# Teaching Through Problem-solving

In Teaching Through Problem-solving (TTP), students develop each new mathematical concept by solving a problem that illuminates it.

### What is Teaching Through Problem-Solving?

In Teaching Through Problem-solving (TTP), students learn new mathematics by solving problems. Students grapple with a novel problem, present and discuss solution strategies, and together build the next concept or procedure in the mathematics curriculum.

Teaching Through Problem-solving is widespread in Japan, where students solve problems *before* a solution method or procedure is taught. In contrast, U.S. students spend most of their time doing exercises– completing problems for which a solution method has already been taught.

### Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

- Learn to reason mathematically, using prior knowledge to build new ideas
- See the power of their explanations and careful written work to spark insights for themselves and classmates
- Expect mathematics to make sense
- Enjoy solving unfamiliar problems
- Experience mathematical discoveries that naturally deepen their perseverance

### Phases of a TTP Lesson

Teaching Through Problem-solving flows through four phases as students 1. Grasp the problem, 2. Try to solve the problem independently, 3. Present and discuss their work (selected strategies), and 4. Summarize and reflect.

Click on the arrows below to find out what students and teachers do during each phase and to see video examples.

Grasp the Problem

#### WHAT STUDENTS DO

Understand the problem and develop interest in solving it.

Consider what they know that might help them solve the problem.

#### WHAT TEACHERS DO

Show several student journal reflections from the prior lesson.

Pose a problem that students do not yet know how to solve.

Interest students in the problem and in thinking about their own related knowledge.

#### EXAMPLES

Try to Solve

#### WHAT STUDENTS DO

Independently try to solve the problem.

Do* not* simply following the teacher’s solution example.

Allow classmates to provide input after some independent thinking time.

#### WHAT TEACHERS DO

Circulate, using seating chart to note each student’s solution approach.

Identify work to be presented and discussed at board.

Ask individual questions to spark more thinking if some students finish quickly or don’t get started.

#### EXAMPLES

Present & Discuss

#### WHAT STUDENTS DO

Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)

Actively make sense of the presented work and draw out key mathematical points. (All students)

#### WHAT TEACHERS DO

Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)

Monitor student discussion: Are all students noticing the important mathematical ideas?

Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.

#### EXAMPLES

Summarize & Reflect

#### WHAT STUDENTS DO

Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.

Write journal reflection on their own learning from the lesson.

#### WHAT TEACHERS DO

Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.

Ask students to write in their journals about what they learned during the lesson.

#### EXAMPLES

New Learning

The **New Learning **is what students actually take away from the lesson, as reflected in their journal writing, their work and ideas during class, and the exit task, if any. Identifying what students took away, and noticing any ideas that are fragile or partial, allows teachers to plan the next lesson.

### How Do Teachers Support Problem-solving?

Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known *5 Practices for Orchestrating Mathematical Discussions *were based in part on TTP*.* Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics. Classroom routines for presentation and discussion of student work, board organization and reflective mathematics journals work together to allow students to do the mathematical heavy-lifting.