Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:It's kind of ridiculous that MCPS can't point to a standard example test and say "a student with a grade of X in math course Y would score about Z% on this test."

It does. It’s called syllabus, curriculum, and grading policy.

Usually homework is a good indication of what will be on the test.

What’s ridiculous about this?

It's not true. The syllabus is general, and does not mean the tests are standardized across the classes. Even simple things could vary, like 20minutes vs 40minutes for the same test, and also what questions and how many questions the teacher chooses to put on the test, and how they weight the questions. Kids in the same school sometimes report that different teachers assign and score differently.

Homework may or may not be a good indication of what's on the test.

For some courses you can look at past year exams. The information is there in the course materials, most students don’t even bother to look them up.

You want the same questions, time allocation, grading policy, etc for all precalculus courses at the same school in every year? Now that’s being ridiculous! Teachers need the flexibility to teach the materials, emphasize certain topics, and grade based on the students in the class and the course goals.

Read the class materials, they are available the first day of the class or even earlier since they don’t vary much, then decide if the course is for you. Don’t complain that the hard teacher allowed 20 mins for the test while the easy one allows 40 minutes with unlimited retakes. That’s what you’re signing for, a rigorous course so you really master the material.

Where would you look for these? Do you mean the teacher might post them on the class canvas page, or they are available through MCPS somehow?

Does anyone have a syllabus for Hon Precalculus so that my son can start studying early for it?

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:PP, you seem to have skipped the part where they learn and remember the stuff they "already know and learned". Most humans aren't computers who memorize everything at first sight.

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.


https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/

Skip to the part about Jared the bond trader and then Serge Lange the world-class mathematician and professor.

"Serge Lang used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t.
The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."

My curiosity about this origin of this story was piqued , so here's the pdf of the actual article where Serge Lang discussed the test, December 1969 of the Columbia Daily Spectator (page 6 in the pdf):

[url]https://spectatorarchive.library.columbia.edu/?a=is&oid=cs19691208-02&type=staticpdf&e=-------en-20--1--txt-txIN-------
[/url]
Nowhere did Lang say that students who can solve 5 simple problems in a few minutes will pass his calculus class with flying colors. Here's what he actually said in the article:

After receiving Mr. Wyer's letter, I decided to give a short test to check Mr. Wyer's opinions. The test consisted of five problems, and was given to the 1A sections. One can draw some conclusions: a) The test is very easy and students unable to do reasonably well on such a test should not be taking a calculus course....

So your original quote is highly inaccurate. While it's more provocative and makes for a good story, it doesn't change the fact that being fast at doing very easy problems does NOT automatically predict success in math. All we can conclude here is that this is an absolute minimum of what the students are expected to know before taking the class (btw, if you're curious, in the article you can find the actual 5 questions that were asked). So no, Serge Lang did not believe that students who can do easy problems with speed will get As in his class (and why would anyone believe that? Correlation is not causation). The test was simply a low bar filter to identify students who could be at risk of failing because they don't understand fundamentals. Actually doing well requires much more than that, namely working hard and thoroughly learning the material.

Even more interesting in the article, is Lang's take on memorization:

Although a couple of students raised an objection to the time limit on the test, most students felt the time was reasonable, and in any case, part of the test amounts to verifying that students have reasonably fast responses to the type of question involved. Another point raised by some is that the question on sines of angles involves "memory." Which memory? Part of being properly acquainted with the basic facts of trigonometry is to be able to draw the proper triangle for and angle of 30 degrees, or 45 degrees, and determining the sine from that triangle, possibly using the pythagoras theorem. The type of memory involved is that of understanding, not the brute force memory...

It's not about memorization and regurgitation at high speed, it's about understanding. Speed develops naturally as a consequence of understanding. Doing homework (i.e working on stimulating problems) is by far the most important part of learning mathematics. Initially it's a slow painful process, but over time it solidifies concepts in students minds. If done well, it can lead to mastery of the material by the end of the semester/year.

I also take issue with the work analogy you make:

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.

You can't actually compare kids education with adults performing tasks at work. Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning.

Read the part you went to the effort to copy and paste: "part of the test amounts to verifying that students have reasonably fast responses to the type of question involved."

Also:

"Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning."


No, the analogy is that the goal of education is to be able to ANSWER a question that someone asks.

Nothing you wrote rebuts or even challenges the common sense claim that practicing basic problems helps develop valuable fluency.

Why would math be only subject where practicing fundamentals isn't helpful?
Athletes do drills and workous. Musicians do scales and etudes. Painters and sculptors make many iterations of the same project. Chefs prepare the same recipes many times.

The issue isn't that fundamentals are bad, the much bigger problem is that students don't do much more than that, they don't get to actually learn to think for themselves by solving problems. In fact, what you're calling fundamentals aren't even fundamental, they are endless rote drills. For example, from an early age kids are forced to learn long division and do endless computations with decimals, and they don't even have any idea how or why the division algorithm works. They would be hard pressed to even tell you what division means if you asked them! They don't understand fractions, and have no number sense (because of the above endless rote drills, they have not gained any intuition about fundamental things such as place value, the distributive property, simple geometric relationships between areas, and the list goes on and on). Then you race them to algebra and now they really don't understand what they're doing. The lucky ones who can follow procedural steps of how to solve for x, then get lost at the slightest adjustment (i.e putting a simple fraction in an equation, etc).

I honestly don't think you have any idea how bad it really is, since I can tell that you are not a teacher. Here's the kicker: Not only do kids not understand what they are doing, they think math is just a bunch of calculations. So it's no surprise most hate math and find it boring and stupid. And by the way, they are absolutely right, doing endless mechanical drills by hand is boring and stupid and they should use a calculator for tedious computations once they've learned how to do it by hand.. Yet teachers don't allow them to do this, in the process contributing to their disgust with math. Then in middle and high school the tables turn and teachers now force them to use calculators, in fact they're so dependent on them that they lose all basic reasoning skills. This is why kids taking AP calculus can mimic steps better than highly trained monkeys, but if asked, cannot solve basic fundamental problems that deal with whole numbers, multiples, divisibility, fractions, etc. They will instinctively pull out their calculator and stare at it, but then not know what to do next. It's a sad world.

I've tried to describe the symptoms for you, and if you've read and understood, you should be able to realize what the elephant in the room is: Kids have almost no conceptual understanding of mathematics, and even less (close to 0) ability to think and problem solve. They can neither appreciate nor enjoy math because they see it as just a bunch of disconnected rules to memorize, then perform and regurgitate back on tests. The main reason for this is because ALL they are doing is drills, nothing more. When you do that to curious early elementary kids (often all the way through middle school until high school), their brains permanently change.. They no longer ask questions, they don't wonder about things, they don't try stuff to see what happens, they only put effort if they HAVE to... i.e if it's for a grade. And nowhere is this more true than in math, because they've been conditioned to hate the subject. Of course there are lucky ones, many who learned math outside this system, perhaps at home, or at a place that teaches things in an authentic way. But the overwhelming majority of kids are mathematically stultified by the time they leave late elementary.

Anonymous wrote:

Anonymous wrote:PP, you seem to have skipped the part where they learn and remember the stuff they "already know and learned". Most humans aren't computers who memorize everything at first sight.

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.


https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/

Skip to the part about Jared the bond trader and then Serge Lange the world-class mathematician and professor.

"Serge Lang used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t.
The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."

My curiosity about this origin of this story was piqued , so here's the pdf of the actual article where Serge Lang discussed the test, December 1969 of the Columbia Daily Spectator (page 6 in the pdf):

[url]https://spectatorarchive.library.columbia.edu/?a=is&oid=cs19691208-02&type=staticpdf&e=-------en-20--1--txt-txIN-------
[/url]
Nowhere did Lang say that students who can solve 5 simple problems in a few minutes will pass his calculus class with flying colors. Here's what he actually said in the article:

After receiving Mr. Wyer's letter, I decided to give a short test to check Mr. Wyer's opinions. The test consisted of five problems, and was given to the 1A sections. One can draw some conclusions: a) The test is very easy and students unable to do reasonably well on such a test should not be taking a calculus course....

So your original quote is highly inaccurate. While it's more provocative and makes for a good story, it doesn't change the fact that being fast at doing very easy problems does NOT automatically predict success in math. All we can conclude here is that this is an absolute minimum of what the students are expected to know before taking the class (btw, if you're curious, in the article you can find the actual 5 questions that were asked). So no, Serge Lang did not believe that students who can do easy problems with speed will get As in his class (and why would anyone believe that? Correlation is not causation). The test was simply a low bar filter to identify students who could be at risk of failing because they don't understand fundamentals. Actually doing well requires much more than that, namely working hard and thoroughly learning the material.

Even more interesting in the article, is Lang's take on memorization:

Although a couple of students raised an objection to the time limit on the test, most students felt the time was reasonable, and in any case, part of the test amounts to verifying that students have reasonably fast responses to the type of question involved. Another point raised by some is that the question on sines of angles involves "memory." Which memory? Part of being properly acquainted with the basic facts of trigonometry is to be able to draw the proper triangle for and angle of 30 degrees, or 45 degrees, and determining the sine from that triangle, possibly using the pythagoras theorem. The type of memory involved is that of understanding, not the brute force memory...

It's not about memorization and regurgitation at high speed, it's about understanding. Speed develops naturally as a consequence of understanding. Doing homework (i.e working on stimulating problems) is by far the most important part of learning mathematics. Initially it's a slow painful process, but over time it solidifies concepts in students minds. If done well, it can lead to mastery of the material by the end of the semester/year.

I also take issue with the work analogy you make:

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.

You can't actually compare kids education with adults performing tasks at work. Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning.

Read the part you went to the effort to copy and paste: "part of the test amounts to verifying that students have reasonably fast responses to the type of question involved."

Also:

"Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning."


No, the analogy is that the goal of education is to be able to ANSWER a question that someone asks.

Nothing you wrote rebuts or even challenges the common sense claim that practicing basic problems helps develop valuable fluency.

Why would math be only subject where practicing fundamentals isn't helpful?
Athletes do drills and workous. Musicians do scales and etudes. Painters and sculptors make many iterations of the same project. Chefs prepare the same recipes many times.

Anonymous wrote:PP, you seem to have skipped the part where they learn and remember the stuff they "already know and learned". Most humans aren't computers who memorize everything at first sight.

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.


https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/

Skip to the part about Jared the bond trader and then Serge Lange the world-class mathematician and professor.

"Serge Lang used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t.
The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."

My curiosity about this origin of this story was piqued , so here's the pdf of the actual article where Serge Lang discussed the test, December 1969 of the Columbia Daily Spectator (page 6 in the pdf):

[url]https://spectatorarchive.library.columbia.edu/?a=is&oid=cs19691208-02&type=staticpdf&e=-------en-20--1--txt-txIN-------
[/url]
Nowhere did Lang say that students who can solve 5 simple problems in a few minutes will pass his calculus class with flying colors. Here's what he actually said in the article:

After receiving Mr. Wyer's letter, I decided to give a short test to check Mr. Wyer's opinions. The test consisted of five problems, and was given to the 1A sections. One can draw some conclusions: a) The test is very easy and students unable to do reasonably well on such a test should not be taking a calculus course....

So your original quote is highly inaccurate. While it's more provocative and makes for a good story, it doesn't change the fact that being fast at doing very easy problems does NOT automatically predict success in math. All we can conclude here is that this is an absolute minimum of what the students are expected to know before taking the class (btw, if you're curious, in the article you can find the actual 5 questions that were asked). So no, Serge Lang did not believe that students who can do easy problems with speed will get As in his class (and why would anyone believe that? Correlation is not causation). The test was simply a low bar filter to identify students who could be at risk of failing because they don't understand fundamentals. Actually doing well requires much more than that, namely working hard and thoroughly learning the material.

Even more interesting in the article, is Lang's take on memorization:

Although a couple of students raised an objection to the time limit on the test, most students felt the time was reasonable, and in any case, part of the test amounts to verifying that students have reasonably fast responses to the type of question involved. Another point raised by some is that the question on sines of angles involves "memory." Which memory? Part of being properly acquainted with the basic facts of trigonometry is to be able to draw the proper triangle for and angle of 30 degrees, or 45 degrees, and determining the sine from that triangle, possibly using the pythagoras theorem. The type of memory involved is that of understanding, not the brute force memory...

It's not about memorization and regurgitation at high speed, it's about understanding. Speed develops naturally as a consequence of understanding. Doing homework (i.e working on stimulating problems) is by far the most important part of learning mathematics. Initially it's a slow painful process, but over time it solidifies concepts in students minds. If done well, it can lead to mastery of the material by the end of the semester/year.

I also take issue with the work analogy you make:

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.

You can't actually compare kids education with adults performing tasks at work. Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning.

Anonymous wrote:Honors precalc is notoriously the hardest math class most kids will see in MCPS.

+1000. My math-y kid struggled a bit in Honors Precalc - now in Calc A/B and says it's "easy" comparatively.

PP, you seem to have skipped the part where they learn and remember the stuff they "already know and learned". Most humans aren't computers who memorize everything at first sight.

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.


https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/

Skip to the part about Jared the bond trader and then Serge Lange the world-class mathematician and professor.

"Serge Lang used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t.
The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:Some exercise of repetitive calculation patterns is helpful as an aid to long-term memory and improved fluency (reduction in speed and cognitive load) on the mechanical commutational bits. On a hard problem, a lot of the thinking is dead end brainstorming, which isn't the part the kid needs to remember.

What you're calling dead end brainstorming is the most valuable part of learning. Learning how to solve problems by exploring and trying various things is not only important in math, it generalizes to almost everything in life.

It's not about what kids need to remember (which is very little) it's about recognizing, connecting, and putting ideas together.

Come on. People don't have time to re-derive all of math from first principles on every single problem.

They're not rederiving everything. They're using what they already know and learned to solve a problem. Initially they might not know what to do, so they have to try stuff. If the problem doesn't look very similar to a previous problem, then no amount of memory recall will help them. They have explore, find patterns, make analogies, compare/contrast to past problems, etc, i.e they have to actively think. In the process of doing this over time, they slowly build intuition. And this is fantastic training for life, since most desirable jobs (where desirable could mean a combination of stimulating, challenging, high paying, etc) don't have a predefined recipe for success.

pettifogger wrote:

Anonymous wrote:Some exercise of repetitive calculation patterns is helpful as an aid to long-term memory and improved fluency (reduction in speed and cognitive load) on the mechanical commutational bits. On a hard problem, a lot of the thinking is dead end brainstorming, which isn't the part the kid needs to remember.

What you're calling dead end brainstorming is the most valuable part of learning. Learning how to solve problems by exploring and trying various things is not only important in math, it generalizes to almost everything in life.

It's not about what kids need to remember (which is very little) it's about recognizing, connecting, and putting ideas together.

Come on. People don't have time to re-derive all of math from first principles on every single problem.

Anonymous wrote:Some exercise of repetitive calculation patterns is helpful as an aid to long-term memory and improved fluency (reduction in speed and cognitive load) on the mechanical commutational bits. On a hard problem, a lot of the thinking is dead end brainstorming, which isn't the part the kid needs to remember.

What you're calling dead end brainstorming is the most valuable part of learning. Learning how to solve problems by exploring and trying various things is not only important in math, it generalizes to almost everything in life.

It's not about what kids need to remember (which is very little) it's about recognizing, connecting, and putting ideas together.

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:It's kind of ridiculous that MCPS can't point to a standard example test and say "a student with a grade of X in math course Y would score about Z% on this test."

It does. It’s called syllabus, curriculum, and grading policy.

Usually homework is a good indication of what will be on the test.

What’s ridiculous about this?

It's not true. The syllabus is general, and does not mean the tests are standardized across the classes. Even simple things could vary, like 20minutes vs 40minutes for the same test, and also what questions and how many questions the teacher chooses to put on the test, and how they weight the questions. Kids in the same school sometimes report that different teachers assign and score differently.

Homework may or may not be a good indication of what's on the test.

For some courses you can look at past year exams. The information is there in the course materials, most students don’t even bother to look them up.

You want the same questions, time allocation, grading policy, etc for all precalculus courses at the same school in every year? Now that’s being ridiculous! Teachers need the flexibility to teach the materials, emphasize certain topics, and grade based on the students in the class and the course goals.

Read the class materials, they are available the first day of the class or even earlier since they don’t vary much, then decide if the course is for you. Don’t complain that the hard teacher allowed 20 mins for the test while the easy one allows 40 minutes with unlimited retakes. That’s what you’re signing for, a rigorous course so you really master the material.

Some exercise of repetitive calculation patterns is helpful as an aid to long-term memory and improved fluency (reduction in speed and cognitive load) on the mechanical commutational bits. On a hard problem, a lot of the thinking is dead end brainstorming, which isn't the part the kid needs to remember.

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:It's kind of ridiculous that MCPS can't point to a standard example test and say "a student with a grade of X in math course Y would score about Z% on this test."

It does. It’s called syllabus, curriculum, and grading policy.

Usually homework is a good indication of what will be on the test.

What’s ridiculous about this?

It's not true. The syllabus is general, and does not mean the tests are standardized across the classes. Even simple things could vary, like 20minutes vs 40minutes for the same test, and also what questions and how many questions the teacher chooses to put on the test, and how they weight the questions. Kids in the same school sometimes report that different teachers assign and score differently.

Homework may or may not be a good indication of what's on the test.

One important observation that correlates to lots of learning and (usually) translates to good test grades, is the difficulty of the homework assignments. The more difficult the outcome, the better the kids will do on tests (assuming of course they attempt/do the homework). If homework assignments are too easy, kids are not learning much and the scenario is either 1) the test is also easy which means they've had a bad learning outcome and likely get punished in future classes. 2) the test is hard and they're seeing things that they did not see or practice at all on the homework, which is equally bad because they are forced to learn on a timed test and their grade very likely suffers.

On the other hand, if homework is difficult and challenging, and if kids can take the time to persevere through it, the outcomes are usually great: 1) the test feels easier than the homework and the kids who learned by doing the homework can show their understanding and feel good about all the work they put in or 2) the test is perhaps of similar difficulty to the homework, but again those who took the time to learn and understand the concepts should be able to do fine on them (assuming they're given a reasonable amount of time to think). Either way, grades are likely to be good, and most importantly, they've learned a lot and are very well prepared for future classes.

TLDR: Parents should really pay attention to their kid's math homework. If they are not struggling to some degree, they likely are missing out on a prime learning opportunity. In particular, if quizzes and tests are HARDER than the homework, that is a big red flag that grades might suffer without additional inputs. Also, it is the quality (not the quantity) of homework that matters.. i.e working through 40 questions for multiple hours means nothing if almost every question was an exercise (and not a problem). Better to have 5-7 questions that are problems (in that they actually force the student to think from an initial state of "I don't know what to do" to the end state).

Anonymous wrote:

Anonymous wrote:It's kind of ridiculous that MCPS can't point to a standard example test and say "a student with a grade of X in math course Y would score about Z% on this test."

It does. It’s called syllabus, curriculum, and grading policy.

Usually homework is a good indication of what will be on the test.

What’s ridiculous about this?

It's not true. The syllabus is general, and does not mean the tests are standardized across the classes. Even simple things could vary, like 20minutes vs 40minutes for the same test, and also what questions and how many questions the teacher chooses to put on the test, and how they weight the questions. Kids in the same school sometimes report that different teachers assign and score differently.

Homework may or may not be a good indication of what's on the test.

Anonymous wrote:OP here. Just came back to say that every time I posted I identified myself as OP. I am not a magnet parent and I am not puppeteering this thread .My son is far from being a genius and I am taking this issue very seriously. He obviously has gaps somewhere. I am no chucking it up to test anxiety, I think it’s that there is not enough time on quizzes to get through the problems and he is rushing and making mistakes by getting confused between definitions. Not sure how the leap to test anxiety was made. Also, the poster who keeps calling me out for puppeteering is obviously not familiar with AoPS. Even when you don’t master the material you are still advanced to the next level. His Algebra 1 AOPS and MCPS was 100% virtual. The poster is also not familiar with MCPS middle school math where you can retake quizzes so if he was struggling somewhere, I would not even know as his grades were good. Hon Algebra 2 was available at Frost last year, it was not on HS campus, so they were allowed retakes, late submissions etc. That’s why I previously stated that I think the gaps are from Algebra 2. Algebra 1 and Geometry at AoPS ran concurrently with MCPS. Algebra 2 material was different from MCPS. I don’t know how different but that’s what my son was saying. I do believe that the amount of work in Hon Precalculus is ridiculous. To the poster who keeps accusing me of puppeteering….dude, this is a discussion about a math class, not supposed to be a controversial topic. Please chill.

The MCPS material should have been a subset of aops algebra 2 material so if your child mastered Aops there should not really be gaps.