Research Paper: Learning to think like a mathematician: ideas for measuring and comparing complexity in geometric constructions with a circle arc template
The 15th International Congress on Mathematical Education
Sydney, 7-14 July, 2024
learning to think like a mathematician: ideas for measuring and comparing complexity in geometric constructions with a circle arc template
Christopher C. Tisdell
School of Mathematics and Statistics, UNSW, Sydney, 2052, Australia
School of Education, Deakin University, Melbourne, Australia
Understanding how educators can create opportunities for their students to “think like a mathematician” remains underdeveloped. The purpose of this paper is to partially respond to the above challenge by illustrating that the learning and teaching of geometry provides a fertile environment for such development. We share some ideas on how students and teachers can measure, compare, and refine the complexity of geometric constructions. This opens possibilities for students to explore some core values of mathematicians, such as simplification, minimality, accuracy and elegance in geometric problem-solving techniques.
brevity is wit
The famous idiom “brevity is the soul of wit” illustrates Shakespeare’s belief in minimality, simplification and accuracy in language. Minimality, simplification and accuracy are also core values of mathematicians, for example: removing certain conditions in a theorem while maintaining the conclusion; formulating an “elegant” proof; or providing a more efficient way of solving a problem.
Every year, millions of students around the world learn geometry and create “geometric constructions”. Students begin with a given set of data (e.g., a line segment, a set of discrete points, an angle) and use certain tools (e.g. a compass, a straightedge, a circle arc template), and are challenged to produce certain geometric entities (a point, or a set of points). Such challenges are one of the oldest forms of mathematical problem-solving.
The past 2300 years has shown that some of the most well-known geometric constructions (e.g., those from Euclid’s Elements) are not necessarily the simplest (Mackay, 1893), and that the traditional tools (e.g., compass and straightedge) are not necessarily the most accurate (Tisdell and Bee Olmedo, 2022). As such, the learning and teaching of geometry forms opportunities for learners and teachers to think like a mathematician, by exploring simplification, accuracy and minimality.
At its atomistic level, solution strategies to problems of geometric construction involve students performing two basic operations: creating a line segment (or extending one); and generating a circle arc. To construct the solution set, students combine these two operations in strategic ways, via a finite number of discrete steps (much like a terminating algorithm). The finiteness of steps ensures that a count can be given to each solution method. How might students count these steps? As a starting point (no pun intended), students can consider: counting how many lines are drawn and how many arcs are drawn in the solution method; and counting how many preparatory steps are involved to draw such lines and arcs. Once we have several kinds of different solution methods, each with their own number of steps, then learners can compare them and search for simplification, accuracy and minimality.
The above kind of enumeration forms the basis of the theory of geometrography (Lemoine, 1893). Geometrography has been used and adapted by various geometers (Quaresma and Graziani, 2023) and mathematicians (Mackay, 1893). However, it seems to have received little attention within modern mathematics education circles.
Some Case Studies
Bisecting a given line segment is one of the classic problems (Euclid’s Elements, Book 1, Problem 10) from high-school geometry. We discuss and compare several solutions for constructing the midpoint of a given line segment through the lens of geometrography. Instead of a standard compass, we will use a circle arc template tool, see Figure 1, to create arcs that have a fixed radius, due to its superior accuracy, ease of use, and safety (Tisdell & Bee Olmedo, 2002).
Figure 1: A Circle Arc Template
Consider Figure 2. Students may construct this figure with a straightedge and a circle arc template, or they simply may be presented with the final figure. Essentially, this solution constructs an equilateral triangle with base and then bisects the angle at the apex of the triangle to form K.
Figure 2: First Method to Construct the Midpoint of
Let us examine the sequence of moves captured in Figure 2. An arc is drawn with centre so that a new point is constructed between and . This is repeated at the other end of the line segment to construct a new point . Then, two arcs are drawn with centre and so that two new points and are constructed. Two line segments and and then constructed with the straightedge, and they are extended to meet at a new point , forming the equilateral triangle The angle at the apex, , is then bisected by constructing an arc with centre that intersects with two sides at and . Two more arcs are constructed with centres and with the arcs intersecting at a new point . Finally, the straightedge can be used to rule the line segment that can be extended to produce the desired midpoint . Note that an understanding of how to bisect an angle plays an important role for students in understanding this particular construction method.
Students can now proceed to count the total number of moves and points produced in this construction, for instance: 6 circle arcs; 3 line segments; and 9 constructed points. Let us form some alternative solution methods that will enable some comparative judgements.
Consider Figure 3. Here learners have the same problem of constructing the midpoint of , but now with an alternative solution method.
Figure 3: Second method to Construct the Midpoint of
The two opening moves to produce points and are the same as those in Example 1, however if students now draw arcs with centers at and they can construct and , two alternative points of intersection, lying on opposite sides of the line segment The straightedge is then used to construct the midpoint by joining and . Note that, unlike Example 1, no angle bisection was used in this solution method. On the other hand, the number of moves in this solution method is dependent on the length of . If is long compared to the fixed radius of the circle arc template then more moves will be required in order to get “close enough” to produce points and .
Students can now proceed to count the total number of moves and points produced in this construction, for instance: 4 circle arcs; 1 line segment; and 5 constructed points. Is the strategy simpler than that of Example 1? Yes, from a geometrography perspective, provided the length of is not too long when compared with the fixed radius of the circle arc template.
Consider Figure 4. Again, students have the same problem of constructing the midpoint of , but now with another alternative solution method.
The two opening moves to produce points and are the same as those in Example 1, however if students now draw arcs with centers at and they can construct and , two alternative points of intersection, lying on opposite sides of the line segment The line segment can then be formed, with the point of intersection with and forming its midpoint. Note that in this solution method, there are no additional moves to be performed if the length of is much longer than the fixed radius of the circle arc template.
Figure 4: Third Method to Construct the Midpoint of
Students can now proceed to count the total number of moves and points produced in this construction, for instance: 4 circle arcs; 1 line segment; and 5 constructed points. Is the strategy simpler than that of Examples 1 and 2? From a geometrography perspective, the answer is yes.
Our collection of examples viewed through the lens of geometrography provides learners and teachers with a way of understanding simplicity and minimality through comparative judgement. Our use of a circle arc template is designed to improve accuracy. Important questions include, which solution is the simplest? Can you produce a simpler solution, or one with the minimal number of moves? In this way, students are learning to think like a mathematician.
Of course, the shortest solutions are not necessarily the most illuminating, and this is an opportunity for teachers to discuss the idea of “elegance” by asking a question like “which solution methods makes each of us wiser?”.
Mackay, J. (1893). The geometrography of Euclid’s problems. Proceedings of the Edinburgh Mathematical Society, 12, 2-16. doi:10.1017/S0013091500001565
Tisdell, C. C. & Bee Olmedo, D. (2022). Beyond the compass: Exploring geometric constructions via a circle arc template and a straightedge. STEM Education, 2(1), 1-36. doi: 10.3934/steme.2022001
Lemoine, E. (1893). Géométrographie ou Art des constructions géométriques. Paris: Gallica Bibliotheque Numerique.
Quaresma, P. & Graziani, P. (2023). Measuring the readability of geometric proofs: The area method case. Journal of Automated Reasoning, 67(5), 21 pages. doi: 10.1007/s10817-022-09652-0