Confession time: Just as I’ve never really thought of maths as beautiful, I’ve struggled to understand why people talk about maths being about patterns. For me maths is powerful and maths is about abstracting from reality to build models. So I thought I would explore the idea of patterns in mathematics.
Numbers show patterns. The fact that all prime numbers over 3 occur on either side of a multiple of six, is pattern I find appealing. The digits of multiples of three sum to multiples of three. Multiples of eleven are pretty exciting with the way they repeat. The fact that the digits of pi have no discernible pattern is mind-boggling. I guess that is the absence of pattern, but that is interesting too. The behaviour of rational numbers that mean that some recur and some don’t, is a pattern.
Geometry is about patterns. It is pretty cool that the angles inside a given polygon always add up to the same number of degrees. Trigonometry relies on the pattern that similar triangles have their sides in the same ratio.
Frieze patterns can be interpreted with a mathematical lens. It is said that all frieze patterns can be expressed as one of seven varieties of frieze.
Algebra can be used to describe growing patterns – in fact growing patterns are a visual way to introduce the idea of functions and graphs. There are all sorts of patterns around binomial expansions and Pascal’s triangle.
One of my favourite types of problems involves systematic counting. I spent quite some time enumerating all the possible combinations for attackers and defenders in the game of Risk, before we had spreadsheets to speed up the process. The best way to ensure completeness when enumerating alternatives is to tap into the patterns inherent in the problem.
Statistics is TOTALLY about patterns. A wonderful example of statistical patterns is time series graphs. I could spend all day looking at the view counts from my YouTube channel, which shows fascinating patterns. There is a weekly pattern – Wednesday is always higher than Saturday or Sunday. This repeats every week, with slight variations. And every year there is a precipitous drop in views around the 25th December. Oddly enough, people tend not to watch videos about statistics on Christmas Day.
In statistics one of the big challenges is to identify what is the pattern and what is noise. Humans love to see patterns in all sorts of things, so in statistics we need to make sure that what we are identifying as a pattern is not there just by chance. When we smooth graphs, or fit lines, we are attempting to find the pattern within the noise.
The ability to recognise patterns helps in music, art, dance, science and language. At swim training in my teens I would recite Latin verbs as I made my way up and down the pool, breathing on the third person forms. (Amo amas amat – breathe – amamus amatis amant – breathe). I find square dancing mathematically appealing because of its patterns and problem-solving.
There are educational observations about how novices and experts differ. One of the key differences is in the ability to recognise patterns, which reduces load on working memory. Research around chess players found that expert chess players see the chess board more as a whole and can recognise patterns whereas novice players need to think through individual moves. The same is true in various branches of mathematics. For example if I see the expression 14 × 4, I can recognise easily that it is the same as 7 × 8, which is a number fact I know. When I look at fitted lines in regression models, I am better able to tell what the correlation is likely to be than a novice, because I have seen so many fitted lines (from grading thousands of assignments).
Mathematics education research has found that students who recognise patterns acquire deep conceptual understanding. Identifying and explaining patterns helps people to develop their skills of critical thinking and communication. This interesting blog tells us why time is spent exploring patterns and that researchers have found that understanding and being able to identify recurring patterns allow us to make educated guesses, assumptions and hypothesis.
Teachers of mathematics and statistics need to help their students develop their ability to recognise and create patterns. Novices become experts by learning about patterns. In my exploration of pattern I found this website with a whole range of activities around pattern.
I have also found that learners are really good at working out spurious and irrelevant patterns. For example in multichoice questions, the longest answer is more likely to be correct than the others. And questions about faulty tyres are most likely to be about the binomial distribution. I have had students explain to me how the questions had little “tells” in them. If only the students had put as much work into learning the real patterns, they would have done considerably better!
I think I have convinced myself that patterns are an underpinning concept in mathematics and statistics.
What about you? How do patterns figure in your understanding of mathematics and statistics? What are you doing in your teaching and/or learning to build on patterns and pattern recognition?