One of the most common reasons why students are deterred from subjects such as physics (and to some extent, chemistry) is the misconception that “My maths is not good enough.” And perhaps older than this misconception is the litany of “I don’t have a maths brain”, “I can’t do maths”, “I never understood numbers” and “Equations terrify me.”
When I went to Oxford to begin my DPhil at the Scanning Proton Microprobe Unit at the basement of the Department of Nuclear Physics, I was armed with no more than a measly C grade for maths at ‘A’ levels. But numbers and long equations never terrified me.
Thus, after years of reading research about maths anxiety and dyscalculia, I was delighted to read a piece of research recently published in the journal Behavioral and Brain Sciences that debunks “number sense”.
The researchers argued that critical to developing higher maths ability is understanding the relationship between size and number. This made a lot of sense if one were to look at the physical sciences from a bird’s eye perspective rather than the fine details of the syllabi. Physics and chemistry are full of convenient constants invented so that we can make sense of astronomically large or mind-bogglingly small numbers. Suddenly, with these multiplication factors, our limited cognition ability can begin to make sense of the surreal.
In my novel, Catching Infinity, the protagonist lamented at the impossibility of drawing a hippo and a bug on the same piece of paper without losing the scale or detail. Therein sits the beauty of constants: they make this possible. In physics, I would urge students to learn to understand and love the ‘Plancks’; the path will become both easier and more magical once one is firmly conversant with the Plancks. For we have Planck length, Planck mass, Planck time and Planck constant that miraculously make complicated numbers simple enough for a hapless student to grasp. And the universe is abound with many such constants.
The radii of isolated neutral atoms range between 30 and 300 trillionths of a meter. Thus, in a small speck of dust, there are gazillions of those little things. By multiplying the Avogadro constant (6.02214086 × 1023 mol-1) to these dimensions makes the numbers more manageable. We can thus talk about one mole of a particular element being a certain number of grams, for example, one mole of sodium is 22.9898g and one mole of zirconium is 91.224g, which is the approximate mass of a large apple, though the size of each zirconium atom remains mind-bogglingly small.
I think time is well invested if teachers of physical sciences were to engage students in playing with the relevant constants at the start of the course, given that constants are what that links size and numbers to the ‘real’ world. These playful one-hour sessions could simply be a progression from nursery maths where three-year-olds guess the number of tins in a basket or slices of pizza in a box. It is from these simple maths games that we learn to make sense of numbers, namely to correlate between numbers and continuous magnitudes to compare magnitudes.
And we must never forget, when trying to grapple with maths, that numbers were invented in the first place simply because our distant ancestors ran out of fingers and toes to count the items in their increasingly complicated lives as civilisation progressed.
Guess-timating the volume of shapes. Students gain much from working with solid shapes, which lays the foundation for later years.
Exercises with Vernier calipers/ micrometer to enable students to visualise small divisions of a milimetre. Students could have fun measuring very small things that they find in the class and differentiating between internal and external measurements (both have the same zero start positions).
For simple everyday maths magic, the slide rule is it. Invented before the calculator, it was widely used in schools until modern technology rendered it obsolete. But for an old timer like I, the slide rule is still full of magic. I think it’s amazing that just by adding up two lengths on a slide rule, you can effect complicated multiplications, as well as other more complex functions such as exponents, roots, logarithms and trigonometry. Back in the good old days, NASA scientists used this simple tool and paper to crunch numbers.
Tali Leibovich, Naama Katzin, Maayan Harel, Avishai Henik. From ‘sense of number’ to ‘sense of magnitude’ – The role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences, 2016; 1 DOI: 10.1017/S0140525X16000960.
Jacqueline Koay teaches physics and chemistry as a subject specialist.