The Psychology of Mathematics

A Journey of Personal Mathematical Empowerment for Educators and Curious Minds

Anderson Norton

• Publisher: Routledge
• Year: 2022
• ISBN:
  9781000547061
• Edition: 1
• Topic: Bildung
• Subtopic: Mathematik unterrichten

This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures.

Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level—including geometry, algebra, calculus, complex analysis, and more—Norton unlocks students' personal power to construct mathematical objects based on their own mental activity and illustrates the power of mathematics in organizing the world as we know it.

Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared toward current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally.

1 What Is Number?

DOI: 10.4324/9781003181729-2

Suppose you are walking along a sidewalk and notice seven yellow bricks. How do you know they are bricks, that they are yellow, and that there are seven of them? Yellow and brick are subjective descriptors of perceptual experience. Determining whether something counts as a brick depends on language and cultural experience. Determining whether something is yellow depends on the way the cones and rods in our eyes transform electromagnetic waves into neural signals that we then interpret within some kind of color scheme, which again depends on language and cultural experience. Determining whether something is seven has a different nature.

The number 7 does not exist out there in the world for us to see. It begins in our minds, and then we project it out into the world.¹ Seeing 7 depends on the coordination of several mental actions. First, we have to notice something that we separate from the rest of our perceptual field. In the case of bricks in a sidewalk, the yellow ones stand out so we can easily focus attention on each of them. Second, we have to treat the objects on which we focus as if they were identical, even though no two things we see are truly identical. Each yellow brick is unique in some way (different hues of yellow, imperfections in their rectangular shapes, etc.), but we ignore idiosyncrasies for the moment and treat the bricks all the same, simply as objects to count. Finally, we put those identical things in one-to-one correspondence with our number sequence. For instance, we might point to each brick as we recite, “One, two, three, four, five, six, seven” (see Figure 1.1). We might then view 7 as a property of the collection of bricks, when in fact, we have made it so through our own mental activity—even in making them a collection to count.

Figure 1.1 One-to-one correspondence between pointing acts and number words.

Source: © Eleanor Norton

As adults, it’s difficult to recall the laborious (and mostly subconscious) process of constructing number from our coordinated mental actions. We feel like we can simply see numbers, like we see colors, and so children should be able to see them too. We feel like we should be able to point to seven objects and say, “See? That’s seven.” It doesn’t work that way because children have to learn to see numbers. They have to construct numbers before they can project them out into the world they perceive. “There is the mistaken belief that since we, as adults, can see mathematics in the blocks, the students will too. But the seeing requires the very construction the activity is intended to teach.”²

For young children, counting requires great effort and concentration. Counting past 7 can be particularly tricky because the English word seven has two syllables, so children will sometimes point to the eighth item while reciting the second syllable in “se-ven”.³ Thus, children would lose the one-to-one correspondence between pointing acts and number words. Without this correspondence, reciting number words, “one, two, three…,” would be no more mathematical than reciting the alphabet, “A, B, C…,” or singing “Do, re, mi…” It would be nothing more than a memorized sequence of words—not counting.

ORDER AND CARDINALITY

The development of number comes in stages, beginning with one-to-one correspondence. Consider the case of my daughter, Eleanor, when she was 4 years old. Eleanor had been carefully counting out collections of chips when I asked her a question…

Me: Is 9 bigger than 7?

Eleanor: [slapping her hands to her face; see the left side of Figure 1.2] I’m thinking yes.

Figure 1.2 Eleanor comparing 7 and 9.

Source: © Eve Azano

Me: Why?

Eleanor: Because one, two, three, four, five, six, seven [pointing to different places on an empty area of the table as she said each number word; see middle of Figure 1.2] and you haven’t heard “nine” [raising hands in the air; see the right side of Figure 1.2]!

Eleanor’s case teaches us a couple of things about numbers. First, learning how to count involves a careful coordination between pointing acts and number words in order to establish a one-to-one correspondence between them. We can see traces of that activity as children progress from needing physical objects to touch, to a focus on the acts of pointing themselves, to imagined acts of pointing, as indicated when a child looks up and nods for each number.⁴ In Eleanor’s case, she pointed to places on the table where she imagined the chips. It was the acts of pointing, and not the chips themselves, that were salient.

Second, learning to count is not the same as understanding numbers as quantities. Eleanor could easily count collections of 20 or more objects and had done so many times, but she did not know that 9 was bigger than 7 until she recited her number sequence again. For her, one number was bigger than another if it appeared later in the sequence. In other words, she compared numbers based on their order, not their cardinality.

In his studies of child development, Jean Piaget described the construction of number as an integration of order and cardinality, where cardinality refers to size or quantity.⁵ As children integrate these two aspects of number, they begin to understand that one number is larger than another, not only because it follows in numerical order but also because that number contains the other. For example, 9 follows 7 and contains 7. This is not a simple matter for children to resolve.

Consider the case of my other daughter, Caroline. Starting when she was 5, I would ask her questions like the following: “How much is seven 1s?” The question may seem ridiculous to adults, but not to 5-year-old Caroline. She would always respond to such questions in the same way. In the case of seven 1s, she would count out seven fingers, raising them one at a time as she recited, “One, two,…, seven.” Then she would point to each raised finger while reciting the same sequence of number words again! A year later, when I asked her about twenty 1s, she laughed and said, “That’s just 20!” Something had changed. Twenty 1s had become, for her, the same quantity as 20.⁶

As cardinalities, 7 is seven 1s, and 9 is nine 1s. Nine is bigger than 7 because the nine 1s in 9 include the seven 1s in 7. Between the ages of 5 and 6, Caroline seemed to develop an understanding of numbers as cardinalities. Figure 1.3 illustrates what this understanding entails.

Figure 1.3 9 as a cardinality containing 7.

Whereas order involves placing pointing acts in one-to-one correspondence with a number sequence, cardinality involves iteratively building up collections of 1s. As Caroline would later explain, seven 1s is 7 because 7 is 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1. And as Eleanor later learned, 9 is bigger than 7 because it has two more 1s. With cardinality, children begin to understand numbers as quantities measured in units of 1. The number 1 becomes a unit of measure, like an inch or a meter.

UNITS OF UNITS

Once we understand numbers as measures, anything can be a unit for measuring, and whatever we choose as this initial unit becomes 1. This mental action of creating a unit is called unitizing. With the seven yellow bricks, we unitize instances of bricks, which we treat as identical. Their differences do not matter because they are simply objects to count, and it is the counting itself that matters. Likewise, we might unitize the entire collection of bricks or the distance between adjacent bricks. One (1) is simply a choice we make when determining how we want to measure collections or lengths.

Measuring amounts to iterating a unit of 1, where iterating refers to the mental action of making connected copies of an identical unit. For example, we might measure the distance on a track by iterating a meter or a yard. However, in mathematics, units do not need to refer to anything physical. A unit is simply the result of unitizing, producing a 1 that we might iterate to produce and measure other quantities.

As outlined in the previous section, constructing numbers as measures, or iterations of a unit of 1, takes years of experience. Manipulatives, such as fingers, chips, or blocks, can support those experiences by offering children a way of carrying out their mental actions on physical material. They can use the manipulatives to keep track of their activity, so they don’t have to coordinate it in their minds all at once. This is a theme we will return to in Chapter 6.

Along the way, children will learn to count on from a given number to another. For example, they might determine the total number of objects from collections of eight objects and five objects by starting from 8 and counting on “9, 10, 11, 12, 13” while pointing to each of the five objects in the second collection (see Figure 1.4). They will even learn to double count, keeping track of the number of counts in the second collection: “9 is 1; 10 is 2; 11 is 3; 12 is 4; and 13 is 5.” As you might imagine, this coordination of counts involves considerable effort for young children. However, these coordinations support children’s construction of composite units, which are worth the time and effort.

Figure 1.4 Counting on and double counting.

Source: © Eleanor Norton

Composite units are units made up of other units (units of units). For example, the numbers 7 and 9, as illustrated in Figure 1.4, are composite units: 7 is a unit made up of seven 1s, and 9 is a unit made up of nine 1s, including the seven 1s that make up 7. We might also build up 9 from composite units of 3: 9 as a unit made from three units of 3, each of which is three units of 1.

Figure 1.5 Bars task.

With composite units, we can measure the world not only in units of 1 but with all kinds of units. Moreover, we can relate those measurements. For example, consider the bars sho...