Thursday, August 20, 2015

Counting on Day 1

I usually like to take the first day of a course to talk about some interesting and simple math that students might not get elsewhere.  In the past, I've talked about why we write numbers the way we do, interesting facts about large numbers, and a basic overview of math subdivisions (topology, analysis, algebra, statistics).  This year I was inspired by a blog post from Andrew Stadel about counting dots.

The link to my slideshow is here.  Feel free to use as is or save a copy and modify to your own needs.

I begin with having them do a math meditation that I learned from Dr. Edmund Harriss at Twitter Math Camp 2014.  I ask the students to do a visualization exercise (close your eyes if you need to, or stare of into space if that helps).  Begin with a blank canvas -- it could be any color -- then we are going to add a certain number of dots to the scene.  I then call out the numbers 1 through 5 (maybe a number every 10 - 20 seconds).  I pause at five and tell them that this is about where we run into the limits of being able to put dots anywhere.  They might think about starting to organize the dots as we progress.  Then I continue on in the same manner until about 15 to 20.

When I stop, I ask students to describe to me the picture in their head.  It might be a 4 by 5 rectangle.  It might be two 2 by 5 rectangles.  It might be two groups of 3 by 3 squares with 2 dots to the side.  One person from Calculus said, "Three dice with the six side up and one with the two." (D&D player anyone?)

I then proceed with the sideshow and tell them I want them to try to figure out how many dots are on each of the next slides, but they will only appear for 0.6 seconds.  We do a few of these and they love the challenge of trying to get it quickly.  I'll explain what they are doing is called "subitizing" and how many animals (including humans) have the ability to do it with small numbers, but only up to a point.  Imagine if 97 dots flashed up there.  Could you count them in half a second?

I also like to stop after a couple of them and ask them how they figure it out.  I don't know much about the subitizing literature, but my guess is that with small numbers of dots you can make shapes with them easily (three dots make a triangle or line, 4 into some kind of quadrilateral or rows, etc.) then count after the image disappears if necessary.  Making a decagon is quite a bit harder, though.  Even with seven dots, some students will say they saw a row of 3 and a group of 4 (like we chunk phone numbers).

I then show the Vsauce video which talks generally about numbers (including logarithmic number lines as well as subitizing).  We discuss any questions, comments, or other thoughts they may have after viewing and then go back to trying out our counting techniques.

In the next few slides, the dots are organized into easier to think about shapes.  Even if they don't get the number correct after it flashes on the screen, we talk about how they might find the number of dots without counting each and every one.  The students sub-organize the shapes into smaller shapes (eg this section was a 3 by 3 square, then one more to the side) and it's interesting to hear the different ways they group them.  The switch to blue squares is not too different.

Once we do a few slides of blue-square counting, I ask them what percentage of the whole shape is made from blue squares.  In the slides I made, I have it animate to move the squares around and see the same amount in a different way.  By moving a blue square to a white space in an organized way, it's easier to determine the percentage.

To finish, I tell the students I wanted to show them this stuff to highlight some interesting psychology behind our number sense, but also to motivate the need for math as a subject.

By organizing and moving things around in a meaningful way, it is easier to interpret what is quantitatively in front of us.  Especially with my algebra students, I tell them that our notation and algebraic manipulation will cause some growing pains, but once we get some techniques under our belt (like we did with chunking shapes into smaller pieces or moving blue squares to fill white holes), it will make the process easier.

Tuesday, March 10, 2015

Trusting in An Answer

One thing I've been noticing this year as I find "challenge" problems online or give sets to my own students is that the first thing I acknowledge is the trust.

Often, we ask students to "Prove that..." or "Solve" in problems.  All well and good in a classroom setting, but in a more "real-world" situation, the first thing that needs to be assessed is whether that problem CAN be solved.  I think my own problem-solving strategies are sometimes directed by knowing that there is an answer before I even begin the work.

I trust that my teacher or problem-giver is giving me the opportunity to solve the problem when it is posed.  That is, if I am not allowed a calculator, I trust that it CAN be solved without a calculator and that my answer will most likely be a whole number or at least a simple fraction.  If I am in a 6th grade classroom then my problem will be "6th grade appropriate" (ie should be solved without calculus or other advanced methods).

Of course, this is not the case in higher mathematics where the main question can often be, "Is this solvable?"  Certainly less so in every day life when we are faced with innumerable unsolvable problems.  I'm not suggesting we give students unsolvable problems, but it might be interesting to mention this to students that we are in an artificial, safe environment in this classroom where these problems have already been solved and I (the teacher) am giving them to you knowing you should be able to complete them.