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Mathematics Matters Lesson Accounts 22 - Creativity & Solving Equations

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 28 May 2008 by ncetm_administrator
Updated on 16 June 2008 by ncetm_administrator

Mathematics Matters Lesson Accounts
A collection of memorable mathematics lessons that conference and colloquia delegates had observed or taught which they felt were successful.  Each account refers to one or more of the values and principles in the report.

Lesson Account 22 - Creativity & Solving Equations

Written by Joan Ashley
Organisation NCETM
Age/Ability Range 16-20 – FE College – GCSE resit class Very mixed ability – Final grades B-U. No setting at all. Multiple social & medical problems.

How was the session/task introduced?
Students were given a selection of equations to solve, as a benchmark of their current knowledge. They were told that they were not necessarily expected to be able to solve them, but that they should be able to do so by the end of the day.

Atmosphere was completely apathetic. Post lunch slump, plus the introduction of algebra, which from past experience they “knew they couldn’t do”. Very quiet (unusually), probably because of 6 observers and 2 video cameras.

How was the session/task sustained?
The beginning of the lesson was led and guided by the teacher, the students were invited to create a complex equation by applying a number of operations to a simple equation. Initially these were limited to + – × ÷ operating on numbers up to 10, however students asked if they could include √, 2 and even 4 and they were allowed to do this.

Example : jointly constructed by the group starting with x = 4 (every single step was provided by students, including the choice of letter).

  x = 4  
x 2    ÷2
  2x = 8  
-3    +3
  2x – 3 = 5  
÷5    x5
 = 1
+4    -4
 + 4 = 5

Everything except the final equation was covered up and students were invited to say what had happened to x. They could tell me. I was surprised.

They were then invited to unpick the problem to get back to x which they did.

This was repeated once more for a new example, with the pace dictated by the class, as understanding was vital.

Students then worked in pairs to generate equations for their partner to solve. The one who had generated the problem was expected to assist if the solver got stuck.
This resulted in some superb and lengthy discussions where students sorted out their own errors. Surprisingly some of the weaker students had the greatest persistence.

How was the session/task concluded?
The lesson developed a momentum of its own. All were engaged, successful and interested. Algebra was accessible.

A group of 5 stayed behind when they should have gone to an English lesson because they wanted to finish a discussion.

They worked effectively together – magic.

By the end of the lesson most had:

  • Gained algebraic confidence
  • Understood where equations came from
  • Were able to solve linear equations, using their own recording methods which they understood.

What were the critical moments?
I think the main one was when they realised that they could understand the structure of an equation. This gave them the confidence, which they had previously lacked, to have a go and to persist until they were successful.

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?
Creating & solving own equations accurately.

How was that mathematics learnt?
Through discussion, mostly with their peers.

Other memorable outcomes
Although I was asked to be filmed teaching this, I was very sceptical about whether or not it would work. I had never taught it on camera and was very nervous about making a complete fool of myself.

I was wrong. It did work.

As a piece of professional development, being challenged out of my comfort zone to teach something I was unsure of did me the world of good, and changed my classroom practice radically. Do we need therefore to ensure somehow that other teachers are able to experience the outcomes of research evidence on effective learning in their classrooms? If this aspiration is impossible, how do we make a stronger link between research evidence and effective learning?

I was prepared to teach this through joint PD sessions with other teachers. I feel this sort of collaborative support is vital to the promotion of change.

References: Article entitled “Talking in class”, published in “Quality Matters”, journal of the LDSA (Learning and Skills Development Agency), June 2002 LDSA curriculum resource “Learning Mathematics through Discussion and Reflection” reflection”.


Downloadable PDF

Click here to download this lesson account in PDF format.

Values & Principles

Conceptual understanding and interpretations for representations
Strategies for investigation and problem solving
Exposes and discusses common misconceptions and other surprising phenomena
Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work
Encourages reasoning rather than ‘answer getting’

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