The Reflective Educator

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Year: 2019

Planning Sequences of Lessons

I recently had someone ask me this question on Twitter and I think it’s an important question!

How do you plan a succession of lessons for a maths topic, say multiplication or division?

As it turns out, I’ve spent the last 4 years planning sequences of lessons as part of the curriculum work I do with a team of math specialists, so I have some significant experience with this task.

 

Decide what success looks like

The first step in our process was to decide on a logical sequence for the units of study for the year. Given the amount of time we had to devote to curriculum development six years ago, we basically decided to outsource this part of the process to the Mathematics Design Collaborative, but there are a lot of ways to decide on an order to units and to some degree, the choices are arbitrary. It’s what comes after that is critical but this allowed us to decide on an initial apportioning of the content standards to specific units.

We started with assigning a formative assessment lesson from the Mathematics Assessment Project to each unit, essentially deciding on “what does it look like to be successful in the mathematics of this unit?” first before outlining the mathematics of the unit. This decision, to work backwards from the end goal, is described in more detail in Grant Wiggins and Jay McTighe’s book, Understanding by Design.

 

Decide what a unit looks like

My colleague, Russell West, created this Unit Design Schematic to outline the general structure we intended to build for each unit.

The structure of a unit in our curriculum

 

Align mathematical tasks to the unit

6 years ago, we had a partnership with the Silicon Valley Mathematics Initiative, and they have literally hundreds of rich math assessment tasks aligned to high school mathematics. I printed out all of them for middle school and high school and we put them on a giant table in our office board room. My colleagues and I then sorted all of the tasks into the units where we felt like they fit best (or in some cases to no unit at all). Our experience suggested that tasks are a better way to define the mathematics to be learned than descriptions of mathematics.

Once we had descriptions of the units, formative assessment lessons, and tasks for each unit, we decided an initial task and a final task for each unit. The goals of the initial tasks were to preview the mathematics for the unit for students and teachers and to give students and teachers a sense of what prerequisite mathematics students already knew. The goals of the end of unit assessments were to assess students understanding of the current unit and to give students and teachers a sense of the end of year assessments, which in our case are the New York State Regents examinations.

A sample assessment task licensed to Inside Mathematics

 

Be really specific about the mathematics to be learned

With all of this framework in place and a structure for each unit defined, we then did all of the tasks we had grouped into each unit ourselves, ideally a couple of different ways, and made lists of the mathematical ideas we needed to access in order to be successful. Essentially we atomized each task to identify the smallest mathematical ideas used when solving the task, but we were careful to include both verbs and nouns and created statements such as “describe how independent and dependent quantities/variables change in relation to each other.”

By chance, we watched this talk by Phil Daro on teaching to chapters, not lessons, and decided that we needed to group the atomized ideas we had generated into chunks and we labeled these chunks “big ideas”.

 

Group the mathematics into meaningful chunks

The next part of the process took a long time. We wrote descriptions of the Big Ideas and the evidence of understanding that would tell us if students understood the big idea for the week. This evidence of understanding was essentially the result of the atomizations that we had previously created, grouped together into week-long chunks. The process took a while because we wrote and rewrote the evidence of understanding statements so that our entire team and a sample of the teachers we worked with felt like we understood what the evidence of understanding meant.

For example, the first Big Idea of our Algebra I course is “Rate of change describes how one quantity changes with respect to another” and our evidence of understanding, at this stage in the course, include statements like “determine and apply the rate of change from a graph, table, or sequence” and “compare linear, quadratic, and exponential functions using the rate of change”. The last Big Idea of our Algebra II course is “Statistical data from random processes can be predicted using probability calculations” and the evidence of understanding for this Big Idea includes statements like “predict the theoretical probability of an event occurring based on a sample” and “compare two data sets resulting from variation in the same set of parameters to determine if change in those parameters is statistically significant”.

Once we had this evidence of understanding mapped out, we also checked to see whether important ideas would come back throughout the course in different forms and looked to make sure that deliberate connections between different mathematical representations were being made. This way students would get the opportunity to revisit ideas, make connections between topics, and have opportunities to retrieve information, frequently, from their long-term memory.

We also revisited the alignment of the New York State Learning Standards to our units, and ended up adding standards to some units, moving standards around in some cases, and writing some clarifications about what part of a standard was addressed when during the course.

 

Design tasks for each chunk of mathematics

Now, finally, we were ready to make tasks. Well, actually, in practice we started making tasks as soon as we had a sense of the Big Ideas and then occasionally moved those tasks around when we had greater clarity on the mathematics to be taught. But once we had nailed down the evidence of understanding, we were able to map the evidence of understanding for a week to specific activities, essentially creating blueprints for us to design our tasks from, since each of the evidence of understanding statements were linked to observable actions of students.

This is an example of a mapping between the evidence of understanding and activities

We ended up with a final product we called a Core Resource. It’s larger than a single lesson but it’s not just a random collection of lessons either. It’s a deliberately sequenced set of activities meant to build toward a coherent larger idea, while attending to two practical problems teachers encounter frequently – that of students forgetting ideas over time and needing a lot of time to build fluency with mathematical representations. Here is a sample Core Resource for Algebra II.

 

Summary:

In hindsight, the most important parts of this process are:

  • to work backwards from the goals at the end of a year and the end of a unit,
  • use tasks as examples of what success looks like,
  • be really specific about what the mathematics to be learned is,
  • chunk the mathematics into meaningful pieces,
  • and then finally design tasks that match the mathematics.

For multiplication and division specifically, I would be tempted, as much as is possible, to frequently interleave the two ideas together, after identifying the many constituent mathematical ideas that together represent these large mathematical ideas. For example, if students learn to skip count by twos, five times, to find how many individual shoes are in five pairs of shoes, I would want to work backwards fairly soon from there to I have 10 pairs of shoes, how many pairs do I have, so that students can more directly see these two operations as opposites of each other.

Help Wanted

In 4 months, the grant that funds the consulting on curriculum and professional development I do with New Visions for Public Schools ends and unfortunately, it does not look like a new grant is on the horizon. Consequently, with blessings and support from my colleagues at New Visions, I am looking for either another consulting contract or full-time employment.

The work I do is extremely varied. This past month I:

  • Developed two weeks worth of mathematics curriculum for classroom use as part of this collection of resources,
  • Designed and ran two workshops for teachers on using mathematics curriculum aligned to Geometry and Algebra II curriculum,
  • Created interactive dashboards using Data Studio to display historical data from the NY Regents exam,
  • Wrote a script to automate responses to the thousands of Google Doc share-requests my colleagues were receiving,
  • Wrote a script to automate conversion of Google Docs into PDFs,
  • Created two videos examples of classroom teaching in action (example),
  • Double-checked that typesetting we had done is correct for thousands of math and science questions for this Google Sheets add-on,
  • Supported a new assistant principal to make plans for her math team,
  • Other smaller tasks related to supporting 80 or so secondary schools with their mathematics instruction.

Here is a graphic to give you a rough sense of my professional skills.

I live on a small island off the coast of British Columbia, Canada where I grew up. It is beautiful and we are settled. If it is absolutely necessary, I will live apart from my family for at least part of the year, but ideal work for me would allow me and my family to remain where we are. I don’t mind traveling if it is necessary for work.

I’m looking for work that makes use of my skills and allows me to grow and continue to be challenged to learn new things.

You can help by:

  • Giving me encouragement and cheering from the sidelines,
  • Suggesting opportunities and organizations that will meet my needs,
  • Sharing my resume with people and organizations that you think may be interested in someone with my skills.

 

A Benefit of Open Source Curriculum

As some of you probably know, I’m one of the lead designers on an open source math curriculum. Today I had an interaction that reminded me of a key benefit of open source curriculum.

In a traditional curriculum model distributed either on paper or via PDF, Hannah would have to either print and then painstakingly correct the errors above by pasting over them or use some likely-painful-to-use PDF editing software to fix these errors.

We distribute our open source curriculum via Google Docs and as a result Hannah can just make a copy of the document, make the edits she wants, and then print the resources for her students. Hannah is also legally able to do this because our curriculum is licensed specifically for adaptations.

Since each resource we create has its own page on our website (the resource Hannah describes is here), Hannah was able to comment on a specific document and I was able to respond to her transparently.

There are drawbacks of using Google Docs. For example, it is not currently possible with our curriculum to print out the entire set of student handouts. This is a fairly frequent request we get from teachers but we don’t know yet if the loss of editability is worth the increased ease of printing resources.

It would be helpful for me to know what other pain-points exist for teachers when adapting and modifying curriculum for their own classroom use, especially given that a high percentage of teachers make adaptations to curriculum they are given. If you were in charge of how someone shared curriculum, what would make it as easy as possible for you to make thoughtful changes to that curriculum?