*Paul A. Kirschner & Mirjam Neelen*

Sometimes there are heated debates, both online and offline, discussing if in education or learning we should first strive for understanding before teaching and learning the mechanics.

In this blog, we discuss a few examples to show that sometimes you really don’t need to understand something for it to be useful to you. Knowing is sometimes – or even often – enough. We’ve written about this before when we blogged about *the usefulness of useless knowledge*. Now, just take a few minutes to read through the examples and whenever you design a lesson, training, learning experience etcetera next time, think of how this applies in your specific context.

Back in school, we both learned that *pi* (π) was ‘equal’ to 3.14 or 22/7 and that both of these were just approximations. Along the same lines, we also learned that we could calculate the circumference of a circle with the formula 2πr and the area with πr^{2}.

We didn’t, and still don’t understand how mathematicians arrived at the value of π or why the formulae for circumference and area are what they are. And the funny thing is that knowing both without understanding them is extremely useful for us almost every day. For example, Paul is a fervent chef, and, as always, prepared the family Christmas meal this past year. As appetizer he was making a vegetarian quiche and as dessert a cranberry lattice pie. Though he has a really well equipped kitchen he, of course, didn’t have pie tins that were the exact size as those in the recipes and he also needed larger ones because he was not serving 8, but rather 12 people. For those dishes he needed to make more dough for the crusts and lattice top and also needed more filling for both. If he hadn’t known the formulae, he would have had a gigantic problem. How much larger is a 28 cm pie tin than a 24 cm tin? What area did he need to cover with the dough? How much filling did he need? For the first he needed just to know the difference in areas and then increase the flour, butter, salt, and water accordingly. For the second, he not only needed to know the area, but also the volume. Luckily, he also knew that the volume of a cylinder – and a pie tin is just a very short cylinder – was equal to the area of the bottom of the tin multiplied by its height so he also knew how much to increase the recipe for the fillings[1].

https://www.onlinemath4all.com/finding-the-area-and-circumference-of-a-circle.html

Paul could do all of this without really understanding the concepts behind the mathematics, though he did understand the concepts behind cooking. He of course had to recognise that he could apply the calculations and formulae in this particular context but then he only needed to know how to do the calculations and use the formulae.

Paul also does carpentry and it’s the same there, for example when he needs to know the amount of trim that he needs for a round table he’s constructing. He doesn’t need to run a tape measure or a string around a table that’s 180 cm in diameter nor does he need to use trial and error to determine how much varnish he’ll need.

And it doesn’t stop here.

We both have been in situations where we had to fell a tree (actually, have someone fell a tree for us). In this case, we can use the trigonometry we learned in high school. In a right triangle we learned how to use the SOHCAHTOA mnemonic to determine the length of the base, height, and hypotenuse as well as the calculation of the sine, cosine, and tangent.

https://educationisaround.com/sohcahtoa/

SOHCATHTOA tells us that the Sine of an angle in a right angle triangle is equal to the length of the Opposite side divided by the Hypotenuse, the Cosine is equal to the length of the Adjacent side divided by the Hypotenuse, and the Tangent is equal to the length of the Opposite side divided by the length of the Adjacent side. We can then use that plus a little help from an app to determine the angle to estimate the height of the tree.

https://slideplayer.com/slide/764134/

Paul also once had to shingle his roof and add solar panels, and luckily he learned the Pythagorean theorem so he didn’t have to climb the roof with a tape measure to determine its area.

All this could be done without ‘understanding’ the deeper mathematics behind determining the value of π or the theory behind sines, cosines, and tangents. As long as you have gained the required knowledge and have learnt how to apply that knowledge in different, even if they’re not real world (i.e., textbook, fictitious) examples and exercises. Yes, then the trick is to recognise where you can apply it in the real world (so real world examples are useful but you don’t have to ‘understand’).

Back to food, cooking, and science.

Let’s look at how baking soda and baking powder work. We learned in chemistry class that the first (sodium bicarbonate – NaHCO₃) needs an acidic environment to work (i.e. using vinegar or buttermilk and not just water or milk in the muffin or pancake recipes) and the second doesn’t (an acidic component has already been added to the baking soda to make baking powder).

https://bakingsodaandvinegarbonding.weebly.com/bonding-process.html

We have no idea of the actual chemistry behind all this. We don’t know exactly what chemical reaction takes place with each to create the carbon dioxide which makes them rise, but we know what to do if we don’t have baking powder, but don’t want to make flatbreads, namely we need to add vinegar or buttermilk.

We also know that heat is necessary for the reaction to occur (i.e., that heat is a catalyst) so when using baking soda or baking powder, there’s no time stress like when with yeast. Why? Because the first two processes are chemical reactions that almost don’t take place or take place very slowly at lower temperatures so we can take our time preheating the oven or doing something else. But yeast is a different story. Rising with yeast is a biological fermentation process. It works best at or around body temperature (37°C) – which is why we let bread rise near a radiator – but extreme heat in the oven (as well as too much carbon dioxide during the rise or the alcohol produced when the yeast cells metabolise the sugars) stop the reaction! That’s why we do a pre-rise when baking bread.

Mirjam’s husband loves to cook as well and recently, he started using Eke Mariën’s ‘Keukenlab’ [EN: Kitchen Lab]. Through this book, Mirjam learned that it’s better to froth milk for her cappuccino when it’s cold, and only then heating it. Now, the book explains why (different types of proteins play different roles when it comes to creating and stabilising foam), but she wouldn’t be able explain it properly and it doesn’t matter. Knowing the sequence to create the best foam – don’t heat the milk and then try frothing it – is enough to make it happen.

We could go on and on, but will end with a final example, namely knowing that enzymes are broken down at higher temperatures. Doing the wash with an enzyme detergent that works at 40C will not get you the results you want, or think that you’ll get, at 60°C or 90°C (i.e., falsely thinking that if it works well at a low temperature it’ll work even better at a higher temperature). Here too we have no idea what enzymes are at work, why certain enzymes remove certain stains, or what exactly happens at higher temperatures except that the enzymes are magically broken down and rendered ineffective. We just know that it happens.

https://biology-igcse.weebly.com/use-in-biological-washing-powders.html

Don’t get us wrong! We’re definitely not saying that understanding isn’t good or isn’t necessary. What we’re saying is that we shouldn’t simply accept beliefs that in teaching and learning we should always first strive for understanding the concepts and theory before learning the mechanics. We should always ask ourselves: is understanding needed or is simply knowing enough. Our conclusion is that sometimes it’s more than enough to just know!

[1] For the sake of keeping it relatively simple, we’ll conveniently leave out the calculation of increased or decreased baking time as deeper or shallower pies (i.e., the sides of the tin are higher or lower than in the recipe; often needed when baking cakes) also need longer or shorter baking times as this wasn’t an issue here as Paul’s pie tins were the right height. [BTW, bonus tip: This increase or decrease in baking time isn’t necessarily linear.]