## Monday, October 21, 2013

Edited on 10/12

Phil Daro discusses the reasons against Answer-Getting HERE.  Daro's 18-minutes talk gets to an important aspect about teaching.  In this talk, Daro suggests that in the U.S. we (math teachers) focus on helping students get answers.  On the other hand in countries like Japan, teachers focus on the mathematics that can be learned from problems, this simple, fundamental difference leads to vastly different outcomes and perspectives about teaching and learning.  When the focus is on mathematics and not just Answer Getting, then students can engage in doing the kinds of things that mathematicians believe is real mathematics.

Additionally Daro discusses the notion that mistakes and answers are part of the process of learning mathematics.  They are not the ultimate goals.  Answers, while still essential, are only a part of a larger endeavor, and not a signal that there is nothing left to do.

Mistakes should also be valued as useful discoveries.  When we discover a method that does not work, it needs to be fleshed out so that we can be sure that we can learn as much as possible about the related mathematics.  Such a process is not usually part of the standard method of instruction.

IBL instruction is consistent with these ideas.  Instructors in college-level IBL courses use a well-crafted set of problems to provide the context for the learning experiences.  (Course materials for some college-level courses can be found at The Journal of Inquiry Based Learning in Mathematics.)  Students work on these problems without being shown solutions ahead of time.  Part of class time is used by students to present solutions or ideas to the rest of the class, and the audience peer-reviews these ideas.  Logic and reason are used to determine if solutions are correct, and mistakes (discoveries) are used as opportunities for further investigations.

It's a wonderful notion to view mistakes as important discoveries.  This sets up the framework for class discussions in a positive, scientific setting.  Mistakes then become identified as useful explicitly in the running of the class, and this is where diverging from Answer-Getting becomes fundamentally different than doing mathematics.  If the goal is getting answers, then by definition getting non-answers isn't getting us to our goal.   While we mathematicians view mistakes from a healthy perspective (at least when we are doing math), our views and attitudes are divergent from the way some students look at mathematics.  They "FOIL it" or "Butterfly it" or "Cross Cancel/Multiply."

A related point is what new IBLers often say.  A common statement is, "I'm surprised that students have so much trouble with these concept questions."  These concept questions may be true-false questions or questions that ask students to apply an idea or method to a slightly novel (to students) problem.   The results are usually discouraging, and college instructors wonder why this is the case.

Daro's talk sheds some light on this phenomenon.  In the segment where students apply the "butterfly algorithm" to add fractions, it is noted that the trick doesn't generalize (easily or obviously to students) to adding three fractions, and U.S. students perform especially poorly on adding $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ compared to their international peers.   Daro suggests that it is because we spend too much time on tricks and on Answer-Getting, often at the expense of doing the underlying mathematics (in this case equivalence, equivalent fractions, and common denominators).  By the time these students get to college, their limited experiences with logic, problem solving, and higher-level thinking in mathematics leaves them underprepared for rigorous thinking.

What can we do in college?  Focus on mathematical discoveries of all types, and let students inquire together about the meaning of mathematics.  Each new idea is a discovery and we can provide supportive classroom experiences, high-quality tasks, and effective coaching/mentoring to move students towards successful habits of minds and attitudes.

One can start a course by sharing prepared common mistakes and use them as the first experience in learning from mistakes and how the course will view mistakes as important discoveries.  Instructors can state something along the lines of... "What discoveries have you made about this problem? Please work with your partner to write these down and be ready to share your discoveries."

## Thursday, October 10, 2013

### Quick Post: Building Self-Esteem and Confidence

Just a quick post to share a quote by Randy Pausch, who presented and wrote The Last Lecture.

"There's a lot of talk these days about giving children self-esteem. It's not something you can give; it's something they have to build."

Self-esteem and confidence are built from hard work and success.  The overly simplistic model of IBL teaching is that we pitch (some) problem just outside the grasp of students, and through hard work, support, and guidance, students succeed.  Repeat. Repeat. Repeat...

Success breeds confidence like no other.  Good teaching practice can support fruitful struggle that then leads to new knowledge (for students), ways of thinking, habits of mind, and the oft-elusive quality of confidence.

## Tuesday, October 1, 2013

### Being Stuck

Dealing with "Being stuck" is one of the most critical components of IBL teaching.  IBL teaching rests on several factors, such as good content, questioning strategies, setting up a safe and productive environment for learning, having an assessment system that is consistent with the goals and ethos of the course,...    In this post the focus is on handing situations when students are stuck, which can sometimes make or break a course.   A little background first to set the stage.

The short, oversimplified background story is that mistakes are stigmatized (in the U.S.).  Thus struggling in Math is equated by some students as a sign of being dumb or slow or that the teacher isn't doing a proper job.  Traditionally math teachers present nice, clean solutions, and there are very few instances when students can witness the math process that actually is what makes us successful learners.  It can be the case that a student has never experienced or witnessed what mathematicians do regularly.  That is, the process of problem solving, inquiring, experimenting may all be unconnected from Mathematics.

Consequently, the IBL instructor who gets a group of such students must not only deal with the "regular" learning challenges that a math course presents, but also the legacy of underdeveloped/negative attitudes and habits of mind that promote learning.  When these underdeveloped/negative components of the learning process come out is when students get stuck.  Being stuck is both an opportunity and a risk.  It takes courage (at least initially) for students to admit to being stuck and to then engage in problem solving.   The risk to the students and teacher is when students are so frustrated and stuck that they shut down and stop learning.  (In math speak, we want to avoid the boundary.)

What we can do as IBL instructors?
1. Make sure students know and feel that it's okay to be stuck.  "Are you guys stuck?  Great!  It's okay to be stuck!  Let's use being stuck as an opportunity to work on our problem-solving skills... How can we break this problem down to a manageable size?..."
2. Scaffold enough so that the students see your role as their advocate and facilitator in learning. It's better to error on the side or more scaffolding than less, early in the term.  What this means is to provide enough hints/lemmas/basic examples so that students perceive themselves as progressing.
3. Create a positive, relaxed class environment by using group work and visiting groups to check-in with individuals.
4. Use (more) starter problems.  One of the main roles of an instructor is selecting appropriate tasks.  Giving several starter problems, where all students can get traction is important.  Early in the term this is especially important, and the lower the level of the course the more important it is to have good entries into topics.
5. Summarize, restate, and give alternative solutions here and there to provide the expert insight that students often gain from.  When students have finished a section or unit, that is a wonderful opportunity to highlight all of the wonderful insights, ideas, and strategies that students learned.  It's a way to make explicit progress and achievement, as well as review material.
6. Marketing what IBL is and why it is good for students should be steady and ongoing.  What this means specifically is clearly indicating that the goals of the course include handling being stuck, problem solving, communicating ideas with peers.  These are in addition to the content goals.  "The reason why we are doing these activities is so you get better at..."
7. Consider employing reading assignments and journal writing.  Such assignments can be especially useful in addressing beliefs and attitudes.  Burger and Starbird have a book, "The Five Elements of Effective Thinking," that can be used in any math course as a supplement.  Students can read a couple chapters at a time and write a one-page reflection paper on what they learned and how they might use the ideas in the class.  Getting a second or third opinion in writing is very effective.  It's one thing to get the message from class, and yet another when multiple sources support the same messages, thus providing more opportunities for students to take necessary steps towards successful learning.
If being stuck in class is explicitly a good thing, students' struggles are respected, and class activities are designed to take advantage of these opportunities in a positive spirit, then being stuck can be a positive force in learning!

Have ideas?  Send the via email or post them in the comments.