I just read the following online article: Why Use IBL? and wanted to note the link for later use. This is a great summary of the history, use and effects of Inquiry Based Learning (IBL) in the classroom, and its undergraduate-level cousin, the Modified Moore Method.
My approach to teaching has evolved along with the use of IBL. My work is to develop material to guide my students’ exploration, and then to facilitate their mathematical investigations.
In my current work, I teach two graduate level courses. One is an Abstract Algebra course where I guide my students to reconstruct and understand an unusual proof of the Fundamental Theorem of Algebra first developed by Euler and others. This course is a delightful collection of topics connected by several themes, including problem solving, the connections between graduate level mathematics, use of historical mathematical documents and the school curriculum, and of course, the Fundamental Theorem of Algebra. The other course is a Real Analysis course, which makes use of my modification of Mahavier and Mahavier’s Analysis problem sequence. We don’t have much time, so for this course my personal goal is that the students prove the Intermediate Value Theorem from the ground up.
I liked reading the Why Use IBL? article I mentioned in the first paragraph, so now I intend to make it required reading for my students as they start the Analysis course.
The short answer is no, but I think that Wikipedia is an excellent first stop on your way to find better sources.
Move towards the source: Ideally you should cite primary and secondary sources in your work, but Wikipedia might be considered tertiary or even more remote. However, most Wikipedia entries are compiled by knowledgeable human beings who are aware of primary and secondary sources, and prepare excellent reference sections.
One of my favorite sketches in the Monty Python film, The Meaning of Life, is of a middle-aged couple (completely lacking any intellectual curiosity) who have an awkward and scripted conversation.
The waiter tries to interest them in conversations about minorities, football, baseball, and finally manages to start them on a conversation about philosophy.
You can read the conversation here at the movie script site www.intriguing.com.
This sketch came to mind last week when my MAT students and I discussed conversations in class. They told me why they value conversations in math classes, and we considered two math ed articles:
This article which is based on two case studies, seems to be a reaction against the (then) recently released NCTM Principles and Standards which admonished teachers “not to tell”. Chazan and Ball are eager to describe many classroom circumstances where it is completely appropriate for teachers to take the lead and actively direct the class.
Chazan and Ball leave with three considerations:
Mathematical Value in Relation to Students – does the current student discussion have significant value to the students’ understanding of mathematics? Not all conversations are valuable.
Direction and Momentum – is the conversation at the right pace and level? The waiter in the Monty Python sketch recognized that the couple’s conversation was going nowhere, and helped them get started. The results were, well… let’s say better than before the intervention. A teacher can keep a conversation going at the right pace and at the right level of intellectual challenge for the students.
Social and Emotional Tone – part of the classroom culture is how students treat and respect each other. I’ve seen classes where the students are downright mean to each other, and nobody is willing to take intellectual risks. Luckily, the opposite is true in most classes I visit, and a lot has to do with how the teacher intentionally created the supportive and cooperative atmosphere.
The second article we looked at was more recent, and directed towards teachers taking their first steps at facilitating rich mathematical discussions in class.
Orchestrating Discussions, Margaret S. Smith, Elizabeth K. Hughes, Randi A. Engle, Mary Kay Stein, May 2009, Volume 14, Issue 9, Page 548.
This article presents a framework for teachers in the context of a discussion that follows a student activity on proportional reasoning. The five steps to a successful discussion are:
Anticipating – as part of your planning, imagine different ways that your students could successfully approach the task, and what misconceptions or difficulties they might have. Write these down and have it with you during class.
Monitoring – as you circulate around class during the activity, intentionally monitor your students’ work and identify which students are using the strategies that you’ve anticipated. This frees you up from having to think too much in the moment, since you’re prepared for most of what will happen. It also means that you’ll have more energy to focus things that you have not anticipated.
Selecting – the authors recommend against a show-and-tell style discussion. Instead, carefully select which student work you’d like to highlight, and have a reason to do so.
Sequencing – now that you’ve selected which student work to highlight, the order that the class shares it in is just as important. One possible trajectory is from the simplest strategies to the most abstract, but it’s important to plan a sequence.
Connecting – as the class discusses their solutions, the teacher can make explicit mathematical connections for the students and highlight certain aspects of the material. After the activity, the students are primed for assimilating the new knowledge and connecting it to what they already know, so here is where the teacher can maximize the benefit of the activity.
Favorite Quote
My favorite student comment is that all of this is just plain common sense. That’s how I see it too – that once you’ve read through this article, it’s pretty much self-evident that this is a good way to plan for the discussion after an activity. No controversy here.
Common sense may be a revelation to some, but it might be too often overlooked, and it is useful to hear it once in a while, especially as we get our start in teaching! What is your favorite common sense advice in teaching?
One of the many special things about the Bard MAT Program is just how much time students spend out in the schools, allowing for a gentle development of their skills and understandings as teachers.
Now that September is nearing an end, the MAT students will soon scatter around the neighborhood, to the nearby classrooms at some of our partner schools. Here’s a map I made of the placements:
I’m preparing for the fall math teaching lab, which starts on Wednesday, September 7th. Last year, I co-taught this class with my colleague Ben Blum-Smith, and we planned the course on the belief that mathematical problem solving is the heart of mathematics. The guiding questions we used were
What’s Motivating about doing Mathematics?
How do we Thrive and Grow Mathematically?, and
How do we Create a Community of Doing Mathematics?
We also worked with our students to envision, plan and implement the Bard Math Festival. Each of our students selected and developed a short and inspirational math activity for 4 or 5 students, and we invited math classes from the three schools in our building to participate.
Fall 2011
This year is going to be a bit different; most significantly, Ben has started graduate work at NYU. (I hope that he can come visit the lab a few times!) There is also a push to coordinate all the teaching lab courses (Mathematics, History and Literature) and to focus them on the teaching placements. Many of the texts that we’ll be reading are geared towards classroom conversations – developing and maintaining high level discourse in the classroom, and I’ll try to include a small project on math education (the mini-CRP). Some things will have to be dropped, but I plan to keep the Bard Math Festival, which meant so much to last year’s students as well as children and teachers in the schools we work with.
Two interesting blogs that I follow are called, respectively, The Science Babe and mathbabe. No, neither of these blogs are prurient websites that are weak on content. Instead, they both seek to reclaim the terms from websites that are.
The Math Babe is none other than Cathy O’Neil, a Ph.D. Number Theorist who left academia about four years ago for industry. She blogs not only about being a female mathematician, but also the mathematical techniques and tools that she is acquiring right now in industry. I do hope that she blogs about Number Theory too, although it’s exciting to read about how she is mastering Python and R.
A recent post that I find interesting is about Working with Larry Summers. Summers was the president of Harvard University between 2001 and 2006, who resigned in disgrace after the bad publicity generated in part by his comments about women’s aptitude in mathematics. After this, he worked at D.E. Shaw, which is where Cathy got to work with him. His project there was apparently to chase dumb-money. Any profit involved was at the cost to pension funds that many of us hope to retire on. Yuck!
Now, how about the Science Babe? Let’s save that for another post!
A sentence in natural language can often be interpreted in two ways, but in mathematics, we must be precise in our use of language and avoid ambiguity. This often least to humorous interpretations of street signs and the results can be adorable.
Take this street sign example, which I photographed in September of 2010 at the Happy Paws pet store near NYU. It’s a wonderful sign on a messy street that immediately attracted my attention and confusion.
Do Not Litter And Feed The Birds
Please leave a comment below on how you understand this sign:
A) Two separate commands: you should not litter AND you should feed the birds. Why?
B) You should neither litter nor feed the birds. Why?
C) You should either not litter or not feed the birds, or do neither. Why?
Last weekend was the last meeting this semester of the Bard Math Circle at the Kingston Library, and we made magic cubes. A magic cube is a large cube made up of eight smaller cubes that are hinged together in such a way that they rotate through themselves magically, revealing many surfaces. This activity is a little more involved that what we normally do, so you either have to follow directions, or problem solve.
Several museums sell these with artistic pictures on several surfaces, but you can make your own. It’s more fun!
Instructables has instructions that also use wooden cubes, but go a step further with photos: http://www.instructables.com/id/Crazy-Foto-Cube/. The downside of Instructables is that they want you to become a member.
The job market in NYC for teachers and schools has reached a new level of irrationality this year with unreasonable threats of a massive teacher layoff (yet again), and a seemingly eternal (yet porous) hiring freeze that separates a pool of talented but anxiety-ridden teaching applicants from ever more desperate public school principals in need of qualified instructors.
My prediction is that the big winner this year is the charter school. This is not a big surprise; their lack of restriction in hiring gives them unfettered and early access to the candidate pool. I’ve also heard that they have a direct line to bottomless pits of hedge fund monies.
A lack of restriction and unlimited funding is one thing, but last week I saw an unstoppable weapon, the charter school recruiter!
I met my first recruiter last week while observing one of my MAT math students at one of my favorite urban high schools on the Lower East Side of Manhattan. My student was doing a brilliant job: he was motivating, provided excellent scaffolding and carried out a wonderfully planned lesson. Several times throughout the lesson, his students had truly become thinkers and mathematicians.
And then, a stranger appeared. In walked an elegant and confident young lady. She was dressed for success and exuded expertise. In the clearest pantomime performance I’ve seen, she asked my student teacher, “can I videotape your lesson with my flip video recorder?” Then she sat down next to me in the back of the room, and I introduced myself.
It turns out that she is one of two full-time recruiters working for a cluster of four charter schools in Harlem. She was following up on my student’s interview with a visit to his classroom, just long enough to let him know that they were very interested, and to record some footage of his lesson for the charter school principal.
Even in this market, where I expect many of my amazing students to linger on the job market through the summer, and to be excited when any tenuous job offer comes in (after the first day of the school year, and with a long commute), I knew that this student was not going to be on the job market for long.
The best teaching candidates never stay on the market, even in the worst of times. (I suspect that the very best candidates never even step foot on the job market.) But there are exceedingly good teachers on the market right now. This week, the job market is HOT!
Last year an extremely good math candidate contacted me for help finding a job. I sent out emails, made phone calls on her behalf, and offered her encouragement (there was a lot of anxiety even in last year’s market). Five days later, she landed her first interview on a Friday. The school immediately invited her in for a demo lesson on Tuesday and offered her the job on the spot. She wrote to me that “Surprisingly, Discovery HS took three days to carry out the hiring process- interview, demo and decision making.” In all, she was on the market for only 7 weekdays.
How can we compete with professional recruiters, and schools that can turn around a hiring decision in just three days? It may seem simplistic or silly, but I think it is easy:
Confirm receipt. When a prospective hire applies, they should get a personal response promptly. This immediately reflects that they will be treated with respect, from the very beginning.
Explain the process. The worst part about being on the job market, is the experience of limbo. It takes almost no effort to explain how the hiring process will work, and approximately how long each part will take.
Check completeness of the application. Let them know if there is anything missing, and if so, what it is.
Interview them if they are of interest.
Let them go if they’re not. Don’t keep their hopes up, but say it nicely.
Inform them what comes next. How long will it take? How close are they to a hiring decision?
Answer them promptly. Don’t let a decision linger!
If you’re curious how I know that this method works, just check out what the first initials spell. This is exactly why the MAT program brings in students year after year, and is able to compete with the more established schools, as well as the cheaper programs.
The Bard Math Circle is growing rapidly this year. Perhaps this started because the wonderful Kingston Library director, Margie Menard, sent out a press release that was picked up by the local media, or that we’ve worked hard to develop a consistent and predictable schedule. But the fact is that the Bard Math Circle has found its niche: libraries.
This has me thinking of ways to ensure that our activities are safe spaces.
Safe to Take Risks One important aspect of this is that participants should feel safe to take mathematical risks. Considering all the adults I meet whom have experienced some sort of mathematical trauma when they were young, this is crucial. Participants need to feel safe to explore mathematical ideas.
Here’s a quote from a Kingston grandmother who brought her granddaughter to yesterday’s math circle. After hearing this, I know that we’re doing a great job:
Since she’s been coming here, her math has improved. She thinks about things now. This is the most worthwhile thing she’s been involved in.
Right on! We’re definitely going to come back to this topic in the future.
Physical Safety Another important aspect of safety for our Math Circles is physical safety. I traveled to Houston last month to attend the Circle on the Road Spring 2011 conference. Besides some incredible mathematics, the most interesting presentation, by far, was Brandy Wiegers‘ talk, Always Be Prepared. Brandy has been involved in Girl Scouting for over 20 years, and as a result, she is the mathematician I’d most want to be with in an emergency.
Here’s Brandy’s abstract:
Math Circles should be fun and engaging. To keep it this way it is important to be prepared with a box of tricks and some quick plans to ensure safety. In this session we’ll discuss what we keep in our Math Circle Box of Supplies, important legal aspects of working with minors including adult to student ratios and the buddy system. We’ll conclude the session discussing participant waivers and plans for emergencies. With a little bit of work we can all be more prepared to ensure that we never need to use our emergency plan. This way we can get back to math and everyone can have fun!
Brandy’s talk was really a Math Circle milestone. Talking about safety means that Math Circles are established, and that it’s now time to plan for safety. What procedures and guidelines does your Math Circle have in place to ensure the physical safety of your students? I think this is the beginning of a Math Circle discussion at the national level.
Well, Brandy did give us a lot of useful information. Perhaps the best thing she did was point us to the Girl Scouts, who’ve put a whole lot of thought into how to keep girls safe.
Safety-Wise The decades-old Girl Scout safety bible is a publication known as Safety-wise. I just found out that this publication has been replaced with the new publications:
Volunteer Essentials
Safety Activity Checkpoints
Risk Management
Here are links to some specific online Girl Scout safety pages:
These safety guidelines are focused on keeping girls safe, but one thing I know for sure – if it’s good enough to keep the girl scouts safe, then it will keep all of our math circle students safe!
