Tuesday, March 3, 2015

[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+55
· Replace something by its equivalent 偷梁换柱  (~H12, H10) e.g. x ® eln 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)


  1. This is a very nice list, updating the famous compilation by George Polya in HOW TO SOLVE IT.

    In my own experience, it's not so easy to teach a heuristic, as opposed to teaching a specific procedure. My students -- even the very able ones -- have gotten so used to being taught 'facts' and 'procedures' that they are suspicious when I introduce heuristics as something to be learned, perhaps because the exams they take have only questions which can be solved by selecting the right procedures.

    I mentor/tutor a group of students who are preparing for Olympiad (first level)-style questions, and I assure them that there they will encounter questions which cannot be answered by simply applying a procedure that they have already learned.

    1. Glad you like it. Many of my articles in this blog illustrate the use of heuristics at varous levels, and there are some Olympiad ones too. Type "Olym" in the search box and you should be able to find them.