The IBL Blog focuses on promoting the use of inquiry-based learning methods in college mathematics classrooms. Learn more about IBL at The Academy of Inquiry Based Learning
First, let's look at the big picture. The major problems in Math Ed are not at the top end of the achievement distribution, where students are successful no matter what pedagogy is employed. This isn't the first priority when one is thoughtful about the landscape of issues that we face in our system. This isn't to say that we should ignore opportunities here or the students. It's just that the way to handle high-achieving students is clear and straightforward. Let's dispatch this one quickly...
General effective pedagogy includes keeping all your students, no matter the level, engaged in activities appropriate to their level. High-achieving students are a pleasure to work with. These students like math, they are motivated, and want to learn more! So there's no barrier here to deal with other than laziness. All these students need are good problems and some feedback.
The main ideas to keep high-achieving students engaged are listed below.
Problem sets should include problems challenging to your high-achieving students.
High-achieving students can also be given extra problem sets where they submit proofs in writing. These problems would not be presented in class normally.
Additional /supplemental articles or chapters can also be assigned.
So there you go. Give them problems at their level, and then get out of the way. Offer feedback and support as necessary. What's the trajectory? Junior/senior students should be able to do graduate-level work after about a year of full IBL in a particular subject (e.g. a year of Real Analysis, Topology or Abstract Algebra).
Additionally, high-achieving students can become peer mentors and teachers. They can model how to write a good proof, offer constructive feedback in ways that are often better received than from the instructor, and can be deployed to help struggling students, via small group work.
When you have high-achieve, socially skilled students, you have a great, great asset that can positively affect the learning culture of your classroom. Good teaching practices tell us to keep all students engaged appropriately, and then one can redirect all this talent and ingenuity to increase the level of the whole class.
Jackie Jensen Vallin is a faculty member at Slippery Rock University in PA. Jackie has been the co-organizer for the Legacy of R. L. Moore conference, and has been actively involved in mentoring new IBL instructors, the MAA, and Project NExT.
1. How long have you been teaching and what was your teaching style before you started using IBL?
Counting my teaching experiences in graduate school, I have been teaching for 15 years. In the early days, including when I finished graduate school and had to write my first “Teaching Statement,” I was much more of a lecture-based instructor. I even said in that statement “I am more of a sage on the stage than a guide on the side.” This was probably a little misleading since I really had a very interactive style with my students – knowing them all by name and calling on them to further the conversation in class. However, it was pretty unusual to do anything in my classroom except lecture and a few group-work style worksheets.
2. How did you learn about IBL and when did you begin using it in your classes?
The spring of my first year of “real teaching” (in a faculty position), I attended the Legacy of RL Moore Conference in Austin, TX. There were a lot of people talking about the Moore Method (and, appropriately enough) refusing to define that method. But I met some great educators and was intrigued by the idea of putting the responsibility for learning more firmly on my students by guiding them to the answers without pretending that my lecturing them would be enough to get them to understand.
I began simply after that – introducing more worksheets into my classes, doing more group work, and implementing presentation days during calculus. It wasn’t until a couple of years later that I taught a class in which the students were really responsible for presenting almost all of the material. And even then I didn’t do that in all of my classes, but only in upper level courses (intro to proofs and abstract algebra, in particular). It was another couple of years before I integrated IBL into all of my classes, and now I teach everything (from Math as a Liberal Art to Math for Future Teachers to Intro to Proofs) with as many student-centered activities as I can.
3. What was one of your best IBL experiences?
My best IBL experiences actually happen in lower level courses – students take responsibility for their own learning. This means that frequently students who thought that they were bad at math get to master that fear, the material, and convince themselves that they are not “bad at math.” This happened this semester in my Financial Mathematics (for non-majors) courses – students were shown a couple of examples of how to do problems, and then completed problems on their own, in groups, sharing answers at the end of the period. I walked around the room, checking work, answering questions (by asking more questions) and making sure everyone was on task. Since this course is based entirely on applying formulas to word problem scenarios, the only way for students to learn to read word problems is by making them *do* word problems. And they are succeeding! What a great semester!
4. What advice do you have for new IBL instructors?
Start in a way that you feel comfortable – if that means turning over presentation to students one day per week and lecturing the other days, then do that. Don’t let anyone else tell you how *you* have to do IBL. Find your own method. And ask lots of people for ideas, suggestions about notes or activities, or if you feel stuck. I have gotten great pieces of advice from many different people – I steal the parts that work for me and discard the pieces that don’t to have found my own method of IBL. Everyone is willing to help and talk about teaching – we all care a great deal about our students, or we wouldn’t spend so much time developing good course notes!
Note: This is great advice -- "find a level that is comfortable for you." If you want to be "scripted" at the beginning that is also fine, and lets you concentrate on the day-to-day teaching aspects. Most importantly, if you need help, let me know!
"I may have exaggerated somewhat in order to make plain the typical points of the old education: its passivity of attitude, its mechanical massing of children, its uniformity of curriculum and method. It may be summed up by stating that the centre of gravity is outside the child. It is in the teacher, the textbook, anywhere and everywhere you please except in the immediate instincts and activities of the child himself. On that basis there is not much to be said about the life of the child. A good deal might be said about the studying of the child, but the school is not the place where the child lives. Now the change which is coming into our education is the shifting of the centre of gravity. It is a change, a revolution, not unlike that introduced by Copernicus when the astronomical centre shifted from the earth to the sun. In this case the child becomes the sun about which the appliances of education revolve; he is the centre about which they are organized."http://bit.ly/JvR3cG
This passage is from "The School and Society," originally published more than 100 years ago (in 1899). It is relevant today, sadly. Are students at the center of instruction in math classes? Mostly no. Math teachers predominantly lecture at students, and the great change that Dewey saw has not yet come about, though many teachers have made the shift. A question one can ask is "Why have things not changed significantly in all these years?" Sure the books have colors, and we have technology beyond our grandparents' wildest dreams. But when you look beyond mere surface beauty, you can see that the heart of it is still the teacher telling, and the students following.
One major issue behind the lack of change is data. More specifically an issue that persists is in assessing teaching and learning, and more pointedly how this data might change our fundamental beliefs (or axioms) about teaching and learning. What we assess and how we assess it determines our evaluation of student ability and achievement. Herein lies one of our fundamental issues. Lack of good assessments can lead us to continue doing what we have been doing.
To get some insights, let's look outside of education to provide a backdrop for analyzing our own system. One of the unique aspects of baseball is the wealth of statistical information that has been available for generations upon generations of players. It's one of the reasons why baseball is such a wonderfully interesting sport to be a fan of.
Earned Run Average (or ERA) is one of the traditional measures of a pitcher's ability. A lower ERA is considered better, since the pitcher gives up fewer runs per 9 innings. The problem with ERA is that it is a noisy and flawed measurement system of pitching effectiveness. It depends on factors not under control of the pitcher, such as the quality of the defense supporting the pitcher and the effects of stadiums on balls batted in play. Some pitchers are overvalued and some are undervalued in terms of their contributions to team wins, if ERA is weighted too heavily as a measure of ability. Voros McCraken conducted some groundbreaking analysis, establishing the concept itself and subsequently methods to measure pitchers that are "defense independent." This story among others is chronicled in "Moneyball" by Michael Lewis. But Voros didn't expect baseball teams to rejoice when learning about his findings. He knew better.
"The problem with major league baseball... is that it is a self-populating institution. Knowledge is institutionalized. The people involved with baseball who aren't players are ex-players... They aren't equipped to evaluate their own systems. They don't have mechanisms to let in the good and get rid of the bad." (Voros McCraken)
This is a striking insight! Voros essentially identifies why baseball resisted modern statistical methods that could help. Baseball is not set up as an institution to evaluate how it evaluates players. Baseball people normally did not have the knowledge, ability or willingness to entertain ideas developed by people like Voros, who is a baseball outsider.
What is the implication for us in the teaching profession? It should be stated that major league baseball and education are not very similar as institutions. That said, we have some similarities and we can draw conclusions about our shortcomings from baseball's own struggles. Indeed teaching is also a self-populating institution. Students who do well in the current system are the ones who end up become teachers or professors. Some future math teachers state things like, "The reason why I like math is because there's always one right answer, and there's a simple, straightforward structure to all problems." They are good at memorizing rote skills, the rote skills appear on tests, they get good grades, they are labeled as good in math (which may or may not be true), and then they model themselves after their favorite teacher. Thus the cycle perpetuates.
Colleges and universities have in their mission the goal of seeking truth and knowledge. When it comes to teaching, however, discussions among faculty often are about style, "what my students like...," and about delivery of information. The focus is usually not on learning and what students are doing. A major point is that we do not use the scientific method to evaluate teaching, just as major league baseball didn't use any scientific methods to validate their player valuation systems.
Consequently we have several metric problems. Are the usual metrics like skills-based tests and student evaluations the right ones? Clearly the answer is no. Let's consider the typical calculus sequence with a thousand-plus page texts. In the typical chapter on optimization in calculus books, the authors usually highlight in a colored box the steps for how to find relative extrema. What this tells many (but not all) students is that they should memorize the recipe and regurgitate it on an exam. That's how one can get a good grade after all. These students will not walk away with a conceptual understanding of the subject, and probably will forget what they have memorized once the term is over. In short, their education is unintentionally of a lower quality than what we want. Mathematics is reduced to applying recipes that many students do not understand or even care to understand.
If you don't believe this can happen, here's data from Physics by Professor Eric Mazur, Harvard University, presenting at the University of Waterloo. (It's 1 hour long, but worth it!) At Harvard, 40% of the students in freshmen physics who did well on the procedures had inadequate understanding of basic concepts.
The result of traditional assessments is that many students who are traditionally given good grades have major gaps in understanding of basic concepts. Students think they know it, but maintain "Aristotelean understanding of Physics" rather than a Newtonian one. Their education amounts to very little, even at a sublime places like Harvard.
Now let's consider traditional teaching assessments (i.e. student evaluations), and consider data from Physics, based on the Force Concept Inventory (FCI). All of the reddata points are from traditional instructors who lecture. Represented in these data points are teachers who are highly rated and lowly rated on student evaluations -- the red dots contain some star teachers and the teachers on the "oh bummer" list. And they all do about the same on the FCI within statistical significance! Students gain on average about 23% of what is possible in the pre-post test design. Student evaluations are like ERA. They are a noisy, flawed metric. Actually student evaluations are worse than ERA. ERA has some value in aggregate (whole team ERA), and outliers tend to have outlier ERAs. In contrast, the highly-rated, award winning instructors are doing no better than Dr. Boring or Professor Snoozer.
The green data points represent faculty who use Interactive Engagement in their classrooms. One of the main trends is that there is very little overlap between the reds and the greens. The average green gain is double compared to traditional instruction. Thus a better way to measure if an instructor is effective is to know what skills and practice he or she utilizes in the classroom. While crude and incomplete, it at least it tells you whether the instructor is on the red or green distribution. But these qualities and practices are not usually assessed or measured in teaching evaluations, so there does not exist sufficient data or incentive for the system to embrace change. We keep on doing the usual, while the traditional assessments tell us things are okay. And the results keep staying in the "redzone" above. Education has a bunch of Voros McCrakens, so there is hope. Baseball has changed, and I believe education will continue to improve for the better.
What about Math? Calculus Concept Inventory has been rolled out and studies are underway. Thus there is hope that we will embrace new assessments that tell us what is going on. Preliminary results suggest similar outcomes to the FCI. Interactive Engagement and Traditional instruction are different distributions. I look forward to seeing the published results. Moreover, a growing body of evidence in research in undergraduate education also suggests that students in traditional courses are not learning what we want them to learn. (More on this in a future post.)
The MAA's Calculus Study indicates that 80% of college calculus courses are taught in sections of 40 students or less. Additionally, very few institutions have large lectures for upper-level courses. Ample opportunities exist for IBL methods to be deployed courses across the nation.
What can an individual instructor do personally? Looking at data can be demoralizing at times, but one should be optimistic. In particular, one can turn assessments into valuable tools that guides students and instructors in the right direction.
Assessment is more than grading stuff so that you can assign course grades. Assessments should be utilized in ways that provide students with regular feedback (formative), instructors with information about their students (formative), and to evaluate demonstrated achievement (summative). Assessments should provide incentives for the qualities we actually value, including creativity, clarity, exploration, problem-solving ability, and communication.
Ideas for what to assess:
Student presentations and/or small group work
Reading or journal assignment
Math portfolios
Exams
Homework
The items above are not revolutionary. What matters is what we put into them. Exams can be rote skill based or they can also test for conceptual understanding, application of ideas, and problem solving. Homework can be made more interesting.
Student presentations and/or small groups are a wonderful way to assess understanding. When students present their proofs or solutions, it often a rich experience and rich source of information. You see it all in IBL classes: great ideas, small ideas, half-baked ideas, insightful questions, victories and defeats. It's a slice of real life, and it's really great. It's very easy to detect where students are at, and then take action.
Reading assignments can be used to offload (i.e. flip a class) basics to homework, leaving time in class for the harder tasks, where inquiry is useful. Portfolios are like a CV, and can be used to demonstrate what a student has been able to prove on his or her own. Additionally portfolios can be used to create a record of the theorems proved by the class. (I'll write more about portfolios in future posts.)
In IBL courses, one has continuous formative assessment. Instructors are always analyzing whether students understand an idea or not, by giving students meaningful tasks and then working with them to overcome learning challenges. If students are stuck, then there is another question or problem that can be posed, and then students are off on another mathematical adventure. Students are continuously engaged, are monitored, are self-monitoring, get feedback, and so on. This rich, integrated assessment system of actual learning is a core advantage in IBL teaching.
Returning to Dewey... If we continue to teach and assess teaching in traditional ways, we will not gather the data and information necessary that support change for individuals and systemwide. Gathering good data about our students' thinking, which is also a core part of effective teaching, is a key to the way out. Engage your students, collect good data, share it, publish it.
Upward and onward!
"It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so." - Mark Twain
The end of the academic year is a busy, busy, busy time of year. Commencement, grading, fatigue, summer plans, submitting finals grades,... so much going on.
My recommendation is to stop for 30 minutes. Get a cup of tea or coffee. Grab a sheet of paper and a pen, and write about your year in teaching. Write down what your teaching successes were and what you want to improve on the year. The hard won knowledge and experience from the academic year, from all the grading, and class time can slip away with time. But if you just spend a few minutes to write your thoughts, it makes it much easier to recall what happened. This makes it much more likely that you will see big gains the next time.
As they say luck is when preparation meets opportunity. In teaching, part of preparation is having a framework that effectively guides the next round of preparation.
Take notes, then take a break.
Congrats to all new graduates this month and next!
One of the most frequent things I hear from my fellow colleagues is "I like lecture. I love listening to lucid presentations." I do to. In fact, great lectures are wonderful experiences, and many instructors want to inspire students just as they have been inspired. This is a noble sentiment, and shows that instructors care deeply about their students. In this post I discuss the role of inspiration and lectures and their limitations.
When we attend performances by musicians it is an aesthetic experience. We are living in the moment. If we are lucky enough to attend a great performance, we can be swept away by the beauty and passion of the event. Art, music, great scholarship are indeed high points of society.
Yo-Yo Ma plays "The Swan"
Herein lies the complexity of teaching and some of the pitfalls of the profession. We can mistake beauty for effectiveness, and this is very easy to do. Just as Sirens luring sailors with their enchanting songs, beauty, lucidly, and inspiration can lure a teacher to lose sight of the point of school and effective teaching.
I make my point by analogy. Going to see musicians like Yo-Yo Ma perform are an inspirational experiences. It is a necessary but not sufficient condition for studying to become a musician. If you want to be a musician, you have to play. Play lots. Play often, and reflect.
Similarly learning mathematics requires one to see the subject as meaningful. But one also has to do the hard work to build the mind. Listening to a lucid, inspirational math talk can be enlightening. BUT lectures do not build capacity to do mathematics with nearly all listeners, especially novices. Mathematicians already have the capacity to think mathematically -- we can learn what we need from a lecture much of the time. For students who are not yet advanced in mathematical thinking, they cannot take away the same lessons from the very same lecture.
If you want your novice cellist to learn to play, you can't just show a vid of Yo-Yo Ma, and say "There you go. Now go home and practice hard." It's not that simple. Telling isn't teaching. Likewise really understanding calculus, deciphering nested quantified statements in theoretical math courses, and learning to build differential equation to model a physical situation requires more than just following a recipe, processed by the professional mathematician (i.e. the equivalent of Yo-Yo Ma).
Students need a supportive environment, well-matched problems, time to be stuck, and opportunities to figure things out for themselves. It is this long, arduous, and rewarding process that unlocks potential. Good musicians know this. They don't just listen to someone else play. They also practice with intent with teachers, with collaborators, and with new music to keep their minds and hearts fresh. They experiment. They learn new skills, they interpret pieces in their own way. They do.
To put it simply, lectures and IBL methods should be used for different purposes. Lectures can be used to inspire. IBL methods should be used to build students' abilities and capacities to do.
The myth that "Teaching is Art" is one that lets us rationalize "aesthetics = effectiveness." I am not criticizing aesthetics, by the way. Beautiful ideas are why we are here. But there is a difference between showing students something beautiful and helping students become young mathematicians. Horses for courses, as they say across the pond. Or use the right tool for the job over here in the U.S.
What's the right mix of lecture and IBL? We don't know exactly. Data suggest a small percentage of the time should be lecture, and that as instructors talk more, students report less learning gains. Many of the most effective and experienced IBL instructors use IBL daily, and intersperse mini lectures as needed or at opportune times. Examples are (a) when the students have completed a body of work, (b) to showcase for students how certain ideas or techniques can be further used, (c) to summarize big ideas, (d) enculturation, (e) exposing students to things that there is not sufficient time for, (f) summarizing student strategies and proof techniques from the week,...
"Science is 1% inspiration and 99% perspiration." -- Albert Einstein
The current fad in education news in mainstream media is the Value Added Model or VAM. It goes like this. There are a bunch of bad teachers at the cause of our woes in education, and we need to find them. So they are evaluating teachers, and applying business model methods like VAM. There exists a group of people in the United State who presuppose that the problem with education is a bunch of "bad teachers." The purpose of my post is to redirect this discussion into one that is based on reality and one that would actually help us solve real problems. The current black-and-white debate about education today misses many important issues and ultimately, in my opinion, will be counterproductive, leading to inadequate responses.
I point out that essentially every person in our country wants to improve education and help children become good thinkers. We all can agree on this. The problem with our debate is that we are focusing far too much on teacher evaluation, while missing the larger picture. In this post I explain what I mean by this.
First, pointing fingers at one group or another is counterproductive. The current media attention has been to place blame on teachers (and teacher unions) and to promote the use of VAM-based evaluations of teachers. Then using this data, some districts have publicly humiliated teachers who perform poorly by publishing this data. Even business leaders know that public humiliation is not the way to bring about positive changes, as Bill Gates writes.
Second, VAM appears to have dubious intellectual or scientific merit for teacher evaluation. Jesse Rothstein (UC Berkeley) published a paper that indicating the using VAM is not suitable for measuring teacher effectiveness. He states, "My results indicate that policies based on these VAMs will reward or punish teachers who do not deserve it and fail to reward or punish teachers who do." The essence here is that certain assumptions being made are actually false. VAM might work if we randomly assigned students to teachers. In the U.S. we track students. If I know who your child's teacher is in 5th grade, I can infer information about how you did in 4th grade. In order for VAM to work, it has to be the case that know a child's 5th grade teacher has no influence on how the child did in the 4th grade. This condition is false.
To put it simply, we may be using a hammer to paint our walls. Wrong tool. Wrong job. We could realize in the future that we fired a bunch of teachers erroneously. It would be deeply unethical to use a system without knowing it actually works, and then to make life and career altering decisions based on it.
Third, it is assumed that teachers have adequate control and power, and hence should be held accountable. Rhetoric based on this premise sounds good from the podium, but unfortunately is incomplete. Richard Ingersoll (University of Pennsylvania) portrays the climate teachers work in in his article "Short on Power, Long on Responsibility." With great responsibility placed upon then, they are given far too little control or input on issues that would allow them to do their jobs better.
teachers generally have little input or influence on overall school curriculum
teachers have little input on course assignments or class size
teachers generally have little input into schoolwide behavioral and disciplinary policies
teachers rarely have the authority to have disruptive students removed from their classrooms
teachers have little input into hiring, firing, budgetary decisions, and whether to hold back or promote students based on their academic performances
teachers have little input on the content and form of their own on-the-job development and inservice programs
Ingersoll writes "Teachers are akin to men or women in the middle. A useful analogy is that of supervisors, or foremen, caught between the contradictory demands and needs to two groups: their superordinates -- school administrators -- and their subordinates -- students. Teachers are not part of management and they are not workers... although [teaching] involves much responsibility, it involves little real power." If we are to hold someone accountable for a job, then they have to have commensurate power and control over this job.
Fourth, perverse incentives may come into play that are unforeseen at this time. No one can think through every detail of a system that involves millions of people. So here is a hypothetical situation:
Suppose Mrs. Jones has two students with learning disabilities. These students' disabilities have been undiagnosed until Mrs. Jones takes the time and effort to figure this out and help these children deal with and overcome their disabilities. These students succeed, learn, and are set on path that will lead them to a productive and fulfilling life. This is a major victory for education, for Mrs. Jones, and especially for her students. BUT because these students were not able to learn some topics in previous years, due to their disabilities, they do not score well on state tests. Thus Mrs. Jones is labeled a "bad" teacher. If instead, Mrs. Jones ignored the evidence of a disability, and had these students removed from her class somehow (i.e. failed them), she would then be labeled a "good" teacher, but at great cost to the two children, their families and to society.
We know how perverse incentives can lead to terrible outcomes. The financial crisis of 2008 is a prime example. The numerous cases of school districts cheating on high stakes testing is another. If your job depends on a misguided set of incentives, then you will get tragic results. We must carefully think through the evaluation system to make sure it works without creating incentives to cheat or to make unethical choices.
Fifth, it is assumed that test scores measure something of great intrinsic value. VAM is based on test scores. Standardized testing is a long separate discussion in itself, so I will not describe the tests and how they are inadequate in great detail. Standardized tests in the U.S. focus primarily on rote skill types of tasks. These are the kinds of tasks that do not involve high-level thinking or require deep conceptual knowledge. Moreover, these tests do not measure effectively problem-solving ability or other higher-level, critical reasoning abilities. It's not that the tests are useless, but there is some question about what they actually tell us.
I'll argue now by way of analogy. Free throw percentage is a measure of a particular skill in basketball. In math, it could be something like solving linear equations or adding fractions. Here is a ranking of NBA players by Career Free Throw Percentage:
Player 1, 88.9%
Player 2, 87.7%
Player 3, 85.4%
Player 4, 84.8%
Player 5, 83.5%
If I asked you, who is the best player in the list above, you could say player 1. Or if you thought about it a bit, you would say that you don't have enough data. Here are the names of the players:
Scott Skiles, 88.9%
Jeff Hornacek, 87.7%
Mario Elie, 85.4%
Spud Webb, 84.8%
Michael Jordan 83.5%
With no disrespect to any of the players on the list, but I'd choose Michael Jordan to be on my team, even though he's the lowest in rank. In fact, let's consider Shaquille O'Neil. Shaq's career free throw percentage is 52.7%. He's terrible by this measure, yet, in his prime he'd be one of the first players you'd pick for your team.
Let's relate this back to teaching and learning. As free throw percentage is an incomplete measure of basketball ability, so are today's standardized tests an incomplete measure of student mathematical ability. Today we can test for achievement of basic skills, but we do not test for problem-solving ability, proof writing, argumentation, the ability to experiment and explore, and so on. No kid thinks shooting free throws is the end all, be all of basketball. No kid thinks bubbling in scantron forms is the heart of mathematics.
The example goes on. It would not be fair then to compute the team free throw percentage and determine if a coach is good or bad. Let's look once again at the 1992-1993 NBA season. The Chicago Bulls were ranked 22nd out of 27 teams that year in team free throw percentage. By this standard, Phil Jackson should be have been fired as a coach after the end of the season, his name shamefully listed on the internet for all to see with disdain.
What happened that season? The Chicago Bulls won the NBA title over the Phoenix Suns. The Bulls didn't "test" well, but they sure ripped apart the competition. Clearly we didn't use the right evaluation tool here.
Thus, while I agree that we need to ensure high teacher quality, this doesn't excuse using the wrong tools or using a tool prematurely before we are sure it's the right one. I propose that we evaluate teachers in a smart way, and develop a trustworthy evaluation system. This is a big job, and will require a ton of resources. Administrators and teachers must come to an agreement on a fair system that uses scientifically proven methods as well as classroom visitation and other forms of data (such as student portfolios).
Sixth, we have lost sight of the whole system. What we are doing is the proverbial rearranging the deck chairs on the Titanic. If we continue to teach rote skills in passive learning environments, no matter how well we test our students or evaluate teachers, the probability to transform the system is zero. The reason why I believe this to be the case is that focusing only on teacher evaluation misses big pieces of the system like curriculum, learning environments, developing critical thinking, community and school support of academics, switching to student-centered instruction (e.g. IBL) that support higher-level thinking, etc. Teacher evaluation is but one (small) piece.
What should we do?
We should rethink our basic assumptions about education. Now that we live in the google era, it isn't enough to just teach the three R's (reading, 'riting, 'rithmetic). We need superbly critical and creative thinkers. Guy Claxton does a wonderful job of describing what we should do in "What's the Point of School?" Claxton states that the point of school is the great eight qualities. They are:
Curiosity
Courage
Exploration
Experimentation
Imagination
Reasoning
Sociability
Reflection
These qualities are the kinds of big objectives that supporters of student-center or inquiry-based learning methods agree with and pursue daily in our classes. We have seen immense growth in our students in these qualities, when allowed the time and space to give our students challenging math problems to solve and discussion collaboratively.
Consequently, what we should be doing is working as a team or community to help teachers create class environments that support the learning of these great eight qualities. Professional development efforts are the best bang for the buck. Help teachers use effectively the best teaching practices we have available today with better, more modern curricula. Other countries are more supportive of their teachers, especially high performing, modern nations like Singapore, Japan, Finland, and South Korea. Some of these countries view their teachers as "nation builders." Teachers are supported, respected, and given working conditions that allow them to be more effective in the classroom.
Pointing fingers at teachers is counterproductive, and in fact, it is potentially destructive, since we are ignoring more pressing issues. If hypothetically a foreign power wanted to dismantle our educational system, what strategy would they employ? Would they like to see us working together with civility towards a common goal or pointing fingers at one another?
In our rush to judge teachers, we have lost sight of some of the overarching issues that need to be addressed. It's time to shift the discussion to the great eight qualities and how we can pull that off in our classrooms. Teachers are the ones who will implement this, and they need our support and our respect. I encourage teachers, parents, administrators, students, and policy makers to rethink basic assumptions about education, and consider how we can work together, without pointing fingers, to upgrade our system to better educate our youth.
One of the main reasons why IBL produces superior learning outcomes compared to non-IBL teaching methods is metacognition. Metacognition described simplistically is thinking about one's own thinking. Professional academics do this as a habit of mind. We ask ourselves, "What is my approach to this?" or "The way I'm thinking about this is..." It's one of the reasons why we are peculiar.
Students sometimes (often) do not think about their thinking. Most have not had experiences in school that support this. This is easy to think about. If all you do is follow rules and procedures to compute algorithms you don't have to think about your own thinking. All you need to do is follow the thinking of someone else. My intuition about why passive learning fails for most people is that unless you are predisposed to independent thinking, there is little in traditional education that can transform one towards independent, critical thinking. Monitoring one's own thinking is part of the sophisticated set of thinking processes that distinguish experts and novices. And it can be trained!
Where this comes into play in IBL math classes boils down to this: students in IBL math classes must explain their reasoning on a regular basis. I claim that the process of justifying answers engages the metacognitive process. It goes something like this:
Student: "I believe the statement is false." Instructor: "Can you tell us why? Student: "Well... I looked at these examples and then I thought that this one here doesn't satisfy the second condition." Instructor: "Very interesting. What do the rest of you think?..." <discussion ensues>
The support of metacognition in IBL classes is much richer than what is presented in the vignette above. Students are stuck on problems. They are proving theorems from first principles, and are asked to write proofs outside of class. Moreoever, students are required to present their proofs to the entire class for peer review. Students cannot get through class without having to think about their thinking.
Some questions to get you to think about your students' thinking about their thinking:
Do your students think about their thinking and how do you know?
If you are not sure students are thinking deeply about their thinking, what can you do in class and via assignments to encourage this?
What mathematical tasks and class setup could you use to support thinking about thinking?
