Tag Archives: mathematics education

Preliminary Investigation P+C+A

This is the second phase of the research. The idea in this phase is to do 4 main activities.

  1. A focused literature review of relevant theory regarding the topic we are addressing
    1.a) Reading of 3 papers regarding epistemological theory about DR
    Aim: to understand in depth the concept of design research and how it is conducted. In order to have inputs for the design
    Outcome: From this reading the group put in the comment area the outcome of their work in class. Each group assigned one aspect that is relevant from the literature to come to a better understanding about what is Design Research.
  2. Analyze a promising example that will guide us in our design: Tool Use and the Development of the Function Concept: From Repeated Calculations to Functional Thinking  (Doorman, et al., 2012)
    2.a) Reading an article about a research conducted under DR approach.  The aim of the research was designing a learning intervention in mathematics education for secondary school in the Netherlands.
    Aim of the reading: To see how to derive the design principles and how to do the match between the theoretical framework  and the design principles as starting points for designing the learning intervention
    Outcome: The transcription of the interview.  We bold some quotes relevant to our work
  3. Analyze practical context: We did it in our group in an informal way. Looking for particularities of our context (the context is our working group in the summer school)
  4. Talk to experts and practitioners: There were 4 interviews conducted by our group. Experts:
    Frans van Galen : Expert disigner
    Paul DrijversExpert researcher. One of the authors of the inspiring example. Link to his interview. You can hear his interview also opening the file in the box.net widget at the end of this page.
    Michiel DoormanExpert researcher and designer. One of the authors of the inspiring example. Link to his talk in class or opening the file in the box.net widget
    Paulina  and Daniel van den Bergen: Master students at FIsme
    Outcomes: The extracts relevant to our interests and from there we are taking decisions regarding the design of the product. It has been interesting hearing about the difficulties doing DR.  It is our intention to see in which way our product can help to overcome such difficulties.

We extract some relevant quotes to the learning of DR and its difficulties:

From P. Drijvers’ interview:
Difficulties in DR:
1. Generalizing: Are the findings context independent?  I would say not really.
2. Communication: Making explicit what is implicit in the researchers’ mind so others can understand, check, agree or disagree, criticize, improve his/her work. A researcher can have nice theoretical ideas in her/his head but she/he has also many other things to do with teacher, students, gathering data, analyzing it, etc. How do you match this 2 aspects? When working alone all is going on in the head of the researcher and he needs to make it explicit so others can collaborate.
3. Some tension between the goals of the teacher and the one of the researcher.
4. To connect theoretical ideas with the rest of the research ( math content, tasks, observations, analysis of data, reflecting, refining, retesting). When all those aspects are well aligned the researcher knows what to look at beforehand, having the reasons for doing so.
Design Research as well as theory should work for us, not the other way around. 

Michiel Doormans’ talk:

Carolina: Is designing an issue of steps, the principles show you the how to design, the sequence to follow?

Doorman: It is not as you have the principles and you design the task and then you go back to the principles and design the next one. It is not as linear as that. You need to know the kind of process-reasoning you like to evoke and think the kind of principles you are based with, and then you design something accordingly and at a certain moment you show your product, because it is very difficult to be critical on your own design…

To criticize your design you need to get the answers from your principles. You need to be very careful that what they do not understand is referring to one of your DP and not to some external variable (Use of complicated sentences, using symbols that they did not learn yet, etc)

Christo: Resuming about the design principles from what I have heard so far:

1. Generate thinking that makes them receptive towards what you want them to learn later. With the intermediate step activities
2. Emergent Modeling: You provide them with the model of arrow chain because it is a step in between of what they know and where you want them to go.
3. Tool instrumentation: You do paper+pencil activities so yo can be sure that when they are working with the computer tool they are not showing you that they understand because of what the tool does but because of their understanding of the concept.
The design principles determine the results of your design? 
Doorman: Not completely! I cannot quantify it, but I would guess 20% is guided by the main design principles and 80% comes from your creativity.
In your hypothetical learning trajectory you describe your hypothesized learning process you want them to achieve. There you have to be as precise as possible in connecting what you expect to this principles as they guide your design and hope this principles are explanatory for what is happening and for will happen. So when it goes different then from what you hypothesized, you can see what the design can learn from it, how it informs your design regarding those 3 starting points, the 3 design principles. Is it connected to them? Or did they not understand because of something totally unrelated to the design (complicated use of language, a large group, etc). So this will not say much about the accuracy of your design principles.
Connecting the understanding or not understanding to the design principles
The understanding of the students must inform you about the good use of the design principles. They are important for the way they describe the hypothesized learning process.

Bonus activity: Doing a critical reading of an article

I have come across with an interesting list of questions to ask to my self while reading an article. I think it will be useful for us to do this exercise as a group and come to a final product of the reading. Be creative in what form will this final product look like. It could be a video, a tutorial, a wiki space, a page or what ever you feel inspired to do.
The article is one in mathematics education, it is about a research that was done here in the Freudenthal Institute by a group of researchers.  Both documents, the list of questions and the article (Doorman et Al, 2012) are in our box.net.
Outcome:  The outcome was a critical review of the article done by the group, but there was no time in the sessions to do it.