Learning to Listen (and thence to write)

I have been struggling to read Jere Confrey's paper called Learning to Listen.  It is an account of how to apply constructivist epistemology to mathematics education. It describes an incident of a college student trying to create a timeline for a list of events spanning many millennia, to illustrate how to look for what the student may be thinking.

It should be easy to read - after all I am part of a team right now that is struggling with how to interpret what we see during our data structure interviews. In this paper, it does exactly what our situation is:

A one-one encounter between researcher and student where the student is solving a problem selected for her by the researcher. The problem consists of capturing unorganized information and creating an information organization. The researcher is observing the entire process and gathering the think aloud comments of the student. The researcher later attempts to analyze these comments to see if the students's solution confirms some idea that the researcher went in with. This is the exact process that Confrey describes.

The long initial section on comparing and contrasting the discovery learning  theories and constructivist theories, is dense and most of her references are from the area of education research that is too theoretical for me.  After four failed attempts to proceed past the first section, I decided to go to the end of the paper and work my way back from there. Unbelievably, that seems to have worked. (Though there are still sections that are hard and who is Lakatos(I think I sort of know), and Hegel and Popper these are names I recall from my classmates college Philosophy course!).  Ok there is a lot to be learned here and it is worth the hard slog!

Trudging on - I found five assumptions that Confrey scatters throughout the paper that we maybe can use as a framework to analyze participant data:

1. Constructivists view mathematics as a human creation, evolving within cultural contexts. They seek out the multiplicity of meanings, across disciplines, cultures, historical treatments, and applications. They assume that through the activities of reflection and of communication and negotiation of meaning, human beings construct mathematical concepts which allow them to structure experience and to solve problems. Thus, mathematics is assumed to include more than its definitions, theorems and proofs and its logical relationships - included in it are its forms of representation, its evolution of problems and its methods of proof and standards of evidence.
2. In examining a student’s understanding of a mathematical concept, a constructivist seeks to represent how a student approaches the mathematical content. S/He expects diversity - and idiosyncratic rationality. The interviewer’s knowledge of the mathematical content, complete with multiple representations, competing interpretations, various applications - guides the inquiry, but his/her intent is to examine the student’s use of examples, images, language, definitions, analogies etc. to create a model which may well transform the interviewer’s own understanding of the mathematical content in fundamental ways.
3. Problems serve a crucial role in the construction of knowledge. Problems reside in the mind of the student - not in textbooks or in the mathematics. Problems are felt discrepancies, roadblocks to where a student wishes to be and therefore catalysts for action. To accept a problematic an individual must believe that it is capable of being solved - and act as though the problem and solution were preexistent. The cycle of identifying (noticing) problematics, acting and operating on them and then reflecting on the results of those actions is emotionally charged, motivating and demanding. It is this process of knowledge construction which is the critical site for constructivist researchers/teachers.
4. Problem solving as enacted in interviews or constructivist instruction is an interactive process. The interviewer selects a task for its potential to invite students to engage with a particular mathematical idea. The task will yield to multiple interpretations and resulting approaches. The interviewer must seek out an understanding of the students’ problematic, choices of actions and means of reflection. The interview setting will itself promote more self-reflection and a stronger approach to knowledge construction. The definition of the problem, of what concepts are related and of what constitutes an appropiate answer will evolve over the course of the interview.
5. Students’ responses which deviate from our expectations as research- ers/teachers can appear to he reasoned and well-considered to the student. They may be entirely legitimate - as an alternative perspective, or be effective for a limited scope of application. We must encourage students to express their beliefs, keeping in mind that deviations provide precious opportunities for us to glimpse the students’ perspectives.

Some other glimmers that glow for me:

• the constructivist is engaged in a processs of invention - invention of his/her own models for explaining students’ actions and words.
• s/he(the researcher) begins with the assumption that what a student does is reasonable and then seeks to describe it from the student’s perspective.
• A problem is an intellectual desire ... and like every desire it postulates the existence of something that can satisfy it... (so sez Polanyi)
• Labeling a student’s model as a misconception fails to take in consideration the perspec- tive of the student, for whom the belief may explain all instances under consideration and fail only in cases to which s/he is not privy. 
• Much of the success of the constructivist instructional or research model depends on how willingly the teacher 1) seeks to imagine how the student might be viewing the problem; 2) hears mathematical notions which differ from her/his own but possess internal consistency; 3) examines his/her own mathe- matical beliefs and 4) witnesses and describes the student’s choice of operation (action) and method of evaluation and recording (reflection).
• fit vs match: a conception must fit an experience not necessarily match the researcher's conception
• It is at points of contact, at moments of discrepancy, that we have the highest probability of gaining insight into another person’s perspective.
• Finally, I argued that in examining students’ problems and methods of solutions, one has an opportunity to reconsider the mathematics involved.

On the whole, I made a lot of progress when I decided to skip the arduous theoretical descriptions and the long description of "Suzanne" creating a powers of ten timeline. I read the part focusing on Suzanne later and it was much easier to read once I was no longer looking for deeper meaning.

Same blog, new phase

Since I have always assumed that this blog has exactly one reader - me, I am candid and quick with my thoughts. Should you not be me, let me know if you want me to spend some time cleaning up my thoughts before they make it here.

That disclaimer out of the way, this post is a quick one to say that the original purpose of the blog was well-served especially given that life in the way of GSP5 intervened and my only memory of the classroom observation was buried in these pages. It was these pages that stored the spots that glow and aided the process of both writing the paper(6 months later) as well as preparing the presentation(another 6 months later). "Jackie's" struggles were well received and much recognized. But the most gratifying response was the appreciation for the honesty in the presentation - the acknowledgement that I did not see what I set out to see but that it was not a waste and I did walk away with insights for the team. In my role as an educational researcher, this was a setback in that I didn't see "what happens when" because nothing much happened as planned. However, in my role as a software developer(well ok a project manager for a software development team then!), I did "see" important insights.

Moving on, this blog is now no longer the process of writing an ethnography, but rather a place to capture the work I am doing in the classroom, my thoughts on APS(OpenIDEO) work in India, my reading notes, and any other random moderately relevant thought. I do now firmly believe in the power of gathering random dreamy thought and the power of search to be able to use them when I need them to collate them into a coherent whole.

So the tagline for this blog will now read - "Only those attempt the ridiculous can achieve the impossible".

Till the next time...

Surprise! - tracking within the classroom

It was the day of new seating arrangements. Based on test scores from Tuesday - Jackie divided the class into three groups - passing(6), can pass but not there yet(6), need a lot of work(10). she did this, she said to make sure that she can give some undivided attention to the third group.

This is to continue for as long as it takes for them to change their work style. The first group looked pleased, the second was fidgety, the third mostly passive. They worked on taking the test as groups, working through their wrong answers, comparing what they got correct and sharing that with the group as a whole.

I worry that the second group may slide further down instead of rising up? The third appear to be split down the middle into those that know and are willing to engage when called upon in this new situation where they are under a magnifying glass and those are just not willing to take any visible effort.

Spots that glow:
* Thats why you are in this middle group - because you don't want to learn new things. You only want to coast in your comfort zone. You are not willing to put in the effort needed to be in the passing group, but you aren't in the last group becasue you are using what you have.
* No they are not the smartest. They are the best prepared. They don't pass everything they do - they just never stop trying. They are always asking for help. They turn up for office hours. This group has near perfect homework submission rates. That's what will help them succeed in the course.

We collected measures!!!

Second time in a row, the class was in a lab. This time, Jackie appeared more sure of herself. She had had time to pick a problem and prepare a Fathom based worksheet to solve the problem. It was one of their homework problems. The activity that the class was going to model was a simple discrete variable based simulation of creating a sampling distribution. This is problem 7.43 from YMS third edition. It is in the section pertaining to the Law of Large Numbers.

Students were modeling the process of solving this problem on a calculator. In all such problems, the first step is to "assign digits". What that means is given the table of probabilities, you have to create a list in your calculator corresponding to a random variable X (in this case discrete). It is a hard process for them to think through because they haven't practiced enough. Fathom is one more way to model this process.

Class strength was at 17 in place of 25, the rest were on a junior field trip to the South for following a Civil Rights trail. Yet again, a class that was far less disruptive than before. Amazingly the screens mask ratehr than provide the distraction.

Spots that glow:

• "Oh! I get it..." - a boy looking at the way they created the collection - by "assigning digits".
• Get good at Fathom - it is a tool for homework - see how much more we could do - said Jackie
• Pointed out where the law comes into effect, by asking everyone to take one sample of 5. Then asking them to compare how many times they got the number "5" with it's occurence in the distribution of the population. After that, she colected number fo "5"s from the entire class(which was a sample of size 85) and pointed out how much closer it was to the theoritical probability of getting a "5".
  
• In our desire to promote exploration based learning, I realized that our Fathom activities rarely model the typical(and endless number of) AP Stats homework problems.
  

3 collections and a graph!

Today there was a scheduled lab session in which they were to work with Fathom. Jackie said later that she wished she had had time to work with this before hand and create a student work sheet. She mentioned to me that the reason she decided to go ahead and do this session was because she knew I was coming - so there I have influenced the running of the class and in particular with regards to the technology use! Anyways...

The activity that the class was going to model was a simple discrete variable based simulation of sample means. This is activity 7b from YMS third edition. The gist of the activity is captured below (will try to add it from the book later)

1. Let X be a random variable whose values are drawn from {1,1,2,3,5,8}
2. Take a sample of two values from this set. (Treat the two 1s as separate values).
3. Compute the mean for this sample.
   
4. Repeat this for all possible samples of size 2.

This session took place in the lab. Each student had an individual iMac to work on. The have assigned computers but they can access all their documents from any computer as they have networked logins. The lab is set up as 6 rows of 6 computers each. Each row is divided into two parts by a vertical aisle with four computers on one side and two on the other. There is a screen in the front that is visible from all computers.

Jackie didn't have a student handout for them to work from. She had only that very morning decided that she would use Fathom after she confirmed my attendance. She reported her worry that if she had tried this activity solo then in case she ran into a snag she would get nowhere. Whereas with a physical simulation, she may only do a few runs but she had the confidence that she would get the point across. This appears to be a common worry with teachers who didn't grow up with technology as an integral part of teaching. As it turned out, the only real Fathom help I gave her was to point out that you could escape out of animation when collecting a 1000 measures. At other times, when she looked a bit puzzled, I just waited for a moment and she figured it out (An example being - which menu to choose from and which collection to have selected when wanting to collect measures)

(Development Note : We know this is a problem - trying to figure out which collection is which and when to work with which inspector)

On the whole the students seemed to not have a hard time with this activity and with Fathom. Jackie modeled the process of "putting together a Fathom document". The began with a collection, she called randomVariable. She called the attribute random_var. And created 6 cases using the case table {1,1,2,3,5,8}. She didn't ask the students for input when creating the document. She asked them later "Why do you think I put in those numbers?"

At the end, she asked them to look at their neighbour's computer and see if their graphs looked similar. This led to a bunch of looking around and comparing but not too much engagement or talk around why they may have slight differences.

About three of them were working with a sample of size ten (the default) when the activity asked for a sample size of two. One of them did sampling with replacement - and realized it only because his graph looked so different from others and asked for help figuring out why that was the case.

On the whole, an exciting and satisfying class. even the usual disruptive suspects had less effect on the class - maybe it was the fact they were in front of a computer and all doing something. The individual computer in fact to some extent lessened the disruptive effect because it was easier to stay on task with this screen blocking the distractions.

The person who had caused quite some disruption in the earlier class this time spent a lot of time fooling around but was also one of the first three to finish. So is he disenegaged because he is bored adn this material is too easy? Hard to say...

At the end of all this - they had fun, were more engaged than in any class before this, but did they learn better? Now that is a question for a later time...
Spots that glow:

• Even for a bright and dedicated teacher like Jackie, it is hard to get over the mental hurdle of doing an entire lesson based on technology.
  
• All it took in this case was the knowledge that a "technology expert" would be present for her to take the brave step of plannign this lesson
• "Why does Fathom create the third box?". "I don't know; for fun?"
• "Why do you have different numbers?" "Oh because we got different samples!"
  
• Fathom was very well suited to modeling this simple activity. There was none of the cognitive baggage associated with putting together a probability simulation involving cards or a "real-life" situation. Perhaps for students, whose big hurdle on a AP like test is reading and parsing the question, there are two simultaneous needs fighting for their attention. One is the need to understand the language - the other is to model the situation.
  

This is sooo hard!

Date of class- Jan 29th.

Today's class was a very unsettling experience. Jackie spent large parts of time just dealing with expectations, behaviour patterns and habits of mind. Plenty of spots that glow in terms of how much effort she puts into the class but hard for me to write up in terms of what it meant as class that will work with technology.

Before I forget too much about what happened in this class, I will record a few main themes and then elaborate on them later

• Get Help - use wikipedia, parents, teachers and peers
• Think before you speak and know when you are conjecturing
• Fill holes left by first semester teaching
• Teach new topic for 20 minutes - planned to do Fathom demo but no time
• Use your time - be intellectual - plenty to keep you busy even if you know nothing on quiz.
  

More encounters of the high school kind

Jackie's class is my first close encounter with a high school classroom in the United States. It is a non-typical situation as discussed in my earlier posts about Memorial High School. But for me, I have no baseline for what is a typical classroom. The interactions that I observe in Jackie's classroom are not very typical I guessed but wanted another experience.

I decided to visit an AP Statistics class in a large comprehensive high school in San Francisco. The school draws it's student population from all over the city. The teacher appears popular (enrollment in AP Statistics has almost doubled since he joined 4-5 years ago). The school itself has undergone a self-feeding upward trend with its scores int he same time period. The student population has undergone a demographic change.

The first difference I observed in this class as compared to the Memorial HS class is the more uniform dressing style. The are about 20% of the accessories that I saw in MHS - on both boys and girls. The other is that out of 25 students, there were 2 Hispanics, 1 Caucasian, the rest are all of East Asian origin. The community appears to have much to do with dressing styles. As someone who came from wearing uniforms to school and liking it, I struggle to come to terms with how much time and effort appears to go into dressing for school. I cannot help feeling as if they would regain a good hour or two each day if only they didn't spend so much time on dressing. This will be the last mention of this issue because this ethnography is not about what they wear but what they do; else I will be guilty of the same time waste.

This class apparently has 60-70% ELL students. But I can't help wondering if that just means that their first language at home isn't English - not necessarily meaning that they struggle with English. In addition, the similarity(with MHS) that here again more than a third will be first generation college aspirants seems amply offset by the difference parental expectations. As Jackie has expressed before one of the big challenges facing her students is an almost complete lack of parenting. Jackie also pointed out that even if they don't speak English at home, the language that they do speak, they speak at a much higher level of sophistication sot hey know what it it to communicate at a "college-level" so to speak. The other significant differnec ein this ELL characterization is the language you hear at break (that is their language of choice for social interaction) is English at this school whereas at MHS you hear Spanish in the break mainly.

So on the whole the students appeared on task, prepared to learn (in the sense that all of them had books and calculators and writing materials), non-disruptive but also jaded. There were situations in which one of them was napping. one was eating and another texting - but the teacher appears to have very strict classroom behaviour expectations and in general thery were met.

That said, the class was in a listening mode and not really in any type of active learning situation.
It was a lecture - they did about 5 problems but all of them were done by him on the board and they took notes and answered when he called on them. I definitely need to go again when they may be doing an activity. They did enter data about penny ages into a Fathom survey during break. They were passign around a bag of pennies during class and had been instructed to take 2 samples - one of size 5 and another of size 10; compute the mean; enter it into the survey. He used this data at the end of class to lead up to the Central Limit Theorem. It was the standard demonstration of how the distribution of sample means approaches the normal as we increase the number of samples - irrespective of the shape of the population. He then moved onto using the CLT document that comes with Fifty Fathoms. He also modified the uniform population in that demo to a bi-modal distribution to better illustrate the "normalization" that takes place as the shape changes int he sampel mean distribution is very dramatic moving from a bi-modal to tri-modal to normal by the time you get to about 25 samples. Even this felt more like a demonstration as it was not preceeded by any active work by the students in the area of building the distribution - they computed the means but that was all.

I took detailed notes of what actually happenned in the class in terms of the activities and use of Fathom but what appears here is a distillation of those notes using a framework of comparison rather than of reporting. I did learn how to answer questions like the ones that appear on the AP exam :-).