Written by Lecturer in Mathematics Education, Vinay Kathotia
Here’s a mathematics puzzle: Suppose you walked up a hill at a constant speed of 3 kilometres per hour and you immediately walked down the hill taking the same route but at a constant speed of 5 kilometres per hour, then what is your average speed for the full round trip?
You can work on this using your preferred approaches, but if one asked you for your quick or intuitive answer what would it be? And how confident are you about it?
Spoiler alert! We share solutions to this puzzle below, so if you would like to work on it for yourself first, please do that before returning here.
Walking downhill. Image thanks to KimJaesub (Pixabay)
Many people when posed this puzzle propose the answer 4 kilometres per hour (kmph), though simultaneously, they are not quite sure about this answer, say compared to their answer for 3 + 5 = ? There are good reasons for their answer, and for their scepticism, and we will explore that here.
Nobel laureate Daniel Kahneman and his colleague Amos Tversky have done substantial research on human judgment and decision making. In his book, Thinking Fast and Slow, Kahneman shares some of their research and approaches. They hypothesise that our minds have two ‘systems’, System 1 produces fast, automatic/instinctive responses with seemingly little effort, whereas System 2 is slower, effortful, allocates attention and can regulate our (default) use of System 1. Moreover, System 1 exhibits ‘no sense of voluntary control’, while System 2 operations relate to ‘agency, choice, and concentration’ (Kahneman, 2011, p.4).
We need to bear in mind that these ‘systems’ are models or metaphors, we don’t fully understand the workings of the mind, but there is substantial evidence for the patterns of thought and judgment that Kahneman outlines. What does this have to do with our puzzle? Our hypothesis is that reaching out for 4 kmph as the average of 3 kmph and 5 kmph is an instance of doing mathematics fast (System 1 thinking), whereas any unease you may have and the desire to work it out at your pace could be an expression of System 2 thinking. So how would you or did you go about solving this puzzle. Below we share two approaches, both of which align with what we would call Slow Mathematics – our preferred approach!
Make it real or realisable
Speed, the measure underlying this puzzle is a compound measure (it sets up a relationship between two more basic measures, distance and time). Research tells us that learners can find reasoning with compound measures difficult but, as mathematics is often about studying relationships between variables, working with compound measures can be critical for developing mathematical reasoning. One approach is to use numbers and units that you are comfortable using, that are realisable for you and/or make sense to you. Notice the problem does not tell us how long (in kms say) the trek uphill is. So could we choose a distance that is easy to work with. It need not be realistic but something that makes the mathematics more accessible and realisable. I would suggest a 15 km trek. Why? Setting that question aside for a moment, verify that our trip uphill would take 5 hours and the faster downhill leg would take 3 hours. Notice this approach somehow does away with the ‘compoundness’, having us work with simpler numbers and units one at a time. So our total trip of 15 + 15 = 30 kms has taken 3 + 5 = 8 hours. We can now work out that the average speed is 30/8 kmph or 3.75 kmph. So not far from 4 kmph but lower than that. Is there some way we could have assessed or estimated that earlier?
Going to extremes
One approach when grappling with mathematical problems is to vary the problem, seemingly making it even more difficult, but the variation can help clarify underlying relationships. So suppose you went uphill at 3 kmph but came down at an excruciatingly slow pace (almost 0 kmph), then what would your average speed for the round trip be? Would it be close to the average of 0 and 3, say 1.5 kmph, or closer to 0 kmph, or closer to 3 kmph? Similarly, if you were to whizz down at close to the speed of light, what would your average speed be? Do explore these thought experiments and it may help you see that average speed over two equal-distance legs need not be the mean of the speeds over each leg. Which leg will you be spending more time on, which will therefore be your dominant experience – the fast or the slow one? And how will that impact the average?
Fast or slow mathematics
Traditionally, mathematics seems to reward quick answers. We assess students on timed exams, we search for and value quick algorithms. On the other hand, mathematics with understanding requires noticing what we do, using one approach and then another (… and another), using different representations and making connections. These may slow things down but may also provide a longer and more fruitful journey.
How do you think of fast or slow mathematics, or doing mathematics at all in the emerging era of generative AI? Do leave your thoughts below, and we will pick that up in a future blog. Meanwhile, take it slow!
References
Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux.