3 - Teaching

Though I hate being forced to say it, I believe I am an excellent teacher. Continued improvement requires healthy humility, but when I consider my teaching path I am happy, though not content, with where I am. When I was happy saying I was an excellent teacher I wasn't half the teacher I am now. My improvement has come as a direct result of working with excellent colleagues here, having opportunities to plan or teach together, and sharing frequent, immersive discussions of teaching. Though lately our departmental needs have strongly dictated what I teach, I have a broad variety of teaching interests and have taught Math 096, 097, 110, 202, 210, 227, 327 and 341 in addition to the more math ed flavored Math 221, 222, 322, 229, 329, 603, 629, 641 and Ed 331. I've led or shared independent studies in 408 and 495 also.

I write the vast majority of the activities that I use in class, and my classes are designed to be places of active learning. Most of the other activities are modifications of colleagues' work. Students work in groups, usually with options and choices to involve them in choosing their direction. I structure classes using a workshop model, that begins with students making connections, followed by direction or focus from the teacher, the bulk of the class in groups working cooperatively on a problem or problems and ending in a reflection period that may be individual, group or whole class.

Two structures that greatl

  • in class? Am I creating opportunities for students to work on their conditions?Engagement requires learners believe they are potential doers of what is being taught, what they are doing matters to them, and they are safe to try.

  • Immersion. Learners need to experience real and rich mathematics of all kinds.

  • Demonstration. Learners need many and authentic demonstrations of what they are learning.

  • Expectations. Teacher belief that learners can meet the goals.

  • Responsibility. Learners make their own decisions about when, how and what to learn.

  • Employment. Learners need time to try out their learning in authentic situations.

  • Approximation. Learners need to be able to make mistakes without fear of punishment.

  • Response. Learners need real and meaningful feedback about what they are interested in working on.

y influence my teaching are the ideas of gradual release of responsibility and the Conditions of Learning. The idea of gradual release is part of why my assignments and class work becomes less structured with more choice as the semester goes on. The Conditions of Learning guide me to a great extent as I reflect on my teaching. Am I creating these conditions

  • I also often reflect on my teaching in light of the Teaching-Learning Cycle. (Link is to a post on lesson planning.) It has four phases:assessment - gathering data from students about their understanding

  • evaluation - understanding where students are and what they need to meet objectives

  • planning - design of lessons and assessments based on evaluation information

  • instruction - support of students in active learning situations

Assessment

Assessment is important not only because it's how we gather crucial data about our students' understanding, but also because in our culture it is a truism that students value what is graded. This has been a problem for me in the past. When I felt like I was emphasizing processes but my tests hadn't changed, students still focused on the same old same old. When I started putting problems on a test that could not be finished in time, it finally drew students' attention away from the product, and supported them in sharing their thinking, which better helped me detect what they understood and what they could do. I think assessment is one of the principle ways that students find out what my expectations of them are, and I want that to be throughout the course and in every class period.

While some people think about tests and quizzes when they hear the word, for me assessment is primarily informal and formative. Since my classes are active I can wander and listen to students' conversations, observe their writing and ask them probing questions. It is common to end class with an exit ticket, where there is a short question for them to answer before they go. These can be about their own self-assessment, such as "with what aspect of this do you need more time?" or "explain the concept of ____ in your own words" or a sample problem of the type we're working on. Often i don't have to even collect it as I can peek over shoulders while they write, or listen in as they share what they wrote. This formative assessment begins right away, as I often do a course preassessment that helps me design the course. The preassessment can be on disposition and on course content. (Blog post one and two about this for a Geometry for K-8 teaching class.)

In terms of the summative assessment that I do, the main two items are portfolios and Standard Based Grading.

Portfolios

The portfolio collects all the student work from the semester. It's the main source of current complaints about workload from my student evaluations. I assign 5 hours of homework a week for a three credit class in the form of hour long workshops. (As mentioned above, class work also uses the workshop structure.) They turn in the portfolio a few times a semester, getting feedback on workshops they select. Feedback indicates what grade they would have received, and a focused suggestion or two for improvement. Grades on the portfolio are not set until the end of the semester, and are based on the percentage of workshops completed and a few exemplar workshops that they select and are graded on completeness, correctness, clarity, coherence and consolidation. The portfolios have been successful on several levels. Students like seeing all their work together, they benefit from having to organize and structure it, they are more relaxed with the idea of exemplars. In terms of the Conditions of Learning, it is a good opportunity for students to practice responsibility and employment and for me to allow for approximation and provide response. The amount of choice in the workshops increases throughout the semester until they are responsible for finding worthwhile uses of their time; I do always provide suggestions or possibilities, though.

Issues with the portfolio are students who see choice as meaning 'don't have to do it,' and leaving it until the last minute with the monthly turn in. (It turns out that 20 hours of work is a lot in a few days.) I think this feeds the feeling that shows up on evaluations that the course is too much work or felt like busy work. I understand they may not be able to do 5 hours of work per week, but do not feel like I can cut that at all. I struggle understanding a comment that some work felt like busy work, since they have so much choice in making it worthwhile. Collecting portfolios makes for a long weekend of reading, and it does sometimes take me a full week to get them back.

Standards Based Grading

While the the portfolios were clearly successful in meeting my goals for that portion of a course, I was still struggling with how to best assess content understanding. I gave tests that allowed for student choice of problems and that helped. I moved to a workshop structure on the tests, and that helped. But ultimately that part of students' grades came down to a momentary snapshot, and as my grading focused more and more on the mathematical processes, I felt like I was no longer accurately assessing their understanding of specific mathematical content.

A large part of my professional reading is now inservice teachers' blogs, and a few teachers that I respect (Sam Shah and Shawn Cornally, for example) wrote about their shift to Standards Based Grading (SBG). The idea is that their students' grades are determined solely by measured success on the objectives of the class. There are multiple opportunities for students to be assessed, and they can be re-assessed on standards to improve their score. I thought that this was exceptionally good practice for my preservice teachers, which led to me thinking that it would be good practice for me also. I first tried this in Fall 2010 in one class as an optional way to be assessed, and it was a mess. Having two systems was clunky and confusing. In Winter 2011, I ran the math assessment in Math 229 and 329 as SBG and it was much better, though I am still learning. SBG works better with frequent smaller assessments, and I think in a college class where I see the students so much less than in a secondary class, there need to be ways to demonstrate content understanding in portfolios also. I feel like the conditions of response, expectations and approximation are being better met with this practice.

Evaluation

The structure I often use to evaluate students is inspired by Vygotsky's model of the zone of proximal development. Information on what the students can do is important because their new learning has to build on what they can do, and research shows that knowledge that connects to previous knowledge is longer lasting. Information on what students are ready for now is crucial to planning for instruction and knowing when to move on. Information about what the students can not do yet is helpful in terms of planning demonstrations and plotting a course towards the material that is currently a struggle.

I also evaluate students' work using a communication rubric. I'm looking for work that is:

  • Clear

  • Coherent - moves towards the objective in a sensible way

  • Complete - meets all the requirements of the workshop or problem

  • Consolidated

    • spontaneous work: includes reflection that ties work together

    • planned or revised writing: exhibits structure suitable to the assignment

  • Content - demonstrates understanding of mathematics or pedagogy from the objective

In Standards Based Grading I use a rubric that evaluates the understanding demonstrated, which encourages explanation.

The area of evaluation I'm working on the most right now is how to write constructive feedback that is focused and valuable to the student. This is the crucial condition of learning of response. One of the reasons I am moving away from grades as feedback is that it is of little use to the student in making improvements. This also includes being sure to comment on areas of strength. Specific positive feedback is something many students have not experienced in a K-12 math classroom. Timeliness of feedback is important also. I have tried to address students' past concerns about slow turnaround by better structuring assignments, managing due dates amongst all my courses and making assignments more interesting for me to read. I am greatly curious about student approaches to problems and portfolio assignments, and they are often a pleasure into which to delve. More frequent SBG assessments are also quicker to turn around than longer tests.

Planning

I plan for courses using backwards design. First think about the goals and objectives, then think about how I can assess whether they have met these objectives. Then I look for a logical sequence of activities to equip the students to meet the objectives. For math ed classes this includes thinking about which pedagogical topics will pair best with which content topics, but even in a pure math class I like to consider the five mathematical processes from the National Council of Teachers of Mathematics (NCTM) and when would be best to focus on which process. No one would accuse me of over-planning a semester. The positive side of this is that I can respond to students' understanding by moving on sooner than I would have thought possible or dwelling longer than I would have guessed necessary. The negative side of this has sometimes been perception of a lack of organization from students, though not much recently. One of the benefits of math education courses is that it is germane to share my teacher thinking about scheduling and planning, so I do this and also clearly give students an idea of the upcoming schedule. The use of daily agendas in class has also helped organize me and the students.

I am prone to taking leaps that I don't always know how they will work out. In fall 2010 one class came in while they were filming the Grand Valley lipdub. This led to a class discussion, showing a few lipdubs, and analyzing why students would be willing to put forth so much effort for a project like that, when the conventional wisdom is that students are lazy and unwilling to work. A great discussion that led to a request to make our own class video. We had strong support of about 90% of the class with the rest quite against it. We worked as a class towards making it acceptable to all, and the preliminary work was done in choice workshops. But not enough was done ahead of time, and as the day of the class drew near I was quite depressed about how much effort and course time went towards something that was going to be a flop. But when the filming day came (and one of the two cameras failed) it all came together. The video was fun, but not the most significant part to me. It became a project that was the students', and changed their perception of our course and, for several of them, what they envisioned doing with their students.

I plan for individual lessons by looking for problematic situations that offer a good opportunity to address the objectives. This is about engagement and immersion to me. Taking into consideration the information from the students' informal assessments, I try to make the activity in the sweet spot of Vygotsky's center zone. I usually err to the side of too challenging, but that can be rectified on the spot with more instructional support. I do try to build in extensions and open-ended problems for groups that need more challenge, but I'm willing to try that on the spot also. For the most part I use heterogenous groups so that there is usually a need for table discussion for everyone to understand. I will use homogenous groups when doing centers or when there is a particular group of students with whom I want to work directly myself. Most often homogenous groups come about when students have a choice of activity and rearrange themselves by their choices. I typically have more planned for a class period than could possibly fit in, which can get me into trouble if I try to fit it all in.

Instruction

I love being in the classroom, with students of any age. From preschool through teachers with decades of experience. The problem of creating the conditions of learning for a diverse group of students is endlessly fascinating to me and deeply rewarding.

Both of the classroom observations capture pretty typical classroom periods for me - though we're not always that successful in hitting the learning objectives. The reports capture the cooperative nature of class, the emphasis on active learning and how much I value reflection and discussion. What they don't capture, being in a late-semester class and a graduate class of pretty independent learners, is how much I strive for that independence and cooperation as the semester goes on. It is definitely not always successful with every class, but every class does make progress. I struggle responsibility as a condition of learning, because students have so little experience being in charge of their own learning. The workshop structure has been invaluable for this, as it is a flexible structure that allows me to loosen the reigns while maintaining a continuity of form. Students recognize the value of reflection for consolidation of learning and the vast majority get better at it throughout the semester.

Two common features of my classroom that can be a bit unusual are games and technology use and games.

Technology

Of course, technology in and of itself is not unusual, especially in our department at our university. But I do probably involve it more than most, because I am interested in it as learner support, barrier removal and activity sponsor. As learner support, the ability to automate formerly time-consuming tasks has a dramatic impact on class time use and student ability to make up for deficiencies. For example, graphing calculators allowing us to talk about what we learn from graphs instead of using all our time just making them. Or students being able integrate a function to solve a problem by computer even if they do not recall the by hand method. Barrier removal is related - technology like GeoGebra or Wolfram|Alpha (links to the programs) makes so much more mathematics available to the student. I find technology often to be an invitation to activity because of this. Whereas a student might feel intimidated by the mathematics the technology is appealing and enabling. Or it enables them to make contributions that are more lasting; for example, my 229 students are building a wiki for high school math teachers.

Teaching future teachers raises other issues because technology should impact what we teach also. Do we need to teach taking square roots by hand or using logarithm tables? No. Should we teach how to graph functions or integration techniques? Less clear. And that's just the technology we have now. Five years from now is difficult to even imagine. Twenty five years? Impossible. So I feel a responsibility to teach students how to learn how to use new technology as well as the technology itself.

There is a glog of my technology use (a kind of internet poster). I regularly use (meaning have students use) Twitter, blogging, wikis, Google Apps, Wolfram|Alpha, GeoGebra and more. Of course, this means I do actually use these things also

Games

I value games for engagement, skill practice and context. Even in classrooms where educational games are more common, student interest jumps when a game is introduced. In classrooms where they are uncommon, it can be shocking to the teacher just how much student interest jumps. At all levels there are skills that are necessary for mathematics, and games will keep learners practicing longer than they would without. As these move from clunky paper games to neat electronic ones, this will be an even better effect. Games also make an interesting context for mathematics, and I will use them to get students to explore a mathematical structure or to gather data to analyze later.

Professional Development

I feel like I am constantly improving as a teacher; as my assessment improves it reveals that more of my students are achieving more of what I care about. The main sources of this are my colleagues and the internet. Work with my colleagues has been crucial, and having a department with such amazing teachers has been a real gift. Some of those I've learned the most from include (but are definitely not limited to) Char Beckmann, Esther Billings, Steve Blair, Dave Coffey, Georgi Klein, Gary Klingler, Jan Shroyer, Rebecca Walker, Clark Wells, Pam Wells, Matt Wyneken, Paul Yu and more; these are the people with whom I can remember specific conversations that impacted my teaching. I do have to single out Dave for being a particularly challenging and frequent source of growth. In addition, connection to other teachers through blogging and Twitter has been amazing. I started blogging as a way to record and link to resources I found valuable, and that drew me into a fascinating collection of writing by teachers and mathematicians who shared their thinking generously and authentically. The reading recommendations on my blog give a sense of this material. I began using Twitter because one of those writers, Maria Anderson, described how it impacted one of her online classes. I thought that what she described would be very helpful with some of the challenges we have with student teacher assistants, whom we see infrequently. I always try to pilot technology before using with students, so I signed up for Twitter, feeling a little silly. However, it has been great. The only problem is really too many connections, conversations and resources are available and it's easy to fall down the rabbit hole. (Some of my most recent favorited tweets.)