teach problem solving with/out a calculator 1st/2nd

One of a number of tried and tested technique for teaching work that learners find hard to learn found in teaching tricks and tips

It is also helpful to unlink in our minds as teachers, the difficulty of the numeracy involved in a problem and the difficulty of mathematical process involved in a problem. I'm sure most maths teachers can think of 

• a learner who is especially weak at numeracy but especially strong at maths problem solving and conversely
• a learner who is especially strong at numeracy but especially weak at maths problem solving.

If we assume that low attaining learners are atypical of most learners, then we shouldn't wait for our learners numeracy skills to be at a certain level before we introduce them to maths problem solving however we should wait for our learners to be confident at a certain level of maths problem solving before we introduce them to harder maths problem solving; since holding back from teaching harder work before pre-requisites are mastered is the essential for mastery learning.

In learners early problem solving in maths, using a calculator doesn't help learners answer word problems, as learners will not be confident which operation will be the correct one, and the very act of working through a method without a calculator can help learners firmly establish the correct operation. In timely practice we ensure learners meet many problems where they have to decide, "which operation should I use?". In blocked maths teaching (see retrieval practice or spaced learning in a nutshell), learners can more easily decide which operation to use by the learning objectives of the lesson or the chapter title in the text book, and which numbers to use by finding the two numbers in the question. So in blocked teaching, learners are not likely to be building up a repertoire of situations to match with the 4 operations. 

Once the learners begin to approach word problems with more than one stage - which should only happen once they are confident at matching single stage word problems with the correct operation - then using a calculator can help a learner to learn harder problem solving skills. If a learner can put their full efforts into deciding on the stages and order of calculations, and use a calculator to do the calculation part, then the learner is more likely to "have a go".  

Providing a learner is confident in deciding which operation is correct from a word problem, we need not wait until the learner can, for example - multiply a 3 digit number by a two digit number accurately, before we introduce learners to word problems where the learner has to decide for example

q₁ x f₁ + q₂ x f₂   or     f (q₁ + q₂)  or     q₁ x f₁ + q₂

where q is a quantity and f is a frequency

If a learner has used a calculator and shown the calculations and answers ... I realise this is a "big if", however if, when we start to teach problem solving the first two stage problems the learner meet are like the following example, we can expect learners to show their calculations and answers, we say to our learners that they have not achieved mastery on these type of problems, until they can show sufficient calculations and explanation.

Mia and Lizzy are designing a flat.

Here are some of the items on their wish list.

How much would 2 tablets, a DVD player and 2 penguin cushions cost?

So if a learner has used a calculator and shown the calculations and answers they have found, then when feedback is required, a more meaningful conversation between teacher and learner is possible. Teachers will have available in their repertoire of questions not just "what do you think you could work out first to help you answer this question"  but also "what have you worked out here?" or "I can see you have worked out the number of ... how can you use this to help answer the final question" or when the learner is coming close to mastery "I can see you have worked out .... and you have said ... doesn't have enough, but what were you comparing to know that Mia does(n't) have enough? ... how could you write that down?"

Our word problems which involve more than one stage of problem solving our found in how much enough NC and how much enough CALC  strands. In most circumstances teachers would expect learners to master the calculator skill before the corresponding no calculator skill.

The plan for other pages like this, is to build up examples as above and include links to "teaching Higher work to Foundation learners" videos and more easy on the eye learner "remind-me" videos. However when deciding whether learners should use a calculator or not to solve problems is a more subtle skill and very much depends on the skills and personalities of our learners.