**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.

- We do not use examples to teach your child steps or algorithms.
- We don't let your child remember steps.
- We don't tell stories about why an operation is done, for example if you want to remove 4 from the left side of an equation, then you subtract it from the left and add it to the right of the equation. This is algorithm story and not math.
- We don't use time to make children do sums over and over again until they remember the steps.
- Teach maths in abstract ways.
- We do not teach your child to think like us (the teacher).
- We do not teach your child math sums in isolation (silos) which prevent your child from seeing and drawing connections.
- We do not refer your child to an example if your child does not understand the sum.
- We don't teach your child rules, rhymes and other tricks to do maths.
- We do not teach the curriculum we teach your child.

- We educate your child and build relationships and self-esteem.
- We teach your child to reason mathematically.
- We teach your child to ask: “What do I have and what can I do with it?
- We teach your child to solve problems.
- We teach your child to build smart maths strategies.
- We teach your child to notice relationships, connections and patterns.
- We let child’s brain build new neural pathways to change and broaden their thinking.
- We teach your child in concrete ways - you can even touch the 3 times table.
- We teach your child to think like great mathematicians.
- We teach your child how math concepts build on each other and need each other throughout the curriculum.
- We teach your child to see connections between different sums.
- We teach your child to wrestle with sums and win.
- We teach your child to work out sums independently.
- We use time to let your child do everything above over and over again to get their brains thinking.

Within a Thinking Tools setting learners are exposed to the following skills needed for the Fourth Industrial Revolution:

- Creativity.
- Emotional intelligence (EQ)
- Analytical (critical) thinking.
- Active learning with a growth mind-set.
- Judgment and decision making.
- Interpersonal communication skills.
- Leadership skills.

Thinking Tool teachers enable learners to contemplate the following types of thinking in holistic and integrated ways:

- Additive spatial relations thinking,
- Multiplicative pattern thinking,
- Proportional judgmental logic thinking.
- Functional computational thinking,
- Geomatical relations thinking,
- Sequential analogy thinking,
- Comparative and contrasting thinking, and
- Abductive reasoning (taking your best shot) and alike.

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**