With only so much amount of time in the day, I have felt the need to evaluate what my priorities have to be as I sit through my lectures. Although each lecture is only about 50 minutes in length, to dissect and engage with every 15 minutes of content takes me around an hour on a good day! So each lecture takes me about 3 to 6 hours. While I am okay with 3 hours to each lecture, the 6 hours per lecture tend to set me back and create scheduling issues for the future. I would like to find a way to break down 50 minutes of content consistently within 3 to 4 hours, but I do not know if its possible. However, I do know that if I’m going to be spending all of this time engaging with math lectures, I might as well learn deeply, and hopefully efficiently, by keeping some principles in mind. Thus, I sought out advice on how to deepen and better my experience with math from How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston.
Luckily for me and other readers out there, the chapter Reading Mathematics offered just what I was looking for. Before I demonstrate my list of principles from the advice offered in the book, I want to acknowledge that I struggle to be consistent with large lists. Therefore, I have chosen the four most important principles that I believe can keep me best engaged and efficient in my learning.
“Before reading decide what you want from the text. The goal may be as specific as learning a particular definition or how to solve a certain type of problem.” – page 16
Trudging through a lecture for hours with no end near in sight is incredibly intimidating. I have found it to be very helpful to skim through the lecture or notes (if provided), to seek out important definitions, equations, or ideas. At least by initially having a roadmap of the content, I can anticipate how long certain topics will be and if they will be related to other presented topics beforehand. Therefore, this principle allows me to consider the relationships of ideas as I learn about them. Also, if I notice that there are few topics to cover, I am not as discouraged when I am taking hours to get through one topic, knowing in advance that there are fewer topics to cover.
“The first reason for using pen and paper is that you should make notes from what you are reading – in particular, what it means, not what it says – and to record ideas as they occur to you.” – page 17
For me at least, I benefit from writing “what is says” and then “what it means.” By slowly writing down what I see in a lecture video or a textbook, I understand better what the author is trying to teach. As soon as I reach any hesitation or vague understanding, I make sure to comment my raw thoughts and confusion in my designated commenting area with a contrasting blue pen to my main black pen. This principle has helped me in recognizing and tracking and addressing my patterns of confusion.
“The second reason [for using pen and paper] is more important. You can explore theorems and formulas by applying them to examples, draw diagrams…Physicist and chemists have laboratory experiments, mathematicians have these explorations as experiments” – page 17
Following this principle, I make sure to not concern myself with fully understanding a theory or idea before looking at an example. Although I have a stubborn habit of rereading something and thinking about it until it makes sense, it has also been helpful to attempt related problems which can then reveal how and why the theory works. Although I would think that understanding the theory is vital to solving a problem about that theory, sometimes solving the the problem is vital to helping me understand the theory. I guess you could say that you can read about a subject as much as you want, but finally interacting with it is when you learn whether you truly understand it.
“Ask ‘what does this tell us or allow us to do that other work does not?'” – page 19
This principle pairs well with the first principle as it forces me to be in the driving seat of my learning. If I happen to get caught in the lull of simply memorizing without understanding, constantly asking this question throughout the lecture can help me stay engaged as I relate it to something I already understand. In other words, build your concept images!
Thanks for reading. (:
Question to the audience: How long does an “hour’s worth” of reading/lecture take you to understand on average, and why?
The Learning Code 
What up Steve! Wow. Very nicely done. I love that you are picking up reading as a way to propel your thinking and skills. If at first you don’t succeed, try, try again. But if what you’re doing isn’t working, try something different. To figure out what different somethings to try, I can think of no better habit that picking up books that are related to your subject. You get to leverage hundreds or even thousands of hours of the author’s research and work on a time scale that can be measured in minutes… That is so powerful. I see that practice in action as you write this post. I commend that. Please keep that up.
In terms of the questions you posed in your post, here are some immediate thoughts that pop into my mind. First, when I was in graduate school, I came to develop an expectation that 1 page of math would take me a minimum of 1 hour of deep reading. This was more of a conservative guideline than a hard rule. On a given day in lecture, the teacher would “cover” about 10 – 15 pages of math. Each lecture occurred once every two days. So, that is about 6 hours per day per lecture, minimum. There are ways to speed some of this up, for sure. But at its heart, learning math for me is a slow, arduous process. The more I went through the slow burn, the easier my life got in the future. This is partly because at the beginning there was so much I didn’t know. But as I got more and more advanced, I could map new ideas back to ideas I had already masters.
That 1 hour = 1 page of math rule was one of the major reasons why I never liked taking more than 2 math classes in a single quarter. There just aren’t enough hours in a day to study math in 3 classes per quarter. This is particularly true if the teacher assigns homework with stringent due dates and gives in-class timed exams.
One really useful habit that I found would often speed up the pace at which I could learn was to study in the QA (quantitative analysis) section of the UC Davis library. I actually came to memorize the location of my favorite topics (real analysis, abstract algebra, linear algebra, etc). As I studied, I would pull 6 – 10 textbooks on the subject off the shelves and open all of those to the relevant sections (yes: for sure I was doing this on the largest tables I could find. I often tried to find two huge tables put together and to lay out all books open to the appropriate sections on those tables. I had no shame in being a space hog, except during midterm season when a bunch of people I had never seen in the library all of a sudden felt entitled to the space).
I would start with the book I was assigned in class. Then I would cross reference the ideas in the other books. A few things happened. First, the differences in formal statements of theorems, definitions, example, etc helped me develop a more nuanced understanding of the idea. Second, very often at least one or two homework problems in my textbook were written as examples in other textbooks. This allowed me to go much deeper on the problem and really push myself to learn, rather than just get the right answer.
Another idea that comes to mind as I read your writing is the idea of developing your concept images using multiple categories of thought. Take a look at pages 10 – 12 of my Math 2B syllabus for a refresher here (http://www.appliedlinearalgebra.com/s/Anderson_MATH2B01V_31030_winter2021_syllabus_draft1_20210104.pdf).
In the verbal description, I would really push you to develop both the abuelita language and the nerdy language. Abuelita language is a description of the idea in words that your abuelita can understand. Nerdy language is the formal math language to encode the idea (the one used to present the original concept definition you’re studying). Out of the two of those, the abuelita language is much harder to create. For a new definition or theorem, that usually takes me at least an hour of struggle, failure, and reflection. I liked to run by my ideas with a trusted team of students and also with the professor in office hours. They could give me feedback and guidance that would hone my thoughts. Once I had a good working definition in abuelita language, then map that onto the nerdy language. That process takes time up front but saves so much pain on the back end.
This is why routine office hour visit and self-advocacy/leadership are so important. Our system puts the responsibility to form this type of learning team on the student’s shoulders. I know you have this in you and I know see that you are doing great things. Keep up the hard work. Si se puede.
I’d love to chat about this during our next zoom chat. Just today, I was learning some new math that reminded me of your post.
What comes up for you as you read this? What do you think about explicitly targeting your development of both abuelita language and nerdy language? How might you do that?
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When it comes to abuelita language, I’m reminded of what my professor Soshnikov repeatedly says to us. To talk math as much as you can! There is so much to be gained by transcribing the book’s understanding into our own verbal understanding. I think that with abuelita language, we follow the same idea as we have to find a way to communicate the idea thoroughly but simply enough to be understood. It’s almost like talking math but with yourself.
I already designate a section in my notes (1/3 of left side of paper) where I write comments, but I have been thinking about considering perhaps a summary section so that I can perhaps summarize using this abuelita language. Although there is only so much room on each paper haha. I guess I might just have to spread my notes across even more pages and include around two inches at the bottom for a simplified summary when appropriate.
Sorry mother nature, I will plant some more trees!
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Now we are getting to a topic that is near and dear to my heart: how to write mathematics using pen and paper. I look at this process the way a religious person might look at prayer: this is a very sacred practice that lies at the heart of deep learning in mathematics.
Edward Tavernetti (https://www.math.ucdavis.edu/~etavernetti/), a good friend of mine in grad school, used to say: “The best gift you can give a person is a blank sheet of paper.” I’d like to add to that line. My version would be: “some of the best gifts you can give to a person include a blank sheet of paper and a recommendation for some good reading.”
This topic of how to write math using pen and paper is very much related to our TLC post from last week entitled What is deep learning (https://thelearningcode.school.blog/2021/01/24/what-is-deep-learning/). In that post, we define deep learning and explore what it means to learn deeply. One idea that I think I could communicate better is that deep learning in mathematics involves two separate stages.
Stage 1 is to understand an idea at a very deep level. This is the process of constructing a neural network to encode the idea inside your brain. I like to use the language that understanding foundational knowledge in mathematics is equivalent to constructing a robust concept image that includes multiple categories of thought to encode that idea.
Stage 2 is to remember that idea, to train your ability to recall that idea quickly and without effort. I remember when I was in Math 201B with Becca Thomases (https://www.math.ucdavis.edu/people/general-profile?fac_id=thomases) in winter 2009. She had this line that I will never forget: “Recall is necessary in mathematics. It’s not sufficient but it’s necessary. In order to do math well, you have to be able to remember the ideas you study off the top of your head.” The process of remembering involves strengthening our neural networks and building myelin sheaths to help the electric signals that are brain activity propogate more quickly. This process requires deliberate practice a repetition. Last week’s TLC blog post on What is deep learning focused on strategies to train our memory muscles.
I posit that your current post focuses on stage 1 of this process: building deep understanding by doing active reading. And you have converged on a very deep insight that all professional mathematicians come to understand: in order to understand math, we have to write out the math in our own handwriting. Textbooks are limited by economic considerations like how much it costs to print and how much they can charge the end user. The authorship and editing teams that help produce each textbook necessarily leave out so many details for every concept included in the book, otherwise the books would be too long and thus much more expensive. Moreover, the author writes the book to capture key ideas to study. But the real value is not on the written page. The real value is in the invitation to you, the reader, to recreate and enhance the work by creating your own version and filling in a lot of detail not in the original pages.
Here are some practices that I find very helpful and that have taken me over 15 years of practice, failure, reflection, reading, and growth to develop:
1. I purchase large boxes of blank white paper from Costco anytime I see I am getting low. I try to buy relatively inexpensive paper but also paper that is easy to write on and strong/sturdy.
2. I have built multiple layers of organizational systems. One of the first things that happens when you learn to write math is that you have to have a place to store your work so that you can find any ideas you captured quickly. At this point in my career, I can produce for you in less than 1 minute any math that I have written in the last 8 years of my life. In other words, I’ve created an organization system at home, including many filing cabinets and binders, in which I routinely file all new written work. Because I am a teacher, I also upload all my notes to my online website using SquareSpace and store digital copies in Dropbox. I also have an external harddrive that routinely updates a back up copy of my files. This ensures I have four layers of redundancy for my notes.
3. I tend to write on just one side of the page. This ensures that I always have extra space to add new ideas later in my life. In that case, I can just grab a new sheet of paper and add it to the appropriate place in the stack. Hardly ever will I write on the back of a piece of paper. The benefits of trying to compress ideas into a small space seem less important than the benefits of being flexible with my writing.
4. I’m very liberal with how much space I use. I used to try to write in the margins, as you say. But the longer I live and the more math I write, the more I realize that physical space is a proxy measurement for mental space. When I try to write in the margins, I cramp my own exploration space. Let me share an example of what I mean here. Take a look at my notes on Green’s theorem (https://jeff-anderson-wru2.squarespace.com/s/M1D_Lesson_14_Jeffs_Handwritten_Notes_UPDATED_Draft.pdf). When I taught that class a few years back, I had approximately 3 days to prepare. I hadn’t seen that material since 2008 so I had a ton of work to do. The notes you see in the link for Lesson 14 are my process of writing notes to learn Green’s theorem for myself. In truth, I would have written a lot more but I had to show up to class (so I was limited in a way that I wouldn’t have been if I was doing this just for my own learning).
Try to look for evidence in those notes of the verbal, visual, and symbolic categories of thought in my concept images. Also look for application-based thinking. Notice how much space I take. I didn’t time stamp the bottoms because I ended up sharing this with my students and I didn’t really want future generations of students to see when I wrote the first draft.
5. For math I do for myself (not for my students), I time stamp and number each page that I write. You asked the beautiful question: How long does it take you to learn for one hour of lecture? One great way to get real data on this question is to write the date and time on the bottom of every page of your work. A few years back, I coached a TA in my Math 1C class to do this. Here is an example of that work (https://jeff-anderson-wru2.squarespace.com/s/M1C_MD_Lesson_8_Draft_TA_Solutions.pdf)
Way back when as I launched Conquering College, I wrote just a little about this subject. Here are two posts that I think you’ve already seen:
How to utilize suggested Problems (https://jeff-anderson-wru2.squarespace.com/s/Study_Skills_HW_5_How_to_Make_the_Most_of_Suggested_Problems.pdf).
How to organize your course binder (https://jeff-anderson-wru2.squarespace.com/s/Study_Skills_HW_4_How_to_Organize_Your_Course_Binder.pdf).
I can imagine you picking up where I left off. This topic is super deep and you are now into the world of professional mathematics study. I have never met a mathematician who gets paid to do mathematics and has not thought very deeply about the exact topic you are thinking about now. Cheers to this Steve!
PS. The last thing I’ll say here as a small treat for later exploration: writing math using pen and paper is all about a capture habit. It is a physical record of your Stage 1 journey. Because writing is slow, the process of writing things out makes us go slower. This is one of the principles in The Little Book of Talent by Daniel Coyle. To learn deeply, slow your learning down as much as possible. Moreover, because we capture a record of our work, we actually set ourselves up for an easier time with Stage 2. It is so much easier to test our memory when we have written notes that are easy to reference, look at, and find. This is one way to “speed up” the process of deep learning. It won’t feel like “speeding up” in the early stage because it takes so freaking long to write these notes. But, once they are written, you can refer back to them at will.
PPS. This is where the due-date and testing policies in most classes actually get in the way. At this point in my math career, I would probably fail most math classes. I can’t have a teacher tell me how much time I need to spend on a topic in order to learn that topic. In fact, I need most teachers to get the hell out of my way. Every time you look at the books in my office, you are looking at the most useful teachers in my life. That is a learned skill and one that you are figuring out now. One question I am very interested in solving in my professional career is how do I create systems in my classes that help students develop this type of learning capacity and how do I support my students in leveraging these skills to learn math deeply, both in my classes and beyond.
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that moment you realize you wrote a blog in the comment field… talk about the advantages of being able to “communicate” off the top of your head… you two are the living embodiment of – anything worth doing, is worth doing well. re-reading this post and comments reminds me of…
Epistemology: investigates the domain of knowledge; what knowledge is, what it isn’t; what is a belief; justification; the structure of knowledge; the ultimate source of knowledge; evidence, perception, introspection, imagination, memory, reason, testimony; skepticism and the limits of knowledge; knowledge, wisdom, and understanding…. and linking that to Axiology which is related to our 5th theme of TLC for values… https://thelearningcode.school.blog/about/
Axiology: the study of values; ethics and morality; the nature of right and wrong; duty and obligation; virtues; care; justice; beauty; moral responsibility; metaethics; normative ethics; applied ethics.
ps. you two are awesome? b/c not only do you live with this compass Brian presents here: https://youtu.be/UQoEeGJ_Swo?t=1456 , you share that with others on a daily basis.
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