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Build a Mathematical Mind - Even If You Think You Can't Have One

Albert Rutherford

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mathematics

Chapter 4

  • Paul McCartney claims the melody for “Yesterday,” one of the Beatles’ most beautiful songs, came to him in a dream: ‘I woke up with a lovely tune in my head,’ he told author Barry Miles for the biography Many Years From Now, which was published in 1998. ‘I thought, ‘That’s great, I wonder what that is?’ There was an upright piano next to me, to the right of the bed by the window. I got out of bed, sat at the piano, found G, found F sharp minor th – and that leads you through then to B to E minor, and finally back to E. It all leads forward logically. I liked the melody a lot but because I’d dreamed it. I couldn’t believe I’d written it.’[ii]May 25 2024 11:56AM
  • Similarly, some mathematicians have claimed that critical mathematics discoveries have come to them while they were sleeping. Srinivasa Ramanujan, an Indian mathematician, believed equations were brought to him in his dreams by a Hindu goddess[iii]; Rene Descartes, the French mathematician after whom the Cartesian coordinate system (our standard way of graphing on two axes) is named, allegedly had his best ideas while lounging in bed in the morning, halfway between sleeping and waking.[iv] Something about the relaxed state of sleeping or being barely awake allowed these people’s brains to create, visualize, and dream up ideas related to what their waking minds were focused onMay 25 2024 11:57AM
  • Recent neurology research has proven math is processed in different parts of the brain than language. A 2016 study by two French neurologists found that people process mathematics in the same parts of the brain where they process problem-solving, which are separate from where language is processed. This can help explain why Einstein allegedly said, “Words and language, whether written or spoken, do not seem to play any part in my thought processes.”May 25 2024 11:57AM
  • The authors called for a radical shift in mathematics education, so it focused on the habits of mind mathematicians use rather than the specific facts they have deduced. They proposed teaching students to think rather than teaching them the thoughts that mathematicians have had. They wrote: We envision a curriculum that elevates the methods by which mathematics is created and the techniques used by researchers to a status equal to that enjoyed by the results of that research. The goal is not to train large numbers of high school students to be university mathematicians. Rather, it is to help high school students learn and adopt some of the ways that mathematicians think about problems.[ix]May 25 2024 11:59AM
  • To clarify what it means to teach mathematical thinking, the Common Core State Standards for Mathematics, first published in 2010, include eight standards for mathematical practice.May 25 2024 12:03PM
  • The eight practices are: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.[x]May 25 2024 12:03PM
  • In a 2008 speech, the remarkable Sir Ken Robinson likened the education system to factories. We churn out children in “batches” (based on birth year) and expect them all to function identically. This worked well enough when our economy was primarily based on factory work, but our society has changed and continues to change rapidly. “People are trying to work out: how do we educate our children to take their place in the economies of the st century?” asked Sir Ken. “How do we do that given that we can’t anticipate what the economy will look like at the end of next week?...The problem is that the current system of education was designed and conceived and structured for a different age.”[xii]May 25 2024 12:16PM
  • Sir Ken’s answer: schools need to foster divergent thinking or “the ability to see lots of possible answers to a question, lots of possible ways of interpreting a question.”May 25 2024 12:18PM

Chapter 5

  • Humans are born with a propensity to look for structure. Neuroscientists have proven through functional MRIs that human brains naturally look for patterns in a sequence of items. In fact, there’s a term – apophenia – for “the human tendency to see patterns in meaningless data that may involve visual, auditory, or other senses.”[xiv]May 25 2024 12:19PM
  • Patterns can be visual, auditory, or “the regular way something happens or is done” (as in how babies learn language).[xvii]May 25 2024 12:20PM
  • Pattern sniffing, in all ways, is how we learn. Early humans (and animals) learned which plants were safe to eat by watching what happened when their peers ingested different plants. Mothers learn quickly that feeding or changing a crying baby usually stops its crying. Even weather forecasting is based on pattern recognition. While powerful machines now use algorithms to predict the weather several weeks out, early weather forecasting was based on observing what had happened.[xviii]May 25 2024 12:21PM
  • If it’s a pattern, it must follow a rule; the key is in describing that rule.May 25 2024 12:32PM
  • Let’s look at the same pattern written numerically, with each figure represented by a number:May 25 2024 12:35PM
  • Triangular numbers are numbers that, if represented by dots, would form a triangleMay 25 2024 12:35PM
  • The Fibonacci Sequence is a classic pattern of numbers that begins like this: Can you figure out the relationship between the numbers? Each term in the Fibonacci Sequence is the sum of the previous two terms. After would come 89, since 55+34=89. Fibonacci numbers have been known to Indian mathematicians for over two thousand years. They were first introduced to the western world in 1202 by the Italian mathematician Leonardo of Pisa, who later became known as Fibonacci.[xix]May 25 2024 12:39PM
  • But why does the Fibonacci Sequence exist? What significance does it have? The numbers in the sequence have a proportional relationship to each other. That means one term divided by the term before gives us a constant (or almost constant) ratio. The longer the sequence continues, the closer the ratio of terms gets to the magical number of phi, 1.618.May 25 2024 12:40PM
  • The number phi appears so often in nature that it has earned the name the golden ratio. The overlapping spirals in the head of a sunflower almost always contain two consecutive Fibonacci numbers, meaning they exist in the golden ratio. Snail shells contain the golden ratio in the way their size increases as it spirals out.[xx] The way trees branch out follows the golden ratio, with the number of branches growing proportionally as the tree gets taller. The golden ratio is everywhere.May 25 2024 12:41PM
  • Remember that everything starts with a single person asking a question or noticing a pattern.May 25 2024 12:41PM

Chapter 6

  • Now, look at those outcomes – the sums of the two dice. How many ways are there to roll (as a sum of the two dice) a one? (Zero.) How many ways to roll a two? (One.) A three? (Two.) Keep counting the possible outcomes until you get to seven. How many ways are there to roll a seven? That’s right, six. There are more ways to roll a seven than there are to roll any other totalMay 25 2024 12:45PM
  • In the example of rolling two dice, the denominator of the fraction – the total possible outcomes – is 36. The numerator – the number of favorable outcomes or ways that we could roll seven – is six. So the probability of rolling a seven with two standard dice is 6/36, which can be simplified to 1/6.May 25 2024 12:46PM

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