I wrote an overview earlier about the structure for this conference.
This post will focus on one aspect the research forum that I attended. There were 5 different research forums that took place. Each research forum had two sessions - one near the beginning of the conference, and a second one near the end of the conference. The goal of the research forum is to create dialogue and discussion by offering PME members more elaborate presentations, reactions, and discussions on topics. I attended the session in bold:
RF1 - The challenges of teaching mathematics with digital technologies
RF2 - Mathematical Tasks and the student (attended this one)
RF3 - Mathematical modeling in school education: mathematical, curricular, cognitive, instructional, and teacher education perspectives
RF4 - Spatial reasoning for young learners
RF5 - Habermas' construct of rational behavior in mathematics education: new advances and research questions
The overarching focus of the research forum I attended was: The tension between the teacher's instructional intentions and the students' consequent activity.
I won't attempt to cover all aspect of the research forum - since there were so many good ideas and discussions. Instead I will focus on one aspect of session 1 that got me thinking.
We had session 1 near the beginning of the conference. Before the session got officially underway, one of the facilitators, David Clarke, gave a brief intro on the rationale behind the session.
"In this Research Forum, we focus attention on the task as mediating artifact and address the question of how the resultant socio-didactical tetrahedron (Fig. 1) might structure our consideration of research into the function of tasks in facilitating student learning and into the dynamic between student and task" (David Clarke, Heidi Strømskag, Heather Lynn Johnson, Angelika Bikner-Ahsbahs, Kimberly Gardner, 2014, p.1-118)
Clarke et al. (2014) paraphrased Rezat and Strässer and described how each of the triangular faces of the tetrahedron stands for a particular perspective on the role of tasks within mathematics education. It offers a nice representation of the complexity of classroom teaching and learning. It was interesting to think about the connections, and it definitely prompted me to read about the ideas further.
There were 5 presenters during session 1, and 4 presenters during session 2. The two sessions focused on the following 4 issues.
1. Differences in the function of mathematical tasks and the nature of student task participation in different instructional settings.
2. Utilizing mathematical tasks to promote students' higher order thinking skills.
3. Differences in the theoretical frameworks by which the instructional use of mathematical tasks might be better understood (particularly from the perspective of the student) and thereby optimized).
4. The accommodation of student agency within the instructional use of mathematical tasks.
Each speaker had a chance to present for 5 minutes. Immediately following each presentation, the audience had a chance to discuss ideas, as well as pose questions. After three presentations on Issue 1, we then had a large group discussion of all of the ideas, lingering questions and thoughts. After the large discussion, 2 more presenters gave their presentations on Issue 2. Again there was a discussion after each presenter, as well as a large group discussion after.
At the end of the sessions the facilitators asked us to also write down a summary of ideas (as well as any questions), and to submit it. This was then used in the next session. Session 2 also followed a similar format and had 4 presenters.
The following is the list of the 9 presentations on the 4 issues (please note that this post isn't about the sessions. Instead it's on my thoughts about one thing that came out of the sessions for me. If you are interested in any of the following presentations or issues, let me know and I can make a post about it)
Issue 1: Differences in the function of mathematical tasks and the nature of student task participation in different instructional settings.
Alf Coles (University of Bristol) - Making distinctions in task design and student activity
Joaquim Gimenez, Pedro Palhares (Barcelona University, Institute of education) - Order of tasks in sequence of early algebra
Annie Savard, Elena Polotskaia, Viktor Freiman and Claudine Gervais (McGill University, Université de Moncton, Commission scolaire des Grandes-Seigneuries) - Tasks to promote holistic flexible reasoning about simple additive structures.
Issue 2: Utilizing mathematical tasks to promote students' higher order thinking skills.
Einav Aizikovitsh-Udi, Sebastian Kuntze, and David Clarke (Beit Berl College, Israel, Ludwigsburg University of Education, Germany, University of Melbourne, Australia) - Hybrid tasks: promoting student statistical thinking and critical thinking through the same mathematical activities
Heather Lynn Johnson (University of Colorado) - Designing covariation tasks to support students' reasoning about quantities involved in rate of change.
Issue 3: Differences in the theoretical frameworks by which the instructional use of mathematical tasks might be better understood (particularly from the perspective of the student) and thereby optimized).
Kimberly Gardner (Kennesaw State University) - Applying the phenomenographic approach to students' conceptions of tasks.
Heidi Strømskag (Sør-Trøndelag University College) - The milieu and the mathematical knowledge aimed at in a task.
Carmel Mesiti, David Clarke (International Centre for Classroom Research, University of Melbourne) - writing the student into the task: agency and voice.
Angelika Bikner-Ahsbahs (Bremen University, Germany) - Emergent tasks: spontaneous design supporting in-depth learning.
First up was Coles who spoke briefly about their beliefs in "task design that centres around activities that provoke differences in student response can allow the opportunity for students to make mathematical distinctions and for teachers to introduce new skills. (p.1-120).
He provided an example for the audience which was the following picture:
He also gave the instruction for us to consider what is the same and what is different.
Discussions extended to area, perimeter, definition of "equable shapes" (for the 3x6 rectangle), as well as whether there are other equable shapes.
The task was designed for learning proofs. Coles then used a "pipe" metaphor to describe tasks as providing a constricted place, from which the students can explore open follow up questions.
This pipe metaphor is what I wanted to think more about. The diagram that he provided was the following:
The idea is that we design tasks like designing pipes - where water (student and student ideas) flows. This metaphor was quite a hit. The discussions following the presentation, as well as subsequent discussions, often contained some sort of application of this metaphor.
I thought about this concept a bit more as well. Let me walk through some of my thought process, as well as the diagram that I arrived at.
I initially thought about the structure of the pipe. The length may refer to the duration and length of the task. How big is the task? Does it take 10 minutes? an hour? several days?
The width of the pipe may refer to space within the task, and how restricted the task is.
Then I thought about connections between tasks with respect to task sequencing. what task occurs next? how do we recollect the students to a common place? I then came up with the following diagram (in my head) to illustrate this.
If our idea is to actively move students across concepts, then we need to consider the appropriate length, width, and sequences -- in order to allow students to effectively move themselves across. The water pressure is important. We can't have the pipe be too narrow because it may leave out students. We can't have the pipe be too long or else we would be unable to sustain student interest.
If you zoom in, there is actually a resemblance to what I've heard Dan Meyer spoke about with respect to tasks that have an open middle. I am unsure if he referenced this idea from elsewhere. More on this later.
I didn't like that diagram with the two pipes though. Not completely yet. Besides the fact that it is poorly drawn and ugly, I had an issue with the fact that it was linear. So I thought about it some more and came up with the following. I don't really have a name for it:
Let me go into some of the features of this diagram.
1. The Engagement Vacuum
Likely the most important part of any task is its ability to draw students in and to have them engage in the mathematics itself. Whether it is to "recollect" the students or to allow them to move themselves onto the next concept, engagement is important. I am tentatively calling it the "engagement vaccuum" aspect of the diagram. I don't mean vacuum as in a space that is devoid of matter. Instead, I mean one of those household vacuums that takes up everything (I welcome alternative wordings...). One of the flaws of the previous diagrams (I think) was that there isn't really an explanation for why the water is moving across the pipes in the first place. Are we forcing them through? Are they willingly moving across? It has great implications to other aspects of the process such as length and width. How do we gauge how long or wide the tasks need to be, if we lack the understanding of the pressure of the water moving across? Hence the idea of the engagement vacuum. As students move across, there is an engagement vacuum at the entrance of each task to allow them to carry themselves through interesting problems across different concepts of mathematics. The arrow represents the vacuum The different sizes of the arrows currently represents the strength of the vacuum.
Also note that some of the pipes don't have vacuums. This was intended to represent tasks that either don't have these engagement vacuums, or don't need one. In other words, they are not engaging enough to allow students to move through on their own. Maybe this is because they don't need one there. It's merely there as a guide to get to the next engagement vacuum.
2. Structure of pipes
First, notice the different sizes of the pipes. The idea is that different tasks may be of different lengths or widths depending on the student, teacher, classroom dynamics,...etc. Perhaps the structures would also depend on the strength of the vacuum (how engaging the task is) inherent in the task.
3. Network of pipes
I didn't like the linearity of my previous diagrams, and so this diagram attempts to address this. The diagram is a network moving across different mathematical concepts. Sometimes several outcomes could be concluded from each task, or there may be different directions student may be interested in pursuing. Note also that the boundaries of each of the concepts is not necessarily at the beginning or end of each of the tasks. This was intentional in that I believe students may explore different concepts within a different task - or move from one to another.
4. Internal structure of tasks
I liked the open middle idea from Dan Meyer. This may be up for discussion, but I thought this:
If we zoom into each of the pipes, there are opportunities for open middles -- but ultimately bounded by the structure of the task itself. This is indicated by the green lines. I like the idea of there not being any boundaries, but I don't think it's possible. For example, let's take a look at a quick example from 101 questions
Of course there are lots of direction that students may choose to go, and that's fine! I have advocated for honouring student ideas and questions, and in exploring answers together. But there are still boundaries. Boundaries like the questions that students may ask, the methods that students may use, or the directions that students may go. Maybe we don't set the boundaries (or maybe we do), and it is determined by other factors like student experiences, student interest, cultural dynamics...etc -- but nonetheless there are boundaries.
There were other features that I put thought into... that I have now forgotten... so I will just move on to what I like and don't like about this diagram.
What I like:
1. It provides a useful image for thinking about not only the structure of a task, but also how it fits with the rest of the tasks, as well as how it is situated in the landscape of mathematics. I think it also provides some perspective on describing the nature of what the teacher may (or should?) control in the classroom.
2. It does NOT indicate an importance of speed, but emphasizes the importance of the "engagement vacuum" that allows students to move across different aspects of mathematics.
What I don't like
1. There seems to be a direction of flow that was unintended. We move from one concept to another in the diagram, and always moving outwards due to the nature of the vacuum, and pressure, and whatnot... I don't like this. This may be solvable if we conceptualize a three dimensional model of some sort...
2. The diagram doesn't really provide a platform for discussion student creation of questions and student creation of tasks. Sure, within each task we can interpret the movement of student as being motivated by their own questions... but that only works within the scope of a teacher-provided task (e.g. a picture or video in the style of #3acts). What about larger student-initiated projects? How does that fit with the current diagram?
I think there is a way of mediating the two issues by including other aspects in the diagram... but I think that involves some 3D drawing that I am currently not capable of producing...
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