincing by having students come up with their own examples for measuring/evaluating. Drawing their own triangles, and then seeing that the ratios are correct, is more convincing than just working with teacher or textbook pre-prepared triangles.
Don’t take this to mean that using hands is not important. Writing things down by hand is slower and gives more room for reflection and thought. Writing things down, such as summarizing definitions of concepts or what was done in class that day, is also retrieval practice which, perhaps more so than anything else in psychology, has a well-documented effect on memory. But writing things down is possible with geogebra as well, just ask students to summarize what you just showed them, and they’ll happily jump to it.
Riley Lark
December 8, 2011 at 7:17 am
Interesting distinction, Julia! It makes sense that the element of mysticism is important – even if the students don’t *feel* mystified. I think you’re right about the logarithmic laws – your approach seems very similar to what I’m talking about, even though there’s no drawing, etc. I do think that maybe drawing / using unusual tools is more memorable than plugging in numbers – but I don’t know how to draw an exponent, so…
I’ve talked for a long time about fluency as a vital precondition for learning math – the students have to be comfortable enough with their environment to experiment. In your example that means understanding how to plug in different numbers and generalize to a variable. With geogebra it might mean something more like understanding how to program the thing, or how the computer basically works, etc.
Kate E Farb-Johnson
January 5, 2012 at 9:16 pm
Asking student to do something by hand encourages them to find shortcuts. I had students making modular arithmetic times tables, and I told them they could use a computer if the wanted to. I got some impressive programs, like , but no one noticed any of the shortcuts for making these tables, or really noticed the structure of the tables. Working by hand means you have to pay more attention to detail, and that if there is a shortcut you can use, you have a motivation to find it.
Carroll B. Merriman
January 10, 2012 at 2:56 pm
This MUST STOP.