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Where is this happening?

I don't think I've had a math teacher say "memorize it" since Geometry in middle school, where we had to memorize pretty useful things to generally know, like SOHCAHTOA.

Basically every math test I've had since that has required nearly any type of memorization has been open note.

Also, at the end of the day, some memorization is necessary to advance in mathematical complexity. If you can't remember solutions to basic differential equations or the integrals of trigonometric functions, it will severely limit your ability to build useful things out of that basic knowledge.

I'm not saying I remember what the integral of arctan is off the top of my head, I'm just saying that there are practical reasons why teachers may require certain things to be memorized.

I was attracted to math in elementary school because it required less memorization and rote learning. Spell - that was the subject where rote memorization was required. I essentially failed spelling until it stopped being a subject - somewhere around Middle School.
I remember being so flooded with information that memorizing it would lead to failure. In other words, my classmates that tried to memorize formulas and steps would fail because there were so many of them. Those who internalized the relationship between components and understood why the steps were needed did alright.

One of the best tests to see if rote memorization is taking place, or there is actual understanding is word problems. In ex-Soviet schools they emphasized word problems heavily. In case of a word problem it is very hard to memorize the steps like one would do for a Gaussian reduction.

You're right about making connections, but one difficult part is that sometimes you just have to derive too much if all you have are connections and you haven't memorized things that could have created shortcuts for you.

I actually used to have trouble running out of time on tests sometimes when I had all the knowledge connected, but not enough of the right things memorized.

Depends on how the tests are structured. You could structure the tests so that rote memorizers win or those who have truly understood the relations win. And then sometimes it is not just he tests that are structured mostly one way or the other, but whole educational systems.
This is a great article. I've thought for a while that the way we teach maths is broken - not because we use too much rote or muscle memory, but because we separate teaching from practicing. We spend an hour learning a concept, and only practice using it when we get home. We spend six weeks teaching the basics of differentiation, and then move on to integration. While we do integration, we totally forget how to differentiate.

The article mentioned that memory retention is an exponential curve. Science has also shown that the best time to practice a concept is right before we would forget it. Put together, these two principles allow us to devise revision plans that make learning efficient.

Why are we stuck in the 18th century? We know so much more about how memory works, but we still keep doing the things we've always done.

> We spend an hour learning a concept, and only practice using it when we get home.

This is why I'm a fan of what the Khan Academy[1] is striving to do. It flips this pattern around, with more instruction at home and practice when with the instructor. The teacher can then spend less time introducing a topic and instead working through the process with individual students.

Watching Khan's TED talk[2] is pretty inspiring.

[1]: http://www.khanacademy.org/ [2]: http://www.youtube.com/watch?v=nTFEUsudhfs

The math books by the late John Saxon are a superb example of keeping teaching and practicing better connected. Unfortunately, I'm not aware of any large public schools (in my area, anyway) that use them.
Hey, I googled the textbooks and they sound intriguing. I may have to get one, just to see how it works in practice (I'm at university now, but it's never to late to go over old material). Thanks for sharing!
Here is a much better set of textbooks (I own the series recommended in the parent comment and what I recommend here, having learned about both from homeschooling communities when my oldest son was little):

http://www.singaporemath.com/Primary_Mathematics_Stds_Ed_s/1...

Part of my rationale for recommending the Singapore Primary Mathematics series comes from studies of mathematics curricula by mathematicians who think deeply about pedagogy:

http://math.berkeley.edu/~wu/Lisbon2010_4.pdf

http://www.math.wisc.edu/~askey/ask-gian.pdf

http://mathexperts-qa.blogspot.com/2011/04/math-experts-q-wi...

http://mathexperts-qa.blogspot.com/2011/04/math-experts-q-wi...

http://www-users.math.umd.edu/~jnd/Singapore.Math.htm

http://www.worcester.edu/SMIP/Shared%20Documents/PDFsPowerpo...

I also recommend the Singapore Primary Mathematics series materials, which I have used with all four of my children, because they are known to produce good results. Chapter 1: "International Student Achievement in Mathematics" from the TIMSS 2007 study of mathematics achievement in many different countries includes, in Exhibit 1.1 (pages 34 and 35)

http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf

a chart of mathematics achievement levels in various countries. Although the United States is above the international average score among the countries surveyed, as we would expect from the level of economic development in the United States, the United States is well below the top country listed, which is Singapore. An average United States student is at the bottom quartile level for Singapore, or from another point of view, a top quartile student in the United States is only at the level of an average student in Singapore. I've been curious about mathematics education in Singapore ever since I heard of these results from an earlier TIMSS sample in the 1990s.

After edit: The results of using the Singapore Mathematics materials with our oldest son have been gratifying. He was ready to test into our local state university's accelerated mathematics program by fifth-grade age, so by sixth-grade age he was doing the usual high school mathematics curriculum at double speed (and greater depth) and thus ready for that program's calculus course by eighth-grade age. Moving along steadily in math with good conceptual understanding helped him learn Logo, C, Java, Scheme, and Javascript programming through formal courses while still of high school age, and to self-study Python, Ruby, Haskell, and various other programming and scripting languages. He entered his university course (computer science) with as many accumulated "college credits" as most college students have when they graduate, and is enjoying taking upper-division computer science courses while coding for GitHub projects and planning a start-up. Doing elementary math courses that involve lots of problem-solving gets a learner ready for the problem-solving inherent in programming.

"She has this thing where she makes us do reviews that are about 10 questions and then she calls us up to the board to answer each question putting us on the spot."

My 10th grade geometry/trig teacher did this to us. Confidence issues I had as a result of this class lasted many years. I got over it and have done well for myself, but I still have some resentment.

My AP Calc teacher had us do the same thing with our homework assignments. I cannot say if this resulted in confidence issues among his students, but 58 out of the 60 students in the class received 5's on the AP exam.

It is not clear to me that this is obviously a bad teaching method. It definitely accomplishes 'engaging the student'.

It can be a good teaching method or a bad one, depending on how it's used.

Personally, I enjoy having students come up to the whiteboard. But when we do so, I present it as the student at the board is a representative of the class. The whole thing is collaborative, and I have students who are sitting explain what the student at the board is doing.

What this means is that students who know what's going on are able to take the lead (when at the board), whereas students who are less confident can rely a bit on their classmates. The whole time, I stress "Look, we're learning new stuff and it's going to be challenging. Work together and you'll get it." This makes a different vibe than if we just sat and watched before I determine if the answer is correct or incorrect.

I don't use this for individual assessment, and I don't attach any guilt or shame. It's purely done to get students working in the Zone of Proximal Development and to get someone other than me talking.