I am in a newly created middle school math coach / interventionist role this year. My job description is pretty open ended as my district recognizes the need for support and change in math and wants the role to evolve as we tackle the many issues surrounding math education.
One of my main focuses this year has been universal instruction. As Mark's blog points out, universal instruction should meet the needs of 80-90% of the students. Before determining what interventions should be done, it is important to reflect on our effectiveness in universal instruction.
Having been a math teacher this was not a new topic for me. I have had professional development and spent lots of time with my PLN on Twitter. Rich tasks with a low-floor and high-ceiling help us make the necessary shifts in math instruction. (Love this shifts in math self-assessment.) I have used 3-Act tasks, Robert Kaplinsky's problem based tasks, notice/wonder, and many of the resources identified in this document from Margie Pearse. I quickly realized, however, that it is one thing to fumble through these in my own classroom, taking risks (thanks to Classroom Chef for the encouragement), and improving my craft through experimentation. It is completely different to be in a coach role and try to help teachers navigate this process and encourage them to take risks.
Enter Jon Orr and Kyle Pearce and their amazing project, Making Math Moments that Matter. They have created videos, an online workshop, online communities on social media, and my favorite, a podcast. It was early on in the podcast series, when they were discussing the idea of low-floor/high-ceiling tasks that I had an aha moment. My understanding of these tasks was that they have an entry point for all students. All students can work through the task at their level. I don't remember which episode and I don't remember who said it, but Jon or Kyle made a comment about the purpose of the low-floor in the task is to hook the students. Now they are engaged and more willing to go further into the math because it isn't about the math for them anymore. They are curious and want to explore. This comment has been lingering in my thoughts. I realized that my understanding of low-floor/high ceiling-tasks had students working at different levels. They all get to the answer but some students might use lower level math to get there. The statement in the podcast made me realize that this was not the case. Going back to Mark's blog, he mentions access and equity. While students are not tracked into different classes, my interpretation of the purpose of the low-floor was a type of in-classroom tracking. I had a conversation earlier this year about the purpose of using different strategies. Is the purpose of discussing different strategies a means to hopefully get each student to have at least one strategy that works for them so they can find the answer? Or is the purpose of discussing different strategies to help students think flexibly and to deepen their understanding of a concept?
Not that I ever thought it was ok to have students stuck in some type of lower level math land, but the comment got me thinking about how we can help students with what are often considered intervention skills during these tasks. Pam Harris has a blog post about the development of mathematical reasoning.
Non-classroom interventions at my school tend to focus on the first 3 ovals. Often times in class we give students "access" to higher level math by giving them algorithms and calculators. The interventionist part of my role has me thinking about how we can use the low-floor of these tasks to help with student growth in these areas of mathematical reasoning.
In a Twitter discussion about strategies versus algorithms and the visual above, Pam provided this question.
"Where is the opportunity to build students' relationships?"And it ties back to the low-floor/high-ceiling task. Students enter the task at a place that is comfortable for them. They are hooked. Now we can have discussions that move students thinking forward. We can show them the connection between what they know and the higher level math we are working on. One of the big struggles that teachers voice to me about interventions with students is that they don't see the impact of it in class. Interventions work on lower level skills and the students are still struggling in grade level math even with extra math time build into their day. I believe that there is so much potential here to bridge the disconnect between intervention work and grade level math. But how? What does that look like?
It definitely seems like something that is easier said than done. I would say there have been moments of this in my teaching, but how can I make this the norm instead of a happy accident? I've been mulling this over since my aha moment about low-floor/high-ceiling tasks. As if they were reading my mind, episode 12 of the Making Math Moments that Matter podcast answered my question.
The short answer: Anticipate
The long answer: Listen to the Podcast 😉
The highlights and my big take-aways:
The task is selected based on a specific learning goal, but we can be intentional about making connections to other areas of the curriculum.
Prior to task, anticipate strategies and student thinking. *I would anticipate strategies to highlight, but I don't think I ever took the time to anticipate as thoroughly as Kyle and Jon discuss. *
Consider background knowledge. Think about the different entry points and the levels of mathematical reasoning. Identify if the task needs to be modified so that there is an entry point for all students.
Allow students to see their strategy through. Value their work, stretch their thinking, and elevate the new learning during consolidation.
Plan an extension and challenge students to use the new learning in a related task/problem.
Visuals and tools students are familiar with, such as a double number line, can help students connect to new concepts and ideas.
If we give students time to explore and experiment with tasks they will help us to see the connections between big ideas that we may not have planned or even seen ourselves.
The big ideas are a web. I know this. As I teach, my brain seems to shift into linear thinking and I again need the reminder. The big ideas are a web. Perhaps its the name low-floor/high-ceiling that gives the illusion of a linear progression. Perhaps it's because the students go through standards by grade level that we see student thinking as "below grade level". The big ideas are a web.
Questions to continue asking myself:
How well do I know that web?
What connections do I see?
What visuals and tools have students used that could be used to connect to the ideas in the task?
How might this be solved with the least amount of formal math?
Where is the opportunity to build relationships so strategies become natural outcomes?
It seems like it would be beneficial to have teachers discuss the web together. We use CPM and it does a beautiful job of connecting concepts. Each year I taught I noticed more. What connections did other teachers notice that I did not?
When it comes to tools, visuals, strategies, etc. I feel I need more vertical conversations. I do not know what students used in previous years. I also am not always sure when/how to introduce them and how to get students to use them flexibly. If it is a another students strategy, how do I help other students connect with it without it becoming a procedure (ie. multiplication on a number line, area model for multiplying fractions).