Understanding Difficulties In Learning Mathematics

One of the more common themes a tutor encounters, when working with students experiencing difficulties, is of a disparity between a perceived expectation of capability with fundamentals, and that capability in absolute. This appears particularly prevalent in lower secondary mathematical education following on from KS2.

For example, many a parent in my time as a tutor has lamented their child's inability to handle their times-tables, or arithmetic in general over the course of these lower secondary years, when perhaps such things are readily accessible to themselves. It creates a sense of frustration shared between parent and student and a diminished confidence in the latter. In asking the question, 'What can be done to foster a compassionate view of difficulties such as these?', we can begin to tackle issues of confidence, as well as to appreciate subtle complexities in a student's need as may arise from the applications of techniques that may have become automatic to the majority of others.

Over time, some behaviours can become subconscious processes, as we follow through on actions without consciously thinking what we are doing. What we then do may then seem obvious and easy to us, but what if, in a mathematical context, those most basic of considerations aren't automatic or even yet learned? One could argue that the situation described above is a direct consequence of this.

The following scenario is an attempt to exhibit the occasional profound difference between similar situations in which, in one of those situations, deeply-ingrained memories of techniques that have become second-nature are not at our disposal: that is, to make a struggling student of a capable mathematician, and so 'feel' the difficulties that so arise from a removal of knowledge of the fundamentals.

We count using the numerals 0,1,2,3,4,5,6,7,8,9 in the decimal system. There is no separate numeral for the number 'ten': instead we write 1 followed by a 0 to illustrate 'one-of-the-next-whole-number (i.e. ten!) with nothing further added.'. Similarly, there is no separate numeral for the number eighty-seven: we write 8 followed by a 7 to illustrate 'eight tens and an additional seven'.

Let us consider an alternate reality in which we came up with a symbol for ten and worked with it in common usage: let's call it U.

We'd have the numerals 0,1,2,3,4,5,6,7,8,9,U. So far, so easy.

The next number whole number is eleven. To be consistent with our choice of notation for the decimal system, we would write 1 followed by 0 to illustrate this: 'one-of-the-number-after-the-last-one-we-had-a-symbol-for, and an additional 0'.

Our counting numbers would be 0,1,2,3,4,5,6,7,8,9,U,10,11,12,13,14,15,16,17,18,19,1U,20,…U9,UU,100,…

'20' (two-zero) is our twenty-two: 'two ELEVENS and an additional ZERO, '14' is our fifteen: 'one ELEVEN and an additional FOUR', and so on.

All of a sudden, 35 + 65, and 23 x 14, and 89-55, don't seem quite so friendly. We've lost our sense of placement of number, of complements, of times-tables, and consequently the methods that use them. We are slowed, inefficient, and likely more inaccurate than we've been for many years: and all a result of a single symbol making a tiny adjustment to a familiar system. For many students, the world of 0-9 'feels' like the world of 0-U, because they haven't yet grasped the key techniques or memorised the results that allow effective navigation of the realm: something I hope you, the reader, got a brief taste of when met with the calculations above! (unless you're familiar with working with different bases).

No difficulty with mathematics should present as a negative verdict on a student's capability: there is always a perspective from which the nature of the problem can be appreciated and validated, no matter how seemingly basic. Mastery of the subject has always been about insight. Some of the most complicated things penned to paper have simply been trying to describe the simplest of ideas that when grasped, tell the story of thousands of words.

A good tutor's role is to help a student find the vantage point from which an idea makes sense, a concept possesses clarity, and from which techniques seem like natural avenues of enquiry performed with purpose.

Good luck in finding one here at Tutor Hunt!