# Politicizing the Trapezoid

The trapezoid, is the homeliest of our special quadrilaterals. And it has just been redefined by the Common Core’s David Coleman.

Children will be tested, using definitions that are different from what is in their books

For about two hundred years, when American teachers taught about trapezoids, they were teaching about a figure with four sides, two of which were parallel, and two of which weren’t.

**Last year, only the figure on the left was a trapezoid. Now, they both are.**

We call that a “non-inclusive” definition, because it doesn’t include the special case when both pairs or sides are parallel (a parallelogram). Readers who are less familiar with math may have encountered the inclusive vs non-inclusive debate when considering squares: are they special rectangles? or are they something of their own kind? Math people say squares are rectangles (special ones), the inclusive definition. Rectangles have four right angles (that fits a square). Rectangles have both pairs of opposite sides equal (that fits a square). We can ask: “Are there four right angles?” If the answer is yes, we say “Rectangle!” – we don’t say, “well might be a rectangle. Let’s first make sure it’s not a square.” And if we like the inclusive definition for rectangles, why not for trapezoids?

There are conventions in mathematics. Agreements that we have made, that we stick to. We should be aware when something is conventional (electrons flow from the negative terminal) (north is at the top of the map) (x is left right and y is up down) (we use an inclusive definition of rectangle, but an exclusive definition of trapezoid) versus a decision we make with a mathematical reason (1 is not prime is a good example). And we should talk about conventions, with each other, with the public, with our students.

I am not arguing that the change is wrong. I’m saying that change without warning, with no discussion is wrong. And I’m saying that the person making the change should not be a vendor, should not be David Coleman.

We stuck with 200 years of the non-inclusive definition. It is worth changing. But that requires changing every major textbook (geometry, but also many middle school books. It shows up sometimes in algebra too. It’s a lot of books…) It requires a national discussion. Teachers need to know, need to think about it. Teachers need to realize that the really neat figure called an “isosceles trapezoid” needs a brand new name, if we even think it is worth talking about. And honestly, this discussion is worth having. The change is worth making.

But the change happened with no discussion. Common Core’s testing left arm, PARCC, redefined trapezoid. Without permission. Without telling anyone. Without talking to teachers. Without initiating discussion. Without changing textbooks. Without even giving an opportunity for any of these things to happen. Children will be tested, using definitions that are different from what is in their books, different from what their teachers taught them this year, and every year previously.

When we overtest kids, there are players who can object (parents, kids, teachers, even principals and superintendents). When we cut funding, there are players who will object. When we close schools, there are players who can object. But who stands up when a vendor screws around with a mathematical definition?

Do math teachers have organizations? Well, yeah, but… The National Council of Teachers of Mathematics (NCTM) was so ed-reformy that I lapsed my membership over a decade ago and never looked back. The Mathematical Association of America (MAA) and American Mathematical Society (AMS) are for professors and graduate students, not teachers (I should rejoin MAA, they have a journal that I find accessible, and I like the challenge, and I prefer paying the member price for their books). There is a NYC group, but they hold one conference a year, and put out one journal, and nothing else as far as I can tell. The UFT group has some nice people, but not in my area of interest (if you are interested, they did and probably still do some very nice art/math combination stuff).

But I have long maintained connections with the New York State group: The Association of Mathematics Teachers of New York State (AMTNYS). And they are an active group. They have direct connections to the New York State Education Department. They have sometimes campaigned to change rules about calculators, dates for exams. Here, in New York, if there is a place for math teachers to go, it is AMTNYS. Because of this, their listserves, especially their high school level listserve, buzzes when the state screws up a regents exam or a schedule or a new ruling or a roll out. One of AMTNYS stated purposes is “To serve as liaison between the State Education Department and the field.” This sometimes means communicating SED decisions to us, sometimes conveying our concerns to SED, sometimes organizing teacher voices for or against a change, and sometimes seeking clarification on behalf of their members.

Here is an AMTNYS listserve discussion from last year on the new definition of trapezoid. And here’s one from this year.

In this case, a vendor was imposing a change in definition with little warning, and no discussion. Teachers haven’t heard about it (the majority probably still haven’t), textbooks aren’t updated. And the decision was made implicitly, by accepting a test, not by people responsible to the public. AMTNYS should have worked to stop SED. Instead, they told teachers about the new definition. AMTNYS’ leaders can claim they are not an advocacy organization. But when they accept the new testing regimen without complaint, education reform, including in mathematics, without complaint, new curricula without complaint, changes made to a mathematical definition by a vendor, without complaint, and in fact transmit each of those to mathematics teachers – they are in fact adding their weight to the direction of change advocated by the Ed Reformers in Albany, they are advocating, and they are advocating against teachers and against students. I am disappointed.

Of all the bad things ed reform is doing and has done, changing a definition without discussion is not high on the list. And, in this case, its not that Coleman is out to screw kids (he is, but does he care about trapezoids? He has millions of children who he treats with disregard, what’s a silly shape to him?) It’s not that John King and Andrew Cuomo were out to get kids to get one more question wrong (they don’t care). It’s not that Barack Obama and Arne Duncan, when they pushed Common Core on the states thought about ways to change one definition. And it’s not that AMTNYS’ leadership is so badly compromised by needing to be friendly with powerless state bureaucrats that they forgot to speak up.

No, it’s just one more, small, bizarre episode: Common Core redefines Trapezoid.

(Change “doesn’t exclude” to “doesn’t include”, near the beginning.)

Yes! I like helping students understand which definitions are just conventions, and which are mathematically motivated.

Why would an isosceles trapezoids need a new name? (I do see that a rectangle is one example of an isosceles trapezoid, but that doesn’t require a name change.)

I like your thoughts overall here. If a vendor can change what definition gets used, might they make a change that is less mathematically appealing next time?

There are plenty of textbooks that define trapezoid in the inclusive way. All four high schools in which I have taught in the past 45 years do that. It’s a far better definition, as all the theorems that apply to trapezoids now automatically apply to parallelograms (and on down the hierarchy) as well. It allows us to have only two simple area formulas for quadrilaterals: half the product of the diagonals for kites (including everything under kites), height times average of the bases for trapezoids (including…). For a rhombus, being both a kite and a trapezoid, both formulas work; that’s pretty cool. I’ll post a follow-up list of texts when I am back in school tomorrow.

Waiting for those… I’ve dug up The University of Chicago Series – Demana and Waits, and a text by Moise and Downes. You got something better recommended? Those make a pretty shaky table to rest your argument on.

Jonathan,

I agree with Sue that Isos Trap now needs redefining but not renaming.

Snce I haven’t yet seen the official defn here are a couple of thoughts…

1) “A trapezoid with exactly one pair of opposite congruent sides.”

This is concise but not transparent. It’s minimal because “nonparallel” doesn’t have to be stated.

Clearer but wordier:

2) “A trapezoid with a pair of nonparallel opposite sides which are also congruent.”

I prefer an isosceles trapezoid defined as “A trapezoid with a line of symmetry.” This definition is succinct and excludes parallelograms that are not rectangles.

This is a really fascinating question that seems worth exploring — why things are defined the way they are. For instance, say you know that a quadrilateral has at least one pair of parallel sides, and you want to determine if the second pair is parallel. What do you refer to the quadrilateral as? How does this impact the quality of our definition? What properties are unique about a trapezoid that require a non-inclusive definition? How is it different than a square?

These are valuable questions, and legislating definitions from on high takes away from the value of engaging students in their discussion.

I’m not sure I understand why “that thing” needs a name. Is it really that special?

Certainly “isosceles trapezoid” is a very silly name if trapezoid has the inclusive definition – it would include every parallelogram, plus the thing formerly known as an isosceles trapezoid.

If we are giving out names, let’s work on making “lozenge” official (if it is not yet), or naming the 60, 90, 120, 90 kite…

I’m not sure where your information about David Coleman is coming from, but in June of 2012 – nearly 2 years before you wrote this post – the CCSS-M writing team (which does not include David Coleman) posted this in the Progression for K-6 Geometry (draft), pp. 3:

“Note that in the U.S., that the term “trapezoid” may have two different meanings. In their study The Classification of Quadrilaterals (Information Age Publishing, 2008), Usiskin et al. call these the exclusive and inclusive definitions:

•T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides

•T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides.

“These different meanings result in different classifications at the analytic level. According to T(E), a parallelogram is not a trapezoid; according to T(I), a parallelogram is a trapezoid.

“Both definitions are legitimate. However, Usiskin et al. conclude, “The preponderance of advantages to the inclusive definition of trapezoid has caused all the articles we could find on the subject, and most college-bound geometry books, to favor the inclusive definition.”

Here is a link to Bill McCallum’s blog, which is one of the places this progression was posted. (Bill is one of the 3 lead-authors of the CCSS-M.): http://commoncoretools.me/category/progressions/

Notice that I alternately blame Coleman and the contractor, and I’m not sure that it makes a whole lot of difference exactly who pulled this trigger. In fact, I don’t care who Bill McCallum is, or when he wrote his stuff. I don’t care that moving to an inclusive definition makes sense. (It does. I was pretty clear about that.)

I care that a decision that affects every math student in the country was made, essentially, by a private organization and its for-pay contractors, and was made without alerting teachers and students. I care that many students won’t know the definition has been changed until after they’ve encountered it on a high stakes test.

I thought that was pretty clear.

It was very clear. You made it perfectly clear to anyone reading with thoughtfulness that you actually thought it should be changed and that the issue is how it was done by a vendor. It is so frustrating as a teacher. We adopted a curriculum that uses the new definition. Other teachers taught it without saying anything. I was the only one who asked if we should be teaching it when their tests will all have something else. No thought at all to the kids. At all!