Wednesday, July 1, 2015

Knowing What to Do is not Doing

A guest post by Bob Klein, Ohio University

In Summer 2014, I attended the Academy of Inquiry-Based Learning Workshop in Portland that took place prior to MAA MathFest. While I had always come out relatively well in teaching evaluations, I also have been trying to improve my teaching. For the IBL workshop I redesigned my two fall courses (a capstone math content course for secondary education math majors, and a geometry content course for middle school teachers). Later that semester I redesigned my history of mathematics class to follow an inquiry-based learning (IBL) format. Overall, students were more engaged and I was doing far less “delivery” and much more “data collection” and “decision-making.” When students were working at the board or discussing, my role shifted to listening/observing/evaluating so that I could guide discussion. For a while, I’ve adhered to the mantras in my teaching: “Be less helpful” (DM) and “Never Say Anything a Kid Can Say” (SR). But aside from my work in math circles, it hasn’t been until this academic year that I’ve been able to realize those in the classroom.

A side effect of this is that I both thought MORE of my students’ abilities and LESS of their mastery at the same time. MORE because they showed me that they could embrace a learning environment that put more on them—this was a matter of willingness to adapt/change what took place in the classroom. But also LESS because I had a much more detailed picture of what they knew and what they were able to do at any given point. The content capstone course for secondary math teachers-to-be contained mostly seniors who have completed college geometry, three semesters of calculus, number theory, discrete mathematics, algebra, statistics, and an elective. It may just be this lot, but I’m worried about their abilities as they move forward next year to teaching. Worried especially that we may have promoted them through “the system” of courses without an accurate read on their mastery of the content of those courses. That said, I think I caused those students to be worried about their mastery of content because I asked them to PERFORM that content in rather public ways and subject to peer criticism. This is cause for hope and also a good basis for me to make further refinements to the running of the curriculum.

The other course I redesigned for fall was a geometry content course for future middle school teachers. The sophomore level course is significantly different from the capstone course and I think students here were challenged and pushed though in different ways from the capstone course. I was able to push the geometry students to see mathematics itself (as a practice) differently. They began the course trying to convince me that the “(I do)→(we do)→(you do)” formula of exercises (not problems) was what math was all about. By the end of the semester they were asking each other, “How would you defend that?” and digging deeper. I think I probably needed two semesters with them to achieve my stated goals for the course so I might need to be more realistic or more specific about the syllabus goals.

History of mathematics was the hardest to redesign for me. The course enrolls mostly middle-level education majors though secondary math education majors and a few STEM majors also enroll. As such, the group has a wide range of backgrounds and it cannot be a course that focuses in large part on proofs (without significant support) though we do cover some demonstrative mathematics. To adopt an IBL approach, I carefully chose a focus area (in this case Egyptian Mathematics) and we spent a long time engaging in problems and discussions centered on Egyptian techniques. We then spent some time working on Mesopotamian, Greek, and European mathematics (as well as other topics). Having a “big focus” for the course (almost an area of expertise for the students) made IBL techniques of problems and board work something that challenged the disparate backgrounds of the students equally and stimulated discussion about the deep ideas behind mathematics, including the assumptions, leanings, and contextual influences thereof.

All in all, though, this was very eye-opening for me. As an educational researcher (rather than a research mathematician), I KNEW what I was supposed to be doing, but the IBL workshop and community gave me the support and courage necessary to try it and to stick with it, especially in those first few weeks when I was still in the mode of ‘selling the approach’ to the students. The mass of existence proofs for the model, backed up with the leadership team’s smiles and insistence that this was a flexible set of practices rather than orthodoxy, were especially helpful.

One upshot of this work has been my nomination and eventual awarding of the Presidential Teacher Award at Ohio University. I was selected as one of only two faculty on my campus to receive the award after a thorough process involving a teaching portfolio, multiple observations of my teaching throughout the year, and interviews by the selection committee of me and my chair. It was truly humbling and nourishes my belief that the IBL approach is making a difference that is visible to students and beyond. I’m eager to engage in further discussions about approaches that worked for others, and welcome others to contact me at the address below.

The reflections above are from Bob Klein, PhD, Associate Professor and Undergraduate Chair, Department of Mathematics, Ohio University, Athens, Ohio USA. kleinr@ohio.edu

References

[1] Meyer, Dan. http://blog.mrmeyer.com/2009/asilomar-4-be-less-helpful/.

[2] Reinhart, Steven. (2000). Never Say Anything a Kid Can Say. Mathematics Teaching in the Middle School. Vol 5, No 8, pp. 54-57.

Thursday, April 9, 2015

Calculating Community: The Running Equation

How does one do something they perceive to be so difficult that they never thought they would even consider doing it? How does one go from being a non-athlete to running 100 miles at elevation? And, how does this have anything to do with mathematics? I challenge you to read this blog post and replace “running” with “mathematics” for most people.

In my free time, I run. I run a lot. In fact, my new favorite race to run is the 50-miler, but I have completed three 100-mile events. One of these 100-mile events I completed in under 24 hours.

Was I born a runner? No, in fact I was a scrawny girl with chicken legs who would do anything to get out of gym class. Running used to make me cry. It was hard and it hurt. But, I was curious. Why did so many people run and say they loved running? In January 2009, I set out on a mission to find out what this running hype was all about. I trained for my first half marathon (13.1 miles) diligently for 4 months. I followed a training program to the nth degree, ran my race, and found my new hobby. The next day I signed up for a full marathon, 26.2 miles. I again trained and still liked running after completing it. Everyone I talked to told me that after running it I would never want to do another one, but instead I was looking for my next race shortly after I finished my first one. I didn’t want to tell anyone, but I wasn’t all that tired by the end of the race. I was hungry, but not tired. What was going on here?

I had thought I hated running because it was difficult, but I did my homework and ended up liking the feeling of success. With each step of the way I thirsted for more, even though others told me it would be “too hard.”

For a few years, I was able to fill this thirst with trying to run faster, but then I moved to Omaha. Here is where I found the ultra running community – the Ph.D. program of running.

26.2 miles was enough “difficulty” for me for years and I even thought it was insanity to run anything over that distance. Who does that for fun?!?! Not only is an ultra marathon anything over 26.2 miles, but they are often ran on dirt trails following flags on a course that may or may not be well marked. However, they do feed you multiple times along the way and I like food. Think of an aid station as a semester break in grad school. It means you have survived another semester. You get to rest, eat, recover, and forget the pain for a while. Then you go back for more with each leg of the journey providing new challenges and opportunities for growth.

How does one survive such madness? My answer to surviving both mathematics classes/programs and ultra running events is one and the same: community. When people are brought together to complete a task that at the time seems impossible, there’s something special that develops. Here are a few tips I have for creating that community that comes with embarking on a hard task.

The buddy system: Find people to train/work with who inspire you to do more than you think you can. This does not always mean finding people who are better than you. I would say that this should people of varying abilities. For instance, I have three training partners who come to mind when I think of the types of people who help me go beyond my “limits.”

First, there is my friend, Steve Stender. He is fast. He is very fast. He’s also the person I have ran more miles with than anyone I know. By being a friend and getting to know me as a person and as a runner, I value what he has to say. He shows me on a regular basis that I can go faster than I think I can and longer than I think I can with good company. He also has flat out told me to “go out and win something.” This being told to a non-athlete is something I am still not even sure if I believe. However, I have won races now and don’t think I would have even tried without Steve. He would be the friend encouraging you to get an A+ on an exam instead of just passing it.

Second, there is my friend, Kim Moore. She is tough as nails. She challenges me to go beyond my comfort zone. Somehow this lady has gotten me to run outside in the cold all winter and to run in the dark. I hate the cold and am scared of the dark. She would be the friend who encourages you to take the hard instructor or the difficult class. These things make us stronger, even though they may be painful at the time.

The third “type” of running friend I have is similar to my friend, Eric Schelker. He is someone who I mentor/train. He couldn’t run two miles when we met a year and a half ago, but now he is training for his first marathon. There’s something about helping someone and knowing they are looking up to you that makes you not want to quit. This also helps me want to do well and be a good role model. This would be the friend who you have to “teach” while working together, but in return you then understand the material better.

Regular meetings: This one is rather simple. Once you have a core group of people to train/work with you, meet regularly. This helps develop friendships and holds you accountable for doing your training/work. Don’t forget to make these meetings fun! Our running group meeting at a local bar on Tuesdays that has 50 cent tacos. We run 3-10 miles and hang out afterwards eating cheap tacos. My calculus classes meet daily before class to finish homework in a casual setting.

Talk to strangers: Whether there are newcomers in your group or you are seeking out a group, don’t be afraid to make new friends. Everyone feels awkward talking to new people, but you may just meet a life long friend or future colleague by talking to strangers.

Cave into peer pressure: Before you know it you will be running crazy distances (with these people who were once strangers) or taking crazy hard math classes. Sometimes peer pressure is a good thing!!

Impossible goal: Lastly, don’t be afraid to set a seemingly impossible goal for yourself. Break that goal into baby steps and conquer it with the help of your new friends.

For more details watch my TEDx talk at: https://www.youtube.com/watch?v=W6JG9-MWCKA

Tuesday, January 27, 2015

Setting the Stage

Whenever I’m teaching via inquiry-based learning (IBL), it is important to get student buy-in. I often refer to this as “marketing IBL”. My typical approach to marketing involves having a dialogue with my students, where I ask them leading questions in the hope that at the end of our discussion the students will have told me that something like IBL is exactly what we should be doing.

In the past, I would just wing it on day one and it’s been different every time. However, I’ve had lots of people ask me to describe exactly what I do and I also thought it would be a good exercise for me to sit down and think carefully about the activity. So, in the fall of 2014, I created some slides to guide the activity, which I am now calling "Setting the Stage". Since then I have shortened the activity and made some improvements. The current version of the activity is inspired by TJ Hitchman, Mike Starbird, and Brian Katz.

The main idea is that I want to get students thinking about why we there and what we should really be striving to get out of the course. In addition, it helps students understand why I take an IBL approach in my classes. Below is an outline of the the activity.

Directions to the Students

  • Get in groups of size 3–4.
  • Group members should introduce themselves.
  • For each of the questions that follow, I will ask you to:
    1. Think about a possible answer on your own.
    2. Discuss your answers with the rest of your group.
    3. Share a summary of each group’s discussion.

Questions

  1. What are the goals of a university education?
  2. How does a person learn something new?
  3. What do you reasonably expect to remember from your courses in 20 years?
  4. What is the value of making mistakes in the learning process?
  5. How do we create a safe environment where risk taking is encouraged and productive failure is valued?
Each time I’ve run the activity, the responses are slightly different. The responses to the first two questions are usually what you would expect. Question 3 always generates great discussions. The idea of "productive failure" naturally arises when discussing question 4 and I provide them with this language sometime while discussing this question. Listening to the students’ responses to question 4 is awesome. It’s really nice to get the students establishing the necessary culture of the class without me having to tell them what to do.
After we are done discussing the 5 questions, I elaborate on the importance of productive failure and inform that I will often tag things in class with the hashtag #pf in an attempt to emphasize its value. I also provide them with the following quote from Mike Starbird:
“Any creative endeavor is built on the ash heap of failure.”
I wrap up the activity by conveying some claims I make about education and stating some of my goals as a teacher.

Claims

  • An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of.
  • If we really want students to be independent, inquisitive, & persistent, then we need to provide them with the means to acquire these skills.

Lofty Goals

  • Transition students from consumers to producers!
  • I want to provide the opportunity for a transformative experience.
  • I want to change my students’ lives!

Below is the Spring 2015 version of the slides that accompany the activity.


You can always find the current version of the LaTeX source at my GitHub repo located here. Note that I’m using the beamer m theme for the slides, which require the Mozilla Fira fonts by default. Feel fee to steal, modify, and improve. And please let me know if do.

Note: This post originally appeared on Dana’s personal blog.

Thursday, January 8, 2015

The Twin Pillars of IBL

By Dana C. Ernst

As regular readers of this blog know, I am passionate about inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has had a profound impact on my life as a teacher. Actually, scratch that. It has had a profound impact on my life!

When I started teaching, I mimicked experiences I had as a student. I tried to emulate my favorite teachers. Because it was all I really knew, I lectured. And this seemed to work out just fine. By standard metrics, I was an excellent teacher. Glowing student and peer evaluations, as well as reoccurring teaching awards, indicated that I was effectively doing my job. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to: (i) heavily depend on me to be successful, and (ii) retain only some of what I had taught them. In the words of Dylan Retsek:

"Things my students claim that I taught them masterfully, they don’t know."

Inspired by a desire to address these concerns, I began transitioning away from direct-instruction towards a more student-centered approach. The goals and philosophy behind IBL resonate deeply with my ideals, which is why I have embraced this paradigm.

While there is variation in practice, IBL courses typically provide students with experiences that differ from traditional lecture-based courses. 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 IBL approach challenges students to create/discover mathematics. According to the Academy of Inquiry-Based Learning, 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.

I believe that there are two essential elements to IBL. First, students should (as much as possible) be responsible for guiding the acquisition of knowledge, including the pace at which this happens, and second, they should be responsible for validating the ideas presented. That is, students should not be looking to the instructor as the sole authority. In most IBL courses, student-led presentations and small group work form the backbone of the course. In general, the majority of class time is spent on these types of student-centered activities, which provide ample opportunity to discuss and critique ideas that arise from a group problem or student-presented solution.

One guiding principle of IBL is the following question:

Where do I draw the line between content I must impart to my students versus the content they can produce independently?

While instructors might give a mini-lecture to introduce or summarize the day's work, the instructor's main role is not lecturing, but rather to foster a safe environment, facilitate discussion, and redirect as necessary. In an IBL course, instructor and students have joint responsibility for the depth and progress of the course.

A research group from the University of Colorado Boulder lead by Sandra Laursen has conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students’ and instructors’ experiences of IBL. This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years. The courses examined self-identified as IBL versus non-IBL. Classroom observation was used to verify that IBL classes were indeed different from those designated as non-IBL sections of the same course. The following is a list of characteristics that the IBL sections shared:

  • Students solve challenging problems alone or in groups; share solutions; analyze, critique & refine their solutions 
  • Class time is used for these student-centered activities 
  • Students play a leadership role 
  • Activities change often 
  • Course is driven by a carefully built sequence of problems or proofs, rather than a textbook
  • Pace is set by students' progress through this sequence 
  • Course goals usually emphasize thinking skills & communication; content “coverage” is less central 
  • Instructor serves as "guide on the side" not "sage on the stage"—manager, monitor, summarizer, cheerleader

On average over 60% of IBL class time was spent on student-centered activities including student-led presentations, discussion, and small-group work. In contrast, in non-IBL courses, 87% of class time was devoted to students' listening to an instructor talk. In addition, the IBL sections were rated more highly for a supportive classroom environment and students conveyed that engaging in meaningful mathematical tasks while collaborating was as essential to their learning.

Below is a brief summary of some of the outcomes of Laursen et al.'s work:

  • After an IBL or comparative course, IBL students reported higher learning gains than their non-IBL peers, across cognitive, affective, and collaborative domains of learning. 
  • IBL students’ attitudes and beliefs changed pre- to post-course in ways that are known to be more supportive of learning, compared to students who took the non-IBL sections. 
  • In later courses, students who had taken an IBL course earned grades as good or better than those of students who took non-IBL sections, despite having "covered" less material. 
  • On a research-based test of students' ability to evaluate proofs, IBL students showed evidence of greater skill in recognizing valid and invalid arguments and of the use of more expert-like reasoning in making such evaluations. The volunteer sample consisted of only high-ability students; no instructors gave the test to all students during class time. 
  • Non-IBL courses show a marked gender gap: across the board, women reported lower learning gains and less supportive attitudes than did men (effect size 0.4-0.5). Women’s confidence and sense of mastery of mathematics, and their interest in continued study of math, was lower. This difference appears to be primarily affective, not due to real differences in women’s mathematical preparation or achievement. 
  • This gender gap was erased in IBL classes: women’s learning gains were equal to men’s, and their confidence and intent to persist similar. IBL approaches leveled the playing field for women, fixing a course that is problematic for women yet with no harm to men. 
  • When sorted by prior achievement, the grades of most students (IBL and non-IBL alike) dropped in subsequent courses as course work became more difficult. But grades of initially low-achieving students who had taken the IBL course rose 0.3-0.4 grade points, unlike their low-achieving, non-IBL peers, and unlike their higher-achieving classmates.

This work determined that there are two "twin pillars" of IBL that may explain the student outcomes, namely (i) deep engagement in rich mathematics and (ii) opportunities to collaborate. Here, deep engagement refers to individual and group effort in tackling meaningful problem-solving tasks that are not merely "busy work." Collaboration is a key component as students learn from explaining their ideas and trying to understand others. According to Laursen et al.:

"The twin pillars reinforced each other: after struggling with a problem individually, students were well prepared to contribute meaningfully during class, and interested in the solutions that others proposed. Collaboration in turn motivated them to complete the individual work. It also made class enjoyable, encouraged clear thinking, and built communication skills."

IBL is not a magic bullet, but the experiences that I have had watching students transform into independent learners is why I am so passionate about it. I want students to have life-changing experiences! Learning the content of mathematics is just a bonus.

One of my principal goals is to make my students independent of me. I want them to experience the unmistakable feeling that comes when one really understands something thoroughly. In the words of Carol Schumacher:

"When one can distinguish between really knowing something and merely knowing about something, that individual will be on his/her way to becoming an independent learner."

I’m not terribly picky about the particular flavor of IBL or active learning one chooses to employ, but it is becoming increasingly clear to me that if we want to produce life-long independent learners, then the twin pillars need to form the foundation for the pedagogical approach we choose to take.

References

[1]  Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study. Journal of Research in Mathematics Education, 45(4), 406-418.

[2]  Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. Innovative Higher Education, 39(3), 183-199. DOI 10.1007/s10755-013-9269-9.

[3]  Laursen, S. L. (2013). From innovation to implementation: Multi-institution pedagogical reform in undergraduate mathematics. In D. King, B. Loch, L. Rylands (Eds.), Proceedings of the 9th DELTA conference on the teaching and learning of undergraduate mathematics and statistics, Kiama, New South Wales, Australia, 24-29 November 2013. Sydney: University of Western Sydney, School of Computing, Engineering and Mathematics, on behalf of the International Delta Steering Committee.

[4]  Assessment & Evaluation Center for Inquiry-Based Learning in Mathematics (2011). Evaluation  of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics. (Report to the Educational Advancement Foundation and the IBL Mathematics Centers) Boulder, CO: University of Colorado, Ethnography & Evaluation Research.

Further resources can be found here.

Thursday, November 6, 2014

Math Teachers’ Circles: What Makes a Good One?

by Angie Hodge


Math Teachers’ Circles (MTCs) bring together middle school math teachers and professional mathematicians to enrich the teachers’ experience of mathematical problem solving and to build mathematical community. Free math club-like events, MTCs give teachers the chance to have fun doing math three or four times per semester.

  1. increase the confidence of middle school math teachers in their problem-solving ability;
  2. deepen teachers’ content knowledge through exploring mathematically rich problems and developing an arsenal of techniques for solving unfamiliar and challenging problems;
  3. form long-term professional relationships between teachers and mathematicians through regular, highly interactive meetings; and
  4. provide support for teachers who want to bring richer mathematical experiences to their students.
Teams interested in starting a Math Teachers’ Circle in their area should contact AIM at circles@aimath.org. Six teams of four or five teachers attended workshops on How to Run a Math Teachers’ Circle in 2014. At the 2014 workshop in Washington, D.C., the teams were asked to answer two questions:
  1. What makes a good Math Teachers’ Circle session? 
  2. What makes a good Math Teachers’ Circle problem?
Workshoppers were asked to brainstorm with a focus on "quantity versus quality," and they came up with quite a list. Just perusing it gives even someone unfamiliar with MTCs a pretty good idea of what they’re all about:

What makes a good Math Teachers' Circle session?

Snack break
Good snacks
Engaging problems
Aha! Moment
Leader ready to scaffold/backfill/support
Leader ready to give next challenge
Focus on math
Out of comfort zone
All participants feel comfortable with math and other participants
Safe environment for failure
Discussion and collaboration
Entertaining/enjoyable
Buy-in for participants
Community
Group of common professionals
Relaxed, non-threatening atmosphere
Classroom connections without focus on classroom
Interesting presentation of problems
All levels of mathematics
Overplanning
All participants are involved
Generate enthusiasm
Participants explain and present
Variety of participant backgrounds
Pacing good
Participants in the workshop in Washington, D.C. (photo Hana Silverstein)
Memorable
Wine
No whine
Different presenter personalities
Appropriate amount of room
Good number of participants
Humor/laughter
Make friends
Noncompetitive
Supportive
Multiple strategies
Include failure
Hook
Not lecture-y
Celebrate discovery
Good flow
Participants sharing discoveries
End loving/wanting more
SWAG (Sell your MTC by advertising it!)
Climate of respect
Knowledgeable leaders
Focus
Critiquing mathematics/solutions (safe for people)
Providing resources to learn more
Inter-workshop closure, info, etc.
Leader's love of math is transmitted
Plenty of time
Time flies
T-shirts
Time to explore
Time to fail before seeing solution
Good entry/exit
Freedom to digress/follow tangents/not too fixed a goal
Individualized closure

What makes a good Math Teachers' Circle problem?

Hands on
Engaging
Knobbifiable (problems can be made harder or easier)
Low-level entry
Multisensory/multimodal
Mystery
Leads to more questions
Out of the box
Initially simple
Folkloric
Variety of strategy and/or tactics
Minimal lecture
Lets participants get to board
Novelty to participants
Not textbook
Good lead-in
More than an hour to solve
Interesting to different groups
Some element of fun
Joy of math
Physical/"crafty"
The list in its original form (photo Hana Silverstein)
Abstract/thoughtful
Has a hook
Clear parameters
Real world
Not too intimidating
Challenging
Easy to generate data
Strategies embedded
Associated with a lesson
Moral to the story
Cognitive dissonance
Surprise
Some closure
Some open endedness
Group or individual
Multilayered problems
Opportunities for discussion
Little intro prep/setup
Patterns
Connections within mathematics
Multiple pathways
Gives participants something to bring home
Reasoning/argumentation
Memorable problems
Spatial
Games
Intro fun
Not too much tedium
Aha! moment

For the MTC veterans out there, do the items on the lists above square with your experience of what makes a good Math Teachers’ Circle?

Note: This exercise was given as a way to create closure for the How to Run a Math Teachers’ Circle workshop. Participants had been working problems in MTC sessions all week and this gave them a chance to reflect on the experience. I think this exercise of thinking about good problems and a good class session could also be used in other mathematics courses. Imagine your own classes generating lists about what makes a good student, a good teacher, a good exam, a good problem set, etc. The possibilities are endless! Happy list generating!!! 

Friday, September 12, 2014

A First-timer’s Experience with IBL

a guest post by Ellie Kennedy

Prior to the spring of 2013, I taught my discrete mathematics course via a traditional lecture style. I used to bore myself with those lectures. Counting principles seemed like a topic that students could develop on their own with maybe just a little problem-solving help. With induction I felt like I was lecturing and showing the students the same examples over and over and it wasn't sinking in. I needed to try something new and fresh, and inquiry-based learning (IBL) seemed like a method that might work for me. So, last spring when I taught discrete math, I used a modified Moore method. I'd like to share my experience as a first-timer and some of what I learned.

Ted Mahavier started me off with a set of notes, and Dana Ernst helped me sort out the logistics of the course. I was so thankful to have such great resources. Students would read definitions and theorems in the note packet and work on problems at home and then present the problems in class. In a traditional Moore method classroom, students are not allowed to collaborate, but I encouraged students to work together.

The counting and graph theory parts of Ted's notes were fantastic, but I did modify them a bit to fit the topics taught in our course. Ted's notes focused on strong induction and our course has a weak induction focus. This was not a difficult change to make to the notes. Ted's notes did not have anything on recursion, so I wrote an entire section myself. I was surprised how challenging it is to write IBL notes! I found it hard to build questions leading to the main idea when, to me, the main idea was an algorithm to solve recurrence relations. It made me realize how much I personally rely on knowing how and why an algorithm works but not the history of how it was developed in the first place. Very eye-opening for me.

I used the felt tip pen idea that Dana has written about, and it was a true success. While in class with other students presenting, the students would use only felt tip pens to mark up the work they had done at home. This allowed them to produce a solution set of sorts, and it made my grading super easy. I did not grade for correctness. I graded only on the math they produced at home (non-felt tip pen work). This method also allowed students to constantly self-assess, which can be an effective learning tool. The students felt it was easier to do homework when they didn't have to worry about getting it right at that very moment. They found they could just concentrate on the math that way.

I tried to make the class a comfortable place where students could make mistakes freely and without embarrassment. I made a list of "dos" and "don'ts" so the students were aware of some positive ways of pointing out that they thought someone was wrong. The first time a student did a problem wrong, I made a big deal out of it (in a positive way). I thanked the student for having the guts to put up something that was wrong. Then the class discussed what parts were correct and I had the students work on the problem for another night and we came back to it the next day. I always pointed out the learning experience that came from each mistake that was made. A student commented in my end-of-semester evaluations that I "showed respect for the students and encouraged [the students] to fail early and often (this is a good thing)." Mission accomplished!

One of my students struggled to switch to this new learning environment all semester. It is an adjustment for most students, but he never quite got it. During an exam review I realized he was always trying to jump to the end result without any thought of how to get there. I suggested that he write down a list of steps for each type of problem. "Like a map," he said. This comment made me realize an analogy that helped me understand what my students were going through.

 In a traditional lecture-style course we start with a city and then tell the students what road to take to get the next town. We expect them to repeat the same route we just showed them. Yet in an IBL class we give the students the cities and states and then tell them to find their own roads and build their own map. Creating our own paths makes it much easier to remember how to get there the next time.

The number one thing that I learned from this new experience (other than that IBL is amazing and really does contribute to deeper student understanding) is that it is important to understand that being confused is not just okay but a really good feeling to embrace in mathematics.

When I first started going through Ted's notes, I found problems where I didn't understand the question. I realized that this was purposeful, intended to promote conversation. Confusion leads to questions, and it is in those questions that true understanding and learning occurs.

This current technological age pushes us to find the answer FAST. Even I am guilty of just "Googling it." It is challenging for students to work on a problem or question for an extended period of time. They don't understand that some questions go unanswered for centuries and that is normal for us mathematicians to keep trying. I feel that as educators we need to let our students know that as long as they get the important parts before the exam, it is okay (and fun) to be confused and search for an answer for decades or years. Let's encourage the struggle and show our students that struggling in math is really exciting! Hopefully students can realize the incredibly rewarding feeling of solving something after much thought and time!

Friday, August 1, 2014

Teaching Math to Non-Math Teachers

by Angie Hodge

I know how to sell freshman calculus students on math and, in particular, on math taught using inquiry-based learning (IBL). Undergraduate math majors also buy into IBL pretty easily. They like math no matter how it is taught.

The same cannot be said of my students this summer. I've taught graduate courses to secondary mathematics teachers before, and my summer students were teachers, too. They were mostly elementary teachers, though, elementary teachers who had enrolled in a two-year master’s program focused on learning middle school math deeply. They recognized weakness in their mathematical preparation and wanted to learn math better for the benefit of their students.

photo credit Lindsay Augustyn
This is very brave of these teachers. To recognize that you are not the best at something is one thing, but to face that fear head on and enter into a master’s program focusing entirely on your fear takes courage.

I spent five days (8 a.m.-5 p.m.) in the last couple of weeks with 29 teachers who were taking their first math course in a math master’s program for teachers. During these five days, I witnessed the teachers undergo amazing transformations in both attitude toward math and knowledge of it. And I learned a lot, too.

Here are some reflections on the lessons learned over the five-day period. The course was taught in partnership with two middle school math teachers, one elementary school teacher, and two grading assistants. When I say "we" or "the instructional team" that is who I am referring to.

Day 1: Be firm and friendly
We all tried to be firm, but friendly on Day 1. Setting the tone for an entire master's program is a big task, and our instructional team didn't take this lightly. We spent lots of time discussing the importance of working together, having a positive attitude, knowing the importance of productive failure, taking chances, thinking outside the box, and learning to communicate in a mathematically correct manner. We dug right into the mathematics and the mathematical habits of mind. On Day 1, the teachers were "good students," but they were still very timid. They made lots of negative comments about math and moaned when asked to justify the "whys" rather than just memorizing rules. Despite the moans, we kept pushing: friendly but firm.

Day 2: Persevere (pep talks are a must!)
"Ugh" is all I have to say about the first hour of this day! Imagine being swarmed as you walk into your classroom by 15-20 upset people, all of them near tears. "Tough it out," I told myself. "Things will get better for you as an instructor if you persevere as you want the teachers to persevere." 

Teachers wanted to quit. Teachers were really not happy about math or the amount of time it took to think about the homework problems. They did not understand why an answer wasn’t good enough and why they had to "show us" their thinking process.

For all the tears and griping we somehow pulled together and even bonded as a team/class on Day 2. How? There was a lot of pep talking. We talked about productive failure. We talked about the importance of struggling. We cheered for progress. We gave praise for positive attitudes. We rallied and threw energy around like it was going out of style.

Day 3: The calm before the storm
Day 3 was one of our best days and one of the days when we saw the teachers grow the most. Teachers who were barely talking earlier were taking risks to present (but only if they knew they were correct) and were talking more to the instructional team and to their group members. Although it worried us (the instructional team) that some groups weren't talking as much as we had hoped, some were working really well. We debated switching up the groups and decided to try it for Day 4. We didn’t have to do much on Day 3 other than teach and continue to compliment progress, positive attitudes, and good work ethic.

Day 4: Beware of your first "hit" of hard material
We switched up the groups. Maybe not the best idea on the hardest day of class. Fractions!!! Need I say more? :) Wow. At the end of this day I truly wanted to cry. Our evaluations were lower than usual (still pretty decent, but we all strive for perfection). The teachers were frustrated. We were frustrated. How could we use this as a teachable moment to help them persevere?

Day 5: Make every moment a teachable moment
We started Day 5 as we started every day, talking about the evaluations. We discussed evaluations daily to make sure the teachers knew that we heard their voices. We commented on why we were or were not changing things based upon the feedback they gave us. We used negative comments about confusion as a teachable moment. We talked about what it meant to be confused and how it was part of the learning process. We also discussed the importance of speaking up if you are confused or stuck. We stressed the team aspect of the course again and emphasized that even though it was hard we were here to learn together. Since these were teachers, we were able to do this in an IBL manner, asking the teachers how they would respond to students who were confused but did not tell them this until after the fact. This discussion set the stage for a new tone. No matter how clear you think you are with your expectations and no matter how approachable you think you are, you need to remember that your students come pre-programmed to try to get the correct answer quickly. "Unprogramming" this takes time. Be a broken record about this and praise your students when they finally believe you!

Boom! (as Dana Ernst would say) On Day 5 I saw remarkable growth in nearly every single teacher. They made it over many mental hurdles and they realized they made it. Somehow making it past that tough hurdle on Day 4, they had become a team. We let the teachers sit anywhere they wanted to on Day 5. Some sat with their original groups and some paired up with new people they met from switching. Some sat in pairs, some sat in triples. Honestly, I didn’t care how big the groups were. What was important was that everyone had found someone whose learning style complemented his or her own. The day could not have gone better. The teachers did some really tough problems really well (even showing multiple solution paths). They were even asking each other to "prove it" and asking why things worked. Teachers wanted to know when they would get to see the team again. Right then it was clear that the connections made were ones that would extend beyond the one course.

Somehow, 29 individual teachers (and the instructional team) went from strangers to a team in five days. We all problem solved together and bonded—IBL Style. Boom.