CMTP: Changes in Mathematics Teaching Project

Examples of lesson structures used when teaching project students

Lesson A

T says how the ideas in the lesson sequence are progressing and what this lesson will be about and how it relates to last lesson;
Interactive recap of definitions, facts, and other observations.
T introduces new aspect & asks what it might mean.
T offers example, gets them to
Identify its properties
T gives more examples with multiple features; students identify properties of them.
Students have to produce examples of objects with several features,
Three concurrent tasks for individual and small group work:
- describe properties in simple cases;
- describe properties in complex cases;
- creating own objects.
T varies variables deliberately
They then do a classification task & identify relationships within groups
T circulates asking questions about concepts and properties.

Lesson B

T gives object with multiple features & asks students to "think about..."
T indicates a classification by focusing on variables and variation;
Students use prior knowledge to identify properties of diagram;
T summarises what was done before and what problems to avoid; gives format sheet has two way classification to classify shapes,
Students create objects with multiple features and classify them using the format, exemplifying different combinations of properties.
Students induce generalisation and new definitions
Students transform existing ideas in light of new experience
Students 'research' relevant facts.
Predictions and conjectures are written on the public board.
Trefers to things said by students which demonstrate characteristics of mathematicians
Students do informal inductive reasoning.
T leads association of ideas & generalisations.
Students have to compare examples, explain and justify their comparisons and conjectures

Lesson C

Students are asked to visualise spatial movement
Students create object with given two features
T names the object
T draws objects with imagined features
T says what the lesson is about and how this fits in sequence.
T shows multiple objects with same feature
Students describe a procedure, in own words,
T asks for clarification;
Students think about how procedure will give them the desired outcome;
Students then practise procedures
T demonstrates new object with multiple features.
Students make shapes using with given procedure, and then with a choice of procedures, by varying variables.
T indicates application to more complex maths which will come next
T shows one object which is nearly finished & students predict how to complete it by identifying missing features
Students deduce further facts.

Lesson D

T offers one mathematical word & asks students to 'brainstorm' possible connections to word
T identifies properties of what students offer
T and students describe, define and redefine this new concept together
Students use their own words, to describe & define, then create objects to illustrate the new idea
T asks for clarification & examples of what they say
Students and T draw on knowledge, techniques, facts
T introduces conventional formal notation
T classifies students' examples.
T summarises what has been done.
Worksheet to find answers with and without known procedure - seems to be discontinuous with earlier part of lesson
Students adapt procedures & identify relationships by deduction from results
Students explore variation in methods on the sheet, classifying methods
T leads discussion of implications of their ideas, varying notation.
Students are expected to informally deduce meaning.
T shows how generalisation links the worksheet to the earlier part of the lesson
T tells how this applies to harder maths
Final example from teacher has new features, left to think about

Lesson E

T introduces 'learning about equivalence'
T introduces one example and then asks them for examples with certain characteristics
T summarises so far, identifies variables in their examples, and compares selected examples which become more and more complex.
Students solve some equations made by other students and compare methods
T leads public deduction of how methods relate, with explanation and adaptation.
T summarises ideas, and shows application to more variables,
Students work in groups to express in own words the meaning of equations

Lesson F