A Quick Reminder: Small Grants Applications Due Next Week

This blog post is for college math instructors.   Small grants proposals are due next week.  Follow the link to find out more about the program.  There is still time to apply!

http://www.inquirybasedlearning.org/?page=Small_Grants

"I Have a Dream": MLK Day, Math, Art, Inquiry

Here we are on the 50th anniversary of the "I have a dream speech."  I thought I'd share something related from the IBL Math world.

In 2012 I was fortunate enough to listen to a talk by Bob Bosch, Oberlin College, at the MAA Pacific Northwest Section Meeting, held at the University of Portland.  The conference was superb, and I learned much from my experience there.  Here's a LINK to Bosch's art.

One of the ideas I came home with was to develop a unit based on Bob Bosch's Domino Art work.  Fast forward to January 2013 around MLK day, I was working with Pacheco Elementary, San Luis Coastal USD, and the 6th grade teachers let me throw a math + art unit into their curriculum for about 4 days.  Thank you Mr. Deutsch and Mrs. Irion, and to Jamie Coxon at the Linker Workshop for preparing the materials and framing it for final display!!

The setup is that the 12 complete sets of double-nine dominos are used to make a mosaic of MLK.   (Bob Bosch has other images besides MLK.)  The mosaic is created from a B&W image, where the image is turned into a gray scale image of squares with 10 grey-scale levels from black to white.  Double nine dominoes have approximately these same 10 steps from black to white.   Thus an image can be made into an array of numbers, where each "cell" has a number from 0 to 9.  That's the mathematization of the image into a mosaic.

The big question is "how do you arrange the dominos to approximate the perfect mosaic?"  This is a challenge, because one of the rules of the game is to use all of the pieces in the 12 sets of dominos.  No replacements are allowed.  Generally, the perfect mosaic is not attainable, so we are left to find the best approximation of it.

This means we have to accept some error.  One way to try and find the best possible approximate mosaic, is to find a way to measure error and then to minimize that error.  That's where modern math comes in, because the number of possible combinations is astronomical.  Bob Bosch sorted that, and I'll send you to his site (and papers) to get the nitty gritty details.

The MLK Domino Mosaic the 6th Graders Built

Sixth Grade isn't the best context for a short course in Linear Analysis ;)   So my challenge was to find smaller problems that are doable by 6th graders.  The math unit is in a beta stage, and needs development.  (More work to do!)  What I have so far is enough to get across the idea that minimizing global error requires increasing local areas in some parts.  Sixth grade students worked on measuring error, and on a task that required groups to cooperate to minimize global error, while accepting greater error in their section of the image.  Even in math, we are better when we cooperate!

Finally there was the phase of cutting, gluing, and putting together the mosaic.  While this activity isn't math or art per se, it's a nice activity that culminates in a tangible finished product.  Sometimes it's important to do something to mark your achievement and appreciate something aesthetically pleasing. (And it only took one or two periods to assemble.)  The tiles were connected and framed up by a local craftsman.  I'll also note that there are extensions from this unit to other subjects (and more math).  This unit could extend to MLK's biography, History, writing, Art (color and gray-scales), etc.  I can see quite a long list of possibilities.

One of the best learning outcomes was something I heard from one of the Moms.  She said that her son was so motivated by the project that he was really excited to go to school and do math that week!  Moreover her son wanted her to take him to the public library so that he could learn more about Martin Luther King, Jr. 

Working Together in Groups

To me, inquiry-based learning or more generally teaching isn't fundamentally about teaching techniques or skills.  These are important and necessary of course, and I'm not trying to minimize them.  I focus and work on them daily!  But teaching techniques, skills, and practices serve a larger vision for education, where students are deeply engaged as explorers and doers.

Real, meaningful education lies at the foundation of modern civilizations, and Martin Luther King, Jr. realized this.  Here's one of his more famous quotes.

"The function of education is to teach one to think intensively and to think critically. Intelligence plus character - that is the goal of true education."

The Sense-Making Continental Divide

Frequently I am involved in good discussions about what is IBL, whether it is the strict Moore Method or something else.   This is a good topic for discussion, and I'd like to share my thoughts on the issue.   I'm going to approach this topic with a simple, but useful model that highlights a major structural component of IBL instruction.

Math courses are taught in a variety of ways, and even within the IBL community there exists numerous differences.  This is something that should be expected, because environments and goals differ across institutions.  Just as we would not expect Michelin star restaurants to be identical across the world, we should expect that successful instruction will be different and varied to suit the needs across different institutions.  This honors the diversity of humanity, and allows a teacher to be true to her or his personality

Here's the model I have in mind:

In this model I use an idea I call the "Sense-Making Continental Divide."  A key feature that defines IBL instruction is that students regularly are encouraged to do the sense-making tasks, including validation of solutions or proofs, understanding statements of problems, and working from definitions and first principles.  IBL instructors set up courses to get students over the Sense-Making Continental Divide, where students are regularly doing activities that require students to think, decide, explain, evaluate, and reflect.  You have crossed over the SMCD if your students are (a) deeply engaged in rich mathematics and (b) have opportunities to collaborate and discuss ideas and solutions.  (This succinct characterization of IBL is from Sandra Laursen, University of Colorado Boulder.)

It's easy to see there are numerous options for implementing sense-making activities.  From the palette of teaching options is derived the multiple variations of IBL.  In my perspective, this explains why IBL comes in so many different forms.  We have more choices, and the "correct" teaching decision depends on real-time conditions in the specific class setting an instructor is in.

What typifies traditional instruction is that the instructor does the processing and sense-making through presentations.  "This is the proof of theorem 3.6..."  Students do not get many opportunities in class to do the structuring or validation.  In such classes, students might be unintentionally encouraged to memorize facts rather than make sense of the ideas.

The Hybrid IBL zone contains the different forms of IBL methods that are often a result of practical limitations instructors face.  There may be a required syllabus, or a course may be predominantly procedural in nature (e.g. calculus). A course may have large enrollment, or an instructor may not have the requisite skills or experience to comfortably run a full IBL course.

It could also be the case that the (full) IBL class instructor shares a solution on occasion in the event that students are floundering, and moving ahead would be more beneficial mathematically for the students.  Flexibility and adaptability are key traits of effective instruction.  An IBL course may change during the term to adapt to specific needs.

Is your course an IBL course of some kind?  One way to see is if your students are regularly over the Sense-Making continental Divide.

For more about your personal IBLishness, see also this post on IBL Levels.

Link: "Harnessing Your Personality"

I'm pushing out a blog post by my good friend and colleague, Matt Jones, CSU Dominguez Hills, who has a blog (Math Switch).

Harnessing Your Personality

"Ignorance Isn't All That It's Cracked Up to Be!"

In this post, I have invited my good friend and colleague, Ed Parker, to be a guest blogger.  Ed sometimes mentions, “Ignorance isn’t all that it’s cracked up to be,” when we have work talked.  Typically this quote comes up when we are working in workshops, and the discussion is about prior knowledge and content coverage.  Ed attributes John Neuberger for the quote, which he discusses in detail below.  (John is Ed's thesis advisor.)

Ignorance in teaching comes up in several ways, and I'll comment on one way specifically here, and then turn it over to Ed.  One way “ignorance” comes up is in the difference between (a) teaching a topic that students have never seen before vs. (b) teaching a topic students have seen in a prior course.  Let’s take Topology vs. Euclidean Geometry, and focus on Euclidean Geometry first.  College students have been exposed to Euclidean Geometry in K-12, and often their incomplete knowledge of Euclidean Geometry gets in the way of new learning, because they have memorized some things correctly and others incorrectly in a way that is disconnected from the axioms and first principles.  When we ask students to prove a fact in Euclidean Geometry, they often don’t know where to start, because it is hard to distinguish between what is allowed as a fact or axiom.  Many of the statements are “obvious,” and thus the teacher has to navigate around teaching and learning obstacles through all this, which is no easy task!

On the other hand, Topology is a topic students have not usually seen before.  Hence starting with some assumptions and definitions provides a clean slate.  Students are on a level playing field, and the focus is on taking the definitions, understanding them deeply, and then moving to proving theorems.  Students can do better sometimes when they are completely ignorant, because they can then do math in a way that is exactly the way mathematicians do math.  They have to work with definitions and grapple with mathematical definitions, logic, and all that.

The fact that Euclidean Geometry can be more difficult to teach than Topology lies to a significant degree in the fact that something learned poorly has serious unintended consequences.  Students come in with prior incomplete knowledge, expectations, and perhaps unhelpful habits of mind.  The same can be said of subjects like Calculus, where college students are likely to have seen many of the ideas at least once before.  Thus, in such courses IBL instructors have a more complex set of challenges to know about, learn about, and address.  All of this underscores the weight and significance of what students walk into a course with.  We'll dedicate future blog posts to share some strategies for dealing with these kinds of issues.

Now back to "Ignorance Isn't All That It's Cracked Up to Be!"... Take it away, Ed!

-- By G. Edgar Parker, Professor Emeritus, James Madison University

Ed Parker, James Madison University

In my last year of graduate school, I went to the national meetings to interview and spent my "spare" time going to talks. I heard one by Herbert Amann that had an idea in it that intrigued me and I followed up on it, and, after I had done enough background work to put the ideas firmly enough in mind to be able to articulate my conjectures, I went to talk to John Neuberger, my thesis advisor, about it. Since my take on the problem (it was in dynamical systems) was topological in nature, John suggested that I go talk to Bill Mahavier about it. When I got my audience with Dr. Mahavier, he listened carefully, thought about it a few minutes, got up from his desk, and walked over to a stack of off-prints stacked on the floor to about shoulder height (there was a time before the Internet when correspondence consisted of exchanges of letters on paper and off-prints of publications!). As he worked his way about a third of the way down from the top, his first remark was, "This is why old mathematicians know so much more than young mathematicians.". He pulled a paper from the stack, went back to his desk, peered at the paper for a few minutes, and then proceeded to remind me of some work I had done, gave me some definitions that he said should be accessible, and suggested a couple of problems that he indicated might be of some use if I could solve them. Mind you, he didn't show me the paper nor did he even give me the reference; I still don't know what it was!

But he then offered me some professional advice. He told me, "Ed (I had already defended, so I was no longer Mr. Parker), to my mind, when to go to the literature is the hardest decision a researcher has to make. If you take on a problem and the first thing you do is go find out what everyone else knows about it, your thinking is almost certain to be channeled by their approaches to the problem. The only way you'll be able to improve the results that way is to be smarter than they are since they probably know the literature well. (Not so subtle between-the-lines message: and you're not smarter than they are.) On the other hand, if you can understand the problem and go to work on it, normal minds running free can get unique perspectives that sometimes allow us (sic) to see things that others don't. But suppose that you really want to solve the problem, but after three months you aren't getting anywhere. Is it time to go to the literature? And, if you choose to, do you stay there until you are well-informed, or just until you have a new idea to work with?" And he left it at that.

I don't know whether this advice was intended to have global applicability or whether it was intended just for me. My history as a student was that I was a painfully slow learner of other people's mathematics and couldn't tell the "hard" problems from the "easy" problems so I had about the same success with both and usually my proofs, I found when I later became familiar with the literature, were consistently not made along the lines of thinking that produced the proofs that made the books. Trevor Evans, from whom I took my algebra courses at Emory, used to say after I would present, "Yes, Mr. Parker, I suppose you are correct, but (as he would take the chalk away from me) WHY DIDN'T YOU THINK OF THIS?", and he would proceed to show the class a "reasonable" proof. Most of my work in the literature my first decade out of graduate school was spent teaching myself enough to be able to teach my courses or going to the literature to make sure that what I wanted to write up hadn't been done.

Fast forward about 25 years. I was talking to Dr. Neuberger about a problem on power series connected to wherever Jim Sochacki and I were at the time in our exploration of polynomial projection and in passing, made the comment, "You know me, I didn't know hardly any of this stuff..." and kept on describing the idea. When I came up for air, John said, "You know, ignorance is highly overrated." I don't think he was being critical; the message I got was, you could be using the same brain cells that you are using to figure out stuff that other people don't know that you now have to expend to figure out stuff that many undergraduates know.

I'm guessing that the tension between Mahavier's advice and Neuberger's admonition has something to do, by analogy, with "the coverage issue". Hopefully, we put together our course notes well enough so that our students not only experience the growth of solving their own problems, but also come away with a corpus of "facts" that at least prepares them to be a few hours reading away from being literate about any one thing that they might "need to know". I continue to believe that if one is forced to choose between students "doing" and "being shown so as to achieve 'literacy'", one chooses "doing". I believe that the power associated with "learning to learn" is far greater than any level of mastery of someone else's bag of techniques. On the other hand, I'd like to be a whole lot better at communicating that, "now that you have done this, don't stop", is what we're after rather than "you are now certified to repeat some pre-packaged dose of curriculum neatly packaged and ingested".

Ed Presenting at the Legacy of RLM Conference, June 2013

Nevertheless, we should, by intention, be purposeful in trying to make the “curriculum” demarcated by our course notes as informative of the “curriculum” as dictated by the contents of a standard text in the subject as is feasible.  In my own case, I have spent considerable time trying to optimize the fit between the mathematics my students will try to make in my courses with the curricular expectation I perceive that my teaching colleagues have.  I would like what may appear as ignorance to be a foundation for “I’ve seen that idea before.”.  If our students learn to learn, our colleagues will be happy with them, regardless of whether the students can knee-jerk responses.

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Special thanks to Amy Ksir, US Naval Academy, for suggesting I create an RSS feed!

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The 2013 IBL Workshop at Cal Poly

Last week a team of IBL practitioners, and 43 participants met at Cal Poly San Luis Obispo for 4 days at the MAA PREP IBL Workshop.  More information about the workshop is available HERE.  The next workshop will be held at Kenyon College (OH) in June 2014.  Stay tuned for more information!

The 2013 workshop was a blast!  We worked together discussing articles, talking about the nuts and bolts of running an IBL classes, watched and discussed IBL classroom videos, and developed IBL course materials.    Thanks to all who attended and helped out at the workshop!

It is always wonderful to see a group of dedicated math professors, who take a big chunk of their time (and in some instances their own money) to attend a workshop.  They do so because they want to improve their teaching and help their students be more successful in math classes.  Unsung heroes, indeed!