[MathEd] Heuristics in Mathematics

Heuristics are guidelines or rules-of-thumb for doing some task or solving a problem.  They work most of time, but are not meant to be hard-and-fast rules.  It something does not work, try another approach.

The Singapore Mathematics Curriculum includes 13 heuristics (of which 11 are for primary school) that teachers are supposed to train their pupils for use in mathematical problem solving.  For convenient reference, this article lists these heuristics and some others, most of which are from my personal learning and teaching experiences (and a few by searching the Internet).  Making observations, connecting facts and using heuristics to solve problems is part of a good mathematics education.  Actually, there is nothing to stop a primary school pupil from using the more advanced heuristics, it is just that pupils should at least be taught the 11 listed.

Primary School (Elementary School)

H01. Act it out

H02. Use a diagram / model

H03. Make a systematic list

H04. Look for pattern(s)

H05. Work backwards

H06. Use before-after concept

H07. Use guess and check

H08. Make suppositions

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Secondary School and beyond

H12 Think of a related problem

H13 Use Equation / write a Mathematical Sentence  

Other / Related Heuristics

· Look for clues, make observations & connections

· Considering the meanings / definitions

· Draw construction lines

· Identify important variables, use effective notation.

· Identify variable to eliminate

· Keeping one variable constant and observing changes

· Compare what you have currently with what you want (goal)

· Compare and Contrast Data

· Consider extreme cases.  Consider special cases.

· Generalise.  Solve by solving a more “difficult” problem.

· Work Forward (~H01)

· Check for plausibility (~H07)

· Formulate an equivalent problem (~H09, H012)

· Try to simplify & cancel.  Avoid complications. (~H10)

· Create something out of nothing 无中生有 (~H12, H10) e.g. x  ®  x+5–5

· Replace something by its equivalent 偷梁换柱  (~H12, H10) e.g. x ® e^ln x

· Recognise Chunks (Chunking), substitution (~H10)

· Check for parity (e.g. + or - sign, even or odd). (~H10, H11)

· Divide and Conquer, Divide into cases, Break set (~H11)

· Exploit symmetry (~H11)

· Remove denominator from complicated fractions 釜底抽薪 (~H11)
· Think of the opposite problem (~H12)

· Argue by contradiction (suppose the opposite is true)