Reply to Options for child who aced Geometry but is struggling in Algebra 2

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[quote=pettifogger][quote=Anonymous][quote=pettifogger][quote=Anonymous]As a tutor I can say that Algebra Ii is a very common place to fall apart, and it’s usually because kids haven’t learned to keep strictly to a linear organization of algebraic manipulation and haven’t learned to write equations.  It also can be [b]because they are not memorizing formulas and key concepts[/b] (sometimes not even able to recognize what is “key”).  And, in Algebra 2 it can be that[b] their underlying memorization[/b] and application of fractions and exponents/roots and graphing is weak, so [b]because they don’t know or remember[/b] they can’t apply the simple rules to more complex ideas.

Some kids really need a good tutor to do homework with them and explicitly enforce the linear organization/manipulation, re-teach earlier concepts and [b]identify what current concepts and formulas have to be memorized.[/b]

Some schools have some kind of after school help program but often there’s no higher level math teacher there.  Sometimes peer tutors are good but sometimes they can only explain an answer not identify the underlying student problems. Asking the teacher for help at lunch isn’t a great long term solution. (Sometimes the teacher is actually the problem.)

Dropping down from honors is OK, but can be frustrating for bright students because the non-honors pace is much slower and the kids have even more, different kinds of problems with the math.  Those students just need better or different explanations than they get from class. [/quote]
I'm quite surprised that as a tutor, you put so much emphasis on memorization. I've found that the exact opposite is true; the kids who try to memorize things have not understood the ideas, and as a result can only remember very little the following year. They continue to try to do this, but memorizing becomes harder and harder to do, due to the faster pace of new concepts introduced in higher level math classes. In their world, math is just a huge number of facts to be memorized. But in reality, everything is quite tightly connected by a [i]small [/i]number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because [b]whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking.[/b] In addition, practicing problem solving by wrestling with problems that they don't initially know how to approach (as opposed to mechanical/procedural exercises that are solved by straightforward application of rules) also gives them the techniques and ability to quickly reconstruct things whenever they don't remember the details.

Math is very much like a sport or a hobby; in order to do well and improve, one has to actively practice and challenge themselves in order to be able to extend the range of their abilities and become more skilled at it.[/quote]

Memorization doesn’t exclude understanding.  Yes, hopefully you are not memorizing a lot.  But, at the end of the day, if you are in Algebra 1 and [b]you are about to take the coordinate graphing unit test and you can not write down from memory the point slope equation of a line, the standard form equation of a line, and the equation for slope,[/b] you are not going to do well on the test. You can’t keep re-deriving a formula every time you need it, especially when your performance is time-constrained.  I find you bolded expectation highly unrealistic.

The idea of memorization is core to almost every class in MS, HS, college and beyond.

I don’t think requiring so much memorization is necessarily a great idea. Everyone’s working memory capacity is different, and some very bright kids, highly capable of mathematical reasoning can’t even memorize the multiplication table.  Working memory is not necessarily correlated with intelligence, and it would be better, in the name of neurodivergence, if school allowed open book tests or gave out formula sheets, but they don’t. Since school doesn’t (currently) work like that, and many kids don’t understand the implicit rules of the classroom, I explicitly tell students when they have to memorize a formula. 
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The above is a great example of how not to memorize. Take point slope form: Most kids have to force themselves to remember the cryptic looking formula y - y0 = m(x - x0), and hope that they don't mess it up on the test. But how many teachers actually show them that this is simply a restatement of the definition of slope, an idea which students are intimately familiar with at that point in the course as e.g 'rise/run' or change in y/change in x, etc. Yet the above is nothing more than just that, divide both sides by x - x0 and we get:

(y - y0)/(x - x0) = m

The left hand is the expression for slope (change in y divided by change in x), and the right hand side represents the actual slope of a particular line. So if we draw a line, label a fixed point (x0, y0) on it, label it with a slope m, and label any other arbitrary point on the line as (x, y), every student knows how to compute the slope of the line using the two labeled points. And that's all there is to it.[/quote]

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