Post 3

I very strongly echo Hayley's sentiments, and only have one major to add: I wish my students had a better mental filter.

1. I want them to actually engage their brain when taking notes. I can't count the number of times I've seen things written sideways on *their* paper because I had run out of room on the board and written it sideways. They write down things that make no sense and copy stray marks from the board. If they skip a word when writing, they don't catch it, and they think that whatever I say or write is the absolute truth. They write down only what is exactly on the board or what I tell them to, even though I often write more than they need to and say more than I've written. I could say a sentence like "take the cosine of the hypotenuse" and it would be written down, verbatim, without them actually evaluating the validity of what I said. Sounds good, teacher said it so it must be right, my brain isn't engaged to tell me that that statement is a mathematical impossibility. (It *does* make catching cheaters a whole lot easier though, because they copy exactly from their friends!)

I have emphasized time and time again that I do not care about the math definition, the glossary definition, the dictionary definition, or the exact words that come out of my mouth. What I do care about is their articulation of a *concept* no matter what words they choose to use. I wish that teachers took emphasize off of memorization.

2. I want them to use their mental filters to predict and evaluate answers. What *should* the side of this triangle be? Around 10 inches, maybe a little bigger or smaller but if I get 150 inches, that can't be right. Does the answer I get match what I thought it should be? I did it and got a negative number...but how can a side of a triangle have a negative length? (Most common response: Oh well, that's what the calculator told me!). I try to emphasize the real-world use of math in the sense that, in real life, it doesn't matter how you got the answer. What matters is that you got the right answer, and that you can explain and *justify* how you got it. After all, we're past the days of grades based solely on effort. Use the calculator, don't use the calculator, compare your answer with someone else (compare, not copy!), discuss, ask questions, look at your notes, I don't care, just predict, execute, and evaluate.

3. I want them to be extremely critical of what other people tell them. Don't accept the word "seems." This seems like the right answer because the teacher said it. This seems like a good investment because the graph is going up. This seems like the correct conclusion because somebody told me it was. When someone says "this seems..." you say "prove it." They are completely aware of things like bias and the fact that numbers can be manipulated any way you want, but they don't apply that knowledge. Someone says something is good, you say "how good? Give me a number." Ask questions, be critical, and be aware of fuzzy concepts and solutions that are presented as absolutes.