Thursday, May 19, 2016

New AIBL Website

A quick post to say the AIBL Website has been updated with a new look! Check out

Monday, April 25, 2016

A Practical Solution to "What We Say/What They Hear"

“In art intentions are not sufficient and, as we say in Spanish, love must be proved by deeds and not by reasons. What one does is what counts and not what one had the intention of doing.”  -- Pablo Picasso

It's my belief that teachers at all levels have good intentions and genuinely want their students to learn.  What this post is about is the notion that instructor intentions are not sufficient, and getting students to do what is needed for authentic learning is what counts.

The starting point is for the ideas I want to get across from a the Launchings Blog of the MAA.  David Bressoud does a wonderful job of describing a recent paper by Kristen Lew, Tim Fukawa-Connelly, Juan Pablo Mejia-Ramos, and Keith Weber.

Below are links to Bressoud's twin posts.  They are worth the time!
"What We Say/What They Hear I"
"What We Say/What They Hear II"

The short version is presented in this diagram

Despite multiple passes through the proof and explanations, students in the study have a difficult time pulling out the instructor's intended messages in the proof and the instructor's spoken comments.  At first glance this makes sense, as the "information transfer" model doesn't work so well when the goal is "developing critical thinking."

As mentioned by Bressoud, Annie and John Selden and others have documented the difficulties students have with analyzing, proving, generalizing, packing/unpacking statements, etc. Learning higher mathematics is challenging, and most math majors struggle with learning proof.

Bressoud suggests options like flipped classrooms and clickers can work, but he notes that such methods have high initial investment in time and requisite knowledge and skill.  While these methods are learnable, many instructors may be in situations, where implementing them is not practicably feasible.  Other options are needed.

Another Option Exists!
Now to the main point of this post. There exists an "easy entry, high upside" method to help students come away with the intended messages.  Put simply, instructors can take their list of intended messages and turn them into math tasks. These tasks can be deployed via small group work, homework, etc.

Let's take a closer look. The intended messages of the instructor in the study are
  1. Cauchy sequences can be thought of as sequences that “bunch up”
  2. One can prove a sequence with an unknown limit converges by showing it is Cauchy
  3. This proof shows how one sets up a proof that a sequence is Cauchy
  4. The triangle inequality is useful in proving series in absolute value formulae are small
  5. The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small
      Each of these five points can be turned into "after the proof" problems.  They can be reformulated as
      1. Explain using sentences and diagrams why Cauchy sequences "bunch up."
      2. True or False and Explain:  One can prove a sequence with an unknown limit converges by showing it is Cauchy.
      3. In the proof, find the part that proves the sequence is Cauchy.
      4. In the proof, find the part where the triangle inequality is used, and then identify a general strategy based on this specific instance that you can put in your mathematical toolbox.
      5. Explain how the geometric series is part of a mathematical toolbox to keep some desired quantities small.
      One way to deploy these in the classroom is to present the proof on the board or pass out a handout with the proof, and then have students work through some of the tasks in pairs and then share (i.e. Think-Pair-Share).  If time is an issue, doing one or two and assign the rest for homework.  Alternatively instructors could ask students to read the proof before class, and the instructor could highlight the proof and spend more time on student-centered tasks in class.

      Several advantages exist with this option compared to flipped classes or using clickers.  The first advantage is that it requires the least experience and least amount of pre-class planning or classroom equipment.  One could be in classrooms like I sometimes teach in with no technology, old boards, and desks built 50 years ago.

      Another advantage is that it does not require deep knowledge of common misconceptions.  Instead, the instructor asks students to explain their thinking (in one way or another) and that's how the instructor gets insights into student thinking.  That is, use an activity and gather formative assessment.

      A third advantage of this method is that it does not deviate from the conventional class prep process used by most instructors. Preparing presentations of a proof with explanations is something that instructors have done many times. The method presented here tweaks the process by transforming the intended messages that instructors would normally say to students into concrete mathematical tasks for students to work on. This change is practically feasible and doesn't require a significant alteration of an instructor's workflow.  It's also a step towards active, student-centered teaching and can be built upon over time into forms of teaching that more deeply engages students, such as IBL.

      Instructors have good intentions and intended messages. I claim that it is how the intended messages are deployed in class that can be addressed in practical and effective ways. We can turn those amazing insights into amazing learning experiences, and "let problems do the talking!"

      Friday, April 15, 2016

      IBL Workshop Model and Real-World Results (Wonkish)

      This post is about professional development, and is intended for those interested in faculty professional development.  There may be some applicability to K-12 PD, but I'm not promising that. Presented here is a model that I argue can be used outside of Mathematics in higher ed. My experience in K-12 PD tells me that changes can be made for K-12, but that there are other aspects (such as working with school districts, parents, testing, etc.) that add more layers that need to be carefully and thoughtfully handled (that may require much more resources).

      This post is also from the perspective of people on the front lines. Our team works with teachers, and we are teachers.  Our work is grounded in the hard efforts of teachers who have thought carefully about the issues.

      The evaluators for the project I've been working on are Dr. Sandra Laursen and Chuck Hayward, at Ethnography and Evaluation Research, CU Boulder.   We have run a set of IBL workshops (funded by NSF SPIGOT) in 2013-2015, and have some results to share.

      The main results:
      • Total number of participants is 138
      • ~ 75% implementation rate
      • 61% of the participants are in the first 5 years of their teaching careers.
      • In just the first year after the workshop, participants taught 180+ classes to 4600+ students (in the real world)!
      • Ongoing support via Email mentoring is a critical feature
      • An inclusive or "Big Tent" definition of IBL is critical for allowing participants to find their own way, suitable for their situation.
      The short version is that uptake is happening at a high rate!  This is an encouraging sign, as it provides evidence that effective PD can make a difference.

      How we got to these results is a very, very large undertaking spread over about a decade of work by a team of people.  I'll briefly highlight the main facets in the rest of this post.  An embedded slideshow appears farther down this post with more details and diagrams.

      First, we identified major obstacles that faculty face when learning to implement IBL methods.   If an instructor has not had much or any firsthand experience with IBL, it is hard to convey through discussions what an IBL class is like and how to pull it off. Additionally, there are skills, practices, curricula, and assessment to consider, all of which are different or changes.   Hence the need for a highly specialized professional development experience that directly addresses the challenges and needs new IBL instructors deal with.

      Obstacles or challenges include lack of experience with IBL,  the need for instructors to learn IBL teaching skills, the Math Ed Knowledge Gap, IBL course materials, assessment in IBL courses, and organizing the structure (i.e. syllabus) of an IBL course. With a list of obstacles in hand, much effort went into designing workshop components to direct address and reduce the identified barriers.

      It it emphasized that the workshop is designed as a single composition, where all the parts of the workshop are interconnected and the main goal is getting participants to a point where they can teach an IBL class successfully.  It is definitely NOT about trotting out our favorite activities.  Indeed, teaching is a system, and teaching is a cultural activity. Hence, the workshop is about providing a broad IBL framework that participants can adapt to their situation. The main focal points are (a) IBL teaching skills and practices, (b) understanding the evidence for IBL and how students learn, (c) linked teaching choices (assessment, problem posing, and content), and (d) building a practically feasible (to the participant) course.

      Another important feature of our work is data driven decision making. The staff used evaluation data and research to inform decisions about all aspects of running and designing workshops, from recruitment, building workshops materials, adapting sessions to better meet participant needs, and also to main components that are successful.  It's through this iterative, self-assessment process that the workshop model has improved over time.  Sandra Laursen and Chuck Hayward head up the evaluation and research efforts for our projects, and their regular insights helps use make small and longer-term positive changes.

      Our current NSF funded project, PRODUCT, has the main goal of expanding the profession's capacity to offer a version of the 4-Day IBL Workshop and developing short workshops that can be "sent" around the country and increase awareness of IBL.  The goal is to build the PD network up to a point where a much larger capacity PD exists and a larger variety of workshops (weeklong and short workshops).

      More details are in the slides below.

      OR click this link to Slides About the IBL Workshop Model

      Edit: Click here to link to the evaluation report.

      Tuesday, March 29, 2016

      Engaging More Students Framework

      One of the common struggles for instructors is engaging a larger portion of students.  A typical scenario instructors find themselves in is that a small group of students do the bulk of the talking.  The Usual Suspects, The Fab Five, The Fantastic Four... they chime all the time.  There's another, larger group that sits quietly.  What we really want is for the quieter students to talk more and the Fab Five to listen more and pick their spots.  Instead of a polarized class of talkers and non-talkers, we would like to push the poles towards the center, where everyone is engaged.  How can we accomplish this?

      Enter the engaing more students framework:
      • Pairs or small group discussions to get people talking
      • Instructor visits
      • Not asking for volunteers
      • Calling on groups to "Share what you discussed."
      Let's say a student or group just presented or shared an idea or solution to the whole class.  Instead of asking the class for questions right away, the instructor can prompt students to discuss what was just shared in pairs or small groups. They should be instructed to come up with at least one comment, question, or (substantive) compliment.

      Groups can be assigned to be the initial commenters.  I sometimes ask two groups to start the class discussion portion of the presentation.  I let them know ahead of time so they are primed and ready (and not cold called), and I rotate these duties around the class to spread opportunities around the class evenly.

      Next, while groups are discussing, I make instructor visits to listen in and check in.   If there are 10 groups in a class, I plan on visiting 2-4 per presentations, and remember where I left off.  In the next round I pick up where I left off.  The purpose of visiting is to check in to see how students are thinking so I get a sense for where people are at.  I can also help groups sharpen their questions or comments, and get groups used to talking about the topic so that it is easier for them to chime in.  I also make it a point to talk to the quieter students in my classes in a friendly way, usually 1-1.  "Hey Pat, what did your group talk about?..."

      After a short period of time, it's time for a whole class discussion.  One of the main things to avoid is asking, "Are there any questions?"  Instead, I have a plan to call on the pre-selected groups or ask specific groups to comment, where I spread the opportunities evenly over time.  An easy way to do this is to have a seating chart or diagram of the groups and go through the list in order and then repeat. Another simple organizational strategy is to call on the groups who you visited first, and then open the discussion to anyone else.

      A key idea here is that you are not asking people to opt-in, by asking for volunteers.  Opt-ins allow students to do nothing to opt-out.  Passive students can stay passive to opt-out, by not responding to "Are there any questions?"  That is, it takes zero effort for students to do the thing you don't want them to do.  Instead by using this framework, all students are prepared to say something. Calling on students in a way that is not stressful and spreads the work evenly ensures equal opportunity for students to participate. Small group discussions also encourages students to engage every round.  (Hence, creating an environment that supports learner agency.)

      One of the positive effects of this framework is that the quieter students engage more and the "Fab Five" are not dominating class discussions.  All students can be engaged, and this offers a greater sense of the class working as a team to learn together.   Class discussions are more useful, and more questions can be asked and addressed.

      Flexible Framework:  The example above can be adjusted in many ways.  I'll just mention a couple ideas, and let readers take it from there. Instead of a student presentation, the instructor could present a problem or ask students to work on an example or review what has just been presented.  The framework can then be used at that point.  It's also worth pointing out that the framework can also be used by instructors who primarily lecture, and and can be a launching point for instructors to start the journey towards active, student-centered instruction.

      Details: some important details include using quality math tasks, avoiding instructor questions like "Are there any questions?", using good, professional technique when doing instructor visits, student buy-in, and setting up a class culture where students feel comfortable discussing and sharing their ideas.

      Contrast: Let's look at the opt-in model, where we ask, "Are there any questions?" In this case, students can passively opt-out by doing nothing, and we get the common split where only a small group of students talks and the rest sit back.
      The final takeaway is that something useful and doable can be done in essentially any setting to get more students or even all students involved in class daily.  All instructors need is to start with the engaging more students framework and find ways that work for them to implement some form of it.  


      Friday, March 11, 2016

      More on Productive Failure (#PF)

      Another term, another round of productive failure (or #PF) implemented in my classes.  Last term I used #PF in Calculus 1 with freshman.  Previously I included it in a course taken by math majors in the teaching option (i.e. preservice secondary math teachers).   This term (Winter Quarter 2016), productive failure was implemented in a course for future elementary teachers.

      This term, students were asked to share two instances where they learned from being stuck or making a mistake.  Sharing would be done via short presentations to the whole class.  Learning is what is emphasized in productive failure presentations, and students were asked to share what they were stuck on and more importantly what they gained from it.

      The purpose of highlighting productive failure is to de-stigmatize mistakes.  Most all students in the class commented in their reflective writing assignments that they were taught that mistakes were bad. Mistakes were to be avoided. The goal was to be "perfect." The problems with this type of mindset is clear.  Students who are afraid to make mistakes are not fully engaged in their learning process. They experiment less, are reluctant to try something new, and focus on getting right answers instead of primarily focusing on why things work the way they do.  This is all the more important for future elementary school teachers.  If young children are taught at an early age that memorizing (and only memorizing) is the key to success in math, then they will learn far less than they are capable of. It's a crippled way of doing mathematics really.

      Productive failure ties in with Dweck's notion of a growth mindset.  Early in the term, I share a short video of Carol Dweck interviewed by the Khan Academy to help set the stage.

      Additionally students are given reading assignments (outside of class) based on Jo Boaler's book, What's Math Got to Do With It?   This book is about major issues in K-12 math education in the U.S., what doesn't work, and some research-based claims of what works.  Boaler's book adds another layer of support for #PF, offering more reasons why productive failure is an important aspect in successful learning.

      As the course progressed, more students shared their experiences of being stuck, and after each presentation, they earned a sticker.  Stickers are not necessary, but they add a fun way for students to keep track of how many #PFs they have presented.  Stickers also served as a model for how they could implement #PF in elementary school classes.


      More progress!

      Productive failure is fundamentally about unlocking a robust learning process.  When math classes, especially in K-12, are overly concerned about answer-getting, then students (children) learn to focus almost exclusively on memorizing steps and worrying about not being wrong.  And that's when seeds of math anxiety are planted! Education is about learning to become a powerful learner.  Far beyond memorized algorithms or basic facts lies a much more valuable and useful education.  Society needs independent, critical thinkers, who are filled with curiosity, creativity, and tenacity.  You can't develop these traits, if the predominant concern is not making a mistake.


      Monday, February 22, 2016

      Vader's Revelation, New IBL Instructors, Formative Asssessment (and Some Humor)

      In the episode V of Star Wars, Darth Vader makes the big revelation.  (Spoiler Alert!)

      "I am your father."

      How does this relate to IBL teaching?  Before I started teaching via IBL, I primarily lectured.  I thought I was getting through to students, but didn't really know until exam time.   When I started teaching via IBL, then harsh reality slammed open the door.  I realized what students didn't know, their misconceptions, how much logic was missing...  It was quite the revelation.

      It's in this stage, where I see a possible turning point.  How we think about student learning affects our teaching decisions.  The pathway I hope instructors take is to look at the situation, evaluate the learning issues, and then design tasks, assignments and classes to address the misconceptions.  This is one of the values of informal assessment, and you get tons of it in an IBL class.  You get info about where students are at, and then can take action to help students learn.

      Another path I see taken is what is called the deficit model.  The way this sounds is, "My students can't do this.  They don't do that.  They can't..."   And it ends there.  It's akin to the fixed mindset in teaching, where teachers view students' abilities as fixed, instead of malleable.  In this path, it's the students' fault for not knowing, and the courses move on unimpeded.

      Now I realize that this is an oversimplification, and students can fail or succeed for a variety of reasons.  I want to get across a pragmatic point.  When you use active-learning, you will get more info on how students think.  This may be a surprise.  An unwelcome surprise.  The misconceptions have always been there, though, and the difference is that now you know what they are and can do something about it.

      Instructors could (should) view these as opportunities, and do as much as they can within the limitations of their situation.  The pragmatic viewpoint (and decidedly non-ideological viewpoint) is to help out as much as time allows.  If an instructor has control over the syllabus and the course content (e.g. upper division courses), then there are more opportunities to deal with identified learning issues.  If an instructor is teaching section 16 out of 30 of calculus, then there may be fewer chances.  But there exists ways to carve out a bit of time to deal with the most pressing issues (such as making handouts, screen casting, and using small group work).

      Feel the formative assessment, and inform your teaching it will!

      Tuesday, February 9, 2016

      Evidence for IBL (via Bressoud)

      The short version of this post is "Check out David Bressoud's writing from the last year."

      I am finally catching up on some reading blogs...  David Bressoud in the October edition of Launchings has a useful and compelling overview of research evidence supporting IBL.  I thought about writing my own version of this kind of post with the updated results from recent articles, but David Bressoud states things well, and send you to his blog.

      Evidence for IBL has been increasing over time. One of the ways I like to phrase this is "all the vectors are pointing in the same direction."   The nature of education research is that there will not be some gigantic airtight proof that method A is better than method B.  It's the aggregate data that points us towards what is most likely to be true. Over time the strength of the signal has increased.  What has been compelling is the aggregate evidence from across the levels of education and across disciplines.  

      On that note, I'd like to continue to encourage people to consider the "big tent" IBL framework.   Engaging students and giving them opportunities to think, share, and do mathematics like mathematicians have value.   There are many forms of IBL, including those that incorporate lectures. It's not a binary (on/off) choice. Hence, there are many entry points to active-learning pedagogies. Think Pair Share is an obvious place to start, and does not require wholesale changes in teaching practices.  Find something. Find your IBL comfort zone!

      Link:  David Bressoud's Blog Post on Evidence for IBL