One of a number of tried and tested technique for teaching work that students learners find hard to learn found in teaching tricks and tips
write long-windedly is the opposite of simplify - it helps learners both understand and remember their work and also helps learners "get" why mathematicians are always compacting.
Here are a number of situations where learners writing out their maths "long-windedly" helps.
simplify
When a question asks learners to simplify it at first glance seems strange to ask learners to write out the problem long-windedly. However here are a number of situations when it becomes apparent that it does
- help identify the structure
- help explain "the why" of some short cuts
- helps reduce the working memory requirements
- helps improve learner accuracy
simplify in index form
"Simplify b5 x b3"
Initially we get learners to "write it out long-windedly first" that is b x b x b x b x b x b x b x b now when we ask a learner to write this short-windedly or simplify
We are putting our learners in a position where they
- can create a generalisation for themselves (be a mathematician)
- can answer similar questions accurately
- be more likely to deal with "odd" cases such as "Simplify b5 x b"
- be more likely to remember a method, rather than half remember a rule
Ultimately we want the learner to imagine the method of writing the problem out "long-windedly" - to be sure of their answer, but first actually asking the learners to write out the expression "long-windedly" helps making this method concrete helps. We should continue to ask the learner to write out their working "long-windedly" until the learner achieves mastery learning on these types of question.
The plan for this page is to build up a list of examples, as above, and include links to "teaching Higher work to Foundation learners" videos and more easy on the eye learner "remind-me" videos. For now though here is a list of situations where we have found asking learners to "write that out long-windedly" has helped.
simplifypq: b^5 xb^3 and b^5 x b
simplifypq: b^5 div b^3 and b^5 div b
simplifypq: 3a^2b x 4a^3b^5 and ...
simplifypq: (c^2)^3 and (ab^3)^4
simplifypq: (3c^2)^4
simplify algebra simplifypq: 2g x 3g
simplify a fraction
have only 2nd part of 7timesFull etc - so best to record another
simplify a ratio
ratio: simplestform
factorise
factorise: 3x + 12
factorise: 4x^2 + 6xy
factorise: remind me 2 different letters
long multiplication
multiply: partitioning
multiply: 28 x £4.70
long division
divide: divide double digit
divide: divide 5Q
find the mean, median, mode or range from a frequency table
(not available yet)
solve
solve: 4e + 22 = 70 (where the function machine is writing out all the things that happen to e long-windedly)
solve: m - 6 = 8 (where the function machine is writing out what happens to m long-windedly)