Q and A
Have a question that's not on here? Feel free to contact me.
Do you tutor? Can you recommend a tutor? Generally speaking, I no longer tutor/teach. However, I have made and am still willing to make availability for extreme cases of mathematical / computational talent development in exceptionally capable and motivated youth.
Regardless of whether you fit that niche -- if you're serious about learning extra math outside the classroom (i.e. enrichment), or filling in missing foundations (i.e. remediation), or preparing for a credit exam (e.g. AP Calculus BC), then check out mathacademy.com. We've created a fully automated and personalized online learning system that emulates the behavior of an expert tutor, and our monthly price is even less expensive than a single hour with an expert tutor.
Can you tell me more about Math Academy beyond what's on the website? Yes, in fact, I wrote a book on that. And if you want to know more context about the story of how Math Academy came to be, then see here.
I'm a software engineer who wants to work on cutting-edge ML/AI, but the math in those papers is all gibberish to me. How can I get to a point where I can actually understand it? Here is a detailed 4-stage roadmap with links to the best learning resources that I’m aware of.
My child is mathematically gifted. Can you recommend any resources/advice? Yes, see here. Also see here regarding coding. And if your kid spends a lot of time doing little math puzzles, try encouraging them to take on some larger projects. Note that lots of mathematically gifted kids get funnelled into competition math, which is typically not as productive as learning more advanced (i.e. higher-grade) math subjects. Also note that if they want to pursue math seriously at university, then it's not good enough just to get A's in the typical high school courses.
How did you get started with calisthenics? See here.
How can I learn effectively (and how do I do this if I'm on my own)? See here and here. Also, try to make your study sessions short and frequent as opposed to long and sparse.
I self-studied a bunch of university-level math. ...
- ... How can I demonstrate this to colleges and use this to strengthen my college applications? See here.
- ... How can I place into appropriate courses at university? If what you studied is beyond what's covered on the university's placement exam, then contact the head of the math department, give them context about what you studied (and how you studied it), and request to chat with them to see if it would be approriate for you to place out of those courses. (Make sure that you're fresh on all the material you claim to have learned -- if you can't talk fluently about those subjects and solve problems on the board, then you won't make a convincing case.)
What template does this site use? Minimal Mistakes with lots of customizations. Hosted on GitHub Pages. (If you're getting started with a personal website, here's my advice.)
Do you have an RSS feed? Yes, see here.
Can I skim through all of your content (books, blog, Q&A, etc) on a single page? Yes, see here.
I'm a new teacher. Any advice or knowledge you can share? Ah geez. Where to start? Here's a dump of some relevant Q&A I've written related to that topic.
- Don't try to teach like Feynman or Moore. Don't over-rely on projects.
- Expect most things to be centered on political ideology rather than the science of learning. You'll be shocked at how many students come into courses with missing prerequisite knowledge; here's why that happens and what the remedy is. You'll also be surprised at how many things are the subject of massive debates, e.g. whether to lower math requirements and even whether education should seek to maximize learning. And you'll occasionally encounter wishful thinkers who have somehow managed to convince themselves that doing things other than actual math will help their students learn math.
- Read up on cognitive learning strategies, deliberate practice, and the science of learning in general. Lots of people in education resist the idea of measuring learning and being held (or holding others) accountable. But as a general rule of thumb, if you're not measuring learning, then it's not happening -- and if people are not being held accountable, then they're not being accountable.
- Be aware that most students (and most people in general) over-rely on reasoning by analogy as opposed to first principles. Also be aware that nature and nurture are both important for learning -- it's not one or the other, even though lots of people wishfully believe that it's all nurture. Nurture (i.e. upbringing) can provide or withhold opportunities for practice, and nature (i.e. genetics) can lead to faster or slower skill acquisition from practice.
How to explain / understand (insert topic here)? Beyond what I've covered in textbooks, here are other math topics I've written about:
- Arithmetic: improper fractions; commutativity of addition; the order of operations; why is zero even; undefined vs infinity
- Geometry: what's the difference between the segment addition postulate and the partition postulate
- Algebra: closed vs open interval notation; how to graph a linear equation; the order of operations in function transformations; sigma notation; various absolute value inequalities
- Logic: implications; equivalence "if P then Q", "Q if P", and "P only if Q"; equivalence "P unless Q" and "P if not Q"
- Calculus: when it's okay to manipulate differences like fractions; why two numbers with arbitrarily small difference are equal; epsilon-delta limit proofs
- Linear Algebra: why any set of data points with different inputs can be perfectly fit by a polynomial; why the determinant controls the number of solutions to a linear system; how to create a matrix whose eigenvalues have specified algebraic and geometric multiplicity
Miscellaneous Q&A
I generally try to organize Q&A by turning it into blog posts and/or linking it to a higher-level question on this page. Below is Q&A that remains uncategorized.