Many of us were taught mathematics in elementary and
secondary school as rote memorization of mechanical procedures, *period*. We sat quietly and watched as
our teachers demonstrated a new procedure on the chalkboard, overhead
projector, whiteboard, document camera, or smart board. Then we attempted to
replicate this procedure with problems copied from the board onto paper or
pre-printed on ditto sheets, workbook pages, or photocopied worksheets.* Notice how the instructional technology
changed but the instructional routine did not.*

Teachers tend to teach the way they were taught (Reeves, 2011). This is part of the reason reform of mathematics instruction has been so difficult to achieve. The National Council of Teachers of Mathematics (NCTM) Standards were originally published in 1989 and yet the instructional scenario described above can still be witnessed on a daily basis in classrooms across our country.

When teachers have only experienced mathematics learning as “watch/listen and then practice” they need support re-envisioning mathematics teaching. Coaches can use the three-phase problem-solving lesson format to help teachers learn to design mathematics lessons that build conceptual understanding and procedural fluency while engaging students in problem-solving experiences.

The three-phase problem-based lesson format is promoted by
respected math education leaders including Marilyn Burns and John Van de Walle.
In their book *Elementary and Middle
School Mathematics: Teaching Developmentally, *Van De Walle, Karp, &
Bay-Williams (2016) explain that “designing a lesson that engages students in
problem solving looks quite different from a traditional lesson that follows an
‘explain, then practice’ pattern.” They state that the three-phase lesson plan
structure provides opportunities to for students to learn mathematics content
as they build proficiency with critical mathematical practices including the
abilities to reason and justify thinking, represent mathematical ideas, make
connections, and communicate mathematical thinking.

**Planning Lessons Using the Three-Phase Structure Template**

From *About Teaching
Mathematics* (Burns, 2015)

**Introduce**

- Present or review concepts.
- Post a part of the problem or a similar but smaller problem.
- Present the investigation.
- Discuss the task to make sure students understand what they are to do.

**Explore**

- Observe the interaction, listening to how groups organize working together, the ideas they discuss, and the strategies they use.
- Offer assistance when needed.
- Provide an extension to groups that finish more quickly than others.

**Summarize**

- Have pairs or groups review their work and think about what to report in a classroom discussion.
- Initiate a classroom discussion. First have groups report how they organized working together.
- Next have groups report their results or solutions, explaining their reasoning or strategies.
- Generalize from the solutions.

While the intent of weaving together mathematics content and practices reflects best practice in mathematics instruction, when teachers have not themselves experienced mathematics learning in this way, it is difficult to think about instruction through this lens. The three-phase problem solving lesson structure supports teachers in learning to integrate mathematics content and practices within a single lesson. It is recognized as an effective lesson format for use in mathematics classrooms aligned with current curriculum standards and research-based instructional practices.

**References:**

Burns, M. (2015). *About
teaching mathematics*. Sausalito, CA: Math Solutions.

Reeves, A. R. (2011). *Where
great teaching begins: Planning for student thinking and learning.*
Alexandria, VA: ASCD.

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M.
(2016). *Elementary and middle school
mathematics: Teaching developmentally. *Boston, MA: Pearson.

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