About cookies

The NCETM site uses cookies. Read more about our privacy policy

Please agree to accept our cookies. If you continue to use the site, we'll assume you're happy to accept them.

 

Personal Learning Login






Sign Up | Forgotten password?
 
Register with the NCETM

Secondary Magazine - Issue 123: From the Library


Created on 02 July 2015 by ncetm_administrator
Updated on 21 July 2015 by ncetm_administrator

 

Secondary Magazine Issue 123'Pages' by Alexandra*Rae (adapted), some rights reserved
 

From the Library

Why do some pupils find algebra so mystifying and difficult, but some just seem to “get it” and quickly develop both procedural fluency and conceptual understanding? Dietmar Küchemann carried out research in the 1970s examining how 14-year-olds interpreted the use of letters in algebraic contexts. His findings were reported and discussed in K. Hart, 1981, Children's understanding of mathematics: 11-16 and Children’s understanding of numerical variables in Mathematics in School, September 1978.

Küchemann identified six different modes of understanding. Recognising the way we use these different interpretations might help us illuminate where our pupils’ difficulties lie, and then to support them to overcome these. In the first three interpretations the letter is considered as a concrete entity (see The ‘algebra as object’ analogy: a view from school by Kate Colloff and Geoff Tennant, discussed in Issue 116) while the next three levels, on which we focus here, involve aspects of abstraction.

  • A letter may be used as a specific unknown: “if e + f = 8, give an expression for e + f + g.” This requires the letter to be understood to represent a specific but unknown value. Common incorrect answers to this question were 8g, 12 and 9.
  • The letter may be used as a generalised number where the letter could take a number of values: “what can you say about c if c + d = 10 and c is less than d?”
  • Letters may be used as variables where there is a (second-order) relationship between the expressions: “which is larger, 2n or n + 2?”. In his research, Küchemann was looking for answers of the form “it depends” with some evidence of substitution of, say, n = 1 and n = 10. A commonly seen answer was “2n is bigger because multiplying makes things bigger”. This is a very prevalent and hard-to-eradicate misconception (and is coupled with “division makes things smaller”).

As mathematicians, we move adeptly between different interpretations of notation, suiting our reading of the variables to the context. But are our pupils as familiar, implicitly, with these interpretations? Are they accomplished in recognising which is applicable in different contexts? Küchemann used a number of carefully crafted questions to probe and clarify what pupils were thinking; these are available on the CSMS Tests area of the Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) website. You could use these items as an effective diagnostic tool for identifying the interpretations with which your pupils are comfortable and confident, and hence plan precisely the questions and activities which will first consolidate and then deepen their algebraic fluency, understanding and reasoning. ICCAMS, a major research project, was undertaken in 2008 to 2012 building on Küchemann’s research. Information and trial teaching materials can be found on the ICCAMS website: well worth reading before planning your teaching of algebra next year.

Image credit
Page header by Alexandra*Rae (adapted), some rights reserved

 

 

 
 
 
 
Download the magazine as a PDF
 
Secondary Magazine Archive
 
Magazine Feed - keep informed of forthcoming issues
 
Departmental Workshops - Structured professional development activities
 
Explore the Secondary Forum
 
Contact us - share your ideas and comments 
 

 


Comment on this item  
 
Add to your NCETM favourites
Remove from your NCETM favourites
Add a note on this item
Recommend to a friend
Comment on this item
Send to printer
Request a reminder of this item
Cancel a reminder of this item

Comments

 


25 September 2015 12:51
The article on algebraic reasoning in issue 109 might also be of interest; see: https://www.ncetm.org.uk/resources/44331
By KeithJones
         Alert us about this comment  
Only registered users may comment. Log in to comment