On learning mathematics

[EKS]

I'm reading this book called Vector Calculus, Linear Algebra, and Differential Forms that starts out by talking about learning mathematics.  Since there is a lot of talk on the boards about whether treading water to get a stronger foundation in math is better than moving forward and backfilling as needed, I thought this quote might be of interest:

Quote

If you're having trouble, skip ahead to examples.  This may contradict what you have been told - that mathematics is sequential, and that you must understand each sentence before going on to the next.  In reality, although mathematical writing is necessarily sequential, mathematical understanding is not: you (and the experts) never understand perfectly up to some point and not at all beyond.  The "beyond," where understanding is only partial, is an essential part of the motivation and the conceptual background of the "here and now."  You may often find that when you return to something you left half-understood, it will have become clear in the light of the further things you have studied, even though the further things are themselves obscure.

Many students are uncomfortable in this state of partial understanding, like a beginning rock climber who wants to be in stable equilibrium at all times.  To learn effectively one must be willing to leave the cocoon of equilibrium.  If you don't understand something perfectly, go on ahead and then circle back.

In other words, learning mathematics is an iterative process.  In fact, I'd argue that whenever you do complex intellectual work with novel elements, the only way to deal with it is iteratively because it's impossible to wrap your mind around the entire thing at once.  

So many homeschoolers are terrified of moving ahead in math prematurely, but done right, moving forward can actually help bolster shaky understanding in a way that treading water cannot.  I've seen this with all of the kids I've worked with on high school math, and I see it now with my own study of higher math as well.

[mathmarm]

This has been more or less my experience--both in learning mathematics and in my many years of teaching mathematics.

 

 

[Clarita]

I think I would relate it in the homeschooling world, like when you read books to your kids that are beyond their actual reading level. Mathy parents automatically just talk about math beyond what their kids are learning in their lessons. My kindergartener isn't doing division in his math lesson but I talk to him about rates, fractions and dividing things. To me it looks a lot like when I read The Lion, the Witch and the Wardrobe to him, I don't expect him to read at that level and there are a lot of things he needs to know between where he is now and being able to read that book, but you give him exposure to where reading and writing can go.

21 hours ago, EKS said:

If you're having trouble, skip ahead to examples.  This may contradict what you have been told - that mathematics is sequential, and that you must understand each sentence before going on to the next. 

I remember when someone finally told me I wasn't suppose to read math (or a textbook) from cover to cover. You browse the chapter, take a look at the problems at the end and try and do those. While you try your hand at the problems go back through the examples to help you figure out how to do the problems and if you need more information then read the paragraphs surrounding the examples. Then when you encounter a problem in real life and refer back to the textbook again read the sections that are pertinent to seek your answers. That's why the college math textbooks have answers to the odd problems, so you know if you understood, or you need to go and read the section again. 

[daijobu]

I remember Richard Rusczyk describing his time as an undergraduate at Princeton.  He would attempt to study a semester's worth of material in the first few weeks.  He wouldn't always understand everything with 100% clarity, particularly the hardest material at the end.  But because he had an early exposure to it, by the end of the semester he was learning it for the second time, and that made a big difference.  

Also, don't college professors often spend a greater part of the time on the easy stuff at the beginning of the term, then rush through the most difficult material at the end?  

[EKS]

1 hour ago, daijobu said:

Also, don't college professors often spend a greater part of the time on the easy stuff at the beginning of the term, then rush through the most difficult material at the end?  

Yes!  It drives me crazy.  In my more cynical moments I think that it's because they don't expect anyone to invest enough time to understand it anyway. 

[mathmarm]

5 hours ago, daijobu said:

Also, don't college professors often spend a greater part of the time on the easy stuff at the beginning of the term, then rush through the most difficult material at the end?

No, not in my experience. I guess it matters the course, as well as the individual professor.

The departmental-wide pacing guides were pretty uniform at one of the schools that I've worked in, though in the event that some section has to get cut we're more likely to cut material that won't be required in a following course. So, for example, that college algebra course focused on foundational polynomial arithmetic and graphing by hand. The precalculus courses was more  advanced polynomial manipulation, graphing by hand, working with polynomial, logarithmic, and exponential equations, graphing  by hand and with a calculator and lots of trigonometry.

Precalculus had to touch on all the obscure bits of math that students would need for the calculus sequence--Matrices, fraction-decomposition, analytic geometry can be in Precalc or College Algebra in a college, it just depends on the state.

Many students complained because our courses started exactly where they were supposed to start and moved at a pace that assumed that each student was well prepared and understood all the background information.

[EKS]

11 hours ago, mathmarm said:

No, not in my experience. I guess it matters the course, as well as the individual professor.

I just took a course that started by spending four semester weeks on one section of the textbook that had already been covered in a prerequisite course.  Then at the end it devoted four weeks to the last four sections of the textbook which were much (much!) more complicated.  I honestly have no idea why the professor decided to pace the course the way he did.

Hopefully this turns out to be the exception rather than the rule.

[mathmarm]

8 minutes ago, EKS said:

I just took a course that started by spending four semester weeks on one section of the textbook that had already been covered in a prerequisite course.  Then at the end it devoted four weeks to the last four sections of the textbook which were much (much!) more complicated.  I honestly have no idea why the professor decided to pace the course the way he did.

Hopefully this turns out to be the exception rather than the rule.

Oh that's terrible! 

[daijobu]

I think the difference is I attended an R1 school in the 1980s, where the professors were definitely not interested in developing something called a "department-wide pacing guideline" let alone teaching.  

I suspect teachers of remedial college classes like precalculus are more focused on making sure their students were learning.  For us it was sink or swim.  

[Clarita]

For me it was dependent on the individual professors, some were very evenly paced and stuck with it, some tried to be evenly paced but would somehow run out of time at the end and the end materials would be rushed. Also, I felt like semester system reviewed stuff in the first week(s) of class whereas quarter system always just took off with the new stuff. Depending on the class and how long ago I took the prereq it was either helpful to review or a better use of my time to just dive into the new stuff. 

[Green Bean]

1 hour ago, daijobu said:

I think the difference is I attended an R1 school in the 1980s, where the professors were definitely not interested in developing something called a "department-wide pacing guideline" let alone teaching.  

I suspect teachers of remedial college classes like precalculus are more focused on making sure their students were learning.  For us it was sink or swim.  

So precalculus is considered remedial now in college? I’m surprised at that as not every kid is going to get there before then.

To the OP- your thread is so timely as I have a teen struggling on. It has helped me see we just need to push on, not drop back. Thank you for this thread.

[mathmarm]

41 minutes ago, Green Bean said:

So precalculus is considered remedial now in college? I’m surprised at that as not every kid is going to get there before then.

To the OP- your thread is so timely as I have a teen struggling on. It has helped me see we just need to push on, not drop back. Thank you for this thread.

As far as I know, the universities that offer College Algebra and/or PreCalculus do not count them as remedial.  As far as I know, all of the schools that offer College Algebra and/or PreCalculus count them as credit-bearing courses for all students. However, the math requirements for STEM majors pretty much always begin at Calculus I--so the classes don't affect a major GPA for a STEM student.

At my University there's a pretty consistent (at least on paper) pacing for most of the courses within the STEM math sequence beyond Calc III (ie, Numerical Analysis, Linear Algebra, Differential Equations, Complex Variables, Topology, Abstract Algebra, Advanced Algebra, Foundations in Analysis, Discrete Math I and II, etc). I personally manage to stick fairly faithfully to my pacing schedule and spread the material out as necessary. Its easier to cover 10 chapters of Linear Algebra in 15 weeks if you move at a steady pace.

So, what @EKS described hasn't happened in any of my classes.

But we're a STEM university and the Math Dept. has to cover all of the material that students will need during their years in their perspective colleges.

[Clarita]

4 hours ago, mathmarm said:

At my University there's a pretty consistent (at least on paper) pacing for most of the courses within the STEM math sequence beyond Calc III (ie, Numerical Analysis, Linear Algebra, Differential Equations, Complex Variables, Topology, Abstract Algebra, Advanced Algebra, Foundations in Analysis, Discrete Math I and II, etc). I personally manage to stick fairly faithfully to my pacing schedule and spread the material out as necessary. Its easier to cover 10 chapters of Linear Algebra in 15 weeks if you move at a steady pace.

I do know some of my professors will tack on extra "just for fun" things. Things that are not necessary for the course officially but the professor just thought it's an interesting topic. "Just for fun" topics are not necessarily easy in fact most of the time they were challenging; some professors would divulge that this was extra material, I also felt like some didn't divulge this information - aside from not seeing it at all on the final. 

[ieta_cassiopeia]

My understanding is that if a professor realises they risk a pacing problem in a course where the material builds up on itself in advance, they're asked to slightly favour earlier rather than later content. This is for three reasons:

1) students who feel left behind at the start of the course are less likely to catch up than a bunch of confident students suddenly realising mid-course that there's going to be an unusually large difficulty curve to climb ahead

2) students who are lost at the beginning of the course can and often will transfer. Students who lose track towards the end are more likely to stick it out and make the best of it than those in a similar predicament in the first few weeks.

3) professors who get everyone who diligently attended class, paid reasonable attention and did adequete outside preparation/in-course work to a modest pass level fare better than those with a very uneven profile of star students and people who failed despite their best efforts and a reasonable chance of passing on paper.

 

Obviously, this is with the caveats that even pacing, accurately conveyed in pre-signing-up materials is always the best policy. Also, the sort of lopsided EKS describes would never be considered good policy for any professor, even one running a deliberate "weeder" course.

[kiwik]

On 2/28/2022 at 5:41 AM, EKS said:

I just took a course that started by spending four semester weeks on one section of the textbook that had already been covered in a prerequisite course.  Then at the end it devoted four weeks to the last four sections of the textbook which were much (much!) more complicated.  I honestly have no idea why the professor decided to pace the course the way he did.

Hopefully this turns out to be the exception rather than the rule.

Maybe they are finding a lot of holes due to covid.

[EKS]

9 hours ago, kiwik said:

Maybe they are finding a lot of holes due to covid.

This was vector calculus, so probably not.  Everyone in it was a math major, and many of them were actually seniors.  The program is all online, so there was no disruption in that regard.  I mean, there may be holes, but I seriously doubt that covid was a major contributor to them.

I think that the professor just designed the course to be that way.  

[kiwik]

On 3/4/2022 at 3:39 AM, EKS said:

This was vector calculus, so probably not.  Everyone in it was a math major, and many of them were actually seniors.  The program is all online, so there was no disruption in that regard.  I mean, there may be holes, but I seriously doubt that covid was a major contributor to them.

I think that the professor just designed the course to be that way.  

Odd.  From what I can remember of university it was paced.  I did have one semester with a lecturer I couldn't really understand though.