Mathematical tools – Part 3

On the morning of June 20, 1944, two weeks after D-Day, New York high school students sitting for the Plane Geometry Regents exam were asked to do the following construction:The task posed almost 70 years ago was not difficult, but it also wasn’t rote.  It required the test-taker to make connections and to solve a problem: to apply basic construction techniques to notions of similarity and its theorems.  Additionally, it required visualization skills: before actually drawing the lines and arcs necessary to complete the construction, students needed to envision the final picture.

The separately administered Solid Geometry exam of that era is long gone, and the remaining hybrid, like math exams in general, has devolved into mostly one-step questions.

What kinds of construction questions have survived?  In its most recent iterations, the Geometry exam has rotated through a repertory of 4 rudimentary constructions: a perpendicular line through a point, an angle bisector, a perpendicular bisector, and construction of an equilateral triangle given a segment.  None of these require students to actually solve a problem.

What educational goal are we achieving by requiring students to regurgitate verbatim on an exam a basic skill that was practiced in class?

Even New York State is giving up the ghost: when Common Core computer-based summative assessments replace Regents exams, there will no longer remain a practical test of the use of these mathematical tools.

What kinds of questions will fill the void?  Consider the following:

This question was given to eighth graders on the NAEP in 1992, the last exam year that a construction question appeared.  It was bad enough that only 24% got the correct answer, slightly better than guessing, but that more students (27%) chose answer A shows a lack of understanding of basic geometry and the meaning of constructions.

In a further devolution of skills, what will likely emerge is a series of true/false questions, such as appeared on the NAEP Grade 12 exam the same year:

This last question looks eerily similar to the ``selected-response items’’ format that will become a mainstay of Common Core computer-based assessments.  Although 51% of twelfth graders got all 4 choices correct, we are not impressed (1) with true/false questions, (2) with the level of difficulty of this question and (3) that 49% got one or more parts wrong.  (Does one wrong out of four deserve half credit?)

We here at ccssimath.blogspot.com do not want to see constructions degenerate into nothing more than mouse clicks.


There is a glimmer of hope.  To CCSSI’s credit, a construction skill set, albeit tattered, survives in three standards:

G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-C.3. Construct the inscribed and circumscribed circles of a triangle....

Let’s parse the standards, beginning with G-CO.12.  String (cat's cradle?), ``reflective devices’’ (mirrors?), and paper folding (origami?) have their place, but we’re not sure how they relate to constructions, and we will never believe that computer graphics can ever substitute for good ol' hands-on manipulation of a compass and straightedge.

Revisiting one of the themes that runs through this blog, let’s analyze the age-appropriateness of some of the standards.  ``Copying a line segment’’ belongs in fourth grade, when students should first learn how to use a compass.  That a compass draws an arc, i.e., the locus of points that are equidistant from a center, so that any segment from the center to the arc can be a copy of an original segment, should be learned when students first study circles, not a half-decade later.

For that same reason, students in fourth grade, who, not coincidentally, should be learning about equilateral triangles, can then proceed to construct an equilateral triangle given just one line segment.  This connection of various parts of mathematics through theory and practice is something that CCSSI completely misses.  Absurdly, the word ``equilateral’’ does not even appear in CCSSI until high school.

Other constructions in G-CO.12 duplicate each other.  ``Copying an angle’’ and ``constructing a line parallel to a given line through a point not on the line’’ are essentially the same; let’s not give CCSSI credit for two separate constructions.

``Perpendicular bisector of a line segment’’ is a legitimate construction, but it encompasses (ha!) ``bisecting a segment’’ and is encompassed by ``constructing perpendicular lines’’.  Those three count as one.

``Bisecting an angle’’ is also a legitimate construction, so how many does that make in total?  Three, by our count.

What’s missing?  At minimum, ``construct a perpendicular to a line through a point not on the line’’ and ``construct the center of a circle’’ should be in the standard repertoire.

G-CO.14 at first glance looks a bit more challenging, until you realize that inscribing a circle simply requires ``bisecting an angle’’ twice, and circumscribing a circle requires constructing the ``perpendicular bisector of a line segment’’ twice.  So there’s nothing new there, except for memorizing which is which.

This brings us back to G-CO.13, and its ambiguous language, a prevalent problem in CCSSI.  Does ``Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle‘’ mean the equilateral triangle and square should be inscribed in a circle?  Poor parallelism; we have no idea.  We do know that inscribing regular polygons in a circle is predicated upon knowing the location of the center.

The upshot is that Common Core does not present constructions in a way that is going to lead to real challenges.  Like the devolved Regents exams, CCSSI’s standards will lead to rote memorization and regurgitation on exams.

Although it has long been lamented that the American math curriculum has been watered down repeatedly, CCSSI as written does not mandate problems that involve mental effort.  We here at ccssimath.blogspot.com believe that, with an arsenal of basic construction skills, students can be challenged with questions that will enable them develop real analytical skills.

We digress briefly to relate an anecdote which we are well aware does not present a unique situation.  We recently had the occasion to meet a Google, Inc., software manager and we were discussing a newly opened public high school that teaches students both C++ and Java programming.  Google is involved as an advisor to the program.  When we asked the manager how many of the school’s graduates Google might hire, the manager scoffed.  The manager thought they might seek jobs at small start up companies that might need to set up a website, but that they were not even close to being skilled enough to work at Google.  The manager said Google did most of its programmer hiring abroad.

Returning to our main story line, is it possible to take basic mathematical construction skills and use them to solve real problems, kind of like an engineering challenge?  Of course, otherwise, we wouldn’t be writing this blog.

Let’s begin simple.

Construction Example 1

It's already harder than anything in CCSSI, but not challenging enough?

Construction Example 2

These constructions use the same basic skills as in Common Core, but are not rote; they involve problem solving.

Construction Example 3

This problem requires thinking about equidistance and visualizing what is not there, among other things.  Even educators, including one self-important huckster who makes a living selling rhetoric, got a bit tripped up on this question; here’s the tweet:

Construction Example 4

Construction Example 5

This last problem is considerably more complicated than the previous examples, but still within the capacity of middle schoolers (with some guidance).


In mid-1944, although the tides of war were turning, the outcome was by no means clear.  We might even argue that high school students who studied insufficiently to understand mathematics and geometry principles and were unable to solve problems could have jeopardized the war effort and unnecessarily increased their own risk.

Although we no longer have thousands of high school students soon to become pilots, gunners or bombardiers who have to correctly calculate or triangulate distances, there are plenty of careers in which it would be advantageous if graduates had some knowledge of the true utility of math tools and the mathematical theories behind them, and had practice solving real problems, rather than simply memorizing rote constructions.  How else will we have any hope of staying competitive in the global economy?