Anonymous wrote:I think that PP was saying that on timed tests things go faster if you have certain things memorized. I don't have a math degree but I do have a degree in a related field and I used to be one of those kids who would derive things on every test and would often run out of time.

I am worried my child is starting to go down a similar route and have emphasized trying to explicitly memorize those formulas.

A very simple example would be that if you memorize the two point slope formula you wouldn't have to take the time to solve a set of simultaneous equations or if you remembered how to factor the difference of squares you could spend your time on other more complicated parts of a problem.

It's like multiplication. Sure I don't need to memorize it really but it make things go a lot faster.

Sure, memorization is fine when studying for the types of tests given in K12. The reason for that is because most, or all of the problems on tests and quizzes are amenable to memorization (as long as students remember formulas along with recipes for solving a set of standard problems, they are likely to do very well on most tests). But this highlights a few major problems with education: 1) In the real world, and at work, nobody cares whether one memorizes or not. A problem exists, and whoever can come up with a good solution for it gets rewarded. In that sense students are effectively wasting precious formative years 'learning' but not understanding. 2) Students graduate high school, many going on to top ranked colleges, and potentially becoming leaders in the workforce, yet they believe that learning is nothing more than just 'absorbing' information and spitting it back out on tests. They are 'A' students, yet it's quite possible they've learned almost nothing from school despite going through all the steps in the current process. Some luckily correct in college, often when they get their first initial shock of 30% on an exam and realize their method of studying isn't the same as learning. Others may never know what it means to understand; e.g to be able to explain something in simple and logical terms to their kids, or to others around them. And sadly, there are some who realize that while they may be capable of thinking, it is far too inconvenient and much easier to get others to do that hard work.

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:My daughter's Alg 2 Honors teacher told the class it is the hardest math class they will take, especially for this year. Because Alg 1 was right after Covid and then they had Geometry and now they are in high school and expectations and speed of the class are high.

I have no thought to pull her out of honors if needed but so far she is doing ok. She said the first week was stuff she never heard, but the last 2 days things were a little easier.

What is this honors society tutoring? Do I contact a counselor about this?

Extremely unlikely. Much more likely is she forgot after 14 months of not doing algebra while going through the poorly structured geometry curriculum.

I'm going to defend that first PP. I think her daughter is probably right that she has not seen certain types of problems before because if she is a 9th grader I think she might have done C2.0 which was terrible. While certain subjects may have been covered they were not covered well enough for kids to actually understand the material. So when they get to Alg. 2 and start getting problems that deal with the same subject but are more complex and require 3-4 steps and not just 1-2 steps you might feel lost.

DD's math teacher also gave a bunch of "review" problems that were much harder than what she saw in Alg. 1. DD did cover exponents in Alg 1 but it was really basic but the review problems had exponents of exponents involving multiple variables and to her that was "new."

Yes, this all too common and the root cause is a lack of problem solving in schools. At its core, problem solving means take something that a student does not know how to initially do, and via logic and reasoning break it down into things that are familiar. Students who are exposed to problem solving are able to do this reduction to known ideas often; the more problem solving they've practiced, the more easily they can recognize a problem as just a more complicated version of a previous problem, but having the same essential ideas.

Great post, thank you!

Anonymous wrote:

pettifogger wrote:

Anonymous wrote: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 because they are not memorizing formulas and key concepts (sometimes not even able to recognize what is “key”). And, in Algebra 2 it can be that their underlying memorization and application of fractions and exponents/roots and graphing is weak, so because they don’t know or remember 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 identify what current concepts and formulas have to be memorized.

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.

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 small number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking. 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.

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 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, 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.

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.

Anonymous wrote:

Anonymous wrote:My daughter's Alg 2 Honors teacher told the class it is the hardest math class they will take, especially for this year. Because Alg 1 was right after Covid and then they had Geometry and now they are in high school and expectations and speed of the class are high.

I have no thought to pull her out of honors if needed but so far she is doing ok. She said the first week was stuff she never heard, but the last 2 days things were a little easier.

What is this honors society tutoring? Do I contact a counselor about this?

Extremely unlikely. Much more likely is she forgot after 14 months of not doing algebra while going through the poorly structured geometry curriculum.

I'm going to defend that first PP. I think her daughter is probably right that she has not seen certain types of problems before because if she is a 9th grader I think she might have done C2.0 which was terrible. While certain subjects may have been covered they were not covered well enough for kids to actually understand the material. So when they get to Alg. 2 and start getting problems that deal with the same subject but are more complex and require 3-4 steps and not just 1-2 steps you might feel lost.

DD's math teacher also gave a bunch of "review" problems that were much harder than what she saw in Alg. 1. DD did cover exponents in Alg 1 but it was really basic but the review problems had exponents of exponents involving multiple variables and to her that was "new."

I do agree about not needing to memorize exponent rules. If you understand what exponents are you don't need "rules." It's just obvious.

I think that PP was saying that on timed tests things go faster if you have certain things memorized. I don't have a math degree but I do have a degree in a related field and I used to be one of those kids who would derive things on every test and would often run out of time.

I am worried my child is starting to go down a similar route and have emphasized trying to explicitly memorize those formulas.

A very simple example would be that if you memorize the two point slope formula you wouldn't have to take the time to solve a set of simultaneous equations or if you remembered how to factor the difference of squares you could spend your time on other more complicated parts of a problem.

It's like multiplication. Sure I don't need to memorize it really but it make things go a lot faster.

I have a math degree and I don't know the "8 major rules of exponents". I know that exponentiation is essentially repeated multiplication, and I can work out the consequences as needed, because I understand multiplication.

I have certainly unintentionally memorized some things, but never intentionally. I've occasionally tried. It doesn't work.

Learning is working with things until the ideas sink in. There is a small collection of vocabulary words to memorize, but that is not why people have trouble with math, and if you "memorize" them intentionally, you are skipping the *doing math* that makes the ideas stick.

Maybe you are using "memorize" to mean "learn"?

PV=nRT is a great example. If you know what P, V, N, rR, and T mean, you don't need to memorize the formula, because it's intuitively obvious. If you try to memorize the formula, you will fail understanding chemistry.

OP - what grade is your kid in. It's really a shame to me that MCPS has so many of the classes of 2026 and later on this path to take Alg 2 in 9th grade.

Anonymous wrote:

pettifogger wrote:

Anonymous wrote: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 because they are not memorizing formulas and key concepts (sometimes not even able to recognize what is “key”). And, in Algebra 2 it can be that their underlying memorization and application of fractions and exponents/roots and graphing is weak, so because they don’t know or remember 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 identify what current concepts and formulas have to be memorized.

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.

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 small number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking. 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.

A thread with a pettifogger comment is always a good thread. Pay attention.

Memorizing equations is like trying to study for an English test by memorizing sentences from the books you've read. It doesn't work, and even it did, it completely defeats the purpose of learning the material.

If your English test is on punctuation, you have to memorize the rules of punctuation and have to practice applying them. You can’t “derive” them during the test. You aren’t memorizing the sentences - you are memorizing rules or underlying principles. Or you are memorizing what is the definition of a metaphor, simile, etc. You have to have the rule memorized and be able to state it. Similarly, if you haven’t memorized the 8 or so major rules of exponents and practiced applying them, you are not going to do well in certain parts of Alg. I & II. If you are taking Chem, you’ll have to memorize ion valences, PV=nRT, Avogadro’s#, etc. If you’re taking Physics you’ll have to memorize trig rules and physics equations like F=ma and much more complicated ones.

Is memorization the only thing that is necessary? No, of course, you also have to practice application. Am I saying that you have to memorize and mimic each individual problem? No. But you can’t do a vector problem if you haven’t memorized SOHCAHTOA and the Pythagorean theorem, and you’d be surprised how many kids - even bright kids - get to Alg II and haven’t got these basic rules firmly memorized.

This thread is exactly why I supplement outside of school. I don't think it's realistic for a kid to take a year off between alg 1 & 2 for geometry. The school district I grew up in did a mix of prealgebra/algebra and geometry each year.

My kid does RSM, which has them do geometry over 3 years (6th-8th) while doing prealgebra, algebra 1, and algebra 2. By the time they get to algebra 2 in MCPS in 9th, it is easy for them.

You definitely cannot count on MCPS to teach math effectively.

Anonymous wrote:My daughter's Alg 2 Honors teacher told the class it is the hardest math class they will take, especially for this year. Because Alg 1 was right after Covid and then they had Geometry and now they are in high school and expectations and speed of the class are high.

I have no thought to pull her out of honors if needed but so far she is doing ok. She said the first week was stuff she never heard, but the last 2 days things were a little easier.

What is this honors society tutoring? Do I contact a counselor about this?

Extremely unlikely. Much more likely is she forgot after 14 months of not doing algebra while going through the poorly structured geometry curriculum.

pettifogger wrote:

Anonymous wrote: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 because they are not memorizing formulas and key concepts (sometimes not even able to recognize what is “key”). And, in Algebra 2 it can be that their underlying memorization and application of fractions and exponents/roots and graphing is weak, so because they don’t know or remember 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 identify what current concepts and formulas have to be memorized.

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.

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 small number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking. 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.

A thread with a pettifogger comment is always a good thread. Pay attention.

Memorizing equations is like trying to study for an English test by memorizing sentences from the books you've read. It doesn't work, and even it did, it completely defeats the purpose of learning the material.

pettifogger wrote:

Anonymous wrote: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 because they are not memorizing formulas and key concepts (sometimes not even able to recognize what is “key”). And, in Algebra 2 it can be that their underlying memorization and application of fractions and exponents/roots and graphing is weak, so because they don’t know or remember 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 identify what current concepts and formulas have to be memorized.

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.

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 small number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking. 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.

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 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, 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.

Anonymous wrote: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 because they are not memorizing formulas and key concepts (sometimes not even able to recognize what is “key”). And, in Algebra 2 it can be that their underlying memorization and application of fractions and exponents/roots and graphing is weak, so because they don’t know or remember 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 identify what current concepts and formulas have to be memorized.

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.

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 small number of key ideas. If they would instead focus on understanding these key ideas well, they would not need to memorize almost anything because whenever they forget, they will be able to reconstruct most of the things they memorize, similar to first principles thinking. 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.