The core of a Teaching Through Problem-solving (TTP) lesson involves presenting students with a problem that requires that they learn some new mathematics. For students to focus their energy productively, it helps them to know what that new mathematics is – at least vaguely. In some schools, teachers are required to write on the board the goal of the day’s lesson. For teachers trying to use TTP, however, writing the goal on the board is like revealing the murderer at the beginning of a murder mystery. How to resolve this?
The analogy to a murder mystery provides the answer. A good murder mystery doesn’t reveal the murderer in the opening pages, but it does make clear that a murder has occurred and that the mystery to be solved is who did it. In a TTP lesson, we don’t want to give away the solution, but we want students to understand how the main problem of the day is different from ones that they have solved before.
I will use a recent research lesson as a hypothetical example. The lesson was with grade 1 (6-year-old) students and was about using subtraction to solve a comparison situation. The situation was:
“Mrs. X and I each have [school coupons]. Mrs. X has 9 [coupons], and I have 5 [coupons]. How many more [coupons] does Mrs. X have?”
Not surprising to anyone who has tried to teach comparison problems, almost all the students jumped into adding the two numbers. There is, after all, nothing being taken away, which is the only context in which they have used subtraction, and they have seen lots of problems that start out just like this and finish with “How many [whatever] do they have altogether?”
To alert the students to the need for a different approach, the teacher could start with the expected question, “How many [coupons] do we have altogether?” and allow students to quickly solve that (as they in fact did). Then the teacher could cross out that question and write the new question: “How many more [coupons] does Mrs. X have?” Now students can clearly see that the problem is different, and therefore they are going to have to do something different.
A related technique is to leave blanks in the statement of the problem and play around with what numbers go into those blanks. A lesson about fractions as quotients might start with “__ L of juice are shared among __ people. How much juice does each person get?” The teacher could initially fill in the blanks with 12 L and 3 people and have students write a number sentence (which they would easily solve). Then the teacher could change the amount of juice to 2 L.
When students see clearly what is new, they are able to articulate the “guiding question” for the day, i.e. the mathematical challenge, and they can focus their thinking on the new mathematics rather than unproductively solving the wrong problem. Furthermore, by the end of the lesson they are more likely to be aware of what they learned (as a response to that guiding question) and you will get better results when you ask them to summarize the lesson.
For the comparison lesson, the guiding question students might articulate would be, “How do we solve a ‘how many more’ problem?” (Or, the teacher could introduce the new term, “comparison problem”.) By the end of the lesson, students might be able to say, “We can solve a ‘how many more’ problem by using subtraction.”
For the juice lesson, the guiding question students might articulate would be, “How do we divide when the dividend is smaller than the divisor?” By the end of the lesson they might be able to say, as a summary, “When the dividend is smaller than the divisor, the quotient will be a fraction.”
The best time for students to understand what is new can vary from lesson to lesson. Usually, it’s when you first present the problem. But sometimes students will need to work for a minute or two before they realize that some technique they know doesn’t work anymore.
So when designing your TTP lesson, think about both how and when students will understand what is new.