Russell Timmins, August 2016

An unselfconscious use of practical equipment at the start of all new mathematical ideas helps to achieve this goal, it also helps to de-stigmatise the use of concrete equipment for all in the classroom.

In other words do not teach any abstract concepts until pupils have understood what the mathematics is doing, where it is being done and where links to other mathematical ideas can be joined.

A generic approach of understanding from the use of concrete apparatus followed by its withdrawal and replacement with diagrams or visual aids. When this is confidently happening with pupils they will be ready to go for a more abstract and fluent approach to their mathematics in each area of study.

“If I multiply 2 odd numbers, the product cannot be an even number. True or false? How do you know?” Mathematical reasoning as a major player in mathematics education enables the learner to make links quite smoothly to other mathematical areas. It also determines whether calculations are necessary for example, “Is 2472 ÷ 9 a whole number?” (A simple yes or no answer.)

If mathematical reasoning is not used then the calculation must be carried out in order

to answer the question. However, if they reason, that if the sum of all digits in a

number totals 9 then it is divisible by 9 and if not it is not divisible by 9, then

mathematical reasoning has saved a lot of work but still given a satisfactory answer

to the question.

The vast majority of people when asked this believe it to be wordy problems, where the mathematics is not clearly defined in the question. The classic example is this type of question:

“If 3 men take 5 days to build a wall, how long would it take 4 men?” However in reality a problem is something that requires a solution and finding the solution is the problem. How the problem is solved is where we are making our assessments. Is the problem a straight forward case of using an applied process like column addition for example? Perhaps, but is it being checked with the inverse application of column subtraction?

In year 6, given a problem involving percentages, are pupils able to convert to

fractions, if this makes the solution easier to achieve? Problem solving involves

mathematical reasoning and the ability to use links to other mathematics in order to

get to the end result. A simple correct answer is not always enough to make

assessment judgements.

Fluency in mathematics is the pinnacle of learning based on mathematical reasoning

and the ability to problem solve through a process that is well understood and easily

checked for errors.

Think back to secondary school trigonometry. I was taught the acronym **SohCahToa** in order to help me to remember how to find unknown sides and angles in right-triangles. Look at the wording. Remember not understand because if I am frank I did not really understand how I kept getting all my answers correct. All I knew was that the process worked. Today that is simply not good enough and quite right too. True fluency of approach is solidly understood mathematics, the offshoot of which is a much reduced requirement of revision because it is understood and not something to remember.

Fluency in mathematics relies on the learner’s ability to see problems from more than one angle and by choosing the most efficient route every time. True fluency.