Category Archives: technology

Authentic Assessment?

My junior level math classes have begun working on a project called Buying a Car. For the past few classes we’ve been problem solving using spreadsheets. They’ve been working in teams, using Google spreadsheets to solve problems like this and this (which I adapted from our Core-Plus Mathematics text). My teaching colleague and I decided to jump into this spreadsheet mini-unit before our students had to turn in their laptops for the year. (We are a one-to-one school.)

Here are some pictures of my students hard at work.

photo7  photo6

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Some things I heard as the students were working:

  • Oh, so the bank pays for the car and then you pay the bank. I get it now!
  • How much does gas cost right now?
  • Where’s the best place to get the loan from? What’s the lowest interest rate we can get?
  • Are we going to buy that truck? What’s the gas mileage on it?
  • How do we figure out the payment? What did we do before?
  • So we have to add the interest and then subtract the payment.
  • We can cut back on the money for entertainment. We can be cheap. There’s only two of us, we don’t need that much food. It’s not like we’re feeding any children.
  • How do we determine how much for a downpayment?
  • Can we afford a monthly payment of $875?

Here’s what I really like about this assessment (having never done it before):

  • There is a high degree of choice.
  • There isn’t a definitive solution.
  • Students have to make (and state) some assumptions in order to solve the problem.
  • They have to think about lots of things that go into a household budget and buying a car.
  • Students working together and helping each other to succeed.

What I’m not so sure about:

  • The quality of their results.
  • If they’ll really apply what they’ve learned during the past 4 classes learning about spreadsheets.
  • How much understanding they’ll walk away with.

It will be interesting to see what they produce as a result.

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Ask different questions

Rather than assess whether my students can do matrix multiplication by having them multiply matrices without a calculator, I decided to ask a different question. After all, the point is about understanding and not computation, right? So, instead of giving them a non-calculator section on their quiz, I changed the question:

Given that
\left[  \begin{array} {cc}  5 & 2\\  7 & 3\end{array}  \right]  \left[  \begin{array} {cc}  1 & -4\\  -6 & 9\end{array}  \right]=  \left[  \begin{array} {cc}  -7 & -2\\  -11 & -1\end{array}  \right]

explain the calculation that gives the entry in the first row, second column of the product matrix.

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The Evolution of a Teacher

When I began teaching high school mathematics in 1988, there were no such things as affordable graphing calculators. A mere four years later, I had a classroom set of TI-81 graphing calculators.

Actually, in my second year, one of my Honors Calculus students showed me his HP Graphing Calculator. It looked nothing like what we would all soon know as graphing calculators. It had this tiny screen that handled about four lines of text – amazing by 1989 standards – and it had two keypads that were connected across a folding spine. Amazing!

So, five years into my teaching career, I have this classroom set of graphers – TI-81’s. What was I supposed to do with them? I mean, I had taught Algebra 1, I knew what the kids were supposed to learn. They had to learn how to draw graphs of lines. They had to learn how to manipulate symbols. How was this new device supposed to help without undermining me? I didn’t have a clue. Sure, it was cool, but the kids were supposed to be able to manipulate a pencil and a ruler – not this new, cool device. It’s not that I was anti-calculator; I was new. And I didn’t want to lose my job. But this was too interesting a tool not to use. So I learned. I read – journal articles. It was 1993, after all. Web? What’s that?

I bought my own graphing calculator: A TI-85. I know a lot of people didn’t like that model, but I did. I liked the menus. I liked what it could do that the TI-81 couldn’t. But it was more expensive, so schools went with the TI-81, which evolved into the TI-82, TI-83, TI-83 Plus, and TI-84 Plus Silver and TI-89 Titanium. That line has been pretty much developed out. It’s where we are right now. We’re comfortable. We know how to use them – as teachers, as students, as test developers. We use them to analyze graphs, to solve systems of equations, to crunch data, and manipulate algebraic symbols, if we have a CAS.

In 2006, I received an invitation to participate in a field test of a new piece of TI classroom technology: The TI-Nspire. Never heard of it; jumped at the chance. It was both computer software and handheld device. Literally. The first exposure my students had to the TI-Nspire was at a computer in the lab. At the time, I couldn’t imagine how this new tool would revolutionize my classroom. Frankly, the first models were so clunky, that I just wanted my TI-84 Plus Silver. But they (at TI) listened to us – we teachers in the field test and my students, too. There were some things were really didn’t like – those green alpha buttons – and things we really liked – being able to grab and move function graphs around, for example. Think about that for a second. We could grab a graph and move it around the screen, changing the slope or changing the y-intercept. As we did that, the function rule would change. That means that I could graph a line, grab it, move it, and see the effect on the rule. Holy cow! That’s a game changer. There was so much that this new device could do, my head was spinning. After all, it’s just a tool. If I can’t use it to teach something, then what’s the point? How was I to make the best, most effective use of this new tool? Still working on that. Every day.

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