IBL Session at MathFest 2013

MathFest 2013 will feature a session called "Inquiry-Based Learning Best Practices" on Saturday afternoon (of MathFest)

Description:

In many mathematics classrooms, doing mathematics means following the rules dictated by the teacher and knowing mathematics means remembering and applying these rules.  However, an inquiry-based learning (IBL) approach challenges students to create/discover mathematics.  Boiled down to its essence, IBL is a method of teaching that engages students in sense-making activities.  Students are given tasks requiring them to conjecture, experiment, explore, and solve problems.  Rather than showing facts or a clear, smooth path to a solution, the instructor guides students via well-crafted problems through an adventure in mathematical discovery.  The talks in this session will focus on IBL best practices.  We seek both novel ideas and effective approaches to IBL. Claims made should be supported by data (student responses, test scores, survey results, etc.) or anecdotal evidence.  This session will be of interest to instructors new to IBL, as well as seasoned practitioners looking for new ideas.

Organizers:
Dana Ernst, Northern Arizona University
Angie Hodge, University of Nebraska at Omaha
Stan Yoshinobu, Cal Poly, San Luis Obispo


http://www.maa.org/mathfest/cps.html#cps5

Click the link below to submit an abstract.
http://convention2.allacademic.com/one/maa/maa13/

The Grey Zone Part 1: Unintended Negative Incentives

It's time to look at some issues in the grey zone.  It's good for us to push ourselves a bit.  Why?  Any strong system must have the strength and courage to look at itself as it is.  We can only grow out of our weaknesses, if we study them and learn from them. Just as we tell our students that we learn most from mistakes, we must also tell and believe in this ourselves.  Let's get on with the learning!

I also point out that the goal isn't to make anyone feel bad or guilty, but to open up a deeper discussion about what we're really in this for.  Further I emphasize strongly that no sane person intentionally wants students to fail or for our system to have the failings that it currently does.

Here's the starter topic for this series:  Atlanta schools are caught in a large, shameful cheating scheme to boost test scores.   

http://www.nytimes.com/2013/04/03/education/atlanta-cheating-scandal-reignites-testing-debate.html?smid=pl-share

Atlanta is but one example, by the way.  Other districts have been caught tampering with test scores.  Why did Atlanta schools cheat?  I can't know exactly all the reasons why, but certainly it does not help when people's jobs were essentially tied to test scores.  When $$$$ and test scores are tied together tightly, then there exists incentives that encourage people to make unintended choices.  We see this on wall street and the banking/financial crisis.  We see this is sports, where the incentives to be juiced are apparently worth the risk to some athletes for the financial and social gain.  School life is a subset of our larger culture and not completely immune to some of our failings.

One notion that we don't talk about in the U.S. enough is balancing accountability (testing) with responsibility.  What I mean by this is that we are overly concerned with accountability, whether it is test scores, covering all the material, or getting good teaching evaluations.  If incentives are too "high stakes" or put another way if there's too much emphasis on accountability, then we are susceptible to opening a Pandora's Box of unintended consequences.

The Atlanta cheating scandal is a big, headline worthy example of unintended consequences of high-stakes testing, but there are other more insidious and frankly unwelcome versions of this at the college level.  One example is the coverage issue in freshman Calculus.  I'm not saying that cheating and coverage are the same issue, but they point to an underlying issue.   Instructors probably gripe about one thing the most:  coverage.  Calculus courses are jam packed with content.  Implementing active, empirically-validated teaching methods is hard work, and one of the main reasons why instructors do not implement modern teaching methods is time.  "I would like to do that, but there's no time..." The analogous issue in K-12 is "I would like to do that, but it's not on the test..."  In K-12 external accountability pressures that are too great push teachers away from modern pedagogies.

It's clear I believe in active, student-centered instruction.  Let's put that aside and think about the larger issue logically.  Whatever your take is on teaching, coverage should not be a core reason why we choose to use a teaching method or not.  We should use the methods that produce the best learning outcomes, based on the evidence available.  Moreover  focusing on covering a list of topics is only the start of a discussion about real education.  It is noted that missing from a list of topics are problem-solving ability, critical reasoning, communication, curiosity, attitudes about mathematics, and so on.

The part that is especially uncomfortable for us in the teaching profession boils down to this.  When we say, "I would like to do that, but I don't have time..." one could argue that this is like saying "I know this would help my students, but it's my job.  It's how the system is set up..."  I don't like the sound of that.  Being pinned down by coverage is parallel to the excessive accountability in K-12 schools in the sense there exists unintended incentives for not doing the right thing.

Let's turn this around to the positive direction...  we can start talking about coverage as a real issue and deal with it through having productive discussions and seeing what we can learn from one another, especially by learning from successful programs and instructors.  Ideas are out there for improving learning outcomes without sacrificing our standards that simultaneously improve areas such as problem solving and attitudes about mathematics.

The cheating scandal and being trapped by overly long syllabi are two examples of a by product of unintended negative incentives.  Cheating comes from excessive accountability, and slow uptake of empirically validated teaching methods is affected by excessive content demands.

David Bressoud: "Good Teaching"

Check out David Bressoud's results about instructor actions highly correlated with positive attitudes with students and highly correlated with each other.  All of the results are in direct alignment with what we have been advocating in the IBL community.  Ask, listen, support, mentor.

Get the details here:

http://launchings.blogspot.com/2013/03/maa-calculus-study-good-teaching.html

Reminder: IBL Workshop June 24-27, Cal Poly

This post is for math instructors who teach at the college level... About half of the spots have already been taken.  If you're interested in learning more about IBL and how to implement IBL, please consider signing up for the MAA PREP IBL Workshop, June 24-27 at Cal Poly.  More information is available at www.iblworkshop.org and www.maa.org/prep.

Dealing with Student Attitudes Through the "Teaching System Lens"

One issue that doesn't get enough air-time is students' attitudes and beliefs about Mathematics.  It is documented in the Math Education literature that students often have beliefs about Mathematics and how to learn Mathematics that is either not helpful or hurtful for their own development.

I also add the caveat that no one intentionally wants the following outcomes.  They are unintended consequences, and they are consequences we should know about.  Here's a partial list of negative attitudes or beliefs that has been documented.  I've mentioned these before.

1. Memorizing facts and formulas and practicing procedures are sufficient to learn mathematics.
2. Mathematics textbook problems can only be solved using the methods described in the textbook.
3. Teachers and textbooks are the mathematical authorities.
4. School mathematics is driven by rules and memorization, and is driven by procedures rather than concepts.
5. If a problem takes longer than 5-10 minutes, then there is something wrong with the student or the problem.
6. The goal of mathematics is to obtain one correct answer and do it quickly.
7. The teacher is the only source of determining whether an answer is correct or incorrect.
8. Students’ role in the classroom is to receive knowledge by paying attention in class and to demonstrate it has been received by producing right answers.
9. The teacher’s role is to transmit knowledge and verify that it has been transmitted.
10. Only geniuses have what it takes to be good at mathematics.
11. Students prefer to have only one way of solving a problem, because it is less to memorize.
12. The processes of formal mathematics have little or nothing to do with discovery or invention.
13. Students who understand mathematics can solve assigned problems in 5 minutes or less.
14. One succeeds in school mathematics by performing the tasks, to the letter, as described by the teacher.
15. The various components of mathematics are unrelated.

So when an instructor teaches via IBL and works with students who have never had an IBL experience, there usually exists a host of default expectations that one has to work through with the students.   

Let's say the students in your class are reluctant to buy into the IBL system.  Then looking at teaching as a system can provide a broader perspective to address the issues, which could lead to a more coordinated effort to get students to become active participants in their own education.

Attack this teaching challenge through content, assessment, and pedagogy (i.e. through a system approach).

Content

Course content is one of the key components.  If tasks are too hard and only the very best students are successful at solving them, then students who are more likely to have some of the negative beliefs and attitudes above will have "evidence" to reinforce them.  "I'm not going to be successful anyways, so I might as well give up after 5 minutes."

Consequently, the tasks presented to students should be matched to their levels, and having a handful of very accessible problems is almost never a mistake.  At worst, the students mow them down, and you've been able to get more students to participate.

If students are having trouble starting up or you have a subgroup that is passive and not engaged, then one tactic available is to breakdown a problem set (or a part of one) into more manageable pieces.  This method can help students get going on problems.  One thing to keep in mind is to keep problems that are more challenging to keep everyone engaged at an appropriate level.

Assessment

If assessment is setup only along traditional lines (homework, midterm, final), and assuming each of these components is implemented in the usual way, then the ethos of the IBL course is mismatched to the course assessment.  If homework is graded based on accuracy, and there are no venues for safe exploration, then students are being told implicitly that "mistakes count against you."  This can inhibit risk taking and undermine the course.  While our viewpoint from the instructor side is that we expect the exploration to take place on scratch paper, this may not be a concept and workflow practiced by students.  It's a curious thing -- students expect to write down the answer in one shot.  This is absurd in art, science, math, etc. when thinking about it.  So how did we get here?

The pristine nature of clear lectures can be beguiling.  The imagery is one of a the brilliant expert doing it right the first time. Every time.   Instructors are never seen struggling, and this is an impossible standard to achieve as a learner, especially for the ones who need the most help. The unintended consequence is that in our effort to make the best possible presentations, the notions that working hard, being willing to explore, learning from mistakes, and having the predilection to work through our own personal learning obstacles can be undermined.

Homework is an opportunity to portray doing math the way mathematicians do it. One option is to provide feedback but not grade the homework in the usual way with numerical scores.  The explicit message that should be given to students is that their effort and the quality of their exploration is being checked, and feedback will be given to them to assist them with the learning process.  I'm not saying that this is how it should always be done, but this is an idea worth considering and adapting for your own courses.   If your goal is to bring about change in student perceptions in math, then grading homework for process and effort is one of the available tools.

Grading presentations is valuable.  In courses where presentations are used regularly, presentation grades should be reflected in final course grades.  Students are being asked to share their ideas, a noble thing indeed, and the quality of this work should be reflected in their course grades.  A level around 25% works well in lower-level courses. As courses become more proof oriented, then raising the presentation grade as a percentage of the overall grade makes sense.  (Grading presentations and some sample rubrics will be discussed in future posts.)

Mastery-based finals are used by some IBL instructors with success.  Such exams are split into two parts.  Part 1 consists of material that anyone who passes the course should know.  Part 1 is contains the fundamentals and essential basics.  Students should be able to complete the entirety of part 1.   Part 2 is an opportunity for students to raise their grades.  Part 2 contains problems that require applying knowledge to problems novel to the students.  (Parts 1 and 2 are given to students at the usual examination time.)  A final exam structured like this allows students opportunities to demonstrate that they have learned the material and can demonstrate that they can use what they learned to solve problems new (to them).  Before the final exam, students are given a (non-final) course grade going into the final exam.  Passing part 1 means students keep their grade OR get at least a C- in the course.  Success on part 2 only improves the final grade and does not count against students (i.e. non-negative help).

Why do mastery-based final exams make sense?  If we value learning, applying ideas, and self-improvement, then mastery-based finals are a form of assessment of mathematics aligned to these values.  Further it provides an incentive for students to work hard through the end of the course, because they have a chance to succeed right up to the very end.   It is also noted that this does not mean one lowers standards.  The idea here is to keep the standards at the usual level, but provides a structure and incentives for students to achieve these standards.

The three main examples (effort/process based homework, assessing presentations, and mastery-based finals) are ways we can align assessment of the usual assessment items to IBL classes.  Additionally instructors can use portfolios, reading assignments, and reflective writing (journaling) as additional forms of assessment.  These assignments do not need to be large part of the final course grade, and are valuable tools for gathering data about how your students think.  Gathering formative assessment is significantly valuable, and all instructors are encouraged to use one or more of these additional strategies to help put together the full picture.

A specific example I want to share is journal assignments.  Specifically this example is about using "The 5 Elements of Effective Thinking," by Burger and Starbird as a vehicle for students to think about their own thinking (metacognition).  When working with a group of students who do not like mathematics (e.g. Math for Liberal Arts), then having students read and write about healthy ways of doing math is a successful strategy for combating math anxiety and buy-in.  Math autobiographies are normally the first assignment, and the rest of the assignments are based on chapters from the book.  A typical assignment asks students to (a) read the assigned chapters, (b) write about 2-3 things they learned, and (c) write a personal reflection about how they are doing math.

For students in education courses, such as "Math for Elementary Teaching," I use a variety of articles from Teaching Children Mathematics, and books like "What's the Point of School?" by Guy Claxton, and "What's Math Got to Do With It?" by Jo Boaler.   Students in math major courses could read autobiographies of famous mathematicians, books like "Fermat's Enigma" by Simon Singh, or other related books.  A vast array of possibilities exists with journal assignments, and you are encouraged to find (and share!) strategies that work for you.

One last facet of assessment is that it needs to be set before the term starts.  Assessment is the least flexible component, once the term gets rolling.  Thus, it is worth thinking about assessment early in the planning phase of a course well before you meet classes for the first time.

Coaching

As mentioned in an earlier post (Link), coaching is an critical facet of teaching.  When students get stuck, then your role as an IBL instructor is to manage the classroom so that students continue to learn, as opposed to giving up.  If the default attitude or belief of your students is to shutdown when stuck, then coaching students through this phase is part of the job.

• Coaching can be verbal encouragement.  "It's okay to get stuck.  Let's see if we can figure this one out together.  Don't give up... Being stuck is a natural, regular part of doing mathematics."
• Coaching can be directive. "Let's try figuring this related example/strategy/definition..." or "Let's pair up and try to see if you can think of a useful idea or draft a plan you can take home and try."
• Coaching can be done through content. "Let's look at this supplemental handout I wrote yesterday after class, when I saw that we were all stuck on number 19.  I think these problems will be really helpful..."

A couple things to keep in mind.  Frustration levels have to be kept below "redline," and students may need to learn specific strategies and habits of mind.  IBL instructors should work to stay attuned with their student and setup a friendly, safe environment, where students are encouraged to ask questions when they are stuck.  If students are quite, then asking them what they are working on is one way to check in without asking for an answer or question.

In summary, a "system approach" to dealing with a complex issue (students' attitudes about Math) provides a richer and likely a more successful approach.  Teaching and learning is complex, and there are many things that affect the day-to-day activities of our course.  At times it can be a daunting challenge, but as my colleague Professor Dylan Retsek says, "Chop wood, carry water."  In this case, we break down our response to content, assessment, and coaching.

Upward and onward!

Research on IBL in College Mathematics at UC Boulder E&ER

A quick post today:  Laursen, Hassi, Kogan, Hunter have produced recent work on IBL in Mathematics.  Their work thus far (and more is coming) is available HERE

Teaching is a System

When I first started using IBL, I didn't realize exactly what I was stepping into.  While I knew I wanted to get students involved with their learning, I thought I would learn about tasks or problem sequences that I could deploy and then students would *do* math under my guidance.  This is a correct model to some degree, but mainly I was thinking about the content.   I didn't completely realize that teaching is a system.

By system I mean that teaching is a set of interdependent components that form an integrated whole.  Instructor actions, course content, what students bring (intellectually) with them, and how students are assessed are some of the interdependent components.  To simplify things we can focus on three main components of the teaching system:

• Teaching methodology (broadly defined)
• Course content and more specifically tasks students do in and out of class
• Assessment

Each of these components are complex, and have several subcomponents.  I'll just touch on the surface of each of these in this post, as it is beyond the scope of this post to discuss all the important details.  What I want to do here is to outline what I mean when I say that teaching is a system.

Teaching Methodology

In moving from teacher-centered to active, student-centered instruction, the instructor in an IBL (or hybrid IBL) course is to shift the focus from information transfer towards sense-making tasks.  The instructor could employ student presentations, group-work, and think-pair-share as strategies in-class to engage students in the learning process.  Teaching changes that involves students more deeply in the learning process implies that usual content has to be reformulated.  Students need different tasks to work on compared to when they receive information from a lecture.

Mathematical Tasks for Students (Content)

In the google era the objective of education is no longer focused on knowledge acquisition.  Knowing how to creatively use knowledge and effective ways of thinking rise in value.  In light of this, merely sharing insights or showing how things work is not sufficient.  Students then need different, more sophisticated tasks, which require them to think, problem solve, and communicate their ideas.  These tasks then must be designed for the specific stages of development and the variance among students in each class.  Selecting appropriate tasks relies on a knowledge of how students think, and tasks must adapt in sophistication as students develop over time.  

Assessment

Measuring success in this new paradigm then needs to change.  Merely asking students to execute algorithms or bubble in answers does not capture more sophisticated learning objectives, and does not provide proper incentives to become a better thinker.  Thus, as one switches more and more towards full IBL methods, assessments must also change in parallel to measure appropriate learning outcomes.  The simple way to put this is "Put your money where your mouth is."  If we value problem solving, communication, creativity and all those wonderful qualities of an enlightened mind, then we need to align assessment accordingly.  Specifically this means we need to add in presentations grades, adapt tests towards mastery (as opposed to showing knowledge acquisition), and course grades ought to move away from a weighted average of tests, homework, and final exam.  Some of this is controversial, and I understand both sides of the argument, especially in the case of using a weighted average.  My point here is really that instructors should ask themselves if whether a weighted average truly captures the learning they are after.  There exists, in fact, an implicit acknowledgment that the weighted average is not always useful among many (or most) faculty.  For example, it is common practice for instructors to weight the final exam more heavily if a student does better on the final.   

Assessment doesn't always have to affect final grades -- informal assessment helps guide instruction.  Interspersing activities in class to check for understanding can provide instructors with the information they need to address the specific misconceptions and learning challenges the students are dealing with.  Not only are active, student-centered activities be better for learning, they provide a wealth of information for instructors to use to guide instruction.

The Textbook is Not Enough  

Viewing teaching as a system can also help put things into perspective.  For instance, textbooks are generally overvalued as a solution method for dealing with the issues of teaching and learning mathematics.  In other words we may fool ourselves into thinking that we can solve our problems by just having "the right book."  Indeed much of the reform movement has been focused on content, and for good reason.  If we want students to know more than algorithms, then they need updated textbooks (or course materials).  That's true.  But let's say we get good materials into classes.  All we've done is change one aspect of the teaching system, and consequently we need to consider if this materially affects the way in which students interact and engage in the mathematics.   While choosing good course materials is important and necessary, it is by itself not sufficient.  And this is clear from the teaching is a system viewpoint.

We can think about this issue another way.  If it were the case that all we had to do is pick the right textbooks and then all our struggles with teaching and learning would be solved, then it would have been done by now.  Throughout history there have been enough brilliant, eloquent writers and thinkers, who have provided us with a wealth of lucid, beautifully written work.  Just because Shakespeare wrote Hamlet (and that we read it in high school or college), this doesn't mean we have all learned, internalized, and implemented the valuable lessons presented in it.

More Advanced Stuff

There are other issues that come into play that haven't been touched on in this post.  One has to consider the specific course (Calculus, Upper-Level Mathematics, Math for Elementary Teaching, Math for Liberal Arts), student attitudes and beliefs, getting students to buy-in, and so on.  The teaching is a system framework provides us a broader approach to addressing some of these challenges.  Teaching methodology, mathematical tasks, and assessment can be adjusted and developed in coordination to shift the focus on student engagement and collaboration as well as addressing issues related to student attitudes and buy-in.  More posts are coming soon that will present specific examples of how this can be done.