Inaugural David Clarke Memorial Lecture Q&A

Below is the response of Prof. Alan Schoenfeld to additional audience questions asked at the Inaugural David Clarke Memorial Lecture held on Thursday 10 December 2020.

Questions from the audienceResponse from Prof. Alan Schoenfeld

Q. Thank you Alan, I was wondering how, and to what extent, embodiment - including interaction with material artefacts - may be incorporated into the TRU model? (from Maurizio Toscano)

In a word, "fully." Within the mathematics dimension, there is the notion of becoming comfortable and fluent with multiple representations. That certainly includes material artefacts, and embodiments! Moreover, a fundamental aspect of TRU (cf. dimension 4) is social. Clearly, some artefacts (and the task demands, and classroom norms) are more conducive to productive group experiences than others. Examining these affordances is key to TRU.

Q. Just wondering what the role of the teacher is in the video. Did she get to listen to the students' responses? The student discussion is rich. What did she make of it and how can teachers respond to different responses in a classroom? (from Vincent Andrew)

The long answer to that is the lesson plan for the Formative Assessment Lesson she was teaching - see for the lessons in general, and the specific lesson plan,

Q. International comparison studies (e.g. TIMSS) show students from countries with high human development index (developed countries) sometimes show low dispositions towards maths but at the same time do well in maths. To Alan, what could be mediators? (from Vesife Hatisaru)

Tests such as TIMSS cover only a small part of what I called "thinking mathematically" in the first part of the talk. Students who are aware of the importance of doing well in math class can "master" typical course content but not like it; they're acting instrumentally. That's very different from doing mathematical sensemaking (which gets a different affective response).

Q. Alan - Thanks for a wonderful and thought-provoking discussion  - in memory of David. - In your experience, what is the most important ingredient often missing or underdeveloped in a maths learning environment? (from Ann Gervasoni)

In a word, "sensemaking." I truly believe that the entire K-14 curriculum can be learned by presenting students with situations and challenges that they can make sense of, with the formal curriculum being the codification of things they've put together themselves.

Q. Hi Alan, thank you very much for your thought-provoking talk. Having worked on the Learner’s Perspective Study with David for a long time, I realized that “teacher’s perspectives on the leaners’ perspectives” does matter for a productive classroom. Do you have any comment based on the TRU framework? (from Yoshinori Shimizu)

Hi Yoshi, I agree with you. A teacher who used the formative assessment lessons told researchers, "I never thought my students could do things like this. I've learned to trust them much more."  Overall, this is a matter of teacher beliefs. Like all beliefs, they change slowly as a result of experience. So, we have to provide scaffolding for teachers to move toward teaching in new ways, and time to develop the skills needed. The development (or change) of knowledge and beliefs goes hand in hand.

Q. What do you think needs to be considered next in terms of the research agenda and associated video-based methods for international comparative studies. (from Natasha Ziebell)

For me, it's a combination of macro and micro. At the macro level, I've learned a great deal by learning about lesson structure (e.g., whole lessons devoted to one problem that takes student thinking into account), the use of the board to record a lesson's evolution, etc. Once these things are noted, the micro studies that examine the impact of classroom practices on student thinking (including how students make sense of the environment) seem to be the right kinds of follow-ups.

Q. Thanks for an interesting presentation. I am curious to know how the current situation and a switch to online education informed your thinking about your own framework. Is it relevant for this new learning-and-teaching environment? (from Igor' Kontorovich)

The framework is robust, in that it's been shown to apply to a huge range of learning environments - including, to anticipate Hilary's question below, PD. Take Dimension 4, for example, which says that participants' development of agency and identity depends on the character of the environment - its affordances for getting in on the conversation, having one's ideas built on, building on the ideas of others. That's simply true - but, the affordances for doing so in online education are radically different than in classrooms. That says we have a massive research agenda ahead of us - I don't know how to support the right dynamics in an online environment, I don't know what the impacts of e-learning are along such dimensions. There's a lot to be learned!

Q. Thank you Alan for an insightful presentation! I wonder how we might support teachers to develop expertise to attend to student thinking in mathematics problem solving rather than just focusing on skills and outcomes of student thinking? (from Lihua Xu)

That's been a lifelong challenge! Externally, we have to work on assessment systems. As Hugh Burkhardt wrote, "What You Test is what You Get." If we have examples of rich tasks and teachers know their students will be asked to solve them, they'll work on them. If not, not. Then, curricular materials and PD aimed at supporting such thinking are essential. Teachers can't do it alone.

Q. Thank you Alan - David would have loved your presentation and this rich discussion. I’m interested to know whether in your work with teacher professional learning, do you translate the ‘Five Dimensions of Powerful Classrooms’ into ‘Five Dimensions of Powerful PL’? I can see how this could be fruitful. (from Hilary Hollingsworth)

Yes! As mentioned above, TRU is a theory of productive learning environments - so it should apply to professional learning, in PD and in schools and school districts more generally. See my 2015 ZDM paper "thoughts on scale," at DOI 10.1007/s11858-014-0662-3. That paper talks about applications of TRU to PD.