Keep, Flip, or Change the Problem Type

There is a debate amongst teachers using Cognitively Guided Instruction in Math (CGI Math) as to whether or not to keep posing the same problem type for a week or two, changing the problem type almost daily, or flipping the problem type (we’ll talk more about that in a minute), let’s examine the pros and cons.

My CGI math work is primarily done in Title I schools with large, diverse urban populations and many English language learners. Full disclosure, I lean towards keeping the same problem type for a week or so for a variety of reasons. Read the following arguments and let me know what you think.

Repeating the same problem type allows students to become familiar with the structure and the language of the problem. For example, The blue string is 12 inches long. That is 6 times as long as the red string. How long is the red string? The language in this problem type (multiplicative comparison) is abstract as is the concept of measurement. Students from a variety of backgrounds often need time to make since of this language and make meaning from it. One or two times may not be enough for English language learners and beginner’s to grapple productively with the problem.

Sometimes when posing a particular problem type, students may be able to come up with an accurate answer fairly quickly, if not immediately. This makes me wonder if 1) is the problem type challenging enough, and 2) are we concerned with getting the correct answer or are we concerned with flexibility and the sophistication of the answer? For example, if a student can quickly solve a multiplication problem by drawing picture, do we want them to stop there? If a student can quickly solve a multiplication problem with the standard algorithm do we want them to stop there? I say, “No.” I want to see strategies develop from pictures to skip counting, to decomposing, to the distributive property and on. How can these strategies become more complex and diverse if kids only get one shot at them? How will teachers learn to extend students’ thinking and increase precision, fluency, and accuracy if they only get one chance at them?

Students who may have no idea how to solve a problem, can learn a strategy from a partner or from the student work share at the end of the lesson. However, if the problem type constantly changes, they have no way of trying out a strategy successfully, learning how to apply a new strategy, or learning from past mistakes.

Changing the problem type daily or every other day. Once students become familiar with a problem type, Join change unknown/addition) they may stop their thinking and go on automatic. They may think ,”Yesterday was an addition problem, therefore today must be an addition problem.” Again, in this situation, I would urge the teacher to extend the students thinking by providing more challenging numbers (3 digit, decimals, fractions, etc.) or have students connect their strategy to a particular tool or model (number line, ratio table, coordinate grid).

Lastly, flip the problem. I am actually beginning to like this approach more and more. Start with a particular problem type for a couple of days and then add in its related inverse operation. This allows students to make sense of a problem type, learn and apply multiple strategies, and see the interconnectedness of mathematics. For example, 2-3 days of multiplying fractions 6 x 2/3 (6 groups of 2/3) then follow it up with 2/3 x 6. Same numbers, but different thinking is required to solve each one. Try the problems out yourself, draw a model to solve each of them, then explain how they are alike and how are they different. What is their relationship and when would you use each one.

The debate continues, we want students to think, struggle productively, and be successful, which approach will you use keep, change or flip?