Non-mathematical skills

Some non-mathematical skills taught in mathematical lessons

1. cancellation
2. change the sign
3. move the decimal points
4. move the term to the other side of the equation

Why mathematics teachers teach non-mathematical stuffs?

Recently, I gave tuition to secondary 1 students, and I think nowadays mathematics teachers are skipping steps, while last time only students skip steps.

For example, 3x + 6 = 18, and the next line we put 3x = 18 - 6.
Mathematically, we subtracted 6 from both sides because we want to single out the variable x.
However, this is usually not being taught, or it was just mentioned without any intention of teaching by assuming that students won't understand. As a results, I saw in student's workings this: above the first equation, there is an arrow from above the number 6 going across to the right side of the number 18. That is, what students have learnt was they just "move"/"throw"/"bring" the number 6 over to the other side, and "change"/"convert" the plus sign to a negative sign. What is this!!! Learning mathematics, or learning magical or pattern rules of manipulating symbols?

There are many more such non-mathematical stuffs that are taught in schools!

Lateral Thinking

Recently I just came across lateral thinking, although it was proposed some decades ago. I found my way of thinking is quite close to it. However, my experience is that not many are used to it. Most of us are more used to problem solving.

In a way, solving a problem is some how the minimal achievement. Much much more could be done with or without the problem being solve, and especially so when in situations where not even an existing problem could be recognized.

If relating to performance, it is like considering doing one's best as the minimal achievement, while performing more than required being the norm.

E.g. most students are happy when the their answers to a maths problem is the same as the given answer. This actually is just the minimal achievement. One may solve the problem in more than a way. One may reason out why solving it in another way is wrong. One may discover any redundant information given in the problem. One may transform the problem into a more difficult and challenge oneself. In so doing, the student is truly fulfilling the purpose of a student, which is to learn, not just with a false "worker" mentality by just "producing the solutions/answers" to their "bosses" aka teachers.

tests and exercises

A friend commented that her kids will get zeros for their spelling tests the next day if she is not free in the evening to coach them. I feel something not proper here. What are tests for? If they were for the teacher to know the progress of their students, then when the results of the students include effort from their parents and others, the teacher will miss out those students in need of more coaching. I think students and parents of young kids should be explained clearly on how the education system works.

What are the differences between various tests, exercises and exams? Exams are basically for the purpose of qualifying someone's ability. Thus, one strive to score, by all sort of ways, including churning out good answers even without understanding it, and hiding any signs of 'blurness' about the subject.

On the other hand, exercises are for the purpose of evaluating the performance of the teaching and learning process. Thus, very different from exams, one should be very honest, reflecting whatever understanding he has gained through the lessons within the teaching system, specifying the source of knowledge if it were not from within the system. At the same time, the teachers are to figure out ways to improve on the teaching/learning process, instead of just wonder what's wrong with the student.

Tests are the confusing ones. These are conducted in a more controlled way than exercises such that a more unbias evaluation can be achieved. However, many tests have their scores affecting the final qualifying process, which introduce bias since student will hid their weakness and give answers they recognise as correct, but don't understand. A way to reconcile is not to view tests as evaluation tools but trainings and simulation practices to prepare students to perform well in exams.

Personally, I prefer more evaluation sessions, exercises and tests, and minimal qualifying sessions, exams and tests. Evaluation sessions aim to improve the whole teaching/learning system, benefiting the school, teachers and students. Qualifying sessions are for practical reasons such that other organisations can use to select their preferred students. However, the sad news is, most organisations find many of these qualifying sessions are not really suitable for them, and the more resourceful ones will conduct their own assessment. I think objectives of an educational institute should focus more on teaching rather than qualifying. It is also fairer for an independent qualifying institute, and more effective for the employer or higher institute to conduct qualifying sessions for the incoming cohort.