Wednesday, June 11, 2014

Developing CCSS Math Transitional Units of Study: Patterns

I had a good time this past year playing around with new resources and teaching methods as I worked with my Intensive Algebra students.  The Formative Assessment Lessons (FALs) were a great help to me, and I used a number of them both as-is, and as inspiration for developing my own materials.

This year, I will again be teaching two sections of Intensive Algebra 1, along with three sections of IB precalculus.  Because the IB curriculum and assessments are pretty clear and established, this will really give me the time and flexibility to focus on the needs of my lower-skilled freshmen, which I am happy about.

My district has been partnering with the Silicon Valley Math Initiative (SVMI), and they provide numerous resources, such as the MARS tasks and Problems of the Month.  They have also created a format, based on the work of Phil Daro, for generating units of study to be used as we transition from California State Standards to CCSS, before high-quality curricula are commercially available (if ever!) and districts adopt new materials.

The basic unit structure is as follows:

  1. Introductory lesson for engagement, to spark curiosity and interest.
  2. Several conceptual development lessons, after which you can expect student understanding to still be fragile.   These lessons are what we typically think of as inquiry-based or constructivist lessons.  (I think this is a good eye-opener for me, because it has always felt frustrating how my students would still not "get it" after we engaged in, what I considered to be, really powerful learning opportunities.  Setting my expectation that students' understanding is going to still be fragile at this point will be key.)
  3. One or two getting precise lessons, in which the teacher attends to precision, definitions, conventions, symbols, etc.  This is often going to be a more traditional "I-We-You" direct instruction approach.
  4. One or two getting general lessons.  The goal of these lessons is not 100% clear to me, but some ideas for generalization are to use concepts across different contexts, generalize with variables and parameters, use different types of numbers, operations, functions, or structures in the same context, and so on.
  5. A formative assessment lesson (which often takes multiple days).  These are intended to be done about 2/3 of the way through the unit.  They all start with a pre-assessment, and a well-defined set of tasks to help students further develop their understanding of the concepts.
  6. Additional concept development lessons, as needed.
  7. One or two robustness and differentiation lessons.  This is an opportunity to do re-engagement lessons with students who are struggling, as well as enrichment for students who are showing solid understanding.  The goal is to move all students from a fragile to a more robust understanding via a variety of rich problem solving opportunities.
  8. An expert task assessment, in which students engage with tasks that have them operate at levels 3 and 4 on Webb's DOK.
  9. A closure lesson to revisit and organize the unit goals and outcomes.
  10. A summative assessment to see if students really have mastered the unit goals.
Clearly, this is intended to create true depth of understanding of a concept, and it sacrifices breadth to a certain extent.  But I am totally fine with this.  After years of focusing on breadth over depth in CST land, and seeing students never really reach mastery of concepts, and need to be retaught concepts again and again, I am happy that we are finally being encouraged to use a new approach.

I did a lot of work with quadratic patterns in the last month of school with my Intensive Algebra 1 students, using this FAL as inspiration:  Manipulating Polynomials.  It started off rough, as I underestimated the complexity of the task.  But I kept at it, I re-scaffolded and pushed my students, and by the end of the year, most of the students could see how to generalize a quadratic pattern.  These students were still struggling with basic integer and multiplication facts, but they could look at a complex pattern and determine that the rule was f(n) = n^2 + 2n + 1, for example.  Moreover, they could clearly explain why it worked and how they got it.

I decided that, instead of ending the year with this, it would be a great way to start the year, and use it to set the context for much of the other learning to follow.  Last year, I also started with patterns (as you can see in a previous post), and at the time, I felt it was successful.  However, I realized later that I made a fundamental mistake in the way I did it.  I only had students work with linear patterns, and they were pretty quickly able to see that the increase after each step was the slope, while the 0-th step was the y-intercept.  This was great!  But, once they made this generalization, they stopped ever looking at the structure of the pattern.  They would go right to making an in-out table, and from there, generate the function rule.    When we got to the quadratic patterns at the end of the year, they had absolutely no access to them.  They kept trying to make in-out tables, would see that the growth was not linear, and then hit a wall.  I had to un-teach them the concept of making a table and, instead, try to understand the structure of the pattern.

So, my plan this year is to start with a general patterns and functions unit, where the key concept will be to focus on the structure of the patterns.  After the first unit, I will then move on to a linear function unit, where we will look at linear patterns as a subset of patterns in general.  Each time, I will ask students not to just give me the rule, but to also explain the connection between the rule and the visual pattern.  For the third unit, we will work on linear equations, which will be motivated by giving them a linear pattern, telling them how many items are in a particular step number, and then having them try to work backwards to figure out what that step number is.

I've already made most of the content for this unit, but please note that none of the following lessons have been taught yet.  They are just my initial stab at creating a unit following the SVMI transitional plan.  Please feel free to look at, play with, and comment on these lessons.  If you see any problems or areas for improvement, please comment.  The introductory lesson is not there, as it is still under development.  If you have ideas for how to excite students about patterns, please share!  Also, the expert task is still under development.  The idea is to first have students work on the Table Tiles FAL, and then create patterns of their own using paper cut-outs, and write functions for the patterns they developed.  I still need to create the project description and rubric.

Here is the link to the box folder with the materials:  Patterns Unit

I hope everyone is having a great, relaxing start to their summers!


David said...

Hi Dan,

I am also partnered with SVMI. I find the unit structure you are working on to be interesting. We are using the Math Design Collaborative units to structure our work, but your within-unit structure seems to be a bit more robust than what we have.

If you are interested, I can send you our website where we are gathering our stuff. It's not 100% public yet, but we are using many of the same resources, so you might find it useful.


Dan Wekselgreene said...

Hi David,
Yes, please send the link to me - that would be very helpful.

cheesemonkeysf said...

Dan – As usual, I love your breakdown and analysis of this unit structure! Looking forward to seeing how this unfolds.

- Elizabeth (@cheesemonkeysf)