Teach Like a Freak, Part 2

Inspired by the latest installment from Steven Dubner and Steve Levitt, Think Like a Freak, in this series I consider what it might mean to Teach Like a Freak. In part 1, I took up the idea of experimentation and how I am currently experimenting with my teaching. In part 2, I examine two other ideas from Think Like a Freak, targeting small problems and thinking like a child.

Dubner and Levitt make the point that it often makes sense to target small problems, even if your goal is to solve a large problem. Thinking as a teacher, I think there are times when we can be overwhelmed by the obstacles our students face, both inside and outside the classroom. I think targeting small problems can help a teacher focus and make manageable, lasting changes. As I described in part 1, I am experimenting with regular use of Interactive Engagement questions in my Transition to Proof class. The reason I am doing this is that I felt that it was often difficult to get discussions of student presentations going, and I have been seeking ways to get more lively discussion and broader participation from students.

Another problem I have targeted is attendance. Students in my classes are absent or late at much higher rates than I would like. Over time, I have tried a number of tactics to solve this problem. I have had maximum allowable absences, which did not work for me, since I did not want to further deduct from students’ grades when (because they missed classes!) they were already in a position where their chances of passing the class were low. Another tactic that I have used with some success is contacting students (via email) when they miss class. Generally, I tell absentees, “We missed you in class,” I may let them know what the next assignment is, and I encourage them to contact me if they wish. My sense is that students get the message that their attendance matters. Of course, I have not experimented (!) to see if I can document the impact of this practice. This semester, I have students submitting responses to IE questions online, and am counting that as part of their grade. Some of the points for those questions are just for submitting a response, so I have effectively made attendance a small part of the grade. I am tracking daily attendance to see if there is an impact. Right now, I still feel like a lot of people are late, but absences seem under control. 

Again stepping back to the larger picture, the main idea of this discussion is to look at teaching not as one monolithic challenge, but as a set of smaller problems, and then to tackle them, either separately or together.

The authors also present the idea that one should think like a child, meaning that a child is not afraid of wild ideas. A child is not bound by the conventional wisdom. As an example of this idea, there is a current movement called Statway, developed by Carnegie (http://www.carnegiefoundation.org/statway) and the Dana Center, that aims to serve students who would otherwise be in a yearlong sequence of developmental mathematics, and instead give them a semester of developmental mathematics plus an additional course tackling issues not directly about mathematics content (for instance, developing students with the mindset that they can get smarter), and then putting them into a college level statistics course. This certainly seems unconventional on the face of it. The most common response to students struggling in mathematics is to blame their prior knowledge. The Statway approach is to treat students within the larger framework of their approach to learning, and to address those issues. Although I have not seen a lot of data, from what I know, Statway is showing promise.

In education, especially higher education, we can be victims of our own success. We are the ones who succeeded in education, so it can be especially hard to challenge the norms that, very often, with which we are enculturated. It takes effort to get outside our own perspective, but it can be done. As a recent post from Grant Wiggins (http://grantwiggins.wordpress.com/2014/10/10/a-veteran-teacher-turned-coach-shadows-2-students-for-2-days-a-sobering-lesson-learned/) demonstrates, one way is to shadow a student. If this is not practical, even carrying on a casual conversation with a student outside of class can offer insights into ways we could be better at helping students learn. Statway is an example of finding a way to make a difference by thinking unconventionally. We in academia are proud of our intelligence, innovativeness, and originality, but we need to widen our focus to those areas that have become accepted, and thus, not questioned, if we are to make strides in helping students.

So, to my fellow educators, get your freak on! Try new ideas, and tackle those small problems.

Teach Like a Freak, Part One

Inspired by the latest installment from Steven Dubner and Steve Levitt, Think Like a Freak, in this two-part series I consider what it might mean to Teach Like a Freak. In part 1, I take up the idea of experimentation and how I am currently experimenting with my teaching.

One of the central premises of Think Like a Freak is that one should be willing to experiment, and to make decisions based on the data gathered. For quite a long time, I have been teaching using inquiry-based learning (IBL), a mode of instruction in which students are the focus of classroom activity, deeply engaged in collaboratively making sense of the content. Evidence has been mounting that IBL specifically (http://www.nctm.org/publications/article.aspx?id=42527) and active learning more generally (http://www.pnas.org/content/111/23/8410.abstract) are more effective than lecture across multiple outcomes. At the same time, I have been reading research about Interactive Engagement (http://www.ams.org/notices/201308/rnoti-p1018.pdf), and have been experimenting with trying to blend IE with IBL. The question for me is how to structure my class to take maximum advantage of these approaches. To force myself to take this question seriously, I promised to speak about what I learned at the JMM 2015 in San Antonio. 

Last year, when I taught Transition to Proof, I had made a handful of IE questions, but class time was spent mostly on students presenting their work, and our discussions of those presentations, and a little bit of pair work. What I am doing this semester is using IE questions every week, which means about 15-25 minutes out of 150 minutes are spent on these short questions, with the rest of the time being spent the same way as last year. So, my goal is to compare the two classes in their understanding and skill in writing proof. The next step is to decide how to assess the impact of blending IBL with IE. This involves deciding what to measure. Another issue is that, once I decide on appropriate measures, it can be difficult to get good comparative data. For purposes of assessing impact, I do not have two classes running simultaneously with which to carefully set up a comparison. The best I can do is to use the data I still have from last year’s class. More specifically, in the previous year's course, I had tried a handful of IE questions, and so I have the results from those as well as exam scores for that class. The key component of my assessment, then, is to compare student exam performance last year to the performance this year, when I am using IE questions on a weekly basis. I do not claim that this will give me a definitive answer, but at least it will be a start. Another thing I have been doing is keeping track of participation in whole class discussions, so that I can compare the number of participants in discussions on IE days with discussions on non-IE days. Although counting the number of participants in whole-class discussion is a somewhat superficial measure, it gives me some quantification of how things run differently with the IE questions. If it seems that students are benefiting from more IE questions, I will keep making time for them in class.

Backing away from the specifics of this question, one thing that I have decided to do with courses that I teach regularly is to keep results of exams broken down by question. The reason for this is that I often modify exams from year to year, and so exam scores from year to year are not directly comparable, but there will be questions that are directly comparable. Another thing that I am learning to do is to keep a log of each class day’s activity. This way, in addition to evidence of student learning, I have a record of the kinds of interactions that occurred in class meetings. Together, these provide two kinds of data that help me to know whether what I am doing is working. Although I have always modified my teaching over time, by Teaching Like a Freak, I can hope to have evidence of whether the changes are making a positive impact.

In part 2, I will take up two other ideas from Think Like a Freak, targeting small problems and thinking like a child. 

6 Responses to Students' Questions About IBL

Teaching is a cultural activity. Whenever students enter a classroom, they have expectations about their roles, and about what the teacher will do. If a teacher decides to do something outside of the norm, students are often confused and anxious about what will happen. Students often express these feelings with statements such as, 

"Why do we have to teach ourselves?" "I don't learn this way," or, "Why not just show me what to do?"

Over time, I have accumulated some ways to respond to these statements. I did not develop all of these myself; many of them have come from other instructors I have met at conferences and workshops. I should also say that these are things that I say to college students. If I had an audience of high school students, for instance, I would probably make these same points, but not necessarily in the same way.

1. Think of me as a coach. When you think of things you learned outside the classroom, things like playing a musical instrument, learning to swim, or playing a video game, you probably learned a lot of it by trying things yourself. That is what we are doing here. By letting you show me what you tried, I can coach you, and help you figure out anything you are missing. To quote a student of Dana Ernst’s, “Try. Fail. Learn. Win.” In other words, people learn from their own efforts. And, like a swim coach, while I will expect you to try to swim on your own, I will also step in to keep you from drowning.
2. Learning happens when we are actively involved. A lot of research has accumulated that suggests that classrooms in which students are actively participating and collaborating are better at promoting learning than classes in which students are passive. (Research on cooperative learning for K-12 is summarized in Marzano, Pickering, and Pollock, 2001, for example. The recent article by Freeman and colleagues reviews research on active learning at the college level.)
3. Do not mistake struggle for “not learning.” We are often not very good at judging how well we are learning. When people watch a well-organized lecture, they rate their learning higher than when watching a poorly organized lecture. But tests suggest that the audience in each case learns about the same amount. In contrast to a lecture, when we participate actively, it can feel uncomfortable to struggle to come up with answers. But that struggle is part of learning. Consequently, students sometimes rate their learning in lecture courses as higher than in active learning courses, even when the data suggest otherwise. More than one study has pointed to this, but this is apparent in a study by Lake, 2001 (http://ptjournal.apta.org/content/81/3/896.full).
4. This course will help you develop the skills that employers want, such as independence, creativity, the ability to work in teams, the ability to learn new ideas, and skill in solving problems for which the solution is not immediately apparent. My goals for you in this class are not just to pass exams; I want you to learn skills that will be valuable to your long-term success in your chosen career. But developing these skills requires doing something different than watching the instructor and practicing similar work on your own time. It will require struggle, as you are learning to use a different set of skills than you may be used to using in math class.
5. A lot of people tell me they hate math, or, “I’m just not a math person.” The kinds of experiences that lead people to make these statements have a lot to do with the way math has been taught for a long time. A lot of classes emphasize following the teacher’s steps, practicing specific procedures, memorizing mathematical facts, and developing speed at execution. While there is a role for these things, mathematics is about a lot of other things, and for the most part, it is these other aspects that interest mathematicians in doing mathematics. The other side of mathematics is about solving problems, finding new ways to understand mathematical ideas, and proving that the solutions we find work, or figuring out the cases where they don’t work. This kind of work does not proceed linearly, from problem directly to solution. Instead, we often take a winding road, hit dead ends, and have to re-evaluate what we are doing. This kind of mathematics is not straightforward, but it is exhilarating when we succeed, and even when we don’t, we often learn a lot.
6. A good teacher is not defined by what he or she knows, but what he or she can get students to learn. It is what the students can do that matters. I can explain a lot of sophisticated mathematics, but that is no guarantee that you will learn it. Instead, I carefully prepare problems that will help you draw out your own ideas, and that are most likely to put you into a situation where you will learn the important ideas of the class. Then, as a class, we will struggle, but you will learn more than you would if this class was organized around me, as the instructor, explaining solutions to problems you have not yet thought about.

I’m sure other instructors have other ideas, and I’d be happy to hear them in the comments. Meanwhile, I hope these examples serve to illustrate the kinds of answers that an instructor can use. I find that both the use of analogies (as in #1) and the appeal to research (as in #2 and #3) tend to be my favorite. I probably lean on #1 the most, but I also mention the others regularly. In classes where the students are not STEM majors, #5 often resonates with the personal experience of many students, and can help open the door to having them consider other ways of organizing the classroom that can still be called teaching, or, better still, to think of the classroom as being about what the students learn, rather than what the teacher explains.  

Who counts?

In this post, I discuss federally defined graduation rates. 

If you have not heard, accountability of the sort that has been in place for K-12 for more than a decade seems to be creeping into higher education. 

In some ways, I am happy that this discussion is taking place. As a professor, I take pride in seeing students learn, grow, graduate, and move on to success in their adult lives. I think it is good that institutions of higher education are looking at graduation as a measure of the accomplishments of an institution, and that they are focusing on ways to support students in reaching their goals. However, as often happens, finding an appropriate way to measure graduation is not as simple as it seems, and the current definition does a great disservice to some institutions.

Much of the discussion of graduation rates is based on the Federal Graduation Rate (FGR), which measures the percentage of first-time, full-time freshmen who graduate within six years of entering their original four-year institution. If you went to college immediately after high school, took classes full-time while working part-time, and graduated within 6 years, this definition probably seems perfectly reasonable on the face of it. If you fit that profile, congratulations. More than half of college students do not.

For purposes of illustration, suppose we are looking at the graduation rate for “Big State University (BSU).” In the FGR:

• Transfers are not counted for anyone’s benefit. If a student begins his/her college career at another institution (whether a community college or another 4-year institution), and transfers to, and later graduates from, BSU, that student is not counted toward the success of either BSU or the original institution. Neither one!
• If a student begins his/her BSU career at less than a full-time unit load (often 12 units), or begins in mid-year, then s/he also does not get counted toward the success (or failure) of the institution. Nothing that happens with that student will impact the FGR reported by BSU.
• I have been told, but have not been able to verify, that even if a student withdraws for a semester, or stops taking full-time course loads, that s/he is removed from eligibility to be counted as a graduate.

Taking these factors into account, the American Council on Education estimates that about 61% of students at 4-year schools are excluded from the calculation. Sixty-one percent! How do institutions have a conversation about their successes and shortcomings when they ignore more than half of the students in their calculation?

To give you another perspective, I looked at my own institution to find out how many of our graduates are being counted toward our success, as measured by the FGR.

Using public Integrated Postsecondary Education Data System (IPEDS) data from my institution, the data and my calculations indicate that 306 students who graduated from the university in 2012-2013 were among the students counted in the FGR, meaning that they were first-time, full-time freshmen in one of the years 2007-2010. Does that sound like a cozy and intimate graduation ceremony? I guess that depends on where you went to school. But in fact, the institution awarded 2,481 bachelor’s degrees in the 2012-2013 academic year. That means that fewer than 1 in every 8 graduates is counted as part of the university’s official graduation rate. How do we define the success of an institution when 7 out of every 8 students receiving degrees, or more than 2,100 of its graduates, do not count toward the accomplishments of the institution?

The university serves a population with a lot of students eligible for financial aid, and serves a lot of non-traditional students who may work full-time. That is part of the institution’s mission, and one I am proud to serve. But the FGR does not measure our success.

So, the next time you hear about graduation rates in higher education, be glad that the individuals care about students being able to complete their degree. And then tell them that there is a lot of success that is not being counted.

NOTE:

I am not the first person to make at least some of the points here. Yesterday, I searched for “federally defined graduation rate” and I got hits that included articles and blog posts making exactly some of the points here. But I don’t think they have looked at the problem from the other end, how many of an institution’s graduates are or are not students who count for the FGR.

Here is one example of an article discussing some of these issues:

http://www.acenet.edu/the-presidency/columns-and-features/Pages/Why-Graduation-Rates-Matter—and-Why-They-Don’t.aspx

The Excitement of September

It’s September, and my wife, my daughters, and I have all started our school years. I am excited, as I am nearly every year at this time. September is the time for hope. I have new students that will be engaged and, I hope, experience the joy of learning. I am teaching a course that I have never taught to a group of students, most of whom I have already had. I am excited about working with new content and seeing the ideas that students will bring to it.

In just two class meetings, there have already been some highlights. In one class, the students were deeply engaged with the concept questions that I used with them. The debate was lively, and this set the tone for good discussions of student presentations. In my other class, we left a question unresolved, but two students turned in two different, but both viable, solutions to the problem, which will lead to a good discussion this coming week. 

I hope your year is off to a good start. 

What is acceptable evidence?

I often try to keep the goals of my course in mind when designing a course, and when making decisions day-to-day in a course. But I have a harder time thinking about evidence of student learning. As it turns out, verbs are helpful.

Through my work with teachers, and in partnering with teacher educators, I came into contact with the book Understanding by Design, by Grant Wiggins and Jay McTighe. In their model for developing curriculum and lessons, there are three stages, embodied in the three questions: What are the learning goals? What is acceptable evidence? What activities, experiences, and lessons will lead to the desired results as evidenced by the assessments? 

I teach a range of courses, some for aspiring elementary teachers, some for math majors, and some for practicing secondary teachers. In planning any of these courses, I generally begin with my learning goals for the course. While the official course syllabus sets a direction for the course, I sometimes find it helpful to rephrase the goals, and to prioritize them. For purposes of illustration, one phrasing of a course goal from the course for future elementary teachers is: Students will understand fractions and their representations, and be able to solve problems involving fractions. My rephrasing of that goal is: Students will be able to explain operations on fractions using models such as the area model and the number line, apply the models in realistic contexts, solve problems involving fractions, and interpret their answers.

Pivoting from goals to assessment, I am now faced with the question, What am I looking for during and after the unit on fractions that will let me know if students have reached the goals for the unit? Whereas the initial phrasing of “understand fractions” does not translate easily to something that can be assessed, the use of the verbs explain, apply, solve, and interpret give more direction to how to assess student learning. It lets me know that I am going to assess the students via problems in context that include a requirement to interpret their answers, as well as problems that require explanation of diagrams or models. I will know students have succeeded if their explanations are coherent, their diagrams illustrate mathematical reasoning, and students are purposeful in interpreting their answers in context, for instance by using appropriate units, or rounding up or down as appropriate to the problem. Although I am sure it is possible to have shallow goals using the same verbs, I find that using words such as explain, apply, solve, and interpret translates an abstract concept like understanding into a measurable quantity. 

Rethinking my goals with assessment in mind has helped me keep a focus on what is important in my classes. Having clear goals phrased with verbs that make them measurable makes it easier to write exams. But more importantly, because I am interacting with students at every class meeting, if I feel students are not reaching the goals, I know what is important to emphasize, or what is important enough that we need to slow down, since it is embedded in the goal statements.

Have you worked with your goal statements? Do you find that you assess progress toward your goals regularly? 

Examining Reasons to Use Technology in the Classroom: Mathematical Modeling of Flight Times

In this post, I explore how technology has made mathematical modeling more accessible.

This summer, I had the privilege of teaching a 3-week institute for eighth grade teachers. One of our aims was to help teachers grapple with mathematics in the Common Core State Standards that is new to (or long forgotten by) the teachers. One of the major changes is the inclusion of a mathematical modeling standard (Standards for Mathematical Practice 4), and in eighth grade, three standards refer to investigating patterns in bivariate data. This includes thinking about whether a pattern fits a linear model, and informally fitting a line to data. Thus, we spent a number of sessions engaged with bivariate data. For purposes of this post, the main point I want to make is how technology has made fairly sophisticated mathematics more accessible, and to briefly describe how we used technology to do mathematical modeling.

As I have written previously, I think the TI-Nspire is worthwhile in spite of its price, and so that was the focus of much of our work. The problem Gate-to-Gate, which I updated and adapted from a problem I found in a book, is a good example of our work. In brief, the goal of the lesson is to build and assess a model that predicts flight times from Chicago given the flight distance from Chicago. 

In that problem, we started by making observations about a map produced by http://www.flighttimesmap.com that shows concentric rings labeled with estimated flight times. I used Chicago as the point of origin. First, we collected observations, such as: the rings appear to be circles, the circles appear to be equally spaced, and the first circle is marked as a 1 hour flight. Next, we discussed the meaning of the observations, and we conjectured that the equal spacing is an indicator of a fairly constant flight speed. We also wondered whether the map was completely accurate with its times. 

The next step was to have the teachers explore a data set. I gathered actual flight times that I had looked up and put the data in a TI-Nspire file. The teachers were then saved the trouble of typing in their own data. Teachers then created scatter plots, attempted to fit their own informal lines to the data, and ran a linear regression on the data. With the line, they then chose flight destinations, looked up the flight distances (via web search), and compared the flight times predicted by the model to those they found on the web. They also tried to think of cities on the map that would be 3 hours away from Chicago by air, and again compared the real data to the predictions of the model.

Finally, we had a summary discussion about the quality of the model fit, the meaning of the values in the linear equation, and considerations about what is an appropriate domain for the linear function. (What does it mean to have a flight covering a distance of 0 miles?) Some teachers graphed distance as the independent variable, and others graphed time as independent, giving two different equations. This led to different insights from the different slope and intercept values. It was a good discussion and led to good insights about both modeling and the meaning of slope and intercept in context.

Stepping back from the problem, here is a look at how technology enhanced this exploration.

• Data is more easily shared. This saves a tremendous amount of time. I shared the TI-Nspire file as a Dropbox link on a Lino board that I established for the class. This is a long way from having to either plot data points by hand, or even sharing the data but having each person enter data into their own spreadsheet for analysis. If I were not doing this lesson with iPads, I would either have to pre-load data onto handheld calculators, or if those were not available, perhaps give the data in a table and an already-plotted graph (or two graphs, with the two choices of independent variable).
• Data is more easily analyzed. Fitting a line informally can be easily explored with touch screens. And, since the technology handles finding the equation of the moveable line, the focus of the conversation is on the quality of fit of the model, rather than a focus on the procedure of finding the line equation. Computing technology to perform tasks such as regression has been available for many years. Nonetheless, it is a powerful tool, and the ability to use regression with a button click means that there can be a discussion of how our informal lines compared with the regression line. If this were a calculator lesson, we might still use moveable lines, but less easily. And, barring that, we would have spaghetti on a paper graph, but then we would not be able to compute the line equations quickly.
• Access to the web helps make it easier to test a model with real-world data. With access to maps and the ability to look up flights, teachers had a lot of freedom to test their models. If we did not have the web, I would have had to preselect a set of cities, listed with distances and flight times, and use that as the basis for testing the model.
• Sharing results is easier. We used Baiboard, and I selected individual teachers, who then uploaded screen shots of their models and results. This meant that when teachers were sharing, either they or I could add annotations to the screen shots. Moreover, as others shared, we could swap back and forth between the current person’s work and the work already shared by others. If this were a lesson on calculators, teachers would have had to keep a separate handwritten record of their work, and switch back and forth between sharing their written work and sharing the work on the calculator. We would probably have to keep a (partial) record of what was shared on a whiteboard for later reference.

In looking at the effect of technology on the lesson, it is not the case that without iPad technology, the lesson is impossible. Compared with, say, having classroom calculators, it is that the technology makes the lesson run more smoothly and quickly, adds the authenticity of finding one’s own data, and improves the way results can be shared.