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LessonStructures

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Lesson Structures

What else works apart from three-part lessons?

There is widespread research to show that it isn’t the ‘shape’ of the lesson that matters but the content and the teaching. However, there is value in having some overall organisation so here are 4 generic (research-based) lesson templates:

Contents

Main Section

Four lesson templates are given:

Direct teaching
Imagined representations
Purposeful games and puzzles
What if?

Lesson template 1: Direct Teaching

Key Lesson Element	Explanation
The teacher offers examples of pre-requisite content, assesses current understandings	The teacher poses some examples and monitors answers, including some examples which are less obvious and more searching.
Teacher uses lively methods to illustrate or model aspects of mathematics or procedures, and gives one or two practice examples, which are reviewed	Some students explain or demonstrate what they have done. Teacher poses some more practice examples. Possibly, teacher engages students in evaluating methods and deciding which to use.
Individually or in small groups, students complete further examples, activities or problems designed to give practice and consolidation of the content that is the focus of the lesson. The teacher monitors the work of the individuals, offering enabling prompts for students experiencing difficulty and extending prompts for students who complete the tasks.	In pairs, one student may be invited to create some problems for the other student to solve, and vice versa. Teacher might focus on one particular example which really ‘tests’ understanding rather than just technique.
The teacher reviews the methods and answers of the students, and attends to particular problems or responses that assist in consolidating the purpose of the lesson. Students query answers they do not understand.	Students might be asked which they found hardest and why, and asked to create further examples with the same difficulties.

Lesson Template 2: Imagined Representation:

Key Lesson Element	Elaboration
After posing and clarifying a problem based on an imagined context, the teacher asks the students to record an estimate.	The teacher asks the students to record an estimate. From now on, it is assumed that the students will be interested to know the answer.
Students are invited to think about what strategies they might use to calculate an answer, first individually, then brainstorming in a group, and the groups report their strategies to the class.	After discussion, groups might suggest a range of methods.
Groups choose (or are allocated) a strategy, they implement the strategy to find an answer, and prepare a report. The teacher monitors the work of the groups, ensuring that all students are involved in the strategy implementation.	The groups of students are allocated to a particular solution strategy, preferably one they have suggested. Using whatever resources are required, they implement the particular strategy and prepare a report.
The teacher leads a review of responses, including attending to issues such as efficiency of a strategy, and appropriateness of the degree of accuracy. Ideally the teacher will select few rather than all groups to report, particularly those that are likely to contribute to the purpose of the activity.	The students report on their strategy including indicating their final estimate.
It may even be appropriate for students to complete some additional problems or exercises that consolidate the principles identified in the investigation.	Some similar tasks can be posed that allow students to practise the skills, or prompt for transfer to alternate situations.
The teacher summarises the main mathematical ideas addressed in the activity. One key aspect of the teacher’s role is to emphasise the “dimensions of variation” inherent in the range of strategies and modes of communication of solutions that arise.	The teacher would emphasis the processes involved, as well as the steps necessary to ensure that any data that are collected are accurate.

Lesson Template 3: Purposeful Games and Puzzles

Key Lesson Element	Elaboration
After explaining the rules and purpose of the PGP, the teacher demonstrates the PGP to the class.	The teacher explains the game and associated mathematical ideas, and might model the process.
Students engage in the PGP for a short while, after which there is a teacher led class discussion of the strategies and or mathematical point of the PGP.	After the students have worked for a while on the task, there can be a class discussion of the processes for deciding what to do, the students can continue with the activity. Students can be invited to describe how they made decisions.
The students are then offered further opportunity to engage with the PGP. There can be additional discussion and activity as needed. The teacher or the students can suggest variations, such as making the PGP more challenging for some, or less complex for others. It is possible to group students based on their success at the PGP, so that, for example, student who complete the activity quickly might be grouped together for the next implementation of the PGP.	The teacher monitors the students’ work as they do the puzzle. Have both harder and easier versions available to be used as necessary.
The teacher leads a discussion of the strategies and mathematics of the PGP. Specific problems can be posed that allow practice to focus on the mathematical point.	It is possible to pose problems, for example, such as asking students to create their own mathematical extensions to the game/puzzle.
Finally, the teacher summarises the main mathematical ideas. Again it is stressed that the teacher has an active role here to find commonalities, patterns, and principles that can form the basis of the formalisation of the intuitive insights developed during the engagement with the PGP.

Lesson Template 4: What if?

Key Lesson Elements	Elaboration
Teacher poses and clarifies the purpose and goals of the task. If necessary, the possibility of multiple responses can be discussed.	The teacher might simulate the task by rolling dice and covering them up. The teacher could explain any mathematical content various ways. The students might be invited to record their answers systematically.
Students work individually, initially, with the possibility of some group work. Based on students’ responses to the task, the teacher poses variations. The variations may have been anticipated and planned, or they might be created during the lesson in response to a particular identified need. The variations might be a further challenge for some, with some additional scaffolding for students finding the initial task difficult.	The teacher monitors the work of the students. For students who have difficulty answering the initial question, the teacher might ask simpler ‘what if..?’ questions. For students who produce one or more correct responses, the teacher might ask: What if you were asked to find all the answers? And harder ‘what if..?’ questions
The teacher leads a discussion of the responses to the initial task. Students, chosen because of their potential to elaborate key mathematical issues, can be invited to report the outcomes of their own additional explorations.	Some students with simple strategies might be invited to demonstrate those to the class. Next, the teacher might choose a student who had produced an organised response to summarise their answers to the whole group. Students who have different responses can be invited to contribute their answers.
The teacher finally summarises the main mathematical ideas.	Finally, the teacher can summarise the successful strategies and the collective responses. Again this is the key part of the lesson for drawing out the patterns, commonalities, and generalisations.

Probes & Prompts

What style of lessons are you most familiar with?

What would a template of your usual lesson look like?

Taking Action

Try adopting an element or aspect of a different style into your own style.

Try adopting a different style lesson according to a different template. What makes this difficult to do? Use this as a topic for discussion with colleagues.

Case Studies

Research Sources

Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer.

Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on Mathematics Education (pp. 243–307). The Netherlands: Reidel.

Doyle, W. (1986). Classroom organisation and management. In M. C. Wittrock (Ed.), Handbook of research on teaching (pp. 392–431). New York: Macmillan.

Good, T. L., Grouws, D. A., & Ebmeier, H. (1983). Active mathematics teaching. New York: Longmans.

Hiebert, J., & Wearne, D. (1997). Instructional tasks, classroom discourse and student learning in second grade arithmetic. American Educational Research Journal, 30(2), 393–425.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: How children learn mathematics. Washington, DC: National Research Council.

Stein, M.K., Grover, B., & Henningsen, M. (1996). Building students capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Stigler, J., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press.

Sullivan, P. (1999). Seeking a rationale for particular classroom tasks and activities in J.M. Truran & K.N. Truran (eds.) Making the difference. Proceedings of the 21st annual conference of the Mathematics Educational Research Group of Australasia (pp.15-29). Adelaide.

Swan, M. (no date). Using percent to to increase quantities. Standards Unit trial materials. UK.

Watson, A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Derby: Association of Teachers of Mathematics.

Authors & Sources

These lesson templates are adapted from Professor Peter Sullivan, University of Monash

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