Monday 21 January 2013

My first blog series: A review of some educational literature

"When good teaching leads to bad results: The disasters of well taught Mathematics courses" by Alan Schoenfeld

A few weekends ago I put together a list of reading materials I wanted to go through for my pcutl second module portfolio. I've started going through them and jotting down some notes for each paper/book. On G+ I posted a pic of some of the books I was taking on a quick trip to Germany and +Theron Hitchman asked that I share anything that I find of interest. I started responding that of course I would and all of a sudden remembered the series of blog posts that Theron had put together when reviewing differential geometry books. I know nothing at all about differential geometry but I really enjoyed reading the posts.

So I've decided to do something similar. I'm going to try and post semi regularly (every other day or so) about one particular piece of educational literature. Some of these posts might be quite shorts (briefly describing what is in a paper) and some might end up being a bit longer (I might actually discuss what I think about each paper). Very worst case scenario is that this is the only post of the series...

My main pedagogic interests are in the use of technology to create student led learning opportunities. In particular I like to look at flipped classroom approaches and more recently (mainly based on the cool posts by +Theron Hitchman, +Dana Ernst and others) Inquiry Based Learning.

My first post in this series will look at the following paper: "When good teaching leads to bad results: The disasters of well taught Mathematics courses" by Schoenfield

This paper was passed on to me by Dana Ernst when I asked if he had any literature he could point me to on the subject of IBL (he did not disappoint, more to come later in the series).

This paper discusses a case study on a particular mathematics course:

- It was a succes based on `classical` critera
- It was a failure according to threshold concepts

The paper points out that in early educational research the actual subject being taught (Mathematics, Physics, English etc..) was seen as a minor variable to teaching (see Doyle 1978).

Here's a cool quote from the paper:
"Learning was operationally defined as performance on achievement tests -- tests which, as we shall see below, may fail in significant ways to measure subject matter understanding."
More recent work (Brown and Burton 1978, Helms and Novak 1985, Romberg and Carpenter (1985)) show the importance of subject specific teaching methodologies.
Here's another cool quote:
"In elementary arithmetic, for example, Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem."
The paper then goes on to discuss 'problems' with the classical approach to mathematical teaching. A neat example is given showing that in general students are capable of calculating $\frac{\sum_{i=1}^na}{n}$ by 'brute force' methods but do not have the deeper understanding of the connection between addition and multiplication to immediately realise that the above is just $a$.
"The predominant model of current instruction is based on what Romber and Carpenter (1985) call the absorption theory of learning. 'The traditional classroom focuses on competition, management and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered'. According to this view the good teacher is the one who has ten different ways to say the same thing; the student is sure to 'get it' sooner or later. However the misconceptions literature indicates that the students may well have 'gotten' something else -- and that what the student as gotten may be resistant to change."
The author links the common 'problems' with classical approaches (such as memorising proofs but not understanding them) to 4 beliefs, which I'll paraphrase here:
  • Belief 1: Formal mathematical concepts such as "proof" has very little to do with "real world problem solving".
  • Belief 2: If a student is going to manage a mathematical problem they will do so in less than 5 minutes (implying that if students don't solve a problem in 5 minutes they might as well stop).
  • Belief 3: Really 'getting' mathematics is only doable by geniuses.
  • Belief 4: Students do well in class by performing tasks and doing well in school (implying that 'getting the work done' will do).
I liked all of these as I can almost hear myself and former classmates saying the same things. Belief 3 I think is a very powerful one if "broken", when a student "gets" something and realises that "getting it" is not out of their reach I think that a huge barrier is broken... I'm speaking mainly from personal experience as it was in high school that an awesome teacher helped me "disbelieve that belief" (I wrote a blog post on that particular teacher here).

The next section of the paper goes to show that whilst the course was well run in a 'classical' sense there was evidence of students developing the above beliefs.

The final discussion is well written and describes how it is important to not only consider learning outcomes (a classical measure of success of a class) but also threshold concepts although these are not mentioned specifically. The author talks about students being able to "think mathematically".


I thought this paper was very well written and I really enjoyed reading it. It was a refreshing read saying what could be wrong with a 'good' course and the importance of threshold concepts. Importantly, the author states the importance of subject specific threshold concepts.

For this blog post series I thought I'd give each paper some sort of mark. My critique is obviously quite subjective so I don't want to mark how good a paper is but instead will offer to each document I read in this series a mark out of 10 of "how useful I think reading this document has been to me with regards to pcutl".

In this instance:
PCUTL Mark: 7 (not too sure where to start, I might revise this later)


  1. I also thought this was an excellent paper. It reminds me of Brousseau's 'didactical contract'
    "This is not a real contract : it is neither explicit nor consented to freely, since it relies upon knowledge necessarily unknown to the students. It positions the teacher and the student to face a truly paradoxical demand : if the teacher explains what he wants the student to do, he can only obtain it as the carrying out of an order and not through using knowledge and judgment. "

    In fact, I'd put Brousseau on your list somewhere!

    1. Thanks for the comment Marie and for reading! I had read a little bit about Brousseau's ideas when looking at critical constructivism but I can't say I've looked to closely.

      I'm not too sure how many more of these posts I'll write, I'm kind of losing momentum and more importantly running out of time but I'll certainly take a look at that link! Thanks :)