The two chapters on which I here reflect deal with informal writing.  The Gottschalk & Hjortshoj chapter starts by comparing informal writing to the rehearsals one in the arts may undertake before a performance in front of an audience.  If we think of a pianist this person does not just play in front of an audience without being prepared.  Hours upon hours of practice, rehearsing the material, improving one’s ability and so forth must precede the performance if one is to do well at it.  They paint the same scenario for athletes and actors as well.  Writing, they argue, is no different.  We cannot expect a solid performance (a high level writing assignment like a research paper) if we do not prepare students through practice (informal writing assignments, exploratory writing, drafts and so on).  Yet, they argue, we often do leaving both the instructor and the student frustrated.  I think they make a very valid point.  In fact, I argued something somewhat related in my class two days ago.  A few students expressed difficulty with the homework problems.  After we reviewed some I suggested that they go home and pick a few out of the book to do on their own for practice.  What resulted was a brief conversation where they explained that their hatred/fear of math kept them from doing extra problems (they stuck only to what was assigned, collected and graded) and where I explained that those who struggle most need the most practice.  It was interesting and eye opening.  We, of course, like what we are good at and because we like it we do more and more of it which makes us get better at it and like it better.  Rarely do we invest the same amount of time and effort into something that frustrates us, makes no sense to us, challenges us.  Though in mathematics (and elsewhere I presume) being able to work through frustration, to stick to a problem over an extended period of time, and to relish in the challenge are what make for successful students.

In my own teaching both at the high school and college level (though I admit more so at the high school level) I have tried to incorporate writing into my classes.  I saw some of the types of informal writing from the texts mirrored that which I had tried in my teaching.  At the high school each of my students had a math journal.  We kept them in a file cabinet in the classroom.  The students wrote at times freely, at times responding to a writing prompt of mine and at time answering specific questions.  Every few weeks I picked a number of them to go through and read and commented on a certain number of their entries.  They received a grade for the journal but mostly for keeping at it and doing it.  I found that giving a precise grade was difficult but really valued the ability to have ongoing conversations with students, to see how they were thinking about the material and i think students (with time) came to enjoy writing.  Initially every year I was met with resistance.  The usual “this is math class – why are we keeping a journal?” came out year after year and sometimes those protests lasted all year but more often than not the vast majority of students came to value their journal and the opportunity it represented.

At the college level I ask each of my students to write a letter introducing themselves to me as their first homework assignment.  In addition they are to write about the experiences with mathematics over the course of their lives.  I have found these to be very enlightening.  Students at times spell out what supports they might need to succeed in class, they speak about how they view the subject, how they view themselves in light of it and their goals for the course (from “I’m taking this because it’s required” to “I hope that I understand and even like math after this because right now I don’t”).  I also learned that this writing eases the worry of some of the students that are weaker mathematically (or else view themselves that way).  A student 3 semesters ago came to me after class.  She explained that she was a visiting students from Queens college and that she had already failed pre-calculus twice.  She said she thought a new school might help and how worried she was about failing again.  She thanked me for asking her to write the letter saying it gave her the belief that I really cared about the students, would tr my best to support their learning, and valued the experiences they had with the subject in the past.  In the end she passed the course mostly because of her persistence.  She came to office hours often, attended tutoring, studied often, asked questions and so on.  I wonder how much of that was a result of believing she might have a chance this time?  Or that she was somehow valued in class?

I have also tried having an online discussion in my class last semester but a lack of structure rendered that quite the failure.  Students’ posts were less and less thoughtful with time as my expectations for what should be included were sorely lacking, I am afraid.  I also had no way of counting the work they did with that and many students took that as an invitation not to participate.  I really would like to do this again but in a more structured way.  I am thinking of presenting math class scenarios to my Math 271 class who are all prospective teachers and having them respond to these scenarios online so that each week we have a discussion about one of these.  I have to think it out a bit but I think there is much potential there.  In fact, I think the Bean text highlighted why my efforts failed.  The expectations were not made clear, examples of good posts were not included, and a reward for completing the task not put forth.  In this respect I am most appreciative to Bean for including the “Explanation of Exploratory Writing for Students” (pp. 101-102) where he shares a handout used by a professor to explain to students what type of writing to submit.  The lengthy directions include (1) a clear definition of what is meant by journal writing (the “What” of the assignment), (2) the reasons for such writing (the “Why”), (3) explanation of how to go about the assignment, (4) an example, (5) answers to frequently asked questions.  More and more these readings are reminding me that directions, instructions and clear expectations affect the work submitted as much as the students’ ability to do the work does.

Also by reading the Bean chapter (especially the 25 ways to use informal writing in a course) I see how simple it is to include some informal writing in class.  I don’t really mean to say that it is simple or that one need not think through how best to do so.  What I mean is that we at times push away changes to our pedagogy  because it’ll require too much work, time, effort, etc.  However, some of these ideas can be implemented easily, without adding too much to one’s work load and in a way that weaves the writing into the course without it feeling like an add on.  If nothing else it made me think of ways I can use some of these techniques.  I like the metaphor games and extended analogies.  In mathematics these would be great to highlight the similarities and differences between two problems.  Sometimes students can do a problem (say find the midpoint of a segment when the segment’s end points are given) but if slight modifications are made to the problem (given the midpoint and an end point of the segment find the other endpoint) it becomes impossible.  Using the metaphor games as informal writing students can begin to analyze problems that while related are different in some way and doing so may help them better learn the material and be able to deal with related problems.

The generic exploration tasks for an argument addressing an issue (p. 114) seems useful to me too.  In fact if we add to the title “through mathematics” that fits right into the idea of mathematics as a tool for analyzing, exploring, arguing about social life, etc.

Well, I am off now to actually modify this very task a bit for use in my course this fall.

In my math course we are presently working on graphs of different functions.  Many students are able to do this by picking values for x (the independent variable) substituting that value for x in the given equation and obtaining a value for y (the dependent variable).  They then take the x they picked and y they obtained and plot the point (x,y).  If you do this enough times you get a series of points that when joined should yield the graph of the function in question.

This approach though comfortable does have its drawbacks.  One is that we may pick points that do not show the salient features of the graph.  In simpler terms, across the street from my apartment there is another building and to the right of it a large house.  If I look out my kitchen window I only see the building.  I might not realize that the house is even there because of where I am looking.  The same thing happens when you graph.  The points you plot might not allow us to see the whole thing.  Next, this method takes time.

I was hoping, therefore, to teach students about the use of transformations in graphing.  Here I know the salient features of a curve. Say I know that y=x^2 will yield a parabola and that off the top of my head I can easily graph that parabola using 3-5 points which I know highlight the salient features of the graph…I then am presented with y=x^2+1 and I know that this graph is the same as the prior except that each point is shifted up 1 making it easy to graph.  Basically I need to know what the “parent graphs” look like and the rules for shifting them up, down, left, right, flipping them over the axes and so on.

Well telling students the rules is no fun 🙂  Additionally it might increase students’ understanding if they found out the rules themselves thus being engaged with the material (as opposed to listening to me recite the rules).  To this end the class is broken up into groups.  Each group graphs 5-6 functions all of which highlight a certain transformation.  Graphing y=x^2 followed by y=x^2+1, y=x^2+2, y=x^2+3 and so on allows students to see that adding a positive constant (call it k) to the equation shifts the graph up k units.  The graphs are up on chart paper and hung across the front of the room once all groups have completed them.  In the past I then asked the class what they notice.  They might say “all of the graphs in that poster moved left” and from that we construct the transformation rules.

This time, in an effort to incorporate writing into the class (and to do my writing seminar homework) I did things slightly differently.  After all the graphs were hung up students were asked to pick two of the posters and write about (1) the transformation observed in each   (2) develop a rule in words for what is occurring in each (3) develop an equation whose graph could be sketched by utilizing this rule.  They were told to write for five minutes though I waited until most pencils had stopped moving before telling them to wrap it up.  During the writing exercise a few students called me over to make sure they were “doing this right” which I really did not expect.  Really we were writing to get us thinking about the transformations and be ready for a whole class discussion on the topic.  While there are ways to write rules that were mathematically inaccurate I was not going to grade this and was just hoping that they were engaged with the material and able to contribute later to the discussion.  I expect that the concern arose from my lack of explanation as to what the writing was for and a concern about being right that many students carry with them.  Perhaps if we do this more often they will be less concerned with doing it right and more able to get their thoughts down on paper.

Another thing that I noticed was that when we started the discussion many more students than usual contributed.  Some students who normally do not speak up in class did so even if they did so as a response to what another had said.  There were a lot of “you can use what he said in posted 4 as well except they go left instead of right” which was nice.  I think (only speculation now) that the writing gave everyone a chance to think about the graphs.  Had I just asked in class aloud right away despite even a long pause only the “quicker” and “more confident” students might have answered.  I think the writing also gave those that struggle a bit the opportunity to put their thoughts in order…to rehearse a bit what they might say in the discussion which may have given them the confidence to speak up.  I really like this lesson and have (without the writing) done it before in other classes.  I am curious as to how the students perceived the writing piece and even the whole idea of coming up with the rules rather than being told them.  I am wishing I had asked them to write about the lesson as a whole at the end of class just to get some feedback for myself about it.  I usually do this at the end of the course (evaluation surveys that I design) but never did so for a single lesson or activity.  Maybe the way to go is to ask them to reflect on the lesson next time we meet and to collect those to see what it is they thought about all this.

Well, that post title is a bit long and perhaps not too useful.  Let me clarify: the Rose chapter sets us a writing assignment about culture shock.  Students read a text that discusses stages that immigrants go through upon coming to the United States.  The text classifies the immigrants experiences into several phases and discusses the key feature of each of these.  The students after reading this text move onto 5 other texts written by and/or about immigrants to the United States…stories, if you will, about these specific individuals and their experiences.  The student is then asked to apply the classification scheme of the initial text to the specific cases of these immigrants they then read about.

The Gittschalk ad Hjortshoj chapter has us consider how writing assignments should be constructed.  What I liked most about this chapter is the authors’ insistence on clarity.  They note how often we are unclear about the writing assignment and yet have very specific expectations of our students that are implicit.  They say that we don’t tell them specifically and clearly what we want and then we criticize their writing for falling short of these expectations we never really communicated to begin with.  That’s the short version but it really resonated with me.  I am thinking of an assignment I gave to a class of pre-service teachers.  I wanted them to organize their semester’s work into a binder with the goal of being able to refer to it in the future, when they were teaching themselves.  I saw it as a resource and as I was asked to do this as a student in my own education courses, and create such a binder for each class I teach I was clear (in my head, that is) about what I wanted….what I expected.  I expected that the games we played in class should be included but that in doing so one would have the directions written out and maybe some extensions and variations of the game.  I expected a section on activities we had done and again I thought students would benefit from noting variations to these activities that we had talked about in class.  I expected some reflection after each activity that might guide the students when reading this years later as to the challenges one might face in implementing some of these activities and a list of the required supplies.  In short I expected them to make a binder like mine 9similar to how the authors say that we often expect students to write as if they were experts in the field).  I got some beautiful binders and I got some terrible ones.  Students had the necessary sections but just threw all their work together.  I doubted they would keep it and years later refer to it when teaching prime numbers and looking for a good activity.  The next time arund I was much more specific about what the binder should entail.  I explained its purpose (to serve as a resource when you begin teaching).  I even brought in a few of my old ones for students to leaf through.  The results were much, much better.

Now that you have some info about the chapters and my own connection to them I should touch on my official assignment which was to consider the Rose assignment in light of the chapter on how to develop good writing assignments.  I think Rose made clear what the writing should entail thought perhaps the purpose might have been a bit more fleshed out.  I suppose it’s clear enough in that it says we are applying a classification scheme but I wonder if some note as to why one would want to do this in the first place might help.  The idea of scaffolding assignments is in there too.  The author noted brief in class writing around this topic that precedes the actual assignment I described.  Rose notes that time is spent on vocabulary and on lower stakes writing first.  Finally I think that Rose considers his students in that he adds a secondary option for students which is to use their own experiences as immigrants (those that can, obviously) and apply the classification scheme to that.  This clearly builds on students experiences, centers the work around their experiences and values these students as experts in the area of immigrants and their stories.  I don’t quite recall that in the chapter on developing good writing assignments but it does seem like a valuable thing to do.

My online search for writing intensive mathematics courses yielded many results.  However, I found the vast majority of them to be statistics courses (there were a few mathematics for business courses and one geometry course).  Further, many of the courses had writing assignments that basically asked students to explain in words a mathematical problem that they had solved.  While I see the value in doing that (the ability to communicate what one does mathematically is important and the exercise might even make clearer in our own minds the work we are carrying out), these assignments didn’t feel like they were enough to me.  We have in our seminar considered the idea of writing to learn in our discipline but also what writing is and looks like in our discipline.  I was hoping for a course with assignments that fostered good mathematical writing, valued revision and taught students how to write as a mathematician would write.  Then I came across a syllabus Math 341: Probability and Statistics II at Carrol College in MT and an article written by a professor in that school’s mathematics department explaining not only the assignments but the motivation behind them.  Writing in this course was not just a way of proving what you knew, not meant only for one’s instructor to read, and actually went beyond reflecting on a problem.  The document notes, “the point of good mathematical writing is to write simply and plainly, to take a complex idea and explain it with clarity” and that the course intends to prepare students to write in this manner.

There are three main writing assignments. The course emphasizes peer review and begins by having students read papers from the prior semester’s course.  The students in that prior semester know that their papers may be read by the next class of students and that since they are doing so early on, their familiarity with the statistical concepts learned in the course will be minimal.  As such they are pushed to write in a manner that is clear and that makes complex ideas simple – simple enough for someone without a thorough grounding in this mathematics to follow. The students in the course read these papers and rank them highlighting the strong and weak aspects of each.  The professor notes that this exercise helps students focus on what good writing in the discipline is not by being told what it is but by discovering it through the examination of these papers.  This assignments also reminds me of the peer review process.  In discussing and writing about these papers students learn how others might view their own papers.

The second assignment is a statistical survey.  Students select any topic of interest to them and create a survey that they then administer to a large population.  Large is left to the student to define as it is important mathematically that one knows when a sample might prove too small or flawed in other ways (ie. finding the average height of york college students by surveying the basketball team).  Students then analyze their survey data using statistical procedures learned in class.  Finally they write up their findings much in the same way as researchers would write up the results of their work to be shared with a broader audience.  The third assignment is similar but involves a controlled experiment.  The student develops a question to study, gathers a sample, randomly divides the sample in two groups, treats these differently in some way and the analyzes statistically the differences, if any, between the groups.  Again, the findings are then written up in a paper.

What seems most useful to me about these assignments is how often researchers do just this.  That is, the assignments ask students to do what those in the field do: to share results obtained using statistical measures in a way that is understood by a wide enough audience.  Knowledge of the mathematics (or lack their of) comes through in the assignments even though these are not limited to having someone write up an explanation of a problem they solved.  The assignments seem authentic somehow.  The syllabus notes “mathematical writing is an essential skill for any mathematician, whether in government, private sector or an academic career” and the assignments seem to push students to develop this skill.

Finally, the syllabus notes that students revise the assignments they write a number of times drawing on feedback from their peers, their instructor and even themselves (there are revisions that are submitted after a student writes a critique of their own work, for example).

The authors throughout chapter 1 highlight what they see as a false dichotomy between teaching writing and teaching the content of a specific discipline.  If we start with a course and “add on” a writing component we’re playing into this dichotomy which separates teaching content from the writing itself.  As an educator if we already feel strapped for time in a given course, the idea of adding on anything (writing, technology, the use of maipulatives) seems daunting.  “I already have no time how can I possibly add x, y and z.”  It is this idea of an add-on that the authors argue against.  They argue that writing should be used as a tool for learning and communication.  It affords students a time and vehicle for considering questions and connections in the discipline, time to reflect, to connect and to question.  We learn the content through writing, we grapple with the central ideas of the discipline through writing, we raise questions and wrestle with ideas through writing and we share with one another through writing.  Though I should be clear in that the authors value writing, speaking and listening as a way of fostering learning and the engagement of students.

I liked the idea of considering not what body of knowledge we might want students to know but what skills particular to the discipline we wish to foster.  Knowing what types of questions we can ask and the valued ways of addressing these in a given discipline will take our students further than the accumulation of numerous facts. Recently (last semester I believe) I walked in on two of my colleagues working on pure mathematical research.  My background is in math education.  I know what types of questions one may study in education.  I can ask questions that have a shot of being addressed and know some methods for addressing these.  The process of research in math education (though I have a lot to learn yet) is somewhat familiar.  In pure mathematics, I don’t have these same skills.  So I asked these two colleagues how they select a problem that may have gone unsolved for ages and yet something they know they have some hope of making inroads into.  I shared that I really would love to explore some yet unsolved pure mathematical problems but know not how to select a good problem.  Interestingly my colleagues did what the authors of this chapter propose.  They didn’t just give me some information but rather invited me into the discussion.  You learn what makes a good problem by engaging in the work itself (with some guidance).  I began attending their informal seminar studying partitions.  It has been a wonderful experience where I am learning not just about the content (partitions though we have branched into graph theory a bit as well) but also to see how observed patterns lead to conjectures (hopefully to proofs) and how these yield new questions.  It is a matter of not just knowing the mathematics but being invited into the discussion; into the work of those experienced in this field which is what we want for our students though we may not always afford them this opportunity.

A few months ago I devised my own rather unrelated mathematics question which after a few months I was able to solve.  I do not know if the content of this problem of mine is of any relevance and/or importance to the mathematical community at large but the experience of devising my own problem, strategies and then carrying out the solution has been incredibly rewarding and I am now considering how to best share this with others.

I feel I have drifted a bit from the chapter.  Returning to the authors’ work, I see the main point of the chapter having to do with this idea of writing as an integral part of the course, as a means by which students grapple with ideas, make connections and grow in knowledge of both the content and the skills related to the discipline.  It should not be that we take a course and throw in some writing.  If there isn’t a synthesis between the writing and the other aspects of the course then it really will be an add-on which is frustrating to the students and professor both.  Now, how exactly one achieves this…well, I suppose I have another few chapters to read 🙂

I will get back to you all though on what I learn.

Rose, M. (2006). An Open Language: Selected Writing on Literacy, Learning, and Opportunity. Boston: Bedford/St Martin’s.

Response to Rose Chapter 14

The Rose chapter, to me, was quite interesting because of the arguments put forth by the author and the implications of these arguments upon teaching and learning.  What struck me most is how similar some of the arguments Rose makes about writing are to those in mathematics and mathematics education.

Rose discusses how in the 1970’s there was alarm brought forth by studies and articles claiming the students in the American school system could not write; or at the very least could not write at the level that was expected of them.  This scare drove a movement of accountability where test scores and quantitative data were overwhelmingly used to justify academic programs and places like the writing center in which the author worked at the time.

We presently find ourselves once more in the age of accountability.  I write specifically with respect to the K-12 public school system as my background is in mathematics education at the secondary school level. The NYC Department of Education, under mayoral control and mirroring the national trend established in part by the No Child Left Behind Legislation, is pushing for accountability and quantitative data similar to what Rose described in the 1970s.  Students are tested using city and state assessments multiple times in every grade level. There are predictive assessments, interim assessments, year-end assessments and so forth throughout the year multiple times in multiple disciplines.  Teachers are expected to keep written documentation of interactions with students and as was just noted in the media a few days ago teacher tenure is going to be tied to student tests scores.  The problem with this is not that the data is somehow not helpful but rather that the volume of these assessments is unnecessary and these eat away at valuable time for classroom instruction and lesson planning.  Testing someone more often does not in and of itself lead them to learn more.

Rose notes narrow focus on test scores and quantitative measures that leave out many realities that numbers alone cannot capture.  A few years ago a national mathematics panel was established to determine the state of mathematics education in the United States and make recommendations for improvement.  Their charge as written allowed them only to consider quantitative studies with a sufficiently large n.  I gather a much different reality may have come forth had they been looking at a broader range of studies.

Another interesting point that Rose brings up is the fact that we are the “first society to expect so many of its people to perform very sophisticated literacy activities” (p. 291) and yet do not have the school system needed to live up to this promise.  This is similar to the “algebra for all” movement and the move to academic high schools and away from vocational schools.  I understand that certain students were incorrectly/inappropriately funneled into vocational tracks based not on ability or interest but factors that highlight the inequities in our schools and society as a whole, but every now and again I am reminded of students who without that option simply walk away from school altogether.  There was a student in a bilingual ninth grade mathematics class that I taught.  His knowledge of English was very poor (yet he was made to sit through the English regents which must have done wonders for his confidence and for what reason I am unsure).  His academic background was also very lacking.  In his country after a few years of school (he could read and write in his native language though not very well) he went to work with his father.  He proudly announced that he was a mechanic and that he could fix all sorts of cars.  He was passionate about it wanting to learn more about it.  In class his head was most always down, he often said he didn’t understand things at school and that was not for him.  His self-confidence when it came to academic matters scrapped along the floor.  I called on him once after working with him on a problem so I knew he knew what was being asked.  He ran and hid in the closet.  I felt terrible.  I spoke to his guidance counselor about getting him into a mechanic school but he was under 16 and so had to be in an academic high school tested repeatedly and pushed by teachers like myself into uncomfortable places.  I wonder how long he had to suffer in a school system that was not helping him before he could do what he was not only talented at but passionate about.

Don’t get me wrong, I do not doubt that in the right environment and with the right teachers this student could have excelled in academic mathematics but our school and our system was not what he needed and this “one size fits all” model of education clearly did not fit him.

Finally (and yes, I like to write and am long-winded both) Rose brings up the idea of mastery and potential.  We criticize students’ writing, he says, because it is not up to our standards instead of looking for growth and potential. We’re looking for finished products without facilitating the process of becoming good writers.  I did well in school and in my mathematics classes.  I remember taking calculus and thinking I had a handle on the material.  I could do the problems.  Years later I taught calculus for the first time.  I had to relearn things I had forgotten and in doing so the subject made much more sense than ever before.  I finally knew why they were done in certain ways.  It was so much clearer.   It’s interesting how we teach something in a semester and expect our students to walk out masters.  One semester of practice or even four years does not a master make especially if time and spaces for structured reflections and practice are not made available.  Rose values errors noting, “Errors mark the place where education begins” (p. 292).