1. Students learn math best when it is "real". I put real in quotations for a reason. I've found that it doesn't matter how realistic the situation is that you've presented. You could be designing an actual school or determining the cost of an actual mortgage payment. If the students don't see the task as real to them, then they will not be engaged and want to learn about it. That being said, the task doesn't have to be real-world at all. Finding out how long it will take to mine down to bedrock in minecraft is far more engaging than finding out how long it took to mine the London tunnel. Don't worry if your task does involve something real but doesn't relate to students. You can get over that hurdle by getting the students invested in the task like Dan Meyer does in his 3-Act Math. (Genius work on getting students interested in real world situations btw.)
2. Students need to use knowledge in order for it to stay in their wee little heads. Even when you're using real tasks all the time, if students haven't used the Pythagorean theorem in three months, chances are only one student will think to use it in situations where we'd obviously use it. The problem with math education is that there are hundreds of little tools (tricks) we can use, students often don't remember them all or try to use them when they shouldn't. I agree with Steven Leinwand in his Accessible Mathematics when he wrote that we need to incorporate cumulative review into every lesson. I'm going to try doing it for the start-up activity this next year unless a better idea comes up. Students also need practice, but not 20 "problems" for homework every day. I like using less "problems" that require deep thought on part of the students.
3. Students acquire a deep understanding of math by analyzing situations, coming up with conclusions, and debating those conclusions with others. There is something magical about when students argue mathematics with one another. Maybe I'm just a sick, demented man, but I love those arguments. It shows that students are thinking critically. It shows that students are thinking about the mathematics. It shows that sometimes (most of the time) in life, there isn't a clear cut best answer. Students should be doing this often. They need to analyze, come up with conclusions, and defend their conclusions. It brings up so many misconceptions that get corrected.
4. Students learn math best when they use multiple representations. I know there is a butt-load (that's a real measurement) of research on this. I agree with it. Students should be using graphs, algebra, tables, words, etc. to come up with their conclusions. It allows students to become fluent in many ways of communicating.
5. Teachers need to connect with students. I like smiling, getting students to laugh, and being their friend too much for this to not be a part of my philosophy. If students like the teacher, they are willing to work at things they normally wouldn't. It is a lifesaver in the classroom when you have a less than exciting lesson planed, but you need the students just to do what they're told. You can fall back on your report with the students and get to the work. I realize that every teacher is different and some personalities don't jive, but teachers need to connect with at least some of their students.
Please, share your thoughts. What would you add? What would you take away? What do you think should change?
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