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.

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? 

9 Books to Read and Reread

In this post, I offer some suggested readings that I find help inform my approach to teaching. The books are listed in no particular order. 

For any teacher, I recommend:

• What Works in Schools? Robert J. Marzano, Debra J. Pickering, and Jane E. Pollock. (Note that there is now a second edition available with a substantially different organization. Either edition is valuable.) This book (first edition) discusses nine strategies shown by research to be effective in improving student learning outcomes. Sometimes the strategies are “obvious,” but it can still be helpful to be reminded that they are important teaching tools. For instance, summarizing and note-taking are effective. However, for me, many of my students have never been taught how to take notes, or have never discussed strategies for taking notes. So I make an effort to tell students when someone states an idea that I think everyone should write down, and I set aside some time for students to discuss what they should write down during class. Other strategies take a more concerted and planned effort to implement. For instance, generating and testing hypotheses is another strategy. While this is a natural part of doing mathematics, this reminds me to include tasks in which students do more investigative work. More than a list of nine ideas, the book has specific recommendations that are helpful. For instance, what are some important features to make cooperative learning successful? These are the kinds of specifics that are discussed in the book.
• Why Don’t Students Like School? Daniel Willingham. Willingham is a cognitive psychologist who poses some key questions and answers them from the perspective of his discipline. There are a few things that I like about this book, and that make me go back to it. One of the things I like is that each chapter closes with implications for the classroom. For example, one chapter discusses our human tendency to prefer and make sense of things as stories. In a course like precalculus, this might be used to frame “telling the story of a function,” where a function has properties like limits as x goes to infinity, asymptotes, periodic behavior (or not), symmetry, and so on. In calculus or analysis, the story idea might be put in terms of the central “conflict,” will a sequence converge or not, or another, is a function continuous or not. Rereading (or skimming) this book and thinking about the implications often inspires me to find ways to improve my day-to-day plans.
• What’s the Point of School? Guy Claxton. Claxton describes what he believes are the core goals of an education. These are big-picture concepts like developing people who are curious and are lifelong learners. While this is not a book that I return to for help in thinking through the details of teaching, I find that it helps to remind me of what is really important in my role as an educator.
• Switch. Chip and Dan Heath. This book inspired the name of my blog. The Heaths describe how to make a switch—a change—either in yourself or others. The single most important idea is that a lot of what we do is driven by emotion, and so we need to think in those terms when looking to effect change. The authors go through several ways of activating the emotions that will enable a switch to happen. I have returned to the book many times, for example, to remind me of how to approach students who are struggling, to help them find the emotion that will drive them to turn around their performance in my classes.
• Understanding By Design. Grant Wiggins and Jay McTighe. This is a book that puts forth a framework for thinking about curriculum design by starting with the end results, then thinking about how those results will be measured, and only then moving into designing the learning activities that will produce the desired end results. I return to this book from time to time to remind myself of how to frame my goals, and how to find ways to measure progress towards those goals.
• Mindset. Carol Dweck. Dweck has done significant research into the power of having a growth mindset, a mindset in which one believes that through hard work, one can get smarter or better. In the book, she describes some of this research and how it can make a difference across different domains of school and life. The book helps to remind me of why a growth mindset matters, and serves up examples that I use in explaining the power of the growth mindset to students.

For math teachers at any level, I recommend:

What’s Math Got To Do With It? Jo Boaler. Boaler has studied high school students experiencing problem-based curricula and compared them with those in traditional curricula in two different countries, the US and the UK. This book describes some of what was learned in those settings, and distills for a general audience—including parents of schoolchildren—some of the key ideas of what mathematics learning is, or should be, about. From the perspective of a math teacher, this book is less likely to offer ideas for day-to-day decisions, but like Claxton’s book, helps to remind me of the goals of teaching mathematics.

For college teachers, I recommend:

What the Best College Teachers Do. Ken Bain. Bain’s book centers how the select group of highly-respected teachers he studied approach teaching, from preparing for class, to setting expectations for students, to conducting class, and so forth. Each chapter holds a wealth of good advice, like seeking the commitment of the students: asking them to consider whether they are willing to do what it takes to succeed in the class, and therefore have them commit to the effort required. I find I sometimes return to the questions he poses in the chapters as a way of gaining a fresh perspective on my courses.

Finally, for college math teachers, I recommend:

The Moore Method: A Pathway To Learner-Centered Instruction. Charles A. Coppin, W. Ted Mahavier, E. Lee May, and G. Edgar Parker. The four authors of this text each describe how to implement the Moore method, as they see it. The book offers the reader a chance to consider various aspects of teaching in a learner-centered environment, and benefits from the approach of the authors, which is essentially to offer their individual responses to the key questions in setting up and operating a Moore Method course. This variations-on-a-theme approach has the effect of providing the reader with a canvas and a palette, rather than promoting a specific paint-by-number prescription. The authors take on a wide variety of issues associated with implementing the Moore method, including such topics as, What if no one has anything to present? How do I grade? and many others. I have returned to the book many times to seek out new ideas of how to handle syllabus construction, or to remind myself of ways to approach managing an IBL classroom. (In full disclosure, I should mention that I am personally acquainted with the authors, and have worked closely with Ed Parker.)

What are some of your favorite or most inspiring reads from the educational realm?

A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins

People who follow my posts to G+ and my tweets may have noticed that I am a reader of Grant Wiggins' blog. Not long ago, he had a post, What is a course? I thought it would be fun to play along. With that in mind, I picked my Transition to Higher Mathematics/Introduction to Proof course. Here are Wiggins' prompts and my responses:

By the end of Transition to Higher Mathematics, students should be able to write proofs and grasp the role of proof as a formal mathematical explanation.

The course builds toward having students prove more logically complex statements and gaining facility with different kinds of proof. The recurring big ideas surround how to attack a proof. We go into depth on key tools like using examples, applying the forward-backward method, using the logical structure of the statement, using definitions.

All of the chapters support these main goals. Students are first introduced to the idea of proof through familiar ideas of number theory and divisibility. They then gain some initial background in logic, and apply it to some number theory proofs before moving on to sets. They are then asked to apply set ideas to sets of real numbers. Then they move to the critical mathematical idea of a function and write proofs about functions and their properties. Finally they are briefly introduced to equivalence relations before moving on to looking at other techniques of proof, and applying these ideas to concepts already seen in class.

Given my priority goals, assessments need to determine whether students are able to demonstrate an understanding of the key approaches to proof, the structure of a logical argument, and to explain the key mathematical concepts of set, open set, function, and properties of a function.

Given my goals, I have exercises and follow-up tasks that should help students gain insight into how the main ideas in key proofs are put together.

If I have been successful, students will be able to transfer their learning to upper division mathematics courses that follow, by attacking proofs with confidence and awareness of the tools available to them, and will persist through difficult courses in the major. If I have been successful, students should avoid such common issues as waiting for the professor to tell them what to do, believing that only others can write proofs, and being uncertain of the role of examples in generating proof ideas (vs. using examples as proof). 

I enjoyed completing this exercise for this course. The course is built as a coherent whole, telling the story of proof and its role in mathematics. To make this work, I have done some tinkering with the emphasis areas of the course, reducing time spent on truth tables (which have a role, but it need not be multiple weeks of a course), and trying not to spend too much time proving things that are too basic. 

What about your courses? Does anyone want to take this up with one of their math courses?