Wording VS mathematical notations

Question

Is it better to write everything in words as the concepts themselves should be known? Or will some teachers in some countries prefer to be able to choose questions which also test the student's knowledge of the notation?

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AB}$ is perpendicular to $\overline{CD}$

Answer 1

I apologize in advance if my answer will be long, but this is a very interesting question and I'll try to give it the answer it deserves. I will refer not exclusively to notation in geometry but in mathematics in general.

The very short answer is:

it is fundamental to use some notation, or we'd be back to the painful description of equations of early algebra, but use too much and you'll cruelly add a burden to your students.

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Now, for the long answer:

Mathematical notation can have on students, especially pre-university or non-maths students (meaning, people that did not choose willingly to attend maths lessons), a double-sided effect:

• it can be extremely arousing for us nerds, as it becomes the key to understanding that mathematics is, after all, a way of manipulating concepts as a language per se, and as a language it has its own "alphabet" (forgive the poor wording in terms of linguistics). It is also like giving us new toys to play with, and we then start having fun with them in external contexts - as when we use existential quantifiers in everyday's life. Furthermore, it speeds up explanation significantly, which means gifted students will be more stimulated;

• it can be extremely discouraging for the self-diagnosed-mathematically-impaired, which see in symbols one further obstacle that the teacher/system/Illuminati have made up with the purpose of making this senseless subject even less understandable.

Because of this, as with many things in teaching, it's not intrinsically better to choose to use as much or as little notation notation as possible, but balance is key:

• use too much notation and you'll make things harder for the second group;
• use too little notation and you'll be missing on a very useful tool, and possibly bore the best students to death.

Given all this, let me then give you some, hopefully useful, practical tips for introducing more notation with minimum damage:

• What is fundamental for every student to understand is that no notation has been given to us, perfect and immutable, by some divinity: notation is an agreement between two parts communicating, and different parts (e.g., texts, fields, teachers..) can give to the same symbol different meanings. Mathematics is more alive than one thinks, as well as constantly evolving! Therefore agree explicitly with your students: if you don't mind, from now on I'd use symbol x instead of writing every time [meaning of x]. This will also make the whole teaching feel less an unidirectional flow of pre-compiled stuff from your head to their notebooks;
• A bit at a time: add new symbols progressively, trying to avoid your students an indigestion for receiving too much new information in one go;
• stress out the importance of minor differences, sometimes students tend to quotient over little graphical things thinking they are only quirks while they actually carry a meaning;
• set a "minimum level" of notation and use it in your questions, but encourage your students to use more, for example by providing a little extra credit to those who can also formulate their answers in a concise way rich of notation - potentially, notation-only!

Answer 2

Testing students' ability to identify notation is a perfectly sensible thing to do if learning the notation is an intended learning outcome.

However, the notation you show is not entirely standard. For example, the 'ray' notation would more commonly denote a vector. Some of them, such as parallel and perpendicular, are really shorthand rather than mathematical notation.

In mathematical writing, symbols should be used when symbols are appropriate, but not as shorthand when words are appropriate. Understanding the distinction is not easy, and is probably beyond all pupils and most teachers at school level.

When writing test questions, the question to consider along these lines is 'what mark would I want a student to get if they didn't understand the notation but otherwise knew the concepts?' If the wording of a question would stop an otherwise knowledgeable pupil from starting the question, you need to be comfortable that that is the aim of the question. Context will play a part in deciding how appropriate that is. An internal test written by one school teacher is much lower stakes than a nation-wide standard exam.

Answer 3

Mathematical notation is there to convey concepts in a very compressed manner, as opposed to the verbosity of the natural language when it comes to expressing the same ideas. Being able to read and understand it fast is a key skill in any profession with require working with anything related to mathematics, such as engineering. However, this is a learned ability and it takes time to learn and master.

From personal experience, I felt (still feel) learning to read maths as learning a new language. At the beginning I read it loud in my mind and then link the concepts to understand the meaning of the expression and, after a while, I could understand the meaning directly from the symbols. I felt the experience quite similar to learning to read music.

Considering this, I think the tests should be designed in such a way that they introduce slightly more mathematical notation every time such that students are exposed more and more to the mathematical language and less to the natural language.