#### Great Math Lesson Series:

 Phase I Phase II Phase III Phase IV Phase V Introduce Stimulus Whole-class Activity Problem Solving Synthesis & Reinforcement Revision & Recap

This is the third of a five-part series of articles on how to teach a great mathematics lesson, using a simple, purposeful template that can be adapted for any math topic and any age level. In this article, the third phase is described, in which students engage with a mathematics topic and practise thinking mathematically.

## Third Phase: Student Engagement & Problem Solving

If this phase of the lesson doesn't happen, then there is no opportunity for students to actually learn. In this part of the lesson approximately 50% of the lesson time is used up as students are presented with problems to solve, on their own if at all possible.

Let me talk a little about what this phase is not. This is not the time for students to practice a routine that has been demonstrated by the teacher, in the sense that it has been used traditionally. In days gone by, which many of us remember vividly, our teachers showed us a trick or a pattern of steps to use for a particular class of problem, and then gave us anywhere from 5 to 100 examples to complete before the bell rang. The problem with this math lesson model is that no-one is doing much thinking, and once students "get it", they only have to follow the required steps as efficiently as they can, and the lesson is over with a minimum of effort. When, however, the same students need to use the knowledge and skills of the lesson in a real-life context, what will most likely happen is that the knowledge and skills will not be recalled and the lesson will not be applied to the situation.

In order to reach different outcomes, we need to plan and implement different strategies. And the best strategies will come from the understanding that students are not robots or pets to be trained to follow patterns of behavior, but are thinking, creative, smart people who deserve the chance to exercise their abilities in the pursuit of greater learning. I don't know of a single teacher who thinks that students are either automatons or circus performers, but I have all too often observed or participated in "lessons" that were really training exercises, which were as boring as that sounds, and as ineffective in developing greater mathematical ability.

The phrase I use to describe this phase of a great mathematics lesson is "engaging students in mathematical thinking"; sometimes I add "and authentic, genuine problem solving". Unless this happens, this phase of the lesson has missed its mark, and the lesson lacks an effective core. Genuine teaching will happen in any good lesson (see Phase II of this series), but unless students get to exercise their mental muscles and process mathematical data and create new answers to problems, students have been observers, not participants.

### Engaging Learners

To maximise student involvement in this central part of the lesson, consider how big your groups are. There will be an optimal group size for each activity, determined by the activity itself and the students in your class. A good rule of thumb for groups is four to six students. Any more than six and the group will often split in two, with a core of two or three doing all the work and the others talking about something else. If there are less than four, there will be groups which cannot proceed on an activity because there is no-one in the group that can get started.
An alternative strategy is to use pairs, rather than groups. The dynamics of this learning structure are obviously different, with little chance for anyone to "slack off" and not get involved in the activity.

### Activities

What sort of activities are appropriate for this thinking phase of your lesson? The short answer is "almost anything that includes genuine thinking". And the great thing about this approach is that it can be used with any age group.

Even the youngest children can think for themselves, given the right motivation. If you doubt that, think of the last time you saw a baby learning to walk - no-one had to explain it to the child in simple words, no-one gave a simple trick for the toddler to practice to make walking easy. No, the child worked it out through trial and error, through determination and and unwillingness to give up until the goal of walking was reached. Think about how to tap into that source of energy and effort when you design a task for students to complete.