October 2017 – Thinking Mathematically

The other day I was asked about my opinion about something called entrance slips. Curious about their thoughts first, I asked a few question that helped me understand what they meant by entrance slips, what they would be used for, and how they might believe they would be helpful. The response made me a little worried. Basically, the idea was to give something to students at the beginning of class to determine gaps, then place students into groups based on student “needs”. I’ll share my issues with this in a moment… Once I had figured out how they planned on using them, I asked what the different groups would look like. Specifically, I asked what students in each group would be learning. They explained that the plan was to give an entrance slip at the beginning of a Geometry unit. The first few questions on this entrance slip would involve naming shapes and the next few about identifying isolated properties of shapes. Those who couldn’t name shapes were to be placed into a group that learns about naming shapes, those who could name shapes but didn’t know all of the properties were to go into a second group, and those who did well on both sections would be ready to do activities involving sorting shapes. In our discussion I continually heard the phrase “Differentiated Instruction”, however, their description of Differentiated Instruction definitely did not match my understanding (I’ve written about that here). What was being discussed here with regards to using entrance slips I would call “Individualized Instruction”. The difference between the two terms is more than a semantic issue, it gets to the heart of how we believe learning happens, what our roles are in planning and assessing, and ultimately who will be successful. To be clear, Differentiated Instruction involves students achieving the same expectations/standards via different processes, content and/or product, while individualized or targeted instruction is about expecting different things from different students.

Issues with Individualized / Targeted Instruction

Individualized or targeted instruction makes sense in a lot of ways. The idea is to figure out what a student’s needs are, then provide opportunities for them to get better in this area. In practice, however, what often happens is that we end up setting different learning paths for different students which actually creates more inequities than it helps close gaps. In my experience, having different students learning different things might be helpful to those who are being challenged, but does a significant disservice to those who are deemed “not ready” to learn what others are learning. For example, in the 3 pathways shared above, it was suggested that the class be split into 3 groups; one working on defining terms, one learning about properties of shapes and the last group would spend time sorting shapes in various ways. If we thought of this in terms of development, each group of students would be set on a completely different path. Those working on developing “recognition” tasks (See Van Hiele’s Model below) would be working on low-level tasks. Instead of providing experiences that might help them make sense of Geometric relationships, they would be stuck working on tasks that focus on memory without meaning.

When we aim to find specific tasks for specific students, we assume that students are not capable of learning things others are learning. This creates low expectations for our students! Van de Walle says it best in his book Teaching Student Centered Mathematics:

Determining how to place students in groups is an important decision. Avoid continually grouping by ability. This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.

Targeted instruction might make sense on paper, but there are several potential flaws:

Students enter into tracks that do not actually reflect their ability. There is plenty of research showing that significant percentages of students are placed in the wrong grouping by their teachers. Whether they have used some kind of test or not, groupings are regularly flawed in predicting what students are potentially ready for.
Pre-determining who is ready for what learning typically results in ability grouping, which is probably the strongest fixed mindset message a school can send students. Giving an entrance ticket that determines certain students can’t engage in the learning others are doing tells students who is good at math, and who isn’t. Our students are exquisitely keen at noticing who we believe can be successful, which shapes their own beliefs about themselves.
The work given to those in lower groups is typically less cognitively demanding and results in minimal learning. The intent to “fill gaps” or “catch kids up” ironically increases the gap between struggling students and more independent learners. Numerous studies have confirmed what Hoffer (1992) found: “Comparing the achievement growth of non-grouped students and high- and low-group students shows that high-group placement generally has a weak positive effect while low-group placement has a stronger negative effect. Ability grouping thus appears to benefit advanced students, to harm slower students.“

The original conversation I had about Entrance Tickets illustrated a common issue we have. We notice that there are students in our rooms who come into class in very different places in their understanding of a given topic. We want to make sure that we provide things that our students will be successful with… However, this individualization of instruction does the exact opposite of what differentiated instruction intends to do. Differentiated instruction in a mathematics class is realized when we provide experiences for our students where everyone is learning what they need to learn and can demonstrate this learning in different ways. The assumption, however, is that WE are the ones that should be determining who is learning what and how much. This just doesn’t make sense to me! Instead of using entrance tickets, we ended up deciding to use this problem from Van de Walle so we could reach students no matter where they were in their understanding. Instead of a test to determine who is allowed to learn what, we allowed every student to learn! This needs to be a focus!

If we are ever going to help all of our students learn mathematics and believe that they are capable of thinking mathematically, then we need to provide learning experiences that ALL of our students can participate in. These experiences need to:

Have multiple entry points for students to access the mathematics
Provide challenge for all students (be Problem-Based)
Allow students to actively make sense of the mathematics through mathematical reasoning
Allow students opportunities to students to express their understanding in different ways or reach an understanding via different strategies

Let’s avoid doing things that narrow our students’ learning like using entrance tickets to target instruction! Let’s commit to a view of differentiated instruction where our students are the ones who are differentiating themselves (because the tasks allowed for opportunities to do things differently)! Let’s continue to get better at leveraging students’ thinking in our classrooms to help those who are struggling! Let’s believe that all of our students can learn!

I want to leave you with a few reflective questions:

Why might conversations about entrance tickets and other ways to determine students ability be more common today? We need to use our students’ thinking to guide our instruction, but other than entrance cards, how can we do this in ways that actually help those who are struggling?
Is a push for data-driven instruction fueling this type of decision making? If so, who is asking for the data? Are there other sources of data that you can be gathering that are healthier for you and your students?
If you’ve ever used entrance tickets or diagnostics, followed by ability groups, how did those on the bottom group feel? Do you see the same students regularly in the bottom group? Do you see a widening gap between those dependent on you and those who are more independent?
Where do you look for learning experiences that offer this kind of differentiated instruction? Is it working for the students in your class that are struggling?

I encourage you to continue to think about what it means to Differentiate your Instruction. Here are a few pieces that might help:

I’d love to continue the conversation. Write a response, or send me a message on Twitter ( @markchubb3 ).

Issues with Individualized / Targeted Instruction

I want to leave you with a few reflective questions:

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