Answer Getting vs. Inquiry

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."  

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.

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.

Long-Term Study Verifies IBL Field Experience (and a Bit of Irony)

I have been active long enough in the IBL community to hear questions from a variety of angles and perspectives.   I enjoy discussing teaching issues with anyone, and it's a good for all of us to ask good questions and to seek data that supports our positions.  Here are three of the top questions that come up in discussions I have with colleagues:

1. Is IBL only good for the good students?
2. Is IBL only good for the weaker students?
3. Does lack of coverage harm students?

If you're pressed for time, the answers are "No, no, and no."

In this post I'd like to focus on dispelling a few of the myths embedded in the questions above with some data.  A recently published article by Kogan and Laursen provides evidence that helps us answer these questions.

Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, “covering” less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department. 

One of the findings in the paper is that the low achieving students in IBL courses did almost a half grade point better (2.41 vs 1.95) in subsequent required math courses.  Basically what Kogan and Laursen did was to split the students into three groups (low, medium, high), based on prior academic achievement.  Then the students took IBL and non-IBL math courses, and then Kogan and Laursen measured students' grades in subsequent required math courses.   Simultaneously, the high-achieving IBL students did the same as high-achieving non-IBL students in subsequent required courses, despite being exposed to less material.  It's a win-win.  There is no compromise in terms of grade outcomes in subsequent courses.

The everyday way to say this is that some of the students learned to learn better, and they carried that with them.  The high-achieving students do the same, and the low-achieving students (and medium-achieving students) learned to be more effective thinkers in IBL classes.

Lack of coverage:  There's an easy fix to this issue that is orthogonal to IBL vs. non-IBL.  Just make reading assignments, give mini lectures, or do a screen cast on your tablet to cover some topics to get the exposure of topics up to whatever level desired.  The coverage issue should be a non-issue going forward, now that everyone has a smart phone or access to the internet at college.

In fact the notion that IBL courses somehow create a disadvantage/advantage is completely off the mark, and if fact it is likely that traditional instruction is guilty of creating bias and imbalances.  Recent data suggests that traditional instruction can be damaging to certain subgroups, particularly women.  For example,  there is evidence of a gender bias in traditional courses from at least two separate data sets.  What we are finding is that women are particularly disadvantaged in traditional math courses, seeing greater declines in confidence, interest in Mathematics, and persistence.  This trend has statistical significance in the MAA Calculus Study (Link See David Bressoud's talk at 25:00) and in related work by Laursen's group (Link See Laursen's talk at 11:00).

Kogan and Laursen also make an interesting observation about traditional instruction vs. IBL.  IBL instructors, especially the first ones in a department to try IBL, have been asked to provide justification or evidence that IBL works.  This is ironic in that there is no equivalent scrutiny of traditional methods. In fact, when you consider the evidence from the past two decades, all the vectors are pointing towards active, student-centered instruction.  This is the case across levels and disciplines.

All along, I have said that I'll follow the data.  If the data said lecture is better, I would lecture.  So far there's isn't any evidence that says this.   Further, I am not wedded to a particular teaching style, but I am deeply interested in basing my teaching methodology on the best, empirically-validated evidence we have.  Personally, I have never had an IBL class as a student.  Ever.  K thru PhD was all traditional instruction, and it worked for me.  But I realize that as a mathematician I'm peculiar (more on that later).   Thus, I encourage all instructors (math or otherwise) to be open to new methods and consider the implications from data gathered by researchers.  By working together and being open to new ideas, we can progress faster and smarter.  At the moment, IBL instructors have data that supports their work, and the data continues to pile up validating IBL.

Keep on keeping on!

A Quick Reminder: Small Grants Applications Due Next Week

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

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

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

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

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

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

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

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

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

The MLK Domino Mosaic the 6th Graders Built

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

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

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

Working Together in Groups

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

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

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

The Sense-Making Continental Divide

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

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

Here's the model I have in mind:

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

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

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

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

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

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

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