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Update. I spent the last 6 weeks teaching “Algebra Basics”–basically, Ch. 1 in an Algebra 1 textbook. I spent about half of it using a handson approach, using Algebra LabGear to make integers more concrete, create and write algebraic expressions, combine like terms, and understanding parentheses. For the second half, I used the notetaking guide that complements a traditional textbook, filling in blanks, writing down definitions, and trying examples. We then did a traditional Chapter Review and then the Chapter Test. Afterward, I asked my students to write reflections on the unit and how they learn. Most of them prefer the activities to the worksheets, but they see the value of the worksheets in cementing the concepts. I do have a couple of students who prefer worksheets and who can get confused when there are multiple ways to solve a problem.
It is clear that our new approach–math as a conversation–has made a difference in the confidence of our students. Every one of them reports liking math and feeling confident this year. They can’t always point to what is different this year, but they FEEL that something is different. They are thinking about what they understand, and they know the difference between following steps and understanding. Why is a negative times a negative a positive? They know that it is, but, because it’s one thing we didn’t model concretely, they aren’t satisfied with that. This is really exciting.
BUT.
1) What do we do when the informal conversation and handson approach does not lead to efficient problemsolving? (An obvious answer is that we’re not consolidating the learning clearly enough, but we’d like to think beyond that–that even when we show the group several methods, moving from less efficient to most efficient, they go back to doing the method they started with. Do we let them stay there–drawing models, for example, rather than using common denominators?)
2) What about the students who get confused in this informal approach? I know that by using the worksheets they are memorizing, rather than understanding. But in some cases that might lead to more success for them. What is my responsibility to the student?
3) What are other people discovering about the time it takes to allow students to discover and understand why something works the way it does? I know we all have a worry about having enough time. I have the luxury this year of not needing to “cover” Algebra 1–I’m just trying to introduce the concepts and lay the foundation.
4) How do you give enough practice with fractions, integers, and negative fractions while using realworld contexts? I love using tasks to bring out the concepts, but then the Algebra text immediately throws fractions and integers into the mix. The kids will need to know those to succeed in high school and on standardized tests–just knowing the concepts and being able to problem solve isn’t enough.
I get to talk to you guys, Jon and Kyle, about these very questions in a few weeks, on your podcast.
Right now, I’m building a unit on Solving Equations, using all handson and taskoriented problems. I’m also going to have the students who just completed the “Algebra Basics” unit teach their younger counterparts, who did not do this unit so that they can explain some of the properties we’ll use.
You can tell I’ve rewritten the plan for this year several times, but my final unit, the Probability unit, is still how I plan to end the year.
Onward.