It's easier to stay "in the box." My cat, Tiki, thinks so at least. But I say, let's help students step out-of-the-box and take some risks. Let's teach them to learn some grit and begin to think critically and not just through rote memorization. Students will rise to the occasion when you expect them to.
I have caused a little bit of dissent in our department over the years...insisting on putting a few out-of-the-box questions on tests and quizzes. Some colleagues have firmly stood with me, saying a test should not just be a "worksheet," but others have been worried that their students would not be able to solve the problem(s) and would therefore do badly on the test.
Most of these colleagues that originally were skeptical of putting problems on a test that were not directly taught in class have come around to the idea of giving them problems that are not in the textbook. I used to say that this type of question "separates the A students from the B students," but that's not the reason I do it anymore. It's to try to get ALL students to think critically and not just regurgitate what I have given to them. It's more about teaching students to problem solve rather than to teach them how to memorize isolated facts and then just spit them back out. I have found that at first, students are nervous about these kinds of problems, but the payoff is great..they learn the power of grit, and they feel fantastic when they get these problems right. Therefore, my review sheets are often harder than the test, because I put several questions on them that cover the same topics, but are asked differently or require several ideas to be combined into one.
Today I came across the article Why our Smartest American Students Fail Math. It is a fantastic read, and I strongly recommend that you read it. Basically, Carol Lloyd, exec editor at http://www.greatschools.org/, writes that our top students who go to college for a STEM (Science, Technology, Engineering, or Math) tend to drop the major because they get terrible grades in these courses. They are not learning problem solving or perseverance in their high school courses and, as Richard Rusczyk, former Math Olympiad winner and founder of the awesome website artofproblemsolving.com stated, “These were kids who had never gotten anything but 95s and 100s on their tests and suddenly they were struggling and were getting 62s on tests and they decided they weren’t any good [at math].” Lloyd writes, "Indeed, traditional math curriculum is to teach discrete algorithms, a set of rules that elicit a correct answer, like how to do long division, say, or how to use the Pythagorean theorem. Then students 'learn' the material by doing a large quantity of similar problems." She continues that Rusczyk says the result is that "students are rarely asked to solve a problem they are not thoroughly familiar with. Instead, they come to think of math as a series of rules to be memorized. The trouble is kids don’t necessarily learn how to attack a new or different kind of equation."
This is what I have been saying, too. We cannot just give students a worksheet, have them memorize all of the topics, and then test on the same material. What does that teach them? I think it teaches them to remember things just for a test and not how to critically think in high school. Students need to fight through problems and talk through them with others. They need to hear all the different approaches others have to solving the problem. I spoke about grit here, and my problem solving students are learning about grit daily. I gave the following problem to them on a problem set:
I was shocked at how many students did not know how to do this. There are three Calculus students, three Pre-Calculus students and one Pre-Calculus regular student in the class, and they did not even know how to start it. Finally someone said, "Oh, it's like a composition of functions problem?" Then they got it. The next week, I gave them this problem on their problem set:
This time, they got it right away. But it is important to give different questions to ALL students, not just those in a Problem Solving course. They begin to recognize the type of problem that it represents, not the exact one on a worksheet.
My Mu Alpha Theta students, which I wrote about here, are exposed to new and different problems each week at our club meetings. I feel like the exposure to these problems is definitely making them better thinkers. Spending an hour on math that is not following the curriculum is SO COOL, and I would love to hear from others of you who are either an advisor to a math club or are just thinking about it...and if you are thinking about it but aren't sure, JUST DO IT! Not only is it great for the kids, but it is also great for you--it truly makes my "cup runneth over."
Attached are two review sheets I gave my Algebra 2 Honors classes that had some problems on it that were out of the ordinary, but were related to what we were learning.
These are from famat.org. I know the font is pretty old on the first worksheet. It's because I am using the oldest tests from the '90s in class, as we are using the most recent ones during Mu Alpha Theta practice. In math club, I have students work in pods on these worksheets, and it's fun to see them do this when they get the question right:
Here are some great places to get problems:
- Florida Mu Alpha Theta
- Art of Problem Solving AMC past contests
- Phillips Exeter problem sets
- NCSSM test bank
- Marywood University past contests
- Kings College London Challenges
- Corbett Math 5-a-day and Conundrums