I’ve been taking Keith Devlin’s Massive Online Open Course (MOOC) “Introduction to Mathematical Thinking,” and it’s been really fun to see how my brain’s mathematical thinking skills line up with the course assignments. It’s especially fun to see the times when ideas are new to my brain and how that’s different from when they’re familiar.
For example, Assignment 1 was about reasoning from definitions, proof by counter-example, proof by induction — those are tricks of the trade I’m familiar with. I can read math problems and see how those tools are applicable, I know what to focus on when we’re proving properties from definitions, etc. I wasn’t flummoxed by stuff like what does it mean to divide 0 by 2, because I could work comfortably with the idea of 0 as being just another number that’s divisible by 2 — I was working on a higher level of abstraction.
But then again, on Assignment 2, I was flummoxed! I didn’t know what to pay attention to, and different stuff popped out at me. The assignment was about connecting natural language to formal logic (like if Statement A is “it will rain tomorrow” and Statement B is “it will be dry tomorrow” then is the plain language statement “It will either rain tomorrow or be dry all day” equivalent to the formal logic statement A v B (A “or” B)?) I got quite tangled up in the question of can it rain and be dry on the same day, and trying to nitpick the words in the sentences and find some hidden trick or meaning — because I didn’t have any idea what the basis of comparison was, or what it would mean to make a logic statement equivalent to a plain English statement. On the homework, I found myself confused and looking for “key words” to translate little parts of sentences into logic. In short, I felt like a student who struggles with word problems because they don’t know what it means to mathematize something, and so they are focused on different details than an expert would, and use different rules and tricks to do the mathematization. For an expert, it feels obvious, how to apply this heuristic, but for the student seeing it modeled, it’s not clear what details are salient to the expert.
I had an ah-ha! moment with Assignment 2, but more accurately, I had two or three ah-ha! moments. Two or three times this week I’ve struggled and asked my peers for help and had someone show me a truth table. Ah-ha! I said — to find out if two statements (whether in logic, plain English, or both) are equivalent I need to find out if they have the same truth table. And I could answer one question. But then when I became stumped again, I didn’t think of truth tables, because the question felt different!
I hope that now that I’ve reflected on truth tables and what it felt like to need one, and the (seemingly) different contexts that they helped me in, I will now have truth tables as a mathematician tool. Just like I have proof by induction, and showing sets are equivalent by proving an arbitrary element from one has to be in the other and vice versa, as mathematician tools that leap out at me.
Reflecting on my experience as a learner, I noticed that:
a) I needed to struggle with the problems repeatedly, without help, before I really cemented my learning.
b) I needed models of explicit tools from people who were smarter than me (about this kinda thing), but I couldn’t make sense of the tools and when to use them the first two times.
c) I needed to compare my ideas with other non-experts, to articulate them to myself and others.
d) I needed to persevere through enmeshment in all the wrong details, and be able to come up for air and entertain ideas about what other people saw, not just the alluring details. And be exposed to people just above my level, and their ideas, not just expert ideas.
I wonder how many of those features my classes and work with teachers have. Are they all necessary? Are they sufficient?
Thanks for the blog.
I am taking the course from Canada.
There are “Problem Sets”, “Course Assignments”, “In Lecture Quizes”, “In Lecture Quiz Duplicates”, as far as I know.
It is hard to tell, but I think people conflate some or all of the terms above, when mentioning their content.
I find this confusing, when trying to determine what is being referred to.
For example, are you talking about “Course Assignment 1″ or “Problem Set 1″ or something else in paragraph 2 above?
For me, “Course Assignment 1″, was far and away about statements, and interpretations of them from context. Where did mathematical induction slip in?
Best wishes.
r.
Hi Roy,
You’re right, I have gotten sloppy about distinguishing among “Problem Sets,” “Course Assignments,” and “In Lecture Quizzes.” Looking back over my notes, I actually used induction and reasoning about elements of sets primarily in the Supplementary Reading on set theory.
Thanks,
Max
Thanks for a very instructive article. I just read KD’s post about how difficult indeed impossible it is to keep up with student queries to him. The quantity of requests and KD’s limited time make it impossible for him to read every post. I am getting bombarded with posts about how old everyone is and it takes time to delete these posts. That time would be better spent by me focusing on the course material. So maybe this is a sneaky way of getting KD or his TA’s to find an easy way for me and other students to avoid getting such trivial posts. I suspect they have all ready done so.
I believe you can “unsubscribe” from getting email alerts from Coursera when someone posts on a discussion board. I know when you post to a discussion, the default is to have a box checked that says “subscribe me to this thread.” If you don’t want to get response messages from all 60,000 of us, you can uncheck that box before you post. But since it sounds like you’ve already subscribed, you can just go to that discussion forum in the course. At the top there is a greenish button that says “Unsubscribe.” Any time a discussion thread is overwhelming your inbox, you can visit that thread in Coursera and unsubscribe using the big greenish “Unsubscribe” button.
Max
Cool!
I’m a student too of this course, and I’m from Mexico City. It’s been a great experience so far and it’s interesting to work both with people like you and with people like me, who are not mathematicians.
i need lesson more
Hi Max,
I like your reflection on learning, and it in turn reflects my own experience so far. I’m really struggling with these sets, perhaps more (or even less), than others, but it is of little consequence, because the journey is what is important, and what we get from it.
Although we are assessed (I was mortified at first when I received 11/21 on the problem sets, and was then penalized for being late for the deadline), I remembered why I joined the MOOC…not to bolster my ego, or receive accolades for academic excellence, but to develop my “mathematical thinking”, which, up until very recently, I thought I was categorically awful at. For me, logical thinking has never been a linear exercise. I now realize that it can be either (linear or non-linear), but that the real thinking happens when you just go through the exercises with an open mind, are willing to stretch, accept and then dispell misconceptions, and then move on.
I am going to try to get up to speed on the vast amount of reading, especially on mathematical notation and symbolism, of which I am woefully lacking fluency. I am a teacher, with a pretty healthy work-load, but I have about a week off coming up (fall break), and I’m looking forward brushing myself off, and moving forward.
How about you? Have you begun Assignment 3? Any advice (mathematically speaking)?
Ronni
Hi Ronni,
I confess I haven’t been able to think about Assignment 3 very much yet. I hope to get cracking on it this weekend!
I really enjoyed reading your reflections on the course and the kind of learning we are doing. I too have found it to be very non-linear… much more like successive approximations than like mastering one skill that leads to another skill that leads to another. I’m glad you’re enjoying it and giving yourself space to not get it right away — I find I struggle with that and get annoyed with myself or the MOOC format when what I need to do is relax and try thinking about the material with others or from a different perspective.
Enjoy!
Max
I am just looking at Assignment 3 tonight…looks “simple” at first glance, but I’m sure it will be anything but. I amy try to tackle it more tomorrow in the Study Room on the Coursera site. Good luck to you, too!
I was struck by both of these insights:
Getting students to realize that they need to struggle with problems repeatedly is one of the most challenging parts of working with discouraged math learners, in my opinion. Discouragement takes hold for many of them when they see their “just a little more advanced” classmates simply “getting” something, then moving on triggers their defense mechanisms like nothing else.
This is one of the reasons why I find it so valuable to get students working individually on some new task or skill before putting them straight into group work. It’s hard to get yourself to wonder about something if your defenses are telling you to hunker down to avoid feeling humiliated.
Anyway, thanks for this food for thought.
- Elizabeth (aka @cheesemonkeysf on Twitter)
I totally agreed with the conclusion you have made.Specially the counter proof.
I really enjoyed reading this. I also have to struggle with the problems over and over without help before I really start to “get” the problems. For me the more It’s that repetitiveness that helps.