Inquiry is Differentiation
Written by: Dr. Christine von Renesse
The worst form of inequality is to try to make unequal things equal.Aristotle
There is a part of me that wants to constantly find out where my whole class is at and what all my students are ready to learn next. What is the next best question for them? The problem is that this task is completely impossible! All students in a class will be different. Different how? I just read a paper by the educational psychologist David Jonassen , in which he explains some of the individual differences that influence problem solving. Among the characteristics he lists are:
- fatigue, anxiety, stress
- familiarity with the problem type,
- domain and structural knowledge (how much does the student already know about the content and how well is it integrated),
- cognitive controls (patterns of thinking that control how learners process and reason, e.g. independence and flexibility),
- metacognition (awareness of how one learns, ability to judge the difficulty of a task, monitoring understanding, assessment of learning process, etc),
- beliefs about knowledge and problem solving (e.g. the belief that “knowledge is either right or wrong”),
- affective and motivational elements (attitudes, beliefs about learner’s ability, persistence, making choices, interest, confidence, etc.)
Given the reality of all these differences, how can I even dream of a functional classroom in which every student is learning at their own learning edge?
Before I describe some of the tools we have been trying out in our classes, I invite you to watch Volker as he is supporting 4 students who are all at very different levels in their learning of the Rubik's cube:
As the title of this blog post suggests I believe that inquiry helps tremendously in reaching learners on different levels at the same time - even if they all work on the same investigations. Why? First of all, students can work at their own pace. This is hugely important since students have to be processing the ideas and concepts at their speed to deeply make sense of them. When students are exploring mathematics they can also use strategies and representations that they feel comfortable with. It is amazing to see in how many different (often to me totally new) ways students solve problems in all the classes I teach, from math for liberal arts through calculus 3. They can first make sense of the mathematics using their own way of thinking and then later learn from their peers other ways and connections. Additionally, since students are in charge of their learning process they can ask for help exactly when they need it, either from their peers or from me.
I believe that most students know pretty well how they compare to other students and that they don’t like having to fit in and be the same all the time. So I do prefer to be honest about the differences. Some students are incredibly persistent, some get quickly frustrated, some are super fast and some like to take their time, etc. It is actually a relief to my students and me when all of these differences are on the table and can be addressed openly. Students tell me how much they value being seen in my class and I can see how they engage and advance in their learning process.
I want my students to be as close to their learning edge as possible, which means they have to work in groups that are as similar as possible. See our strategies for grouping to learn about how to make this happen.
Good (Large) Problems
Ok, by now I know that it is impossible to find one particular task that will be just at the right level for all my students. Instead of trying to find one I now like to look for larger problems that have a “low thresh hold” and a “high ceiling”. In my perfect world every student is able to engage with the problem (and find it interesting) while the problem is complex enough that the really strong students are still challenged. An example of a really good problem would be the straight cut origami from the Art and Sculpture book : Given some polygonal shape on paper, fold up the paper in a way that you can cut out the whole shape in one straight cut.
Small Investigations and Extension Questions
Some concepts are too complex to be just be stated as one large problem, for instance the convergence of a geometric series in a math for liberal arts class. Instead of overwhelming the students with the large problem, we can ask a series of smaller investigations. It is fairly straightforward to adjust the investigations to be easy enough, especially if students can work in groups and you are available as a resource. But now we have the really fast students who want or need to be challenged more. So let’s make extension questions that engage the groups that are done with the core investigations. Yes, this is a lot of work and I am still working on always having good extension questions myself. What makes an extension question “good”? I want it to be interesting, challenging but possible for the stronger students, and not “pre-teaching” any material that will be coming later in the class -- since that would only make my differentiation problem worse later on. In our math for liberal arts materials many sections have "Further Investigations" which can be used as extension questions. At the end of chapter 4 of the book The Infinite you can for instance find deeper questions about series convergence.
Another way of bringing in differentiation into your class is by letting students work on two different topics simultaneously. The students choose when they are ready to switch tasks for a while and then come back later to pick up the first topic again. Of course these two topics could be from different books! Our mixing it up blog shows the dependencies of our book chapters so you see which chapters can stand alone.
Volker's classroom reminds me a bit of how my daughter learns in a Montessori classroom . She can always choose what she wants to learn in the moment but she has larger learning goals that remind her to attempt all activities at some point.
See in the video clips below how Volker gives students in one group different tasks and later explains to his students how to work on the two chapters (Rubik’s cube and Kakuro Puzzles) simultaneously.
Resources for Extra Support
Having resources available is helpful to all students, but it is important that the resources don't avoid the students' learning edges. If the resources make the task too easy or even unnecessary, there is no learning happening anymore. Here is an example: In my calculus 3 class, the students can always use play-dough to model the graphs of the functions they are studying. It is amazing to see how much students at any level appreciate this simple medium to help them visualize, communicate and model ideas. While I also usually allow www.worlframalpha.com and graphing calculators during class there are tasks for which I limit their use. If my students for instance grapple with the idea of convergent or divergent series, receiving the answer from the technology can take the fun and curiosity out of the exploration and discussion. Similarly, I allow students to watch videos on procedures of integration after they developed and made sense of the procedure themselves.
Whole Class discussions
This is kind of funny because whole class discussions actually help make knowledge more equal which makes the classroom experience less differentiated. So while this is not really a tool to help students at their individual learning edges, it is a tool to bring out the differences and spread ideas and questions around the class. As a result student knowledge and learning edges are a bit more similar than before. I like to do this especially in classes like calculus where I need to cover material more quickly than some students or groups are ready for independently. But class discussions also help increase curiosity and motivation and make any class more fun for me.
Assessment and differentiation is a tricky subject, it seems like 4 professors can easily have 5 opinions ;-)
For me it depends on the audience. In my calculus classes for instance I believe that exams have to be the same for all students. But homework can be more differentiated. Students can write up their thinking about their current knowledge edge, which will differ from student to student. Instead of grading the knowledge level I can grade the process, their thinking and writing ability and their effort and persistence. In my math for liberal arts classes I like to assess students differently, using projects and posters as assessment techniques which allow students to bring in strengths outside of mathematics, for example a student's poem or a song .
K-12 and Research about Differentiation
Differentiation in K-12 has different challenges than in college, I think. Carol Tomlinson has published several books to that effort and also started research on differentiation in higher education (not math).. It seems to me that there is a need for more research about differentiation in higher education mathematics.
1. Jonassen, David H. "Toward a Design Theory of Problem Solving." Educational Technology Research and Development. 48.4 (2000): 63-85. ↩
2. Tomlinson, Carol A. and Santangelo, Tanya "The Application of Differentiated Instruction in Postsecondary Environments: Benefits, Challenges and Future Directions". International Journal of Teaching and Learning in Higher Education. 20.3 (2009): 307-323. ↩