Noncommutative Analysis

Category: teaching

Why a “scientific approach” to science education is something I reject

Our Department has a new Teaching Seminar (concerned with teaching mathematics at the university level) which is led by legendary math professor Aviv Censor. The first lecture that I attended this semester was given by Professor Emeritus Avinoam Kolodny (Hebrew abstract here. A link to the talk – works only for Technion accounts – here). In the compelling lecture Kolodny started by mentioning the assumptions that we make when teaching (students come to class, they listen, they understand what we say, they then go home and solve homework problems) and contrasts this with empiric reality (a huge portion of students don’t come to class, the ones that do don’t listen, the ones that do don’t understand, and then they go and copy homework or solve routine problems like robots). Prof. Kolodny – an esteemed and decorated lecturer – said that he was troubled and puzzled by his students’ lack of success, and that at some point he became aware of the paper “Why not try a scientific approach to science education?” by eminent physicist, educationist and Nobel Prize laureate Carl Wieman. Kolodny explained various ideas of how to improve science (or engineering) education at the university level, to a large extent in line with ideas presented in Wieman’s paper.

The bottom line of Kolodny’s talk and Wieman’s paper is that the university lecture as we know it doesn’t work and is a waste of time. They have some ideas how to fix it, an approach that – as a first approximation – we can call “technology driven flipped classroom”. To me, the most disturbing parts of their approach are (1) that they believe that their opinions are “science based”, and therefore (2) they believe in promoting institutional change. These two aspects worry more than any technical discussion whether we should flip the classroom sideways or upside-down.

Kolodny remarked during his talk (I am paraphrasing): “I am not here to bury the concept of a lecture. Lectures are good and important. In fact, I am giving a lecture at this very moment. But you should remember that lectures are no good at passing information. In a lecture you motivate, you stimulate, you do propaganda. I’m here to do propaganda”.

Certainly I was stimulated by the talk, I was motivated to look up and then read Wieman’s paper, but most of all I was angry, I felt that someone was trying to brainwash me to believe in a certain ideology, rather than sharing some insights on teaching. Part of what made me feel this way was the “scientific approach” rhetoric. Another thing that bothered me was the jump from facts (some problems that almost everybody will agree on) to conclusions (a particular pedagogical methodology is the only way that works), disregarding tradition as not much more than momentum. Indeed, it felt like propaganda.

In this post I want to record my thoughts on some arguments raised by flipped classroom enthusiasts, and in particular on two aspects: the “scientific approach” approach, and with it the claim that lectures don’t work and we have to revolutionize the whole structure of courses to make them work.

I wish to recommend reading Wieman’s paper. Not only so that you can appreciate my criticism, but because it is a well reasoned piece of work by someone who has not only thought deeply about, but also researched the subject. I have a lot of respect for his efforts.

I am focusing my criticism on his paper, because it is written and available and interesting. But I am really arguing with talks, lectures, discussions, blog posts etc. that I have seen through the years, and have got me thinking for a long time. Now is just an opportunity to pour all of this out.

So, why not try a scientific approach to science education? Here’s why not:

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Topics in Operator Theory, Lecture 1: Introduction

This is a summary of the first lecture, which was introductory in nature.

H will always denote a Hilbert space over \mathbb{C}. B(H) will always denote the algebra of bounded operators on H. I am interested in operators on Hilbert space; various subspaces and algebras of operators that come with various structures, as well as the relationship between these subspaces and structures; and connections and applications of the above to other areas, in particular complex function theory and matrix theory.

I expect students to know the spectral theorem for normal operators on Hilbert space (see here. A proof in the selfadjoint case that assumes very little from the reader can be found in my notes, see Section 3 and 4). I also will assume some familiarity with Banach algebras and commutative C*-algebras – the student should contact me for references.

We begin by surveying different kinds of structures of interest.  Read the rest of this entry »

Course announcement: Operator Spaces, Operator Algebras and Related Topics (Topics in Operator Theory 106435)

Next week I will begin teaching a topics course “Topics in Operator Theory – 106435”. This is an advanced graduate course, where “advanced” means that I expect students to be familiar with graduate functional analysis.

The official name of the course is “Topics in Operator Theory” but the true title is “Operator Spaces, Operator Algebras and Related Topics”. There are two somewhat competing goals driving this course: the first goal is to give students a taste of the beautiful subjects of operator spaces and operator algebras, broadening their view of functional analysis, and giving those who wish enough tools to delve into the literature in this subject. The second goal is to train students to understand the problems in which I am interested and to get acquainted with the methods of the theory so that they will be able to carry out research in my group. The choice of topics will therefore be somewhat eclectic. In fact, I have several different plans for this course, and I am keeping things vague on purpose so that I am free to change course as the wind blows (and as I see who the students are, what their background is and where their interests lie).

What else? The course will be given in English. There is no official web page for the course – I might open exercises online on this blog. The grade will be based on some exercises that I will give throughout the semester, and a final “big homework” project.


A review of my book A First Course in Functional Analysis

A review for my book A First Course in Functional Analysis appeared in Zentralblatt Math – here is a link to the review. I am quite thankful that someone has read my book and bothered to write a review, and that zBMath publishes reviews. That’s all great. Now I have a few words to say about it. This is an opportunity for me to bring up the subject of my book and highlight some things worth highlighting.

I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of L^2[a,b] as a completion of C([a,b]), and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about L^2[a,b] as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with L^2 in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to functional equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.

Introduction to von Neumann algebras, Lecture 5 (comparison of projections and classification into types of von Neumann algebras)

In the previous lecture we discussed the group von Neumann algebras, and we saw that they can never be isomorphic to B(H). There is something fundamentally different about these algebras, and this was manifested by the existence of a trace. von Neumann algebras with traces are special, and the existence or non-existence of a trace can be used to classify von Neumann algebras, into rather broad “types”. In this lecture we will study the theory of Murray and von Neumann on the comparison of projections and the use of this theory to classify von Neumann algebras into “types”. We will also see how traces (or generalized traces) fit in. (For preparing these notes, I used Takesaki (Vol I) and Kadison-Ringrose (Vol. II).)

Most of the time we will stick to the assumption that all Hilbert spaces appearing are separable. This will only be needed at one or two spots (can you spot them?).

In addition to “Exercises”, I will start suggesting “Projects”. These projects might require investing a significant amount of time (a student is not expected to choose more than one project).

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