My Math Education Blog

"There is no one way"

Tuesday, October 10, 2017

On NCTM, Part 2

In Part 1, I discussed my relationship with NCTM, its Standards, and its journals, mostly The Mathematics Teacher. In this post, I discuss NCTM conferences, and compare them with other math teacher gatherings.

NCTM Conferences vs. Local Conferences

I've had much, much better luck attending great sessions at my local Northern California conference in Asilomar (Pacific Grove, CA) than at the NCTM meetings. This could be because it's easier for me to choose the good ones, and attend talks by sure-to-be-good presenters. Or perhaps my preferences align better with NoCal colleagues than they do with people from other areas. Or it could be because there just happen to be more good sessions on average than at NCTM.

One way to make sure I attend something good at NCTM is going to talks by celebrities. I have not regretted attending presentations by Zalman Usiskin, Jo Boaler, Marilyn Burns, Dan Meyer, and other superstars. Obviously, others agree, and those sessions often take place in huge halls.

Speaking of huge halls: there are dozens of booths in the exhibit hall, only a handful of which are of interest to me. One trend seems to be that the major publishers occupy more and more square footage in there, and there are fewer interesting small exhibitors. In any case, walking pointlessly around the exhibit hall, catching the occasional good presentations, and hearing celebrities doesn't seem to be enough of a reason to attend NCTM meetings. Why do I keep going? When I was young and extroverted, I enjoyed meeting people. Now that I'm old and introverted, I mostly enjoy running into people I already know, people I previously met online, and people who like my Web site. 

But really, the main reason I go is as a presenter, assuming my talk is accepted and I can get someone to pay for my travel costs. Over the past three decades or so, I've been a presenter on average about once a year at an NCTM meeting (some years, I spoke at one or more regional meetings, in addition to the national.) I had only two out-and-out disasters. In 1991 in New Orleans, I was scheduled to give a talk on "Logo Tools and Games", but there was no projector. (The tech dimension of conferences has been much improved since then!) In 2009 in DC, I only had three or four people in the audience. I had been scheduled in the last slot, midday Saturday, in the most tourist-enticing city possible, and my topic (Space, an advanced geometry elective) was not of interest to many.

Those experiences were not typical. Usually, a lot of people turn up, they love my session, and I get the feeling that I'm doing something useful for the profession. Still, I don't enjoy speaking at NCTM as much as I do at Asilomar. I have a bit of a fan base in Northern California, so there's a feeling of continuity from year to year. One time my talk overlapped substantially with a talk I had given the previous year, and attendees were irate! The Asilomar conference is a sort of community, and many regular attendees see it that way. For many years, I went to Asilomar with my whole department traveling together in one van. (Yes, it was a small department.) It was the one time of year we'd all be together outside of school, and it helped build our esprit de corps.

A large percentage of good sessions, bonding with my colleagues, feeling part of a broader math teacher community, that's a lot. For me, NCTM conferences do not and cannot fulfill that role.

Even Smaller Meetings

[recycling part of a post from 2015]
In 2008, I gave a talk about teacher collaboration at the Asilomar meeting of the California Math Council. It was well attended, and well received, but more than a few attendees told me that they had no one to collaborate with. They were the only math teacher at their school, or the only one teaching certain courses, or their colleagues had zero interest in this. This led me to the conclusion that there was a need for a way for people to collaborate beyond the walls of their school.

As a result, a couple of years later, I launched Escape from the Textbook!, a sharing and collaboration network for math teachers. The group grew to over 400 members, and for a while used both online and in-person structures to stay in touch. To quote myself:
As middle school and high school math teachers, we find that almost every off-book activity we plan is well received by our students and leads to greater interest and motivation. Freeing ourselves from the constraints of set-in-stone curricula allows us to better respond to the realities of our classrooms, to better tackle situations such as heterogeneous classes, and to better implement cooperative and hands-on learning models.
However pressures of coverage, lack of time, external mandates, and isolation from like-minded teachers can undermine our efforts. 
We helped each other "escape from the textbook, whether for a lesson, a unit, or an entire course." We did this by sharing ideas and resources online, and with quarterly meetings in the Bay Area. The meetings took place on Saturday mornings. The basic format was to split the meetings into two sessions: math, and pedagogy. Various of us volunteered to lead the sessions, which pretty much always turned out well, although attendance fluctuated. My three blog posts about one of the meetings will give you an idea of their mathematical and pedagogical flavor. Proving Pick's Theorem, a page on my Web site, was a result of another meeting.
[end of recycled post] 

Alas, the group gradually slowed down: the online platform we used was not great, and organizing the in-person meeting was a lot of work. At the National NCTM Meeting in Boston (2015, I believe,) I ran into a participant in one of my summer workshops. He took me to the MTBoS (Math Twitter Blogosphere) booth in the exhibit hall. There, someone I knew and someone who knew me teamed up to convince me to join Twitter and the #MTBoS. They succeeded, and I am so glad I listened to them Through that hashtag, I can offer support to math teachers. When I have a little time, I can go online, and answer a specific request for help: does anyone have an approach to proportional thinking? a good way to introduce the Pythagorean theorem? a good lesson on logarithms? Having 42 years in the classroom, K-12, and having developed a fair amount of curriculum, I can often help. (For the above questions: 1 | 2 | 3.) This is something I was never able to do through NCTM.

The MTBoS does not have a lot of in-person meetings. There's a relatively small national gathering in the summer (Twitter Math Camp), which sells out in one day, and occasional local meetings — a handful in the Bay Area in the past few years. For regular local math teacher meetings, I have turned to Math Teacher Circles (MTCs). Those happen more or less once a month, and are organized by teams of mathematicians and math teachers. They offer an opportunity to get together and do math, often off the beaten path, as they usually do not prioritize discussions of pedagogy, or the K-12 curriculum. I have attended them both as presenter and as participant, and they are always fun. Again, this is something NCTM does not offer.

Whither NCTM?

I mentioned MTBoS and MTCs because they fill the needs of math teachers differently from NCTM, and in fact better. NCTM meetings are expensive, with uncertain return. MTBoS and MTCs provide guaranteed high value, at low or no cost, consistently, with no travel required. Both groups have been embraced, somewhat, by NCTM, which offered them ways to engage with the national meetings. That is a good development, but it does not address the deeper problems facing the organization. I cannot pretend I have a way to reverse the decline in membership and the financial woes, but any solution has to address teachers' actual needs, as teachers see them. This might mean fewer conferences, and redirecting resources to very local meetings such as the ones I described above. It might mean more opportunities for teacher-to-teacher sharing online and in the journals. It might mean taking down the wall between the publications and the rest of the Web. It might mean, in short, putting the "teacher" back in NCTM, and having people like myself (non-teachers who want to help teachers) play a support role.


Sunday, October 8, 2017

On NCTM, Part 1

I've read a few blog posts about the National Council of Teachers of Math (NCTM) in the last few weeks:
  • Three posts by Tina Cardone, which can serve as an introduction to NCTM. 
  • NCTM and the Math Forum by Tracy Zager, including links to other posts (including NCTM's response) about the shutting down of the Math Forum.
  • My NCTM Benefits, by Michael Pershan who explains why he doesn't get much out of his membership.
I encourage you to read those posts either before or after you read this one, in which I join this conversation.

The context, as I understand it: NCTM has been losing millions of dollars,  membership is dramatically down, and recent attempts at turning things around have not worked.

Me and NCTM

I've been a member of NCTM for maybe 30 years. During that time, I have been more involved than most:
  • I have been a presenter at 33 national and regional meetings (not counting 37 presentations for regional affiliates)
  • I had an article ("Delving into Functions with Function Diagrams") published in ON-Math, NCTM's short-lived online journal. (A version can be found on my Web site.)
  • I have had four articles published in The Mathematics Teacher
  • I edited the "Activities" department in that journal for two years (1995-1997)
  • I have reviewed many articles submitted for publication there, or in the Journal for Research in Math Education, most recently a few weeks ago.
Does that make me more qualified to comment on the organization than any other member? Not really. In fact, it makes me a little less qualified, since I cannot be said to be representative of the average member, let alone the ex-members who have not renewed. Still, I will share my thoughts.

Teachers of Mathematics? Really?

NCTM's big problem may be foundational: it is not what it claims to be. It is not in fact an organization of teachers of math, but rather of a wide spectrum of people who are more or less involved in the teaching of math: math teachers, yes, but also academics involved in teacher education and educational research, graduate students, coaches and other leaders in school districts, consultants, curriculum developers, app developers, and so on. I don't know the stats on this, but many, many members are not classroom teachers. Not only that, but the non-teachers seem to dominate many aspects of the organization, for example the authors of journal articles, the Board members, perhaps even the presenters at conferences.

This is not surprising: on the one hand, teachers mostly don't have a lot of extra time. On the other hand, those teachers who are interested in the big picture, beyond the walls of their school, tend to leave the classroom, and join the related professions I listed above. This is in part because of the lack of time to think big while teaching, but it is also because of the low pay and low status of the teaching profession in the US. (I myself managed to both be a teacher and work in curriculum development and teacher training over the decades, but that is only because I had a relatively privileged part-time position in a private school. I have now retired from the classroom after 42 years, and have fully joined the legions of non-teachers who would like to help teachers.)

Maybe a more accurate name for the organization would be "National Council for the Teaching of Mathematics".

None of this is intended to pit one group against the other, or to glorify one group at the expense of the other. I am just describing the situation as I see it. It seems like we are faced with a sociological phenomenon that is intractable. How can NCTM truly be an organization of and for math teachers, when those teachers find it difficult to be involved, and when the organization is dominated by non-teachers?

I might suggest some answers later on, but I will start by following Michael Pershan's lead, and discussing what I get and don't get from the organization. Who knows, this may be useful to the NCTM leadership as they try to sort things out.

The NCTM Standards

I appreciated the various iterations of the NCTM Standards, as well as the previous emphasis on problem-solving, and the subsequent documents on implementation. However, I only know this from skimming the documents. Actually reading them was always out of the question. This is in part because of the time it would have required, but also because in my opinion, those documents are supremely boring. I have not read them, nor do I know a single teacher who has. I do like NCTM President Matt Larson's blog posts and conference talks, as he presents aspects of the NCTM philosophy in accessible chunks.

It is probably a good idea for NCTM to generate these documents, as they reflect some sort of professional consensus on pedagogy, equity, technology, etc., and thus may be used to affect government policy, and to support forward-looking teachers and administrators as they discuss those issues locally. I am also excited that in its next document along these lines, the upcoming Catalyzing Change in High School Math, NCTM is finally taking on two previously off-limits topics: tracking and the overabundance of standards in the Common Core. (Though the latter is by implication rather than explicitly. We'll see how it shakes out in the final version of the document.) I was invited to provide feedback to the authors, and I gladly spent many hours on the project.

Writing for the journal

These are the articles of mine that NCTM published:
Iam sure  they reached a wider audience than they would have otherwise. And the feedback I must have received from reviewers probably improved the articles.  So far, so good.

There are several more articles I'd like to submit, but I'm running into a ridiculous policy.
I recently had an article rejected from the Sound Off! department in spite of the fact it was well-liked by the reviewers, because much of it had appeared in my online critique of the Common Core. NCTM will not publish anything that has previously appeared on a blog or Web site. Apparently, the fact that 100 or 200 people saw my original article, some of whom are NCTM members, and that others might some day come across it means that those members are not getting their dues' worth if they also see an excerpt of it in the journal. As for the thousands of members who will never read the article on the Web, well that's too bad for them.

Articles curated and peer-reviewed by NCTM would certainly be more worth reading than articles randomly found on the Web. Moreover, peer-reviewed and edited articles would certainly be better than their original Web version. In fact, that is exactly what happened with my article on interactive whiteboards, which was solicited by the editor after seeing it on my blog. (He temporarily forgot the policy.) The published article was much improved over the original blog post.

This leads me to suggest that a regular department on "Best of the Web" would be a great feature. Perhaps links to worthwhile blog posts and useful sites, plus a curated, peer-reviewed, and edited selection in each issue.

Anyway, I'm told that the policy against publishing submissions that overlap with material previously seen on the Web is currently under review. We'll see if common sense triumphs. You may be thinking: "Henri, why don't you refrain from publishing items on your blog or Web site, so you can submit them instead to The Mathematics Teacher?" The reason is that I can self-publish instantly, and reach a (small) community of people who I know appreciate my work. Waiting for NCTM to review my submission does make sense from the point of view of reaching more readers, but there is no guarantee the article will be accepted. I am not likely to deprive the current readers of my work in the hope of perhaps reaching others many months later.

Reading the journal

What do I get as a reader of The Mathematics Teacher? Well, to be honest, not a lot. The issues often pile up. I skim them eventually, and read some of the articles. My favorites are the ones in the sweet spot where the math is new to me, and it is presented skillfully in a way that allows me to engage, without requiring an inordinate amount of time and effort. On average, I find one or so such article per issue. Yet, in decades of reading the journal, only a handful of articles actually affected my classroom practice. That is in part because many of my favorite articles are really "teachers' mathematics", i.e. in a content area that is closely related to high school curriculum, but is not directly curricular given the time pressure math teachers are always under when trying to cover essential concepts.

I always read "My Favorite Lesson", in the back of the journal. Whether or not it is useful to me, it is always both short and interesting, and it is usually very curricular. Perhaps the journal could use more of this sort of teacher-to-teacher sharing.

One thing that really ticks me off when I read The Mathematics Teacher is the seemingly obligatory genuflection at NCTM's sacred texts, most recently Principles to Action. I was told this is not editorial policy, this is what people submit. Given what we see in the journal, readers and contributors should be forgiven for believing that this, indeed, is editorial policy, and that citing NCTM documents will improve your chances of being published. One thing I'm proud of is that during my tenure as editor of the "Activities" department, I removed all references to the Standards from the articles I edited. I judged submissions not by adherence to dogma, but by their probable usefulness to teachers, and the quality of their mathematical content. Frankly, readers who care about loyalty to NCTM policy can pass their own judgment about whether the article falls within the orthodoxy. We don't need to be constantly showered with reminders of NCTM's latest publication.

In fact, this is one of the differences between teachers and non-teachers who want to help teachers. Teachers are typically not bound to any particular framework, because in the reality of the classroom, there is no all-encompassing, always-right-for-every-situation framework. There are more things in the life of the classroom, non-teacher, than is dreamt of in your philosophy. Teachers need to be eclectic. I wrote about my own eclecticism last year in a post that is very relevant to this discussion of NCTM, as it addresses the gap between teachers and helpful non-teachers, which I believe may be the key issue for the organization. (This is closely related to how one feels about math education research.)

OK, I've gone on too long. I will wrap this up in my next post, with a discussion of meetings, big and small.

-- Henri

Tuesday, September 19, 2017

Stumped by Euclidea

I've really enjoyed solving the puzzles in Euclidea, a brilliantly designed app for iOS and Android. The basic format is "given this, construct that". You start with just two tools: a straightedge and a slack compass (i.e. a compass that does not remember the radius it was last set to). As you find useful and reusable constructions, such as how to drop a perpendicular, those become available as additional tools. The interface is simple and elegant.

For each construction challenge solved, you "get" a star. If you find a construction that uses the minimum number of steps, you get another star, and yet another one if you find an optimal straightedge and slack compass construction. Searching for these optimal solutions is usually more difficult than merely solving the puzzle, but it can be instructive. For example, puzzle 2.7:
 Given a line l and a point P on l, construct a perpendicular to l through P.
My first solution was based on standard techniques to bisect an angle, or to perpendicularly bisect a segment. However, knowing that there is a three-step construction using straightedge and slack compass got me thinking. Here is what I came up with:

Step 1: Draw a circle through through P, with center O not on l. It meets l again at a point Q.
Step 2: Draw a line through O and Q. It meets the circle at a new point R.
Step 3: The line through R and P is the desired perpendicular to l.

This could lead to a great lesson: why does this work? will it always work? To answer this question students need to know that the sum of the angles in a triangle is 180°, plus the isosceles triangle theorem and some basic algebra. If they have trouble, you may offer the hint: "when working with circles, listen to the radii!" Indeed, analyzing the figure below should lead to Thales' theorem about an angle intercepting a half-circle:

(All figures created in GeoGebra.)
Actually, such a lesson would work best after working through Labs 1.5, 1.6, and 1.7 in Geometry Labs (free download). The reason is that the labs give students practice solving problems of this type with actual numbers before tackling the general case. (These labs require circle geoboards, or circle geoboard paper.)

Anyway, back to Euclidea. I was not always able to find optimal solutions. My first failure in this regard is on 1.7, inscribing a square in a circle, given one vertex on the circle, in seven straightedge and slack compass steps. My best attempts required eight steps.

In any case, I was able to solve all the puzzles one way or another, until 10.6:
Construct a circle through P that is tangent to both sides of the angle
My first reaction was that this was merely combining and extending two problems I am familiar with. First, the classic construction of a circle tangent to two lines. And second: given a line l and a point P not on l, construct a circle through P, tangent to l. Both puzzles are part of the construction unit I assigned year after year to my geometry class, and the second is the underlying strategy for the construction of a parabola with focus P and directrix l.

And yet, I was not able to crack Euclidea 10.6. I did it in GeoGebra:
As you see, there are two solutions. I used the fact that the center of the desired circle is equidistant from P and each side of the angle. Therefore it lies on the parabolas with focus P and the sides as directrixes. So it must be at the intersections of these parabolas:
This was straightforward, as GeoGebra has a Parabola tool, but I still have no idea how to do this in Euclidea. The creators of the app claim this can be done in six steps (using any Euclidea tools*) or 11 steps (using only straightedge and slack compass.) If you figure it out, I would love a gentle hint, as Euclidea is not allowing me to proceed any further until I solve this.

* Available tools for 10.6: straightedge, compass (slack or rigid), perpendicular bisector, perpendicular line, angle bisector, and parallel line.

[Breaking news: The Euclidea developer gave me a big hint via Twitter. The approach involves dilation. I am embarrassed I didn't think of it, as I had used a very similar strategy to construct a tangent to two circles.]


PS: for my thoughts on the mathematics and pedagogy of geometric construction, see this recent post, (and for more, follow the links therein.)