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Secondary Magazine - Issue 57: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 30 March 2010 by ncetm_administrator
Updated on 13 April 2010 by ncetm_administrator


Secondary Magazine Issue 57coloured steps  - photograph by Jurek D used under the Creative Commons attribution licence 2.0


Focus on...generalising

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“Spotting patterns can be an important first step - understanding why the pattern works and explaining why it is appropriate to generalise is the next step, and often the most interesting and important.” Read more in this short article from NRICH.

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“A lesson without the opportunity for learners to generalise (mathematically) is not a mathematics lesson. Mathematics is fundamentally about becoming aware of and expressing generality. No-one expects young children to memorise all two-digit additions and subtractions. Rather learners are expected to reconstruct a collection of general methods which will enable them to carry out not only all two digit additions and subtractions, but any additions and subtractions. So even the youngest of children are expected to generalise.” From the Mathemapedia entry A lesson without….

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"Let us see whether we could, by chance, conceive some other general problem that contains the original problem and is easier to solve." – Leibnitz.

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To encourage learners to generalise in the classroom strategies such as Easy-Hard-General, Particular–Peculiar–General, and Another & Another can be useful. If you have a go with these strategies why not post a comment on the relevant Mathemapedia entry, or let us know by posting a comment on this page?

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The Early Algebra Project at Tufts University in Massachusetts, has identified a set of Number Sentences that teachers used to help students articulate mathematical generalisations.

E X A M P L E S    78 + 0 = 78; 23 + 7 = 23 *
"When you add zero to a number, you get the number you started with."
E X A M P L E S   96 - 96 = 0; 74 - ____ = 74
"When you subtract a number from itself, you get zero."
E X A M P L E S   96 x 0 = 0; 43 x 0 = 43 *
"When you multiply a number times zero, you get zero."
E X A M P L E S   65 x 54 = 54 x 65; 94 x 71 = 71 x ____
"When multiplying two numbers, you can change the order of the numbers."
*denotes a false number sentence
Source: National Center for Improving Student Learning & Achievement in Mathematics and Science. (2000). Building a Foundation for Learning Algebra in the Elementary Grades

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This page from Cut The Knot starts. “Mathematical discovery is seldom a single step process. Often it's indeed the case where answering a more general question is easier than finding an answer to a specific one. Later, on this page, we'll see a collection of examples that I'll be updating from time to time…”

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