# Exemplary research results: extending the landscape of diagnostic perspectives and their relation to fostering

To illustrate perspectives and obstacles during a typical collective discussion of the video clips, we present short excerpts from the PD Session 2 in Cycle 2. Afterwards, we condense the findings into the developed perspective model for noticing and fostering potentials.

Functioning of focus questions and categories

In the following transcript, the teachers watch a video clip - complemented by the transcript - of four students working on the Stair Number Exploration in Figure 9.2 which was recorded in a Grade 8 classroom taught by the participating teacher Katharina. Next, they are asked to comment the scene spontaneously and make sense of the students’ (non-linear) exploration process in groups of two or three. Lastly, the facilitator and researcher (the second author) asks them to focus on mathematical potential, which is then discussed in the whole group. The transcript was translated from German and simplified marginally to enhance the readability. All names are pseudonyms.

• 41 FACILITATOR: Let’s bring it all together. What did you notice? Where are the potentials, which cognitive activities do you see? . . .
• 42JULIA: I go first. The first part, the first 43 seconds. It seems like they try

something systematically. Though, in our opinion, Kai is quite dominant. And Lukas is still stuck at understanding [the task]. In between he yells “no, again, again, again”, he is not there, yet. And in the end “ah yes, you can do it with 7, that’s correct”. So he is a bit behind the others and as I said, Kai dominates it all. Until there is the first conjecture after 43 seconds. . . .

46 SVENJA: In our group, we wondered if the students were on the same page

regarding if they look at stairs with two steps or stairs with three steps. When you only read the transcript, we got the feeling that one thinks about stairs with three steps in the beginning and another about stairs with two steps and they are not on the same page. Later, they agree with each other: when the first conjecture is verbalized, then they find each other and notice that they have to get on the same page. . . .

While the facilitator’s focus question in Line 41 addresses the mathematical potential in the scene, the teachers Julia and Svenja mainly concentrate on understanding and evaluating how the students cope with the problem. Both point out deficits in the group working process as one student is lost and the others are perceived to talk about different things in the beginning. This perspective of evaluating if the task is addressed correctly by the students is quite prominent in everyday classrooms, as the teacher has to make sure that everyone is able to understand and work on the given assignment. Svenja’s last sentence can be understood as a more positive evaluation as she points out how the students overcome their difficulties themselves, thus taking a more non-deficit-oriented stance.

• 48 SOPHIE: I find that they begin quite quickly to systematize and justify. When they find out that all uneven numbers work, they all confirm each other one after the other. They say “then you always have stairs” in line 23 [of the transcript] for example . . . and again in line 33. Confirms “because you only have one step”, which is for them already a justification that all uneven numbers work. And then they continue immediately: next you have to math- ematize the even numbers. Here, they are also very fast to try and find a generalization.
• 55 SVENJA: [Referring to the given categories of typical activities in exploration processes] I think they always jump a step up and then another and then one back and then again one up. They always get a step further and then one step back until they justify it at some point. Not on the first pages [of the transcript], I wouldn’t say they justify mathematically correct yet, written or in any other way. At least they did go many steps up, as you said (to Merle), but also down. It’s a constant movement. Nothing written down and never staying in the same spot, but always moving.

These statements by Sophie and Svenja show that they now apply the previously introduced categories for exploration processes (in Figure 9.3) as a lens to interpret the students’ working process, which leads them to an inquiry of their cognitive activities. Thus, the design element focus question and categories seems to successfully initiate the analysis of a more in-depth level of student thinking. Furthermore, Svenja comments on the complexity and non-linearity of the process which surfaces only by the design element video clip. The diversity of aspects and perspectives mentioned by the teachers shows the potential of the selected video clips for activating teachers’ diverse diagnostic processes.

While these statements above seem to be non-deficit oriented, it remains unclear if the teachers already perceive the situational mathematical potential. Thus, the researcher reminds the group of the focus question.

• 72 FACILITATOR: Did you notice anything else where you would say “There is something I would call potential. Something is happening which is mathematical potential for me”?
• 73 EMMA: Me and my colleague, we first had the impression that Tom just listens in the beginning. And I tried to find out if he works on the task or if he thinks about something else. And then we read the transcript and my colleague found out that we have to look more closely, because Tom is always structuring a bit. He lets others think and calculate and then he gives a small summary of what the others did. I find that pretty interesting. I wouldn’t have thought so while watching the scene, but by looking so closely. ... It is really the case, Tom is not absent or lazy and letting the others calculate, but he seems to do something completely different. He thinks about what is the common thing which they all just discovered and then he summarizes it in a structured way. I find that super interesting to discover.
• 74 CHRISTIAN: He refuses a bit to try out and always stays on a higher level and looks

down what the others do. And then he looks up to elaborate on that. 77 MERLE: And in line 85, when they are at the beginning of finding a term. It

was about having to subtract six, which was true for the stairs with three steps. And then he says “or minus 10” which is a reproduction, because it was said before, but I think it’s really clever to say it in this moment. Because he says you always have to subtract the base form, and the base form is always different. It can be 6, it can be 10. And thus they figure out that they have to subtract [a variable] z.

(Cycle 2, PD session 2, Clip “Stair Number Exploration”,

min. 82:49-99:37)

In line 73, Emma describes how her perception of the student Tom changed: While she was first unsure if he was working on the task at all (i.e. identifying a possible deficit in not being engaged in the working process), the detailed analysis of the transcripts leads her to a quite positive evaluation of Tom as a moderator who repeatedly summarizes, systematizes and thus pushes the exploration process of the group forward. Christian and Merle also point out the high cognitive level of Tom’s contributions. A possible interpretation of these statements is that the teachers were able to achieve a more in-depth level of analysis which leads them to uncover a potential in this situation, which they ascribe to Tom. Again, the application of hierarchical categories support them in their positive evaluation. While this potential might have been hidden in the very beginning of the clip, it becomes apparent in the student’s actions later on.

Accounting for obstacles and teachers' perspectives

The described snapshots from the second PD meeting in the second cycle are prototypical for teachers’ differences and obstacles in changing perspectives: Typically, teachers at first argue from a deficit-oriented mode which is overcome by the focus questions. However, the process perspective does not automatically lead to focusing hidden potentials and searching for strategies to foster them. Instead of thinking about strategies to foster uncovered seeds of situational potential, the teachers identify and discuss mainly strategies to help students to solve the open- ended mathematical problems. In consequence, the noticing focuses primarily on students’ processes of coping with the task (or why they could not cope well). This can be illustrated in an even more pointed way by the following excerpts of data.

After watching a video clip of two female students working on an open task about several derivatives (Grade 12), Sonja, one of the video-watching teachers in the third PD session, says

78 SONJA: Where they have problems is with verbalizing what they found out - especially mathematically correct verbalizing. So, I think they did understand the principle, but [not the relevant pattern behind it).

And well, you have to justify or formulate it in a more differentiated way.

(Cycle 2, PD Session 3, Clip "Derivatives", min. 16:48)

Within her analysis of the video clip, Sonja points out what the girls would have needed to accomplish the problem. She emphasizes what they reached and the discursive obstacles they need to overcome. Sonja’s perspective is an instance of what we researchers later decided to call the process-coping perspective (see later in this section): although already overcoming purely deficit-oriented modes and focusing on processes, Sonja does not yet focus on potentials. As our teachers often adopt this perspective, we needed to include it into the model and consider it as rational choice, since it is teachers’ responsibility to support the students in coping with the task (or their acquisition of competences or knowledge). Hence, it is also a direct successor of the product perspective.

This process-coping perspective often coexists with the potential indicator perspective which we have reconstructed when the teacher implicitly poses to her- or himself questions like “Which situational indicators for students’ potentials can we identify?” For example, the teacher Stephanie analysed a video clip of four students (Grade 8) working on a problem-solving task:

• 45 STEPHANIE: That is really a good way of abstraction. They generalize very well at this point. Also, how they stay at it. They know now, they have the odd numbers and now they think about how to adjust the stairs [of numbers], . . . Thus, they communicate well with one another and then generalize really well. There is a lot of potential.
• (Cycle 2, Individual discussion of video clip “Stair Number Exploration”, min. 18:42)

Stephanie also reconstructs steps in the coping perspective, but beyond that, she identifies the students’ way of abstraction as an indicator for their mathematical potential. At the same time, the way she and some colleagues talk about the students indicates that she conceives potential here as students’ stable disposition rather than dynamically emerging and disappearing in the situation which requires teacher’s efforts to be stabilized.

It was a longer discussion in the research team to reconstruct the backgrounds for these observed obstacles. After having also re-analysed other transcripts, we realized the need to differentiate the process perspective which is still too vague in the hypothesized learning trajectory (Figure 9.4). The result of several reconstructions and discussions was a refined perspective model (Figure 9.5) which allows to take into account the teachers’ perspective and to structuring of the PD content which was not adequately grasped by the earlier learning trajectory in Figure 9.4 (cf. Schnell & Prediger, 2017).

The last perspective, at which the PD programmes aim, is now called the potential-enhancing perspective, looking for fragile situational potentials which are worth being strengthened in order to stabilize them. This perspective allows fostering potentials, but in the beginning, teachers rarely adopt this perspective. Nicole is one of the teachers who adapt potential-enhancing perspective in later stages of the PD, as her following utterance shows:

• 334 NICOLE: I would strengthen their generalizing, because i think, Aishe is practically formulating a kind of rule. “I thought that in this case, you always do such and such . . .” She is on the way to systematize, to find a rule. And I think, well, . . . she has potential in that idea and we could work on that.
• (Cycle 3, PD Session 1, Day 2, group discussion, min. 2:30:00)

FIGURE 9.5 Refined structure of PD content: perspective model for noticing and fostering potentials.

Also, Henry starts to adopt the potential-enhancing perspective and even explains what he should NOT do in order to foster the situational potential:

79 FACILITATOR: Would you have liked to give them an impulse, if you would have

been there?

80 HENRY: Yes, I do find it great. So I noticed for myself that it works quite well

even if I don’t give any prompt. I notice that I, as teacher, would have quickly felt the need to say “oh, look here, what happens here? The three here”. And now I think you sometimes give them too little time so that they can unfold their ideas in peace. That it needs a lot of time. . . . Because I find they gave the right impulses themselves.

(Cycle 2, Individual discussion 2 of video clip “Stairproblem”, min. 14:55)

In total, the research contributed to refining the model as it revealed specifically the following observation on the tight connection between teachers’ noticing and fostering:

• • What teachers selectively notice is highly connected to what they intend to foster. As long as the main goal is supporting students’ actual processes of working on a given task, it is rational to stay in a process-coping perspective (cf. Figure 9.5).
• • The potential-indicator perspective looks at indicators for students’ existing potentials displayed in a certain situation. While it is important in our teaching approach, it cannot help fostering students when potentials are perceived as pre-existing and more stable dispositions.
• • In contrast, sensitive strategies for fostering (still fragile) situational potentials in order to stabilize them in the long run require a potential-enhancing perspective of noticing.

It is this perspective which teachers adopt the least often in the beginning of the course and successively learn to adopt during the discussion of fostering strategies. Rather than linear, teachers’ navigation during the professionalization process is forward and backward, since they need to coordinate different perspectives at the same time.

# Summarizing and combining the design and research results

By the case of the DoMath project, we can exemplify typical design and research results of typical PD Design Research projects as listed in Figure 9.1.

Research results

Although the existing literature provided consolidated knowledge of the general structure of teachers’ noticing and general pedagogical principles for enhancing them (Sherin & van Es, 2009; Blomberg, Renkl, Sherin, Borko & Seidel, 2013), little was known about the specific content, noticing students’ hidden mathematical potentials based on our dynamic and participatory conceptualization of potential and their connection to fostering practices. Thus, the empirical research on teachers’ processes was necessary to iteratively refine a local theory on this PD content and individual pathways to approaching it. First research results are condensed in the perspective model for noticing and fostering potentials (cf. Figure 9.5). It provides a content-dependent language for describing typical professionalization pathways and obstacles. Of course, the reconstructed insights into effects of specific design elements like focus questions and categories are not yet generalizable. Hence, their transferability to other contents should be investigated in further research.

Design results

The research results on effects of specific design elements have iteratively influenced the design of the DoMath PD sessions within the mini cycles and between the big cycles. However, we have only achieved first steps for the long-term goal of designing a PD programme with robust materials that can be used for scaling up, i.e. for facilitators who have not joined our programmes themselves. For this purpose, the theoretical foundation is crucial, and in this sense, the specification and structure of the PD content based on the perspective model is also an important design result which will guide a manual for facilitators. With respect to pedagogical design principles, the project has mainly confirmed existing work (e.g. Blomberg et al., 2013) and found content-specific ways for their realization, a design result which is far from trivial.