Archives for category: Reading Notes

I have learned a lot about writing in the disciplines since I first made my calculus students write in response to poorly planned essay prompts. I’ve learned how to break assignments up into manageable steps and how to arrange those steps so students could engage in a coherent writing process. I’ve learned how to give students feedback that lets them know not just how well they’ve done in their writing but also how well they’re doing as the writing takes shape. I’ve learned how to use writing as more than a means of communicating. For me and my students, it is also a means of discovery, exploration, and reflection, even in my most quantitative classes. (p. 129) … [S]ince we are the experts in our disciplines, no one is more qualified than we are to be teachers of disciplinary writing. (p.130) –Patrick Bahls, Student Writing in the Quantitative Disciplines.

I’ve followed Patrick’s blog for a while and was really excited when his book came out. I finally was able to read it. The quotation above opens the book’s last chapter and neatly describes both Patrick’s learning and the learning that I got from his book. Details:

Chapter 1 discusses different functions of writing in a course, such as ways that the activity of writing can help students learn as opposed to “merely” report what they’ve learned (not that the latter is easy). He clarifies some useful concepts and vocabulary, and tackles the psychology of student resistance to writing in quantitative courses—as well as instructors’ reluctance to teach writing in those same courses.

Chapter 2 focuses on the writing process: pre-writing, organizing, drafting and revising, and reviewing others’ work. He’s learned the hard way that students must be instructed and coached in each phase, and gives many ideas about how to do that.

Chapter 3 discusses assessment and feedback. First he points to several ways for you, the instructor, to give feedback in ways that actually help students get better and don’t take [as] much of your time. He then also has an extensive discussion of how to teach students to do peer review.

Chapter 4 covers “low-stakes” writing. These are generally quick exercises, mostly designed to help students with some other task like synthesizing or clarifying understanding. About a dozen good ideas for things like this.

Chapter 5 is about formal projects, ranging from major projects and how to lay them out over a term, to some creative writing ideas.

Chapter 6 is the conclusion, looking at our roles as instructors in teaching, studying, and advocating for writing in quantitative courses.

What I’ll use

I can imagine using almost everything in the book at some point. I got support from an ELA coach at my school last year that helped me learn to write and use a rubric. This made giving feedback on papers about 8-10 times faster. (That rubric, recently revised based on one year’s experience, is here.) So I feel things are under control in terms of how I respond to student papers.

My main focus this coming year is on peer-review. I’m hoping to combine Patrick’s ideas from Chapter 3 with Mylène’s ideas to develop the “peer review” standards and program for my courses this year. I’m hoping that teaching (and assessing) students on their ability to review will, as a side effect, improve the quality of the presentations and papers.


… one of the most extraordinary experiments in mathematics instruction, based on Dewey’s work, was published in 1938 as an NCTM Yearbook called The Nature of Proof [1]. The author, Harold Fawcett, taught a course in Geometry at the Ohio State Lab School that was arguably one of the greatest courses of all time. –Grant Wiggins

Well of course I had to read it. Here’s what I found:

Fawcett opens the book with the case that teaching geometry in high school “is no longer justified on the ground that it is necessary for the purpose of giving students control of useful geometric knowledge, since the facts of geometry which may at one time or another actually serve some useful purpose … can be learned in the junior high school” (p. 6) and that for sophomores instead, the point is widely assumed to be training students to think logically. “The degree of transfer of this logical training to situations outside geometry is a fair measure of the effectiveness of the instruction” (p. 8). And of course by that measure most geometry instruction is useless. Why? Because “If the kind of thinking which is to result from an understanding of the nature of proof is to be used in non-mathematical situations such situations must be considered during the learning process…. Transfer is secured only by training for transfer and teachers of mathematics can no longer expect that the careful study of ninety or more geometric theorems will alone enable their students to distinguish between a sound argument and a tissue of nonsense.” (p. 13)

So what did he do? He opened class with this: “There is no great hurry about beginning our regular work in geometry and since the problem of awards is one which is soon to be considered by the entire school body I suggest that we give some preliminary consideration to the proposition that ‘awards should be granted for outstanding achievement in the school.'” Students took sides, argued about whether a salary was an award and whether playing on the football team counted as “outstanding achievement.” Fawcett reports that “considerable time was spent in what might appear to have been useless discussion.” Finally one student offered, “Most of this trouble is caused by the fact that we don’t know what we mean by ‘awards’ or ‘outstanding achievement’.” (p. 31). He then introduced various topics and guiding students to analyze their own discussion. Over time, the students had determined the following principles (p. 34):

  1. Definition is helpful in all cases where precise thinking is to be done.
  2. Conclusions seem to depend on assumptions but often the assumptions are not recognized.
  3. It is difficult to agree on definitions and assumptions in situations which cause one to become excited.

He then reports that “there was general agreement that it would be interesting to make definitions and assumptions about concepts which did not stir the emotions and to proceed to investigate their implications”, but the students could not think of any such topic. So Fawcett proposed building a theory about “the space in which we live”, and students remembered that the course was about geometry. (p.35) This was four weeks into the year.

How to begin developing their theory? Students proposed defining things, but quickly hit quicksand. Fawcett gently guided them to start making propositions, defining some terms and leaving others undefined, and so on. He worked them to the point that he could give them a figure, say two intersecting lines, and instead of asking them to prove any particular theorem he just asked them to “state all the properties of the … figure that you are willing to accept.” Statements were listed on the board, discussed, critiqued, improved. Proven. Every student kept a notebook but no two were the same as different students found and proved different things which they shared with the class. By the end of the year, the students did better than ordinary classes on standardized tests despite not having been exposed to some of the material.

More importantly though, Fawcett was training for transfer. His exercises and tests throughout the year included items like asking students to analyze politicians’ platform statements, advertisements, supreme court decisions, and on and on. What are the key words that would have to be defined? What are the assumptions? What is the claim, and is it supported?

All this has left me wondering, and worrying, about what transferable skills I am hoping to teach in Algebra II and Precalc this year. My previous attempt is a good start but if I mean it, I need to defined observable behaviors that I am trying to train, and non-mathematical exercises for students to practice. I suspect I’ll be working on this question for a long time.

[1] Harold P. Fawcett, Ph.D. The Nature of Proof: The thirteenth annual yearbook of the National Council of Teachers of Mathematics, Teachers College, New York, 1938. 146pp.

This is not a review. These are my notes from reading Basic Principles of Curriculum and Instruction by Ralph W. Tyler (U. Chicago Press 1949, 128pp.)

How I found it

Brian’s reference led me to this post by Grant Wiggins. Tyler comes in at the bottom, but it turns out Wiggins writes about Tyler a lot. You can get a lot of what follows just from reading those Wiggins links, but it was fun for me to go to the source. Here’s what I found.


The book describes curriculum planning as a process of answering four questions:

  1. What are the right objectives?
  2. What learning experiences are likely to attain those objectives?
  3. How to effectively organize (sequence) those experiences?
  4. How to evaluate those experiences?

He doesn’t answer them directly but outlines very wide-ranging considerations to use when approaching the answers. Every educational philosophy, trend, and approach I can think of sits neatly somewhere in the framework he lays out.

What I’m taking from it

The things that are sticking with me appear early in the book:

A good deal of controversy goes on between essentialists and progressives, between subject specialists and child psychologists, between this group and that school group over the question of the basic source from which objectives can be derived. The progressive emphasizes the importance of studying the child to find out what kinds of interests he has, what problems he encounters, what purposes he has in mind. The progressive sees this information as providing the basic source for selecting objectives. The essentialist, on the other hand, is impressed by the large body of knowledge obtained over many thousands of years, the so-called cultural heritage, and emphasizes this as the primary source for deriving objectives…. Many sociologists and others concerned with the pressing problems of contemporary society see in an analysis of contemporary society the basic information from which objectives can be derived…. On the other hand, the educational philosophers recognize that there are basic values in life, largely transmitted from one generation to another by means of education. They see the school as aiming essentially at the transmission of the basic values…. The point of view taken in this course is that no single source of information is adequate to provide a basis for wise and comprehensive decisions about the objectives of the school. -pp. 4-5

I tend to get identified with one point of view at a time, then suddenly agonize about what I’m not doing. This helped me realize that all these points of view are legitimate and there’s room for all of them in the curriculum, though maybe not all of them at all times. I can relax and think about how to keep them all involved in my teaching overall. Maybe as I get stronger you’ll be able to see more of them in each individual day.

Education is a process of changing the behavior patterns of people. This is using behavior in the broad sense to include thinking and feeling as well as overt action. When education is perceived in this way, it is clear that educational objectives, then, represent the kinds of changes in behavior patterns of the students which the educational institution should seek to produce. -p.5

This quote didn’t really mean something to me until I’d read the rest of the book, but it sticks with me on three levels. On the shallowest level, this helped me move from “skills vs. content?” to “use skills on content.” Tyler recommends a 2D matrix with behaviors on one axis and content on another. So in the Algebra II planning I’m doing for next year with colleagues, we’ve come up with a chart of 26 skills by 5-7 function families. Together these make a year’s worth of daily objectives; it’s what most people think of as “Algebra II”.

The second level is more broad behaviors, or as Wiggins says, “use content well”. For me this is something like “Given some data, make a prediction.” It’s the broad skill of identifying and then using a functional relationship to answer a question. Can you do that and explain it verbally and in writing? This is what our Algebra II course is about. So the matrix is predict, write, present predictions vs. linear, quadratic, exponential etc.

The third level is life skills – what we learn to do in math that helps us everywhere. I took a stab at defining these for myself here, but haven’t made those goals an explicit part of my instruction. Yet. More soon.