In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Dr. Peter Liljedahl.
Dr. Peter Liljedahl is an Associate Professor of Mathematics Education in the Faculty of Education and the Associate Dean Academic for the Office of Graduate Studies and Post-Doctoral Fellows at Simon Fraser University in Vancouver, Canada. Peter is a co-director of the David Wheeler Institute for Research in Mathematics Education, President of the International Group for the Psychology of Mathematics Education, a senior editor for the International Journal of Science and Mathematics Education, and the coordinator of the Secondary Mathematics Master’s Program in the Faculty of Education at SFU. Peter is a former high school mathematics teacher who has kept his research interest and activities close to the classroom. He consults regularly with teachers, schools, school districts, and ministries of education on issues of teaching and learning, assessment, and numeracy.
First of all, I’d like to thank you for taking the time to have this conversation. To start things off, could you discuss your current research interests and projects? How has your work kept you close to mathematics classrooms?
Almost all of my work is centred around improving the teaching and learning of mathematics. To this end, I work closely with practicing in-service mathematics teachers interested in improving their practice. At the same time, I do research on both the teaching and learning of mathematics and the professional growth of teachers of mathematics.
Some of your recent work has been centered around the notion of a “thinking classroom” (e.g., Liljedahl & Williams, 2014; Liljedahl, 2016b). How would you describe such a classroom? How can classroom norms and the classroom environment contribute, or detract from, a culture of thinking?
I define a thinking classroom as follows:
A thinking classroom is a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion. It is a space wherein the teacher not only fosters thinking but also expects it, both implicitly and explicitly. (Liljedahl, 2016b, p. 362)
Institutional norms (Liu & Liljedahl, 2013) often get in the way of such goals as they are predicated on activities of sitting and watching and listening. Desks are designed for this and the classroom is set up for this. It is not surprising, therefore, that traditional teaching happens in such traditional classroom spaces Institutional norms dictate this design and this design reproduces these institutional norms. Classroom norms are greatly informed by these institutional norms. I found that it is almost impossible to change these norms if we do not first change the physical space of the classroom.
For example, what would teaching look like if students stood rather than sat, wrote on whiteboards rather than in notebooks, worked in random groups rather than individually or in self-selected groups? Teaching and learning would, by necessity, look different if this was the environment. My research has experimented with such environments and the results are revealing that student learning can be fundamentally impacted by these changes to the learning environments.
In many of your recent presentations and workshops with mathematics teachers, you have discussed the potential of vertical non-permanent surfaces and visibly random groups in contributing to the development of thinking classrooms. (Readers who attended the 2015 SUM Conference may have been introduced to these ideas by Ontario teacher Alex Overwijk, who uses both extensively in his own classroom.) What advantages do these strategies offer (and why is it important that the surfaces are vertical and that the groups are visibly random)?
I know Alex well and have worked with him on several occasions in Ottawa. My research on Building Thinking Classroom emerged a set of 9 tools for transforming traditional classrooms into thinking classrooms (see Liljedahl, 2016b). These 9 tools turn out to be most effectively implemented in three groups of three tools each. In the first collection of three tools are Vertical Non-Permanent Surfaces (VNPS) and Visibly Random Groupings (VRG).
The non-permanence of the surfaces (e.g., whiteboards) somehow frees students to risk more and risk sooner. The vertical aspect increases visibility and helps mobilize knowledge around the room. Visibly random groupings help students to feel that there is no teacher strategic reason keeping them away from their friends.
I want to emphasize that these tools, including VNPS and VRG, emerged out of research. They are not simply an idea, but rather empirically proven methods of teaching that are effective for transforming non-thinking spaces into thinking classrooms.
The notion of a “thinking classroom” suggests a departure from a focus on the individual—which, as Davis, Samara, and Luce-Kapler write, has historically, overwhelmingly been seen as the “fundamental particle of knowing” (2008, p. 59)—and towards consideration of the role of the collective in teaching and learning practices. Do you see limitations in focusing exclusively on individual learners in the (mathematics) classroom?
As teachers, we tend to have our learning settings emulate our testing settings. Thinking classrooms differentiates these spaces and seeks to maximize the learning setting. My research has revealed that this is most effective when there are opportunities for collaborations and knowledge is allowed to move freely around the classroom.
Your earlier work (and thesis) focused on the “AHA! experience”: “the moment of illumination on the heels of lengthy, and seemingly fruitless, intentional effort” (Liljedahl, 2005, p. 219). What effect do these kinds of experiences have on students of mathematics, and is it possible to “lead” students to such experiences?
An AHA! Experience can have profound effects on students, transforming their beliefs about mathematics and their beliefs about the teaching and learning mathematics. This is true even for students with previously very negative beliefs. However, these experiences are elusive. We can create opportunities for them to occur, but we cannot guarantee them. I like to refer to this as occasioning AHA’s.
For example, my research on the AHA! showed that AHA!’s seemed to occur on the heels of a period of incubation, after having first worked on a problem in a collaborative environment where lots of different ideas had been tried and discussed. Although descriptive, these qualities can be prescribed. I was able to occasion an AHA! for some students by having them work on a problem collaboratively one class wherein I filled the space with lots of ideas, analogous problems, and manipulatives. This was followed by a forced period of incubation where we worked on something else, took a break, or sent them home at the end of class. The next class, we started with an analogous problem where I again filled the space with ideas and manipulatives relevant to the solution of the initial problem. When I then asked the students to turn their mind to the initial problem I was able to see several students having AHA!’s.
In wrapping up this interview, I’d like to touch on a somewhat tangential topic. Throughout your work, you’ve referenced Mihály Csíkszentmihályi’s notion of “flow,” a positive mental state in which a person performing an activity is completely absorbed in the action, experiencing deep enjoyment and creativity (1990/2008, 1996/2013). What connections do you draw between Csíkszentmihályi’s work and the teaching and learning of mathematics?
“In mathematics education, we have very few ways to think about engagement in theoretical ways. Csíkszentmihályi’s notion of flow is one of the few ways that this can be done.”
As I discuss in “Flow: A Framework for Discussing Teaching” (Liljedahl, 2016a), in the early 1970’s Mihály Csíkszentmihályi became interested in studying, what he referred to as, the optimal experience (1998, 1996, 1990),
a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it. (Csíkszentmihályi, 1990, p.4)
The optimal experience is something we are all familiar with. It is that moment where we are so focused and so absorbed in an activity that we lose all track of time, we are un-distractible, and we are consumed by the enjoyment of the activity. As educators we have glimpses of this in our teaching and value it when we see it.
Csíkszentmihályi, in his pursuit to understand the optimal experience, studied this phenomenon across a wide and diverse set of contexts (1998, 1996, 1990). In particular, he looked at the phenomenon among musicians, artists, mathematicians, scientists, and athletes. Out of this research emerged a set of elements common to every such experience (Csíkszentmihályi, 1990):
- There are clear goals every step of the way.
- There is immediate feedback to one’s actions.
- There is a balance between challenges and skills.
- Action and awareness are merged.
- Distractions are excluded from consciousness.
- There is no worry of failure.
- Self-consciousness disappears.
- The sense of time becomes distorted.
- The activity becomes an end in itself.
The last six elements on this list are characteristics of the internal experience of the doer. That is, in describing an optimal experience a doer would claim that their sense of time had become distorted, that they were not easily distracted, and that they were not worried about failure. They would also describe a state in which their awareness of their actions faded from their attention and, as such, they were not self-conscious about what they were doing. Finally, they would say that the value in the process was in the doing – that the activity becomes an end in itself.
In contrast, the first three elements on this list can be seen as characteristics external to the doer, existing in the environment of the activity, and crucial to occasioning of the optimal experience. The doer must be in an environment wherein there are clear goals, immediate feedback, and there is a balance between the challenge of the activity and the abilities of the doer.
This balance between challenge and ability is central to Csíkszentmihályi’s (1998, 1996, 1990) analysis of the optimal experience and comes into sharp focus when we consider the consequences of having an imbalance in this system. For example, if the challenge of the activity far exceeds a person’s ability, they are likely to experience a feeling of anxiety or frustration. Conversely, if their ability far exceeds the challenge offered by the activity, they are apt to become bored. When there is a balance in this system, a state of (what Csíkszentmihályi refers to as) flow is created. Flow is, in brief, the term Csíkszentmihályi used to encapsulate the essence of optimal experience and the nine aforementioned elements into a single emotional-cognitive construct.
If the way we give hints and extensions, then, is guided by trying to maintain this balance between complexity and ability, we can keep students in flow.
Thank you, Dr. Liljedahl, for taking the time for this conversation. We look forward to your upcoming work in this area and to continuing the discussion in the future.
Csíkszentmihályi, M. (1998). Finding flow: The psychology of engagement with everyday life. New York, NY: Basic Books.
Csíkszentmihályi, M. (2008). Flow: The psychology of optimal experience. New York, NY: HarperCollins. (Original work published 1990)
Csíkszentmihályi, M. (2013). Creativity: Flow and the psychology of discovery and invention. New York, NY: HarperCollins Publishers. (Original work published 1996)
Davis, B., Sumara, D., & Luce-Kapler, R. (2008). Engaging minds (2nd ed.). New York, NY: Routledge.
Liljedahl, P. G. (2005). Mathematical discovery and affect: the effect of AHA! experiences on undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 36(2–3), 219-234.
Liljedahl, P. (2016a). Flow: A framework for discussing teaching. Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 203-210, Szeged, Hungary.
Liljedahl, P. (2016b). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.), Posing and Solving Mathematical Problems: Advances and New Perspectives. (pp. 361-386). New York, NY: Springer.
Liljedahl, P. & Williams, G. (2014). Building a thinking classroom. PME Newsletter, February 2014, 8.