A Reflection about Universal Instruction

After reading Mark Chubb's recent blog post about interventions, I started reflecting on the learning I have been doing this year around universal instruction.
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? 

Other notes:

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).  

I think it is worth noting that if the idea of creating and implementing these tasks every day in you classroom feels overwhelming, Episode 8 of the Making Math Moments podcast is worth a listen.  It is a mentoring moment with Katrien Vance and she shares her realization that these tasks don't need to be elaborate, time-consuming tasks.  This can be accomplished with a simple problem as well.  Be sure to check it out. 

Critical Math for the ACT

I recently went to a workshop on Critical Math for the ACT at the Wisconsin Math Institute.
I wasn’t really sure what to expect.  I had been told that there is a lot of middle school
math on the ACT, so it seemed like I should know more about it.  I was really quite
surprised by the information and it got me thinking, not only about the foundation for
the ACT, but about acceleration practices.  This blog is really a place for me to try to
process all that I learned.

I did not know that the ACT has created its own College and Career ready standards.  

ACT College and Career Ready Standards to view online for most up to date version

PDF version Feb2019 - I like the layout of this one better but could become out of date

For a benchmark score of 22, most of the standards align with 6th, 7th, and 8th grade math.  

There are still many up to a score of 27 and some for scores higher.  

So it turns out the critical math for the ACT is middle school math. Middle school math is foundational.  Not just learning it, but retaining it.  That is an issue of focus, coherence, and rigor, but that is a different blog.

The ACT® Test User Handbook for Educators Online Link, PDF lays out the

Content Covered by the Mathematics Test (p. 46)

Preparing for Higher Mathematics (57–60%)

This category covers the more recent mathematics that students are learning, starting when they began using algebra as a general way of expressing and solving equations.

Integrating Essential Skills (40–43%)

This category focuses on measuring how well you can synthesize and apply your understandings and skills to solve more complex problems. The questions ask you to address concepts such as rates and percentages; proportional relationships; area, surface area, and volume; average and median; and expressing numbers in different ways. Solve non-routine

problems that involve combining skills in chains of steps; applying skills in varied contexts; understanding connections; and demonstrating fluency

This means that approximately 40% of the test requires the application of concepts learned prior to 8th grade.  

Achieve the Core did independent research found here. They found:

In mathematics, fewer than half of items on the assessment were judged to be aligned to the claimed Common Core mathematical content standards for high school.

Clearly there is far more value in middle level math than most people realize. We are not just laying the foundation for higher level math, we are the core. (If someone has a better word choice to offer I would appreciate it.  I struggled to come up with the right words.) The ACT can be a gatekeeper to higher education. In middle school, with preadolescent hormones running rampant, many students don’t see the point of school.  They figure they can pull it together when it is important, you know, high school. I think all the above information speaks to why middle school math is important and why teaching it in a way the promotes conceptual understanding and sense making is key.  

But what about the high students?  The students who skip a year of math or work at
an accelerated pace?  When students are accelerated they have less time exploring
these core concepts and therefore do not have as deep an understanding as they could.
 And what is the trade off? Supposedly they get to high math in high school. Depending
on the options in high school this may or may not be the case.  It is quite possible for a
student to take Algebra freshman year and with the right plan, get to AP Calculus senior
year. It seems to be the mindset of parents and students that the higher you get in
math the better.  I’m not sure if that is true.

I love this statement from the Mathematical Association of America and the National
Council of Teachers of Mathematics

MAA/NCTM Position

Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.

Digging deeper in the background information on the statement it states:

...the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. Nothing demonstrates this more eloquently than the fact that from the high school class of 1992, one-third of those who took calculus in high school then enrolled in precalculus when they got to college, 8 and from the high school class of 2004, one in six of those who passed calculus in high school then took remedial mathematics in college.

Within our schools, there is tremendous pressure to fill these classes, accelerating every student who might conceivably be ready for calculus by the senior year regardless of whether such a student might benefit from a slower and more thorough introduction to the traditional topics of high school mathematics.

It makes me wonder if we are we taking time to consider the individual? Do we provide opportunities for parents to discuss this so they can help make the decision for their child?  I’m not talking about the parents who are obsessively asking that their child be accelerated. I feel confident that those conversations are happening, but what about the parents who simply put their child in an accelerated course because the student met the school’s requirements.  As someone who dropped out of a middle school accelerated program my mother never wanted me in, I feel strongly that there are so many factors the school does not know. And the research around retaking and remediation shows that an experience like mine can alter career plans. We need to reach out to parents to help equip them to better make the decision with their child.  

I like this quote from Matt Larson’s Mathematics Learning: A Journey, Not a Sprint

Should we support acceleration? This question, like many questions in mathematics education, does not have a binary answer. The answer is “it depends.” Sometimes acceleration is appropriate and sometimes it isn’t. What does the answer depend on? Here the answer is clearer: it depends on the student’s demonstrated significant depth of understanding of all the content that would be skipped. If a student demonstrates significant depth of understanding of some but not all the content that would be skipped, then this is more appropriately an opportunity for enrichment rather than acceleration.

There is evidence that students who speed through content without developing depth of understanding are the very ones who tend to drop out of mathematics when they have the chance (Boaler, 2016).  Acceleration potentially decreases student access to STEM careers if it results in students dropping mathematics as quickly as possible, rather than cultivating and developing the joy of doing and understanding mathematics. This is important to point out to parents, as dropping out of mathematics is clearly not an outcome parents want to encourage.

I honestly could have copied and pasted the whole article.  It is fabulous and you should click the link and read the whole thing.  

My goals coming out of this workshop:

• Create a parent communication night for students who selected for our compacted Math ⅞ course to help them better understand their options.  
• Analyze our districts data to see which content students are not retaining and evaluate our instructional strategies to see if there are shifts we need to make to better build conceptual understanding so students retain and can apply these skills.  
• Have vertical conversations with the high school.  The middle school teachers cannot be solely responsible for these standards that students are expected to apply on a high stakes test their junior year of high school.  We need to look at when skills are introduced and when we expect proficiency. ACT provides this helpful tool that I am hoping will focus these discussions.  

I’d love to hear about work you are doing in your district around this topic or if you have articles or links to share.  Please use the comment section below.