Problem-based math

1 Where it came from

Problem solving has been central to Japanese math lessons for decades, and almost every Japanese elementary textbook chapter opens with a problem. In the US, Carpenter, Fennema and colleagues at the Wisconsin Center for Education Research developed Cognitively Guided Instruction around how children naturally solve story problems. TERC wrote Investigations in Number, Data, and Space with National Science Foundation funding; Sherry Parrish's 2010 book set out Number Talks as a short daily routine; Illustrative Mathematics published its open middle-school curriculum and, in 2021, its K–5 one. Peter Liljedahl's Building Thinking Classrooms is a newer set of fourteen classroom practices.

2 The big ideas

• Problem first. Children meet a task before the method, so the method answers a question they already have.
• Children's own strategies are the starting point. Teachers who know how children naturally solve problems can build on those strategies.
• The discussion is the lesson. In the Japanese phrase, the heart of the lesson begins after students come up with solutions, when the class compares them.
• Talk and explanation matter. Children explain how they thought, and learn from how others did.
• The teacher still closes the lesson. A good problem-based lesson ends with the teacher stating the idea clearly.

3 What a lesson looks like

A 6th-grade Illustrative Mathematics lesson opens with a five-minute warm-up: four pictures on the board and the question Which one doesn't belong? Every picture has a reason, and students argue for theirs. Then comes the main activity. In pairs, students work out which of two juice mixes tastes more orange, using any method they like. The teacher walks around, noticing who uses a table, who draws, who finds a unit rate, and picks three students to share, in a deliberate order from the most concrete to the most efficient. In the lesson synthesis, the teacher connects the three methods and names the idea: equivalent ratios. The class ends with a five-minute cool-down, one short problem done alone, which tells the teacher who has it. In a Japanese classroom the same shape has names: hatsumon, the key question; kikan-shido, the teacher circulating; neriage, polishing the ideas together; and matome, the summary.

4 The adult's job

The adult chooses or prepares a problem with care, holds back from showing the method too soon, watches how children work, and picks which solutions to share and in what order. Then comes the part that is easy to skip: making the math explicit at the end. In Japan, teachers improve these lessons together through lesson study, planning a lesson jointly, teaching it while colleagues watch, and revising it.

5 How progress is checked

Teachers check understanding by watching and listening while children work, and through short end-of-lesson checks, such as Illustrative Mathematics' cool-downs. Asking a child to explain a solution, and to say how it is like a classmate's, shows more than the answer alone.

6 Strengths and criticisms

7 What the research says

The debate is real, and the honest answer is that the research does not crown a winner. The National Mathematics Advisory Panel found only eight studies that met its bar on teacher-directed versus student-centered teaching, with mixed and inconclusive results, and said that all-or-nothing recommendations either way are not supported by research. The same panel found that explicit instruction, meaning clear models, a range of examples, plenty of practice, thinking aloud and feedback, has consistently positive effects for students who have difficulties, and the 2021 federal practice guide rates systematic instruction as strong evidence for them. On specific programs, a federal trial found Investigations schools scored lower in the early grades than Saxon and Math Expressions schools. Illustrative Mathematics has high EdReports alignment ratings, which measure fit to standards rather than results, and no What Works Clearinghouse rating was found. Some programs deliberately blend the two; JUMP Math, for example, calls its method guided discovery, with direct teaching in very small steps toward ideas children then find, and a 2019 Ontario trial of it gave mixed results.

8 How to try it

1. Once a day, give a problem before showing how to do it, and ask your child to try it any way they like. Then show the efficient way and connect it to theirs.
2. Run a five-minute number talk: say 18 + 15 aloud, ask for the answer in their head, then ask how did you think it? Share a second way yourself.
3. Play Which one doesn't belong? with four numbers or shapes. There is no single right answer; the reason is what counts.
4. Vary story problems by type: I had 5, got some more, now have 12. How many did I get? is harder than I had 5 and got 7. Give both.
5. If your child struggles, lean toward clear examples and guided practice first, and keep the open problems as a short warm-up.

9 Using Peabody with it

• Number lineA projector number line for showing children's different jumps side by side during a discussion.
• Hundreds chartFor number talks and pattern hunts on the board.
• Virtual manipulativesBlocks, fraction bars, ten frames and counters for children to show a strategy to the class.
• Numbers, on the kids' sitePractice with a worked example first, the explicit side of the balance; it does not run open problems or discussion.

Peabody is free and has no ads. Pages on peabody.kids are for children; pages here are for the adults teaching them.

10 Programs that follow it

Programs and materials often named with this approach: Illustrative Mathematics, Investigations in Number, Data, and Space, Cognitively Guided Instruction, Number Talks, Building Thinking Classrooms.