Traditional maths teaching is based on so-called 'knowledge transfer' which is impossible as knowledge cannot be transferred or BlueTooth-ed from one brain to another.
Teachers teach by explaining recipes and methods. For example, teaching long division has more multiplication and subtraction than division. It is not the division that is long it is the method. Notwithstanding this detailed method, most learners get lost along the way and are left clueless.
Recipes and methods have nothing to do with maths. They are the accumulation of mathematicians' insight as they were developed over time.
This result is that learners' logico-mathematical knowledge is a second hand acquiring from a sosio-cultural source (the teacher) and not directly from the mathematical phenomenon/environment.
The result is that they are deprived from internalising mathematical concepts themselves and have to survive tests and exams by relying on methods they can recall or can get hold of at that point in time.
This is going on for as many generations as schools exist. There is a need to reimagine maths teaching and learning.
The questions are:
Is this how you want to teach maths?
Is this how you want your child to be taught?
Because we uncovered the vertical connections of maths concepts and how they support each other from preschool to grade 12. Therefor we are experts to identify why a learner is struggling with maths in a specific grade. We follow remediation approaches to first fill gaps before we progress.
This led to the bird-eye view concept which enables both teachers and learners to zoom in and out a maths sum. Each level of zooming provides a different view and insight that makes maths easier, more understandable and workout-able.
1) Maths is about relationships and patterns and not about mimicking the teacher's methods.
2) Thinking Tools see maths as an EMBROIDERY with a neat front end and a backend where the loose threads are knotted and tied. The neat represents the correctly set out steps and recipes which relates maths more to a recipe book than to maths. It is common knowledge that all recipes cannot be recalled, and the recipe book must regularly be revisited which is not possible during tests and exams. Thinking Tools focus on the backend where real maths is. Learners are enabled to unravel and unpack the backend before attempting the neat frontend. Each type of sum has its own tapestry backend. In the foundation phase the number line has a tapestry backend. This means a number line is not the beginning of a learning process, but the end. Multiplication tables of which the frontend is traditionally exhausted with repetitions without bearing mathematical insights, has a tapestry backend.
The same with exponential laws, the sine rule, functions, etc.
3) Strategy-wise, one of the most recent discoveries is the PILOT CHUTE concept. This analogy, empowers Thinking Tools teachers and learners to first understand what the sum is about, list the operations, spot pitfalls in advance and then pin down the method that will provide the solution and the final answer.
4) Traditionally self-confidence is seen as the result of good marks. The Thinking Tools approach sees developing of self-confidence and mathematical skills as the same process.
5) Maths cannot be taughed, it can only be learned. Thinking Tools teachers also compares learning maths to learning how to ride a bicycle. There is no manual to learn ow to ride a bicycle. The child needs to get on the bicycle and experience the accompanied failures and successes in a safe setting. The teacher cannot learn to ride a bicycle on behalf of the child. Likewise with parachute jumping, it is the child that needs to jump and make a safe landing. Maths, like riding a bicycle and parachute jumping cannot be taught, it can only be learned.
6) The challenge with the traditional explain approach that the teacher does all the riding jumping and then expect from learners to do solo jumps during the exam. This is an unfair assessment practice.
7) Learners are knowledge creators and problem-solvers when engaging in 0D, 1D, 2D or 3D thinking guided by the structure and numbers of the sum instead of following the same steps or algorisms over and over.
8) Leaners are scaffolded from concrete experiences to establish their own pictorial representation before embarking on the abstract level.
9) Maths is about finding and creating patterns.
10) The above happens in an empathy rich environment where the Thinking Tools teacher has skills to engage in cognitive empathy. This means the teacher has tools to walk in a learner's shoes to determine what the learner understands, and which scaffolds should be provided to ensure progress. Thinking Tools teachers engage in emotional empathy as they can, on the one hand, foresee a learner's AHA coming and, on the other hand, know how to relate to disappointments without exposing the learner. Thinking Tools teachers engage in process empathy because they know all learners do not learn at the same pace and can play around with the teaching gearbox to set their pace of teaching at a specific learner's pace of learning.
Within a Thinking Tools setting learners are exposed to the following skills needed for the Fourth Industrial Revolution:
Thinking Tool teachers enable learners to contemplate the following types of thinking in holistic and integrated ways:
Because learners presented the above in visual modes it makes easier for them to comprehend and make sums workout-able.
This is how a group of grade 10 learners analysed the tapestry back-and frontend of a factorization sum
This is how a group of grade 10 learners analysed the tapestry backend of a simplification sum