I've been taking a close look at geometric constructions in the Nova Scotia mathematics curriculum. There is one outcome involving geometric constructions in the NS curriculum. Grade 7 outcome G01 states that students, "will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors." The curriculum guide goes on to define constructions as "performed using a compass and straight edge (without markings) as tools." This is the only outcome that I am aware of that has a focus on the tool being used and not on the mathematical understanding to be achieved. I'm not sure I understand this focus on the tool. Especially such an archaic tool. I can almost hear the students...
Student: Why do I have to use a compass and a straight edge to draw this equilateral triangle? Wouldn't it be easier to use a ruler and a protractor?
I worked with a bunch of elementary teachers on these compass and straight edge constructions and they found them frustrating and tedious. When we performed the same constructions using patty paper and dynamic geometry software, the teachers found the constructions less confusing, enjoyed them more, and seemed to get a better understanding. So I asked, "why is this outcome so focused on a specific tool?" If we have efficient and accurate modern tools, why do we still teach students to use compass and straight edge? Is it more "elegant and civilized"... like learning Latin even though no one really speaks Latin anymore? Anyway, I love a good Star Wars analogy, so....
In grade 6, students look at transformations of 2-D shapes to create a design. In this outcomes, G04, pattern blocks are used as a tool to explore tesselations and the geometric concepts that result from these designs. The outcome does not talk about the tool being used or tesselations. These discussions are saved for the background of the outcome and its indicators. I recently read an article talking about how students explore tesselations and how their interest can be sparked by creating aesthetically pleasing designs. The author stated that, "tiling provides one of the richest possible studies of geometric concepts as well as a powerful topic for open-ended mathematical exploration, fostering important mathematical practices. For students, mathematical aesthetics can be a guiding force for mathematical inquiry, just as it is for mathematicians." from R. Scott Eberle. ““I Don't Really Know How I Did That!””. Teaching Children Mathematics 21.7 (2015): 402–411.
Ultimately, this is what I hope teachers focus on when teaching geometric constructions in grade 7. I'd like to see them work with students on open-ended mathematical explorations of geometric shapes and properties using a compass and straight edge as an entry into this exploration. Memorizing the steps to do geometric constructions should not be the focus of this unit. The focus should be the understanding that comes from working with shapes regardless of the specific tool that is used, exempli gratia compass and straight edge, patty paper, or dynamic geometry software (notice how I used a bit of Latin there?).
So why are Euclidean constructions an outcome in the NS mathematics curriculum? That is a good question. Neither the curriculum guide nor the textbook mention the history of compass and straight edge constructions or why students should be practicing them. Here is my hunch... Let's start with some history. Geometry was an integral part of the classical liberal arts education going all the way back to the middle ages and the Qradrivium. Much of the geometry education at this time consisted or reproducing the propositions from Euclid's Elements. Many of the proofs in Euclid's Elements were proven with geometric constructions using a compass and straight edge. Because of this, constructions were an integral part of the curriculum. It was the tool by which Euclid's postulates were both reproduced and proven. Over time, many parts of the classical curriculum have disappeared from modern education. Euclidean constructions however have long had their staunch supporters and remain. Here is a quote from D.E. Smith's book The Teaching of Geometry (1911), "Constructions excite students’ interest, guard against the slovenly figures that so often lead to erroneous conclusions, has a genuine value for the future artisan, and shows that geometry is something besides mere theory."
Here is a quote from "Ask Dr. Math" at the Math Forum, that I think best elucidates the reason for constructions in our curriculum.
"Certainly learning how to use the tools is useful. Some of the techniques are useful in construction (of buildings, furniture, and so on), though in fact sometimes there are simpler techniques builders use that we forget to teach. But I think the main reason for learning constructions is their close connection to axiomatic logic. If you haven't heard that term, I'm talking about the whole idea of proofs and careful thinking that we often use geometry to teach. Euclid, the Greek mathematician who wrote the geometry text used for centuries, stated many of his theorems in terms of constructions. His axioms are closely related to the tools he used for construction. Just as axioms and postulates let us prove everything with a minimum of assumptions, a compass and straightedge let us construct everything precisely with a minimum of tools. There are no approximations, no guesses. So the skills you need to figure out how to construct, say, a square without a protractor, are closely related to the thinking skills you need to prove theorems about squares."
If you have any insight into Euclidean constructions and their place in the mathematics curriculum, I would love to hear your opinions.
Nova Scotia Mathematics Curriculum Outcomes
Grade 6 G04 - Students will be expected to perform a combination of successive transformations of 2-D shapes to create a design and identify and describe the transformations.
Grade 7 G01 - Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors.