Patterns Everywhere! - or are they?
Michael Naylor
Western Washington University
Bellingham, WA
Study the following sentence for a moment and determine how many letters are in the next word: I am done studying ________________.
Did you say sixteen? That's what most people say. The answer is actually "three" since the word that goes in the blank is "now." Was this is a trick? Doesn't math class train us the believe that everything is patterns? When we see 2, 4, 6, ..., isn't it a given that the next term is 8? According to Western Washington University math professor Keith Craswell, the next term is 7; the list denotes the ages of his neighbor's kids!
The world is indeed replete with patterns, and one of the greatest delights of mathematics is identifying these patterns and using them to understand underlying structures, make predictions, and find connections between seemingly unrelated phenomena. But when is a pattern one which accurately represents reality, and when is a pattern an accident? We are often so busy finding and symbolizing patterns that we overlook this distinction.
In one of my classes, my preservice K-8 teachers had discovered all 14 tetraboloes, the shapes that can be made with 4 right isosceles triangles placed edge to edge, vertex to vertex (Gardner 1977, Burns 1993), and were asked to prove that they had found them all. Some students wrote convincing arguments based on possible triangle placements starting with two triangles, then all possible placements of a third triangle from each of the three possible two triangle arrangements, and so on. Some of the students, however, struggled with patterns. They found the number of shapes possible with 1, 2, 3 and 4 triangles (1, 3, 4, and 14, respectively), recorded the data and pored over their T-charts (figure 1). Scribbling drafts of formulas, they were hoping to find an algebraic relationship between the columns of figures. Some students decided that more data was needed, so they began trying to find the number of shapes that can be made with five triangles, but soon became lost trying to find all of the arrangements.
Triangles |
Number of shapes |
1 |
1 |
2 |
3 |
3 |
4 |
4 |
14 |
These students believed that if they could find a formula to match their data, then (a) the formula must be correct, (b) they could predict the number of shapes for any number of triangles, and (c) the formula would prove that 14 is the correct number of shapes possible with four triangles.
The next day we started with another look at patterns. Our first example was of a familiar type to them, a pattern of growing tiles, with the object to find an expression for the number of tiles in the nth shape (see figure 2). It took them only a minute to decide on the expression: number of tiles in the nth shape =*. This expression was easily defended - it was clear to see how the pattern changed. The rule - make a larger and larger central square and place two single squares on either side - could be translated exactly into algebraic notation.
Figure 2
The next task, however, was purposefully deceptive. The problem: How many regions are created by placing n dots on a circle and connecting each dot to every other dot with a line segment? I had them start by drawing four circles, placing one dot on the first circle, two on the second, three on the third, and four on the fourth. They connected every dot to every other dot on the same circle, and examined the arrangements (see figure 3).
Figure 3: One, two, four, and eight regions
The number of regions in each circle is, in order: 1, 2, 4, and 8. This was enough of a pattern for some students to decide the expression is number of regions in the nth circle = *. Others wanted more data, so they checked with five dots. Sure enough, five dots produce 16 regions.
"What is the rationale for doubling the number of regions?" I wanted to know. "Does each new dot added create segments which split every region in two? What's going on?" The consensus was that it wasn't clear what was happening with the regions, but they did double with each new dot, and the formula worked, so it must be correct.
Flora raised her hand. "I found 31 regions with six dots." Zach, across the table from Flora, had also tried six dots and found 32. Flora immediately conceded, convinced she had miscounted, since 32 must obviously be the correct answer. The situation was confounded, however, when Megan reported, "There's only 30," and Molly announced, "There's only 57 regions with 7 dots!"
Figure 4: 5 dots make 16 regions, and 6 dots make... how many?
The pattern had collapsed. A recount of the regions with six dots confirmed that 32 was not correct - there are only 30 or 31, regions depending on whether or not the three chords connecting opposite dots intersect at a single point (figure 4). The class was stunned - and on the verge of rebellion. "How are we supposed to know anything?" one student complained, "I thought math was patterns!"
Is there a pattern to the number of regions? There certainly is, but it is far more complex than it initially appears. There is something complicated going on when a dot and its associated line segments are added, and the algebraic relationship between the number of dots and the number of regions will reflect that complexity. Although the initial data suggests a doubling sequence, there is simply no justification for why this sequence should continue to double.
We looked at one more pattern. On the blackboard I wrote the sentence from the opening paragraph, "I am done studying _______," and I asked them "How many letters are in the next word of this sentence?" Despite the fact we'd just witnessed the failure of a pattern which was too good to be true, hands shot up across the room with the "answer:" Sixteen!
The answer, course, is 3, because the word I was thinking of is "now." Was this a trick question? Had I tricked them, or had they tricked themselves? Perhaps the answer to this question, like the patterns we just saw, is more complicated than it appears. Perhaps we, their teachers, had been tricking them all along. We've presented our students with example after example of situations with beautiful patterns, to the point which they believe everything is predictable from the numbers alone, regardless of the structure of the phenomenum that is generating these numbers. Our experiences with patterns in the math classroom can lead us to a superstitious mindset where we believe in the validity of a sequence without questioning the basis of its construction -- or equally harmful, they can lead us to believe that patterns are valid only in math class but otherwise disconnected from reality.
We are all familiar with the type of question: What comes next in this sequence: 3, 6, 9, 12, 15, ___, ___. As Richardson points out (1942), in order to answer this question, one doesn't need logic, one needs clairvoyance! A pattern such as this in which the nth term appears to be 3n for the first five terms can be simulated with an expression such as 3n + (n-1)(n-2)(n-3)(n-4)(n-5)f(n), where f(n) can be any function at all. The second term of this expression is equal to zero for the first five whole number values of n, but becomes something else entirely when n is a different value. With f(n) = 1000, say, the sequence becomes 3, 6, 9, 12, 15, 1200018, 720021, 2520024 ... The problem with these sequence patterns is they are a list of data given without the any connection to a structure, and as such it is impossible to reliably predict their future behavior. This is not to say that these sequence patterns have no value, indeed many students enjoy them and the activity encourages algebraic reasoning. One should be aware, though, that in order to answer these questions we must first make certain assumptions about the nature of the sequence. Each time we do so, we run the risk of becoming so comfortable making assumptions that instead of saying "I am done studying now," we find ourselves admitting "I am done studying interrelatedness"
References
Burns, Marilyn (1993). "Math by all Means" Creative Publications.
Gardner, Martin (1977). "Mathematical Magic Show" New York : Knopf : distributed by Random House.
Richardson, Moses (1942). "On the Teaching of Elementary Mathematics" American Mathematical Monthly, October 1942 (498-505).