Lesson Story: Track Stars

I haven't had my own classroom in a while, so when I got the chance last summer to model a lesson for some math teachers at a summer workshop, I was eager to try a task Bill Penuel turned me on to in a paper by Schwartz and Martin (2004):

Track Stars

Bill and Joe are both on the U.S. Track Team. They also both broke world records last year. Bill broke the world record for the high jump with a jump of 8 ft. Joe broke the world record for the long jump with a jump of 26 ft, 6 in. Now Bill and Joe are having an argument. Each of them think that his record is the best one. You need to help them decide. Based on the data in the table, decide if 8 ft shattered the high jump record more than 26 ft 6 in. shattered the long jump record.

Top High Jumps in 2000		Top Long Jumps in 2000
Height	Number of Jumps	Length	Number of Jumps
6'6"	1	21'6"	1
6'8"	2	22'0"	2
6'10"	3	22'6"	2
7'0"	5	23'0"	9
7'2"	6	23'5"	9
7'4"	7	24'6"	4
7'6"	4	25'0"	1
7'8"	1	25'6"	1
8'0"		26'6"

When I used this task with teachers a few years ago in our task analysis research it was rated quite highly: 5 out of 6 teachers said it rated as "Doing Mathematics" in Smith and Stein's (1998) cognitive demand framework and the task was unanimously judged as a good example of a task likely to engage students in Standard for Mathematical Practice #3, construct viable arguments and critique the reasoning of others.

Context

For the summer workshop I was working with about 20 math teachers who would be grouped by grade band (elementary, middle high) and I asked them to attempt the task using the abilities expected of students at their grade level. I admit, this makes for a somewhat artificial exercise, but I wanted to see if this task would stretch across a lot of different levels of student ability and elicit a very wide range of student strategies (even if the "students" were teachers).

One of my greatest teaching weaknesses has always been in my questioning strategies. Too often I accept quick choral responses to questions in the initiate-respond-evaluate pattern, and I don't do much to (a) push student thinking and (b) promote equitable participation, so for this lesson I used a combination of these resources:

• As a planning guide I used Smith and Stein's 5 Practices for Orchestrating Productive Mathematics Discussions.
• I used CMP's Launch, Explore, Summarize as an instructional model.
• My focus practice was facilitate meaningful discourse from Principles to Actions, and I especially wanted to sharpen my pressing and linking talk moves. The Talk Science Primer from TERC is an excellent resource on talk moves.
• I used a form of "student self-check" in the style of practical meausures to help me reflect.

There is a lot of overlap in the 5 Practices, Launch/Explore/Summarize, and the goal of facilitating meaningful discourse. That's a good thing.

The Lesson

I anticipated (the first of 5 Practices) different strategies across the three groups:

• I expected the elementary school group to focus on measuring distances and visual comparisons, and to bring up struggles around working with feet and inches and the under-developed sense of ratio.
• I expected the middle school group to calculate means and use proportional reasoning (like, "The record is 110% of the average), and perhaps use mean absolute deviation (MAD) as a measure of variability. I expected to see struggles in accounting for the multiple jumps at each distance, in calculating MAD, and debates around using mean vs. median as a measure of center.
• I expected the high school group to be similar to the middle school group, but to use standard deviation instead of MAD.

For the launch phase I avoided giving away any hints or clues about possible strategies. It was difficult to design a launch that connected to prior knowledge because of the artificial nature of teachers playing the role of students, so I took a moment to ask the teachers to think about the knowledge they'd expect students to have given the standards at their grade levels.

During the explore phase of the lesson I monitored (the second of 5 Practices) the groups for the strategies I anticipated. I wanted to use pressing questions here to push people's thinking, such as:

• "Can you tell me why you think that is correct?"
• "What do you mean by 'farther'? Is it because you added? What else might you do to measure 'farther'?"

Questions like this designed to press for student thinking were often met with teacher speculation about student thinking. As solution strategies came together, I noted them on my phone with the goal of selecting (the third of 5 Practices) two strategies per group to discuss during the whole-group summary phase of the lesson. The sequencing plan (the fourth of 5 Practices) was to discuss elementary first, then middle, then high school, with the less sophisticated strategy presented first at each level.

Here are the two posters from the elementary group:

The elementary group could quickly work through multiple strategies, so from this group I got more than just the two strategies I planned for. One set of strategies focused on how much more the record was than the next longest/highest jump, and the other set used a graphical representation of the jumps. Here are the posters from the middle school group:

One set of strategies compared the record jumps to the mean jumps, and the other set used a graphical display and interquartile range. Here are the two posters from the high school group:

There was less to differentiate these two strategies, as both groups calculated standard deviations and z-scores as a way of measuring how far above the mean was each record jump.

In the summarize phase of the lesson I focused my questioning around linking moves, such as:

• "How does your strategy compare to the first one from the elementary group?"
• (Following an explanation by Kathryn) "Tammy, do you have any questions for Kathryn?"
• "Phillip, how might your argument change if you used Dan's method?"

With questions like these, I hoped to draw connections (the fifth of 5 Practices) between ideas, such as:

• Connecting the visual centers of graphical displays with the calculated centers of the data
• Connecting MAD and SD
• Connecting the "measuring stick" idea between proportional reasoning at lower levels and the counting of MAD/SD units

Reflection

I had some hits and misses in my anticipation of the strategies I saw. The elementary teachers didn't share my expectation of focusing on measurement and comparing those measurements. Instead, they made some useful comparisons between the record and second-best jumps. I also didn't anticipate the dot plots and fitted curves in the second poster. I know it's uneasy to underestimate the capabilities of elementary students, but these kinds of graphs were not something I anticipated their teachers producing. The middle school group used proportional reasoning, as I expected, but instead of MAD they used IQR as a reference for judging the two jump records. There was one "student" who quickly worked through some MAD calculations towards the end of the work time, but it was a bit late to fit into my selection strategy. For high school, the work was less differentiated and more advanced than I anticipated. Some of this can be attributed to just labeling the group "high school" rather than "9th grade" or "AP Stats."

I was able to practice my talk moves to some degree, but this artificial scenario was less than ideal. In the explore phase of the lesson my questions were generally met with speculation about student strategies, not answers as students might give them. That was great for us all to think through the task together, but it interrupted the flow of responses you'd expect with talk moves in a more typical classroom scenario.

The discussion in the summarize phase was pretty good. Not only did we compare strategies and connect ideas in the way I anticipated, there was a welcome amount of analysis of the task itself and the different layers of ambiguity in how the data was presented. For example, we don't know if the jumps all represent different jumpers, or if the jumps represent jumps in one vs. multiple competitions. We generally agreed that some amount of ambiguity would be good when using this task in a classroom, particularly to hit the "make sense of problems" part of SMP #1.

As part of the reflection I collected data in the form of a "self-check," created in the style of "practical measures" that we've used in our research projects. In hindsight, this data doesn't focus much on my choice of teaching practice (facilitating meaningful discourse), but I like the idea of asking students for feedback that go beyond mastery of content.


Link to Google Form

The responses are a bit difficult to interpret because I'm not sure how many participants responded as teachers versus the students they were sort-of-pretending to be. The results seem mostly positive, and I agree with the very last comment: While the task had reach across many grade levels, first grade was too much of a stretch.

References

Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129–184. http://doi.org/10.1207/s1532690xci2202_1

Smith, M. S., & Stein, M. K. (1998). Reflections on practice: Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350.

A Menu for Making a Math Lesson Story

Lee Shulman
CC BY-NC Flickr

In my last post I talked about different types of lesson plans and suggested that one type, a lesson plan as a story, might have some benefit as a shareable unit of teaching.

When I think of teaching and what makes (or can make) it a profession, I think of attributes of professions described by Shulman (1998):

• the obligation of a service to others, as in a "calling";
• understanding of a scholarly or theoretical kind;
• a domain of skilled performance or practice;
• the exercise of judgment under conditions of unavoidable uncertainty;
• the need for learning from experience as theory and practice intersect; and
• a professional community to monitor quality and aggregate knowledge.

To support teaching as a profession, I value public displays of teaching that reflect Shulman's list of attributes. For the sharing of lesson plans, we can do better than over-templated, step-by-step, anyone-can-follow scripts. We can also do better than brief, make-of-it-what-you-will ideas that lack sufficient implementation guidance. In the stories we tell about teaching, we should seek some middle ground between an over-designed lesson template and an unstructured narrative. Since lesson stories are arguably more about the planning than the plan, they should focus on teacher decision-making and teacher practice, so that other teachers may learn from them. The minimal amount of structure to a lesson story probably starts with these four parts:

1. A description of the context (grade level, class size, demographics, features of your school environment, etc.)
2. The rationales behind your lesson planning (not just the choices you made, but why you made them)
3. A description of the implementation (a low-inference description, mindful of the students' perspectives as participants, of the classroom activity, discussion, and work produced by students)
4. A reflection (now with more inference, with a focus on how the decisions you made in planning played out in implementation and what that might mean for a lesson revision)

A Menu of Math Lesson Planning Resources

So far this is subject-neutral. In some subjects, rationales in lesson planning might have to be developed and explained from first principles. In mathematics education, however, we're fortunate to have an established body of knowledge related to planning and teaching. To plan a math lesson and then tell its story, I see four categories of resources that form a menu of options.

Planning Guide

For planning and describing the reasons for choices made in the lesson, choose one of the following:

• The book 5 Practices for Orchestrating Productive Mathematics Discussions by Smith & Stein
• The "Thinking Through a Lesson Protocol" from the article, "Thinking Through a Lesson: Successfully Implementing High-Level Tasks" by Smith, Bill, & Hughes
• The Lesson Planning Tool from Student Achievement Partners

Instructional Model

To structure the delivery of the lesson, choose one of the following:

• The Launch, Explore, Summarize model from the Connected Mathematics Project
• The 3-Act Task model from Dan Meyer and others
• The 5E Instructional Model from BSCS

Lecture and "I do, we do, you do" are also instructional models. They have their place but should probably be used somewhat sparingly. Besides, there probably isn't much demand for lesson plans that consist of a lecture.

Teaching Practice

Teaching is complex and teachers are engaged in many practices at once. However, for improving one's practice and communicating that in a story, it's best to focus on only one or two teaching practices described in NCTM's Principles to Actions:

• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning and problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.

For a different list of teaching practices, you could also consider the TeachingWorks high-leverage practices.

Reflection

In addition to using student work/activity in your reflection, choose from:

• The SERP 5 x 8 Card (a simple observation tool designed for principals without math expertise)
• The idea of "Practical Measures" as described by the Carnegie Foundation for the Advancement of Teaching
• The Instructional Practice Guide Coaching Tool from Student Achievement Partners

What We Gain

Suppose we choose resources from the menu above and tell our lesson story. What have we gained? We've built upon a body of knowledge that can help readers. To some extent, we already do this. When I hear a teacher say they taught a 3-Act Task, I immediately have some knowledge about the instructional model they used. When I hear a teacher say they planned a lesson using the 5 Practices, I know that means they took time to (among other things) anticipate student strategies. With a piece from each of these four categories there is still a lot of freedom to tell a lesson story, but the shared pieces communicate a lot about your lesson and provide a foundation for a common understanding across teachers.

Now, to refer back to my last post, let's think about the usefulness of lesson plan repositories again. Generally, lesson plan repositories are arranged by grade level, topic, and content standard. Instead, what if a repository allowed you to search based on the items in the menu? Imagine being able to search or filter by teaching practice, such as "Show me lessons in which the teacher focused on building procedural fluency from conceptual understanding." Or perhaps you're working with a new instructional coach, and you search for lessons in which the teacher had an observer use the SERP 5x8 Card. We stand to improve our signal-to-noise ratio considerably when teachers can look for lesson plans based on more than just lesson content, and the lessons they find are more likely to be a better "fit" if they are known to have a preferred planning guide, instructional model, teacher practice, or reflection tool.

Next post: I attempt to write a lesson story.

References

Shulman, L. S. (1998). Theory, practice, and the education of professionals. The Elementary School Journal, 98(5), 511–526.

On Lesson Plans and Lesson Planning

CC BY Brian Swartz, Flickr

As someone who studies teacher curriculum adaptation, Chris Lusto's post last summer, "Lessons for Other People," did a lot to get me thinking. Despite their imperfections, curriculum materials have a durability, scalability, and portability that many educational tools or innovations can only wish for. So why not try to preserve and share the evolution of curriculum materials as teachers make them less imperfect, using some kind of revision tracking system?

It turns out that this wasn't exactly a new idea (see here, for example) and there are probably sensible reasons we don't have such repositories yet. Dan Meyer gave us one big reason: Teachers don't seem to be keen on using off-the-shelf plans, especially when the signal-to-noise ratio ("just right" lessons to "ugh, move along" lessons) is frustratingly poor. There are also technical hurdles involved. We would need to get past (way, way past) discussions of JSON vs. TOML and other forms of engineering-speak. I see promise in things like Mike Caulfield's Wikity project, but then again, I'm geeky enough to run my own Mediawiki installation.

There are certainly new angles to explore on the repository front, but for them to be useful we need to get a better handle on what exactly we're putting in them. As far as I know, there isn't much in the research literature about teacher lesson planning. When I worked with preservice teachers, I taught them to use a lesson plan template to detail the objectives and activities of a lesson. But as a teacher myself, I'm not sure I ever filled out a multi-page template with a lot of details. There's a good reason for that, and it's not laziness — the context, purpose, and needs were quite different as a full-time teacher than for someone who is just beginning to learn to teach.

Especially useful to me in thinking about the difference in the purpose of lesson plans is the distinction of plans vs. planning, which Dan Meyer highlighted with a quote from Dwight Eisenhower:

Dwight Eisenhower on lesson planning. pic.twitter.com/YE9kCO9fVx

— Dan Meyer (@ddmeyer) June 12, 2016

This compliments my own thinking about design work in education: You must accept that much of the positive outcome can lie in engaging in the design process rather than in the thing or product that is ultimately designed. In other words, it's like the quote attributed to Bruce Joyce: We reinvent the wheel not because we need the wheels, but because we need the inventors. For some, this feels inefficient and wasteful, but I say you ignore it at your peril.

Types of Lesson Plans

So what are some different types of lesson plans? I've thought of three:

Lesson Plans as Scripts

Scripts and scripted lessons are loaded terms in education and the connotation is generally negative. I don't think it has to be negative, even though it certainly can be. When I say script, I'm thinking about a detailed, step-by-step description of what should be happening in a classroom, by whom, and at what times, similar to how the script of a play, TV show, or movie describes who is involved in a given scene, the actions they should take, and what they're expected to say. Just as scripted TV differs in quality, scripted lessons can vary in quality also, and they have the potential to be very good.

The most scripted lesson plans I wrote as a teacher were those for substitute teachers. If I had to be away from my students but I still wanted quality work to be done while I was gone, the best I could do was write a very detailed lesson script and hope the substitute could make their way through it.

Lesson Plans as Ideas or Reminders

When teachers plan for themselves, in the context of teaching a thousand lessons a year, many rely on a sparse set of reminders that aren't intended for use by any other teacher. Because of this, we shouldn't be quick to judge the quality of a lesson by this kind of lesson plan. Just because a lesson plan says no more than "Section 4.5, swap out baseball task for the closer, assign evens" does not mean the lesson will be good or bad. There's not enough there to judge, because the lesson wasn't designed for judging.

In conversations around lesson plans last summer, I saw teachers saying they wanted ideas more than scripts. I think part of this is because lesson plans in the form of ideas and reminders are what most teachers use most of the time, and therefore it feels familiar and flexible. I do wonder, though, how well this would really work in practice. The intent of one teacher's notes may or may not be understood by another teacher, and a repository full of lesson ideas might suffer the same low-signal, high-noise problem we have now.

Lesson Plans as Stories

I think there's a third kind of lesson plan, one that puts the planning at the forefront and the plan in the background. These lessons are written so that the reader can think along with the writer and learn from their decisions, rather than follow their instructions. These lessons take the form of a teaching case study or reflection, rather than a script or set of reminders.

Learning to teach through case studies was described by Shulman (1986), so it's far from a new idea. Shulman proposed case knowledge as a form of teacher knowledge, and he proposed (and later led research on) the development of prototype cases designed for teacher learning. Still, it doesn't seem to be the kind of lesson plan you're likely to find in current repositories. Thankfully, I know of two examples in math ed: The lesson descriptions from Jennifer Wilson and Jaime Duncan.

Take for example this lesson from Jennifer on coordinate geometry. It reads like a story: "I found a task, it relates to a standard, here's what I think students will do, here's some of the work they actually did, and here are some things that did and did not go as planned along the way." Jennifer's post is way more than just an idea, and has the detail of the script without any of the "Step 1, do this, Step 2, do that" feeling. Importantly, the students are not left to the imagination. They are seen, heard, and described. I see a lot of similar qualities from Jamie's posts, such as this lesson on fractions in first grade.

Pros and Cons

Here's a quick recap of what I see in these three kinds of plans:

Lessons as:	Pros	Cons	Effort to Implement
Scripts	Detailed; Greater chance of implementation as intended	Feels restrictive; context-unaware	Lower
Ideas	Short; A seed from which other ideas can grow; adaptable	Interpretations vary widely; Quality difficult to judge; Still requires a lot of planning and decision-making	Higher
Stories	Experience a lesson second-hand; think along with lesson designer	Stories can be long, complex, and inconsistent in form	Between Low and High

I think lesson plans as stories have real promise as a shareable unit of teaching. They focus more on planning and reflection, and they may help teachers who use them plan and reflect on their own lessons. However, it feels to me that the stories could benefit from some structure and common elements. After all, there's been way too much good work in the field of mathematics teaching to think everyone writing a lesson story should start from scratch and make up everything as they go along. A free-for-all approach doesn't help the writer or the reader. In my next post, I'll lay out a plan for telling a lesson story that I think has some structure without feeling too much like a template.