Easy Techniques to Improve your Maths, Part 1: Distinct Symbols


Apart from the obvious “practice more” there are a range of techniques that can dramatically improve your maths, either for exams or for work, by mitigating common mistakes. This is the first in a series of short articles by Dr Andrew Smith, the founder of MathU.


Lettering

This may seem basic but it is a very common problem ...

In algebra you need to ensure each of your numbers, letters and symbols are distinct from each other. Otherwise you risk misreading your own writing, particularly in a time-pressured situation such as an exam.

Everyone’s handwriting is different, and only some of the symbols will be prone to muddling for you. Look at the table to your right to see some of the common lettering muddles that happen in mathematics.

Try it!

  • Write out each of the digits \(0-9\)
  • Write the operands \(+, -, \times, \div\), parentheses \((\;)\), brackets \([\;]\), braces \(\{\;\}\) and proportional symbol \(\propto\)
  • Now write the lower case letters \(a-z\) and the upper case letters \(A-Z\)
  • Finally, write the commonly used Greek letters \(\alpha, \beta, \gamma, \delta, \Delta, \eta, \theta, \lambda, \mu, \nu, \pi, \rho, \sigma, \Sigma, \phi, \omega, \Omega\)
    (You may not have come across Greek letters in algebra yet, but they are common in statistics and scientific formulae)

Look across your letters, numbers and symbols - are there any that look too similar to each other?

The solution is to change the way you write some symbols.

Top Tips

  • Use cursive (running writing) - this works well for many lower case letters, particularly \(x, \ell, u, v, j, t, z/\mathfrak{z}\), by adding tails, loops and/or curves to distinguish from other similar symbols.
  • Add serifs (extra pen strokes) such as a vertical line on the bottom right of \(G\) and the horizontal lines at the top and bottom of \(I\)
  • An alternative for \(Z\) or \(z\) is a horizontal line through the middle: \(\mathcal Z\)
  • Make up your own tweaks – the only criterion is that each symbol is distinct from all others

Any tweaks you use to make symbols distinct will likely not be part of your normal handwriting, but used only for algebra – that’s expected. After a while they become habit, so practice your new symbols at every opportunity. This will reduce errors in your maths exams (and science exams, and others too).


Common lettering muddles:
  • number \(1\), number \(7\), lower case letter \(\ell\), upper case letter \(I\)
  • number \(6\), lower case letter \(b\), upper case letter \(G\)
  • number \(0\), Greek lower case sigma (\(\sigma\)), theta (\(\theta\))
  • number \(2\), lower case \(z\), upper case \(Z\)
  • number \(5\), lower case \(s\), upper case \(S\), Greek lower case delta (\(\delta\))
  • number \(8\), upper case \(B\), Greek lower case beta (\(\beta\))
  • lower case \(t\), plus symbol \(+\)
  • lower case \(x\), times symbol \(×\)
  • lower case \(i\), \(j\)
  • lower case \(u\), \(v\)
  • lower case \(u\), \(a\)
  • lower case \(u\), Greek lower case mu (\(\mu\))
  • lower case \(v\), Greek lower case nu (\(\nu\))
  • lower case \(a\), Greek lower case alpha (\(\alpha\)), proportional symbol (\(\propto\))
  • lower case \(p\), Greek lower case rho (\(\rho\))
  • lower case \(n\), Greek lower case eta (\(\eta\))
  • lower case \(w\), Greek lower case omega (\(\omega\))
  • upper case \(C\), opening parenthesis \((\)

Never Overwrite

Another related problem is attempting to overwrite a symbol with a new symbol when an error is detected. This can cause misreading and hence new errors. A very common example is attempting to change a plus sign into a minus sign. It is easy to go the other way (change a minus to a plus), but changing a plus to a minus is simply asking for trouble – it is always better to cross out the old symbol and write the new symbol separately. This is also the case for numbers and letters.

Missing Negatives

Some students regularly miss negative signs on numbers. A good solution is to surround the negative number in parentheses, which emphasises the negative nature. For example: \(-8 + 7 - -9 = 8\) versus \((-8) + 7 - (-9) = 8\)

Take Space

If your writing becomes cramped, it is easy to misread. Take space to write out symbols clearly. Saving paper is not worth sacrificing exam marks (or miscalculating the strength of a bridge or pressure of a reactor, or …).


For further help, including maths understanding, skill, practice, and mitigations for common mistakes, check out our online maths learning tool MathU at mathu.com.au