Constructing an IBL Unit: An Example Using Integers

On this blog I've written about why IBL can be a useful framework for teaching mathematics.  IBL is a general framework that isn't tied to a specific curriculum or type of course.  Engaging students in rich mathematical tasks is something that can be done at all levels.  How instructors specifically implement IBL methods varies due to classroom conditions, and these implementations ultimately require appropriate problem sets or units.  In this post the focus is on how to construct a coherent IBL unit.

In order to hit a wide audience, I've chosen a 6th grade topic, integers.  The reason for this choice is that it is one of the content areas that is often memorized, but not understood conceptually.  The integers are a mathematically wonderful landscape, and opportunities to explore this landscape are usually missed.  Despite the choice of using integers, the core ideas are applicable broadly.

The first step is to survey the learning landscape.  I'll focus on subtracting integers, which is where things get really interesting.  Typical 6th grader (skill-level) tasks with integers includes subtracting positive and negative integers.

$10-6 = 4$
$4-(-8) = 12 $
$-5 - (-8) = 3$

If I ask a middle schooler (or older student/person), why this work, the vast majority of the time I get, "I don't know. It's just how I was taught to do it."  To answer why questions, one needs to understand the notion of mathematical equivalence.  Equivalence is a major idea that lies at the heart of the matter, not only for integers, but also in many other areas of mathematics. Equivalence shows up in the integers via the idea that every integer can be represented in infinitely many ways using zero pairs.

$$5 = 5 + (1 - 1) = 5 + (2 - 2) = 5 + (3 - 3) = \cdots$$

Written another way,

$$5 = 6 - 1 = 7 - 2 = 8 - 3 = \cdots$$

An important developmental milestone for young students is realizing that every integer can be expressed in infinitely many different, yet equivalent ways.  This idea then allows students the mathematical flexibility of choosing suitable representations to more easily enable them to solve problems and verify why things work.

\begin{eqnarray*}
-5 - (-8) &=& -5 + (8-8) - (-8)  \hspace{.5in} \mbox{ (A zero pair $8-8$ is inserted) }\\
&=& [-5 + 8] + [-8 -(-8)]\\
&=& 3 + 0 \\
&=& 3
\end{eqnarray*}
Adding in a zero pair ($8-8$) allows the expression to be rearranged so that the computation can become completed using subtraction and another zero ($-8-(-8)=0$).  This explanation is the bare math, so a context is useful to support the ideas for 6th graders.

Temperature is a suitable context, since negative temperature can represent degrees below the temperature water freezes (Celsius).  The "Temperature Tank" model, which is based off ideas I learned from Professor John Wilkins, Cal State Dominguez Hills, allows students to model the integers using temperature and counting chips to develop their initial understanding.  Contexts are important in school mathematics, because they provide a concrete base to build off of.  It's not that all problems have to be real-world based.  The idea is to provide a realistic enough situations to provide "handles" to the core ideas, and then let students make the generalizations and abstraction naturally.

Let's see how this works for $-5-(-8)$.

A. The number $-5$ can be represented with five negative chips.

B. Using equivalence, students can choose one (of the infinitely many) representations of $-5$ that also includes an appropriate-to-the-situation zero pair.  In this case, using at least 8 chips for each "partner" in the zero pair is needed.

C. Then $-8$ can be subtracted from this expression.

D. The remaining chips are used to compute the result of $-5 - (-8)$, which equals 3.

The unit starts with modeling integers with the temperature chips, and progresses to using these with operations and explaining why things work.  Part of the unit is presented here.

Temperature Tank Model: A model for the integers is the temperature tank.  The tank, when it has zero chips in it is at zero degrees. If you add a PLUS chip, then the temperature in the tank goes up one degree.  If you add a MINUS chip, the temperature in the tank goes down one degree.

Instructions: Please model what you are doing on the open number line AND express each situation numerically.

1. The tank starts off with no chips inside it and is at zero degrees.  Explain what happens to the temperature of the tank, if you add three PLUS chips.
2. The tank starts off with no chips inside it and is at zero degrees.  Explain what happens if you add five MINUS chips.
3. The tank starts off with 6 PLUS chips and 5 MINUS chips.  What is the temperature of the tank?  
4. The tank starts off with 6 PLUS chips and 5 MINUS chips.  Explain what happens to the temperature of the tank if you remove 3 PLUS chips.
5. The tank starts off with 6 PLUS chips and 5 MINUS chips.  Explain what happens to the temperature of the tank if you remove 3 PLUS chips, and then remove 4 MINUS chips.
6. The tank starts off with 8 PLUS chips and 17 MINUS chips.  Explain what happens to the tank if you take out 3 MINUS chips.
7. The tank starts with 7 MINUS chips in it.  Then Francisco adds 3 more MINUS chips into the tank.  At what temperature is the tank?
8. Definition:  A zero pair is a pair of numbers that add up to zero.
9. List as many zero pairs as you can, but organizing them into a table.  After you create a table of all the zero pairs, explain any patterns you notice. 
10. How many possible different ways are there to represent using zero pairs the temperature zero degrees? Explain.
11. How many different ways can you express the number 5 using the temperature tank?  List at least 7 different ways.
12. How many different ways can you express the number (-3) using the temperature tank?  List at least 7 different ways.
13. The temperature tank is at 5 degrees, and you take out of the tank 3 MINUS chips.  What is the resulting temperature of the tank? 
14. The temperature of the tank is -3 degrees.  You reach into the tank and take out 4 MINUS chips.  Find what the temperature of the tank will be.  
15. The temperature of the tank is 10 degrees, and you take out 11 MINUS chips.  What will the temperature of the tank be? Explain.
16. Using the temperature tank, explain why subtracting a negative is equivalent to adding the absolute value of that number.  For instance 5-(-7)=12.
17. ...

Problems 1-7 are intended to help students build connections between the representations using numbers and the context. These problems are important for building conceptual understanding.

Definition 8 states the zero pair, which is the core idea underlying equivalence of integers.

Problems 9-12 are intended to gently guide students to investigate the idea of equivalence.  The instructor has a major role here when summarizing student discoveries.  When students discover equivalence, then the instructor can jump in: "Look what you have discovered… We can now express any integer in infinitely many equivalent ways.  In mathematics this is the idea of equivalence.  Two expressions that are not identical represent the same number..."

Problems 13-16 hit the topic of why subtracting a negative is equivalent to adding.

The unit continues on to hit multiplication, skills, and caps with division concepts (without remainders).

Discussion
Let's break this all down.  First, the task of explaining why is a much more sophisticated task compared to ones asking students to "compute the answer to…"  In fact, students must put several pieces together: the context, the computation, apply concepts (zero pairs, equivalence), and express all this in writing.  My belief is that this is where we want all students to go!  It needs to be emphasized that picking big goals is important. It's not just about getting answers and making it through the computation.  Students should own the math, and by own the math it is meant that they know how things work, how to compute, why things work the way they do, and be able to communicate their ideas clearly.

Construction of the unit has structural features that are generally present in IBL units.  The beginning is usually a collection of starter problems intended to help students become familiar with a context and the basic ideas.   In this case the unit starts with representing integers with a physical model.

The logical order of mathematical learning always dictates what happens next.  Hence the middle phase varies.  In this specific unit, an idea is needed to address the "why" questions later, thus introducing the zero pair is necessary.  The middle phase is built off of developing an understanding of equivalence of integers.   In general, the middle phase of a unit focuses on lemmas, key concepts, important techniques, etc. that are critical in addressing the major goals of the unit.

The last phase is encapsulated by goal problems or goal theorems.  One of the big goals of the unit is for students to be able to justify why subtracting a negative is equivalent to adding.  Problems 14-16 are the goal problems, and in fact were the first problems I wrote down for the unit.  I start with the goal problems, and then reverse engineered what is needed logically and experientially to form the unit.  This takes a bit of time and scratch work to figure out the first time, but over time it gets easier.   More and more examples of problem sets (units and entire courses) are being developed by the IBL community, and the need to build materials from scratch will diminish over time.

The components of a generic IBL unit:

• Opening: units open with starter problems, definitions, assumptions, and establishing the context/topic.  (Starter problems may need to be interspersed throughout the unit.)
• Middle Phase:  the middle phase is the main building phase of the ideas, techniques, and mathematical perspectives.  The building blocks should contain the core mathematical building blocks, which are usually found by reverse engineering.
• Goal Problems:  the big learning goals of the unit are usually at the end of the unit.  The goal problem phase may also include applications.
• Wrap-up: reflecting and organizing what has been learned is highly valuable, and often a good way end a unit.  Additional applications for formative assessment purposes to ensure the goals have been met can also be implemented in the wrap-up.

More examples are forthcoming from different levels of mathematics.  I hope this post highlights that IBL math is more than just a collection of problems, implemented so that students are required to work on the problems and present them.  That captures only a fraction of the IBL framework.  Units are composed with a story or narrative, and designed to support a natural arc of learning.  Moreover, the units are not fixed.  They can be adapted over time, and it is expected that an instructor must add or subtract problems in class to adapt to specific learning needs with each class.

Does this specific unit on integers work? Yes! The unit is not some theoretical prototype I cooked up in the lab.  It was designed, implemented and updated from actual classrooms.  In fact, the unit has been used multiple times in 6th grade classrooms, where students successfully make it through all of the problems, explaining how and why things work, all the while learning the skills, too.  The full unit is posted HERE and HERE.  You are free to use it, change it, make it better.

In comparison, a conventional, teacher-centered unit, where mathematical equivalence is not included, usually means that answering why questions is out of reach for students.  A conventional framework may include diagrams/schemes (e.g. subtracting a negative is jumping to the right), which is merely a restatement of the rule using the number line.  These essentially boil down to, "Here is how to calculate these things, and here are some heuristics to help memorize them."  There are many limitations to such a learning arc, and one of the main limitations is not providing students the mathematical foundations for making sense of the fundamental ideas and building viable mathematical arguments to explain them.  On the other hand, building upon the bedrock of mathematics, providing entry points for all students, and opportunities to explore, experiment, and explain, gives students a chance to truly own it and develop as young mathematicians.  

Believe in your students!

Learning from History, Warren Colburn 1830, and the Implementation Challenge

More than 180 years ago Warren Colburn presented his work and ideas to the American Institute of Instruction in Boston, MA, August 1830.  It was published in the proceedings, and has been reprinted with permission, by the University of Chicago Press.  LINK

(Disclaimer: In this blog post Colburn is quoted verbatim, and as was the style at the time the gender of students or scholars is written as "he" or "him." In an effort to be 100% clear, these are Colburn's words and not mine, and there is nothing implied by me or the IBL community about scholars = male. My own opinion and the viewpoint of AIBL is of course 100% for equality in math teaching and opportunity for all groups.   I hope we can instead focus on the ideas presented, and put aside the arcane language in Colburn's address. One hundred and eighty years is a very long time.)

"By the old system the learner was presented with a rule, which told him how to perform certain operations on figures, and when they were done he would have the proper result. But no reason was given for a single step...  And when he had got through and obtained the result, he understood neither what it was nor the use of it.  Neither did he know that it was the proper result, but was obliged to rely wholly on the book, or more frequently on the teacher. As he began in the dark, so he continued; and the results of his calculations seemed to be obtained by some magical operation rather than by the inductions of reason."          

It's a very familiar point.  Through Colburn's experiences teaching math (primarily tutoring) and writing his own arithmetic text, he developed keen insights into learning mathematics.  Many of the points he made nearly two centuries ago are relevant and warrant thought and investigation by instructors today.

1. A curriculum with appropriate problems suited to the learner is necessary.  Problems should start with the easiest problems first and be logically ordered.

"…choose the easiest [problems] first, and then the next easiest, and so on.  And where one things depends on another, make them follow each other as much as possible in the order of dependence."

2. Students should be allowed to come to their own conclusions first, even if they are not wholly right.  Initial attempts are a necessary component of learning.

"The learner should never be told directly how to perform any operation in arithmetic.  Much less should he have the operation performed for him.  I know it is generally much easier for the teacher… either to solve the question for him or tell him directly how to do it…. Now by this generally no effect was produced on the scholar, except admiration of the master's skill in ciphering." 

"Secondly, when the scholar does not understand the question or proposition, he should be allowed to reason upon it in his own way, and agreeably to his own associations…. it is the best way for him at first, and he ought by no means to be interrupted in it or forced out of it."

3. Success breeds confidence.

"Nothing gives a scholar so much confidence in their own powers and stimulates them so much to use their own efforts as to allow them to pursue their own methods and to encourage them in them."

4. Understanding student thinking is a critical component of effective teaching.  Teachers should be able to trace the logic of students so as to inform their teaching.  

"…it is very important that a teacher should be able readily to trace, not only his own associations, but those of all his pupils, when he hears them recite their lessons.  When a proposition or question is made to a scholar, he ought to be able to discover at once whether the scholar understands it or not."

5. Exposition (presentations) should be a regular part of class, and as developing the ability to communicate is highly valuable.

"It is chiefly at recitations that one scholar can compare himself with another; consequently they furnish the most effectual means of promoting emulation.  They are an excellent exercise for the scholar, for forming the habit of expressing his ideas properly and readily.  The scholar will be likely to learn his lesson more thoroughly when he knows he shall be called upon to explain it."

This is enlightened work, especially given that Colburn had other non-educational interests,  and was not primarily an educator.   Moreover, Colburn's words are quite alarming for those of us working in education in 2013.  The presented ideas provide a mile markers to compare against, and indeed we have known for a long, long time about engagement and learning.  Seventy years after Colburn, John Dewey and others proposed ideas to engage students, and efforts continue to this day to update how we teach on a broad level.  Despite all this our mission is not accomplished, and classrooms with deep engagement are still in the minority.  Hence, evidence supports the notion that implementation is one of the foremost issues in education, if not the most important issue in education.

I am one of many instructors, who believes that finding new knowledge, new ways of teaching, new ways of understanding how students learn, and all that we work for in education can be used to make the world a better place.  It is clear that knowing is not enough, and we need to do a better job of informing the public, colleagues, policy makers, and students about what the main issues are and what the balance of opinion is by experts.  Teaching innovation is good and necessary, but not sufficient.  We are good at innovating in the U.S.  We are much less successful at broad implementation of our innovative ideas.  Finland, the darling of PISA and international comparisons, imports much of their teaching innovations from american research universities.  If only we could more widely implement our own ideas!

To paraphrase Bertrand Russell, are we just mote of dust floating in a small insignificant solar system, or are we what we appear to Hamlet?  Or both?  We have both the capacity to develop wonderful ideas and wonderful ways to teach them.  This is education's Hamlet moment.

The Implementation Challenge:  Can we solve the problem of implementing empirically-validated, student-centered teaching methods widely?

IBL is Fun!

Something that doesn't get talked about enough is how much fun it is to teach via IBL, when things are working well.  At workshops and conferences we focus on the benefits to students and the benefits to learning.  We bring out research papers from the education research literature, and discuss the benefits of active, student-centered instruction.  All these are appropriate and clearly the right thing to do, since there is evidence from literature on K-12 math, K-12 science, K-college non-STEM subjects, research on how people learn, and undergraduate STEM fields.  The evidence from a variety of sources, across subjects, through a long time span, and across education levels points in the same direction.  That's the argument from the scientific point of view.  But is it fun?  The answer is undoubtedly yes!

One aspects of IBL teaching that gets overlooked in our efforts to be careful and scientific is that at a basic level IBL courses are fun teach (when it's working well).  You get to be the mentor in the middle.  Students, who are initially tentative start to open up.  They do more, they show more, and then there are those special moments when you see students realize that they can be players in the game.  They enter the fray, respond, and contribute.

One of the great experiences as a teacher is working with students as a team with the goals of becoming better, smarter, and more knowledgable.  That's a wonderful process to be a part of.  Working with the students,... coaching, mentoring, celebrating the successes, getting through the brick walls, and seeing the development over time is highly rewarding.   The growth mindset is something you can see, and when the growth-mindset light bulb turns on, and you were there to be a part of it… that's magic! That's what makes it worth all those nameless late Tuesdays nights, grading, mulling, thinking of what to do next...  We might ask what this is all for?  And then you see the results, stand back, and smile.

IBL is fun!

Note: We'll have more posts in the future showcasing the transformative experiences students have had.  These experiences have not be capture in research papers yet, and we are collecting videos of students who can share their experiences in IBL classes.  These videos hit on "IBL is fun" and how rewarding it is professionally to be able to create a classroom culture, where such things are possible.

Active vs. "Active"

Not all active classrooms or activities are forms of IBL teaching.  The nature and quality of active learning experiences can differ widely among instructors.  In fact, two people saying they implement active, student-centered instruction can be as different as it gets, where one is essentially a traditional instructor and the other on the full IBL side.  For instance using handouts/worksheets in class can be implemented in a vast array of different ways from low engagement in memorizable procedures to true mathematical investigations and discoveries.  How active learning is implemented matters greatly.

We can highlight the differences between truly engaging active classes from those that have the veneer of active learning in several ways.  In this post, how instructors ask questions is highlighted as one area where differences can be seen.  The quality of the questioning by an instructor says much about the active vs "active."  Below are two examples, set in a calculus context, but the specific context can easily be from any math course.

Example 1:  Instructor A asks students, "What is the derivative of $f(x)=x^{3}$ at $x=2$?"  Pause. A student volunteers, "12"  "Good!"

In the first example, Instructor A asks a simple question that has a numerical answer.  If the answer is correctly stated by someone, the instructor may feel that the students can do the problem.  An untrained observer would think all is well and that this is an example of active learning.  The teacher asked a question and received the correct answer from a student.  All good, right?  Well, not so fast.  I think we can do better.  This is essentially a closed process.  Once the answer is given, it's time to move.  Instructor A may ask, "Are there any questions?" but that question has lost much of its value in math classes today.

Example 2:  Instructor B asks students, "Take a moment to find the derivative of $f(x)=x^{3}$ at $x=2$, and check with your neighbor… [Instructor B selects a student using some method to distribute evenly who gets called on].  Linda, would you share how you did this example?… Thank you!"

In the second example, Instructor B asks a student to explain how she went about the problem, hence encouraging a discussion of the important aspects of a computational problem.  The focus is on the process, and the answer is a component of the discussion.  Based on the response, Instructor B can restate Linda's ideas, ask follow-up questions, ask another student if he/she agrees, and so on.  It's a discussion that is open-ended.

These two examples are based on a basic, computational task.  Greater distinctions in questioning and ensuing discussions exist when topics become more sophisticated, which highlights the importance of asking good questions.

Let's go back to example 1 to see what else we can learn.  Asking a question to the class and waiting for volunteers usually leads to the situation where only a handful of students regularly contribute to the discussion.  Hence a related trap to conventional questioning strategies is that it may only encourage a few students to participate.

Another potential trap for instructor A is reluctance to give harder questions in class or at least discuss them in class, due to the fact that many questions do not have simple, succinct answers.  If instructor A seeks quick answers (for the sake of time), then it limits the level of the tasks presented to students.  While issues like time and coverage play a role, it is a limited framework for learning if the hard material is covered by the instructor and the students are left with the easy material here and there.

Let's push this idea further and consider a more involved task related to Calculus 1.  The task is "Find the equation of the tangent like to $f(x) = \sin (x)$ at $x =\frac{\pi}{4}$."  It is likely that some (many?) calculus 1 students would get stuck on some aspect of the problem, at least for a short period of time.  Such a task would not be given to students as a quick answer question, since it involves several steps, so Instructor A is more likely to demonstrate it, because "it's too hard" or "takes too long."

We know from math education research that the twin pillars are (a) deep engagement in rich mathematical tasks, and (b) opportunities for students to collaborate in some form.  Collaboration is defined broadly, where collaboration can mean anything from pairs to small groups to specialized whole group discussions (i.e. pure Moore Method presentations.)   Instructor A achieves neither of these goals through the style of questioning employed.

Engaging students requires a different skill set compared to making clear, lucid presentations, and starts with effective questioning.  Here are some examples.

• Could you explain how you got the answer?
• Could you explain how you thought about the problem?
• Talk to a neighbor for a moment and come up with two comments or questions.
• Could you explain Linda's strategy to your neighbor/the class?
• Consider alternative strategy ABC, and determine if it will work on the previous problem?  Think for a moment, and feel free to discuss it with someone...  Now I'm going ask a few students to share their thoughts…

How we ask questions is critically important.  Questions are moments in a class when students are invited to be participants.  Starting with engaging, open questions is a positive first step towards full IBL instruction, and using engaging questions regularly provides opportunities for new IBL instructors to practice critically important questioning and listening/observational skills.  Even if you are just dipping your toes in the IBL waters, you can try these techniques right now with low risk and minimal investment.

Instructors can self assess quickly, by reflecting on what kinds of questions they ask and how these questions are implemented.  Such self assessment is useful and can lead to big changes in the positive direction.

A Characteristic of IBL Teaching: Mediating the Interaction Between Students and Mathematics

A question that comes up frequently is, "What are some of the main features of IBL teaching?"  One of the main characteristics of IBL teaching that I'd like to highlight in this post is mediating the interaction between students and mathematics.  

Education is full of monikers for active, student-centered teaching.  We say things like, "Guide on the side," "Mentor in the middle," "Coach," etc.  These are nice ways to think about the nature of IBL teaching, and it's important to unpack what these things specifically mean.

In IBL Math there exists a particular role for instructors, which is to set-up and mediate the problem-solving process.  Students are engaged in exploring the mathematical landscape via well-chosen, logically ordered sequences of problems.  As students engage in their explorations, the instructor works to support the class so that students' interactions with the mathematics results in learning.  

Here are some ways IBL instructors mediate the interaction between students and mathematics.  Instructors

1. design/adapt appropriate curriculum.  The problem sequences must be matched to the goals of the course, must be logically consistent and coherent, and meet the needs of the specific students in the class.
2. keep students going without getting overly stuck.  That is, struggle is good if the struggle is fruitful.  Being stuck is part of the learning process, and students come in with a "tolerance threshold" for being stuck.  It's important to stay within these tolerances so that students stay in the game.  One positive outcome of an IBL course is that tolerance for being stuck can increase. It's something that can be improved upon.
3. provide scaffolding.  This idea is related to the previous point is being able to control the "being stuck phases" by offering "bread crumbs" (hints, lemmas, sub problems, deploying group work, etc.) in just the right type and size to keep the cognitive level of the task high, while simultaneously avoiding student shutdown due to being overly stuck.  This is where calibrating to specific classes comes into play, and making good choices depends on having good data about students.  Listening, observing, and interacting regularly with students forms the foundation of this piece.  With good data in hand, it's often clear what scaffolding is needed.  "Oh, students are having trouble thinking about supremum.  Students may need to work on a couple of problems about upper bounds…"
4. set the class structure and environment for positive discourse.  Class presentations are of no value if the class environment is not setup appropriately.  IBL instructors set the roles, expectations, and procedures of class discourse, which is used to share and validate ideas.  The ethos of the class must include inquiry, discovery, mutual respect for students and learning, and valuing "mistakes" as important discoveries.
5. provide structure of the material being covered, so that students know where they are in the mathematical landscape.  This is an excellent place to inject small lectures and/or organizational tasks, where students are not necessarily solving new problems, but organizing the information they have recently studied.  

The central focus is on the math and how students are working on the math.  The first major component is creating problems that guide students through the mathematical landscape.  Deploying these tasks determines the nature of the interaction students have with the mathematics.   As students work on solving problems, help or direction is needed.  The instructor provides enough assistance to keep students in the learning zone, without taking away all of the fun and exploration.  When students make progress and discoveries, the instructor provides a forum for the ideas to be shared, vetted, and learned by all.

This is the notion of mediating the interaction between students and the mathematics. It is a central component of effective IBL teaching, and a healthy perspective for instructors to embrace.

Related Posts:

IBL Teaching Levels

Asking Good Questions

Sense-Making Continental Divide

AIBL YouTube Channel

Videos from the 2013 Legacy of RLM/IBL Conference have been uploaded to the AIBL YouTube Channel.  You can find all of the sessions from the conference HERE. 

David Pengelley's Closing Plenary Talk

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.

Link: "Harnessing Your Personality"

I'm pushing out a blog post by my good friend and colleague, Matt Jones, CSU Dominguez Hills, who has a blog (Math Switch).

Harnessing Your Personality

"Ignorance Isn't All That It's Cracked Up to Be!"

In this post, I have invited my good friend and colleague, Ed Parker, to be a guest blogger.  Ed sometimes mentions, “Ignorance isn’t all that it’s cracked up to be,” when we have work talked.  Typically this quote comes up when we are working in workshops, and the discussion is about prior knowledge and content coverage.  Ed attributes John Neuberger for the quote, which he discusses in detail below.  (John is Ed's thesis advisor.)

Ignorance in teaching comes up in several ways, and I'll comment on one way specifically here, and then turn it over to Ed.  One way “ignorance” comes up is in the difference between (a) teaching a topic that students have never seen before vs. (b) teaching a topic students have seen in a prior course.  Let’s take Topology vs. Euclidean Geometry, and focus on Euclidean Geometry first.  College students have been exposed to Euclidean Geometry in K-12, and often their incomplete knowledge of Euclidean Geometry gets in the way of new learning, because they have memorized some things correctly and others incorrectly in a way that is disconnected from the axioms and first principles.  When we ask students to prove a fact in Euclidean Geometry, they often don’t know where to start, because it is hard to distinguish between what is allowed as a fact or axiom.  Many of the statements are “obvious,” and thus the teacher has to navigate around teaching and learning obstacles through all this, which is no easy task!

On the other hand, Topology is a topic students have not usually seen before.  Hence starting with some assumptions and definitions provides a clean slate.  Students are on a level playing field, and the focus is on taking the definitions, understanding them deeply, and then moving to proving theorems.  Students can do better sometimes when they are completely ignorant, because they can then do math in a way that is exactly the way mathematicians do math.  They have to work with definitions and grapple with mathematical definitions, logic, and all that.

The fact that Euclidean Geometry can be more difficult to teach than Topology lies to a significant degree in the fact that something learned poorly has serious unintended consequences.  Students come in with prior incomplete knowledge, expectations, and perhaps unhelpful habits of mind.  The same can be said of subjects like Calculus, where college students are likely to have seen many of the ideas at least once before.  Thus, in such courses IBL instructors have a more complex set of challenges to know about, learn about, and address.  All of this underscores the weight and significance of what students walk into a course with.  We'll dedicate future blog posts to share some strategies for dealing with these kinds of issues.

Now back to "Ignorance Isn't All That It's Cracked Up to Be!"... Take it away, Ed!

-- By G. Edgar Parker, Professor Emeritus, James Madison University

Ed Parker, James Madison University

In my last year of graduate school, I went to the national meetings to interview and spent my "spare" time going to talks. I heard one by Herbert Amann that had an idea in it that intrigued me and I followed up on it, and, after I had done enough background work to put the ideas firmly enough in mind to be able to articulate my conjectures, I went to talk to John Neuberger, my thesis advisor, about it. Since my take on the problem (it was in dynamical systems) was topological in nature, John suggested that I go talk to Bill Mahavier about it. When I got my audience with Dr. Mahavier, he listened carefully, thought about it a few minutes, got up from his desk, and walked over to a stack of off-prints stacked on the floor to about shoulder height (there was a time before the Internet when correspondence consisted of exchanges of letters on paper and off-prints of publications!). As he worked his way about a third of the way down from the top, his first remark was, "This is why old mathematicians know so much more than young mathematicians.". He pulled a paper from the stack, went back to his desk, peered at the paper for a few minutes, and then proceeded to remind me of some work I had done, gave me some definitions that he said should be accessible, and suggested a couple of problems that he indicated might be of some use if I could solve them. Mind you, he didn't show me the paper nor did he even give me the reference; I still don't know what it was!

But he then offered me some professional advice. He told me, "Ed (I had already defended, so I was no longer Mr. Parker), to my mind, when to go to the literature is the hardest decision a researcher has to make. If you take on a problem and the first thing you do is go find out what everyone else knows about it, your thinking is almost certain to be channeled by their approaches to the problem. The only way you'll be able to improve the results that way is to be smarter than they are since they probably know the literature well. (Not so subtle between-the-lines message: and you're not smarter than they are.) On the other hand, if you can understand the problem and go to work on it, normal minds running free can get unique perspectives that sometimes allow us (sic) to see things that others don't. But suppose that you really want to solve the problem, but after three months you aren't getting anywhere. Is it time to go to the literature? And, if you choose to, do you stay there until you are well-informed, or just until you have a new idea to work with?" And he left it at that.

I don't know whether this advice was intended to have global applicability or whether it was intended just for me. My history as a student was that I was a painfully slow learner of other people's mathematics and couldn't tell the "hard" problems from the "easy" problems so I had about the same success with both and usually my proofs, I found when I later became familiar with the literature, were consistently not made along the lines of thinking that produced the proofs that made the books. Trevor Evans, from whom I took my algebra courses at Emory, used to say after I would present, "Yes, Mr. Parker, I suppose you are correct, but (as he would take the chalk away from me) WHY DIDN'T YOU THINK OF THIS?", and he would proceed to show the class a "reasonable" proof. Most of my work in the literature my first decade out of graduate school was spent teaching myself enough to be able to teach my courses or going to the literature to make sure that what I wanted to write up hadn't been done.

Fast forward about 25 years. I was talking to Dr. Neuberger about a problem on power series connected to wherever Jim Sochacki and I were at the time in our exploration of polynomial projection and in passing, made the comment, "You know me, I didn't know hardly any of this stuff..." and kept on describing the idea. When I came up for air, John said, "You know, ignorance is highly overrated." I don't think he was being critical; the message I got was, you could be using the same brain cells that you are using to figure out stuff that other people don't know that you now have to expend to figure out stuff that many undergraduates know.

I'm guessing that the tension between Mahavier's advice and Neuberger's admonition has something to do, by analogy, with "the coverage issue". Hopefully, we put together our course notes well enough so that our students not only experience the growth of solving their own problems, but also come away with a corpus of "facts" that at least prepares them to be a few hours reading away from being literate about any one thing that they might "need to know". I continue to believe that if one is forced to choose between students "doing" and "being shown so as to achieve 'literacy'", one chooses "doing". I believe that the power associated with "learning to learn" is far greater than any level of mastery of someone else's bag of techniques. On the other hand, I'd like to be a whole lot better at communicating that, "now that you have done this, don't stop", is what we're after rather than "you are now certified to repeat some pre-packaged dose of curriculum neatly packaged and ingested".

Ed Presenting at the Legacy of RLM Conference, June 2013

Nevertheless, we should, by intention, be purposeful in trying to make the “curriculum” demarcated by our course notes as informative of the “curriculum” as dictated by the contents of a standard text in the subject as is feasible.  In my own case, I have spent considerable time trying to optimize the fit between the mathematics my students will try to make in my courses with the curricular expectation I perceive that my teaching colleagues have.  I would like what may appear as ignorance to be a foundation for “I’ve seen that idea before.”.  If our students learn to learn, our colleagues will be happy with them, regardless of whether the students can knee-jerk responses.