Mathematics is NOT arithmetical methods, devoid of all logic!
Mathematics is NOT abstract: for instance, ‘7 – 3’ may not be equal to 4; it depends on what 7 and 3 represent. The first lesson is on how numbers represent quantities, always. 7 and 3 mean nothing unless we know the things represented by the numerals 7 and 3. For example, 7 may represent 7 pencils, and 3 may represent 3 erasers, and then ‘7 – 3’ would stand for ‘take away 3 erasers from 7 pencils’. All 3-year-old children know that it is not possible.
Mathematics has been abstracted to make it widely applicable, and that does power math to become an eminent domain of knowledge. But it does not mean that we should teach math this way, i.e., abstractly where ‘7 – 3’ is always 4! It is not. School math education makes math abstract a little too early, profoundly changing the very idea of math to children in a generation and the adults in the next generation. Mathematics has been made out to be what it is not!
Five dimensions of math
I personally like the simplicity of the ‘definition of math’ in the Draft (Indian) National Education Policy, 2019. It defines mathematics as composed of the following five dimensions (the elaborations are my own):
- Counting (and measurement) – The means/rules of quantification of things to generate numbers (the starting point of anything mathematical); for example, the whole idea and practice of counting number names, decimal number system, fractions.
- Arithmetic – The basic rules for using/manipulating numbers – the idea and applications of the four fundamental operations on numbers; for example, how the quotient 4 in 12 3 differs from quotient 4 in 16 4.
- Mathematics – The language-like ability to think and express all quantifiable situations using numbers and arithmetic. For example, ‘mathematics’ distinguishes between ‘8 – 1 – 1’, and Prologue – Introducing math 17 ‘8 – 2’ (there are significant enough differences between the two expressions)
- Reasoning – Math is uniquely logical and rigidly hierarchical. Anything mathematically expressed just needs to follow one other statement – the previous mathematical statement/step. Unlike science, math is endlessly and purely logical. One can complete several PhDs in math with just papers and pens to write! For example, deductive reasoning is mathematically valid, while inductive reasoning is not. Science is observation, experience, experiments, and logic – all founded on what’s real.
- Problem solving – The ability to harness the vast set of tools and resources in the domain of mathematics to define and solve any given real world, scientific, or any research situation/event. For example, conceptualising a surface that has the maximum area for the given perimeter.
Hope you did register that one of the dimensions of mathematics is named “mathematics”. This just emphasizes the core of mathematics – mathematical thinking. Not just number crunching!
Arithmetic of mathematics
This ‘5-dimensional view’ of mathematics has some profound implications for teaching and learning math:
What we learn in the math period in schools is just one category of knowledge out of the five – arithmetic – and that too in a very limited manner: using specific methods, without logic, and without the freedom to use different methods to solve the same problem (severely limited number of methods is also a challenge). It would not be too misplaced to say that the math education in schools does not represent what math really is!
No wonder, the overwhelming majority of children in school struggle in math – their innate power of reasoning and language is crippled by the inappropriate school math content and emphasis on teaching rigid, limited arithmetic. For example, no reason is assigned to why ‘10 is written as 10’, why ‘2 divided by 1/2 is 4’, or how exactly ‘125 percent of 80 is 100’.
The right math education will logically explain every such mathematical expression/step and help uncover Math genius in every child
