Common Core Math: Interesting Article

[JoJosMom]

IMACS linked this article from The Atlantic criticizing aspects of Common Core math standards.  I think it's an interesting read.

[kateingr]

Thanks for sharing this! It has some great insights into the pitfalls of over-explaining in math. 

[3andme]

I just wanted to add the authors also contribute to a blog called kitchentablemath. It has lots of interesting posts about math instruction and learning in general such as Daniel Willingham's cognitive research. 

[birchbark]

I have several of Barry Garelick's articles bookmarked. I really like his perspective on math instruction.

[Tsuga]

I agree that procedural fluency and the need for repetition are understated in many CC curricula, but I disagree that knowing why is not as important as knowing how. When you are writing math equations to describe the world it is critical to be able to explain why you chose the variables and values you did and it is a problem in analytics and other firlds when engineers don't do that.

[JoJosMom]

I agree that procedural fluency and the need for repetition are understated in many CC curricula, but I disagree that knowing why is not as important as knowing how. When you are writing math equations to describe the world it is critical to be able to explain why you chose the variables and values you did and it is a problem in analytics and other firlds when engineers don't do that.

 

I think the authors' point was that being to explain why is not the same as knowing why, and that the verbal and writing skills necessary to explain why are far beyond what is necessary and appropriate at the elementary school level.  They are not talking about analytics or engineering education.

[Tsuga]

Well, all math will eventually lead to engineering for some of them (one would hope). The whole impetus behind this was that we aren't producing enough engineers ourselves and we need better education from the ground up to do that. But while I agree that being able to explain is not the same as knowing why in some sense, and therefore that students shouldn't be held back, I also think that explaining why--visual, verbal, or logical proofs--have a place in math education.

[JanetC]

I agree that procedural fluency and the need for repetition are understated in many CC curricula, but I disagree that knowing why is not as important as knowing how. When you are writing math equations to describe the world it is critical to be able to explain why you chose the variables and values you did and it is a problem in analytics and other firlds when engineers don't do that.

 

I don't see the article as saying "knowing why isn't important" so much as "the way that we are making kids show that they know why is overly rigid and complicated."

 

The standard,

 

One way is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, or where a mathematical rule comes from.

 

 

Should really be, in a way that is appropriate to the student's level of metacognition, written expression, and verbal abilities, rather than mathematical maturity.

 

We went through this with the NCLB Washington state math assessments -- if your writing skills were behind your math skills, such as the examples of autistic and ELL students in this article-- it was basically impossible to test at grade level in math, even if you could calculate the correct answer.

 

Demanding in depth explanations of simple procedures is not necessarily developmentally appropriate. Producing stylized "explanations" to pass a standardized test can be learned by rote just as much as calculations can be. We might not be testing what we think we're testing, and the kids might not actually be learning anything by going through this exercise.

 

I do think discussing why in class (where misunderstanding can be corrected immediately) is much more valuable than writing out explanations on homework and tests, especially before middle school.

[regentrude]

As so typical with any "standards", this is a problem of implementation.

Of course it is necessary that students can explain how they obtain an answer. mere procedural fluency is insufficient; as a college instructor, I see the fallout of a math instruction that drilled rote procedure without understanding on a daily basis.

And of course some teachers and curricula go completely overboard and require long explanations in specific formats for things that are absolutely trivial.

The problem is usually that the education people who devise these requirements are not the people who are actually using math or who have any first-hand knowledge what mathematicians actually do (they seem to have a rather vague understanding), and that the teachers who are using those materials are themselves often incapable of providing the explanations if they are not provided to them in a teacher manual in a specific format - and thus are incapable of judging whether a particular answer is reasonable and acceptable if it is not a cookie cutter replica of the answer key.

 

The goal of math instruction should be procedural fluency AND conceptual understanding, and one has only really understood something if one is capable of explaining it to another person (all instructors know that, in order to truly understand something, you have to teach it). The problem is not in this requirement, but in the assinine implementations.

[Tsuga]

I guess my issue with the point, "they aren't carrying this standard out well" is that, of course not. We've underfunded and outsourced education for decades. Who thought private companies would do this well for a profit, on the cheap?

 

What I see in my kids' schools is very different from what he describes. My kids write word problems, draw proofs, and have math challenge packets created by the school in which they solve more advanced issues and get specific instructions on what constitutes a good answer. They rank themselves according to their own use of critical thinking and use of visuals, words and logic. This all costs money and time. This is why I donate as much to the PTA as many of you spend on curriculum over 2 years, and what we get out is excellence.

 

If his complaint is that proving concepts should not be in there, then let's talk about that. If his complaint is that some states have outsourced education to friends' and relatives' fly-by-night education companies or Pearson and haven't trained teachers, then let's talk about that.

 

Let's not confuse poor program delivery with fundamental problems in a curriculum.

 

I agree we have massive issues with delivery. I do not agree that the standard is wrong.

[regentrude]

What I see in my kids' schools is very different from what he describes. My kids write word problems, draw proofs, and have math challenge packets created by the school in which they solve more advanced issues and get specific instructions on what constitutes a good answer. They rank themselves according to their own use of critical thinking and use of visuals, words and logic. This all costs money and time. This is why I donate as much to the PTA as many of you spend on curriculum over 2 years, and what we get out is excellence.

 

You have, in several threads, explained that you live in an extremely good school district with schools performing far above the national average. This is not the situation elsewhere.

I would suspect that you get excellence because your district can afford to hire competent teachers - not because parents donate to the PTA. Donating to the PTA in an average school may get the kids a few field trips or some extra materials, but those donations do not go towards hiring qualified instructors. Anything "enrichment" and "parent involvement" and "challenge packets" is only a drop in the bucket if the math teachers themselves are not proficient in mathematics, or if the few proficient teachers have to remediate what their incompetent predecessors mucked up in the preceding grades.

THAT is the root cause.

 

Curious: can your PTA raise funds that go towards attracting teachers with subject expertise? Because here, PTA funds go for "extras", but cannot be used for salaries (for obvious reasons, since those are not budget line items that can be relied on)

[Farrar]

I was discussing this elsewhere the other day.

 

I think the authors downplayed the importance of being able to explain what you're doing, but I also thought they hit on one of the key problems with CC math standards (and I do mean the standards here, not the implementation or the curricula for once) which is that there are very few ways to really do what they're asking and provide that proof without turning early elementary math classes into writing classes. I think it means you've double penalized kids who lag behind (behind being not writing full sentences at age 5 in CC terms) with writing. They can't even be allowed to be good at math because math is as much writing as numbers. Symbolic representation would be a good way to show understanding as well and oral language is probably the best way for these early grades, but both provide problems.

 

The other piece is that I think it's really a pendulum swing. The CC people saw that all the testing emphasis in classes led to only doing drill and trying to game the test and not to teaching for real understanding. So they put that in the test. Except, it's still a game, just a different one now - one that means second graders need to type out explanations for why 2+5 doesn't come to 8 or something - with bonus point, of course, for mimicking the wording used in a Pearson textbook to explain yourself. I don't think that's an improvement. The real improvement would be teachers actually spending more time on this stuff, enough to actually check for understanding. But to do that, you'd have to move away from the test prep model and they're not willing to do that.

[Tsuga]

Regentrude, they pay for extra teachers. Technically, they are not supposed to, but what they end up doing is moving funds around to pay for the school's #1 need:

 

People in the classroom, skilled teachers and assistants.

 

The facilities and salaries are good but not compared to COL. What teachers do tend to like are resources. We also have good schools of education locally and a high emphasis on social justice, which does produce good teachers. But ultimately, we need to fund education. The PTSA is not supposed to fund basic education but there are ways. Like ensuring schools and districts spend a higher % of funds on instruction by funding other things conditionally. If the PTSA does X, y and Z, the other money can go to assistants for 1 - 2 hours each day in the classroom.

 

Also we fund things so the principal can fund teacher training.

 

The state legislature literally won't find staff salaries so yes, parents pay out of pocket for aides (w/Ed degrees) so budgets go to salaries. They also pay out of pocket for enrichment after school for remediation and for ESL.

 

You couldn't raise this $ for a trip to the farm, I'll tell you that.

 

If the legislature continues to defund education (relative to need) and then cracks down on wealthy areas and high-Ed area PTAs paying for teachers out of pocket, we will see what happens. Right now the legislature is still underfunding education.

 

It's pretty disgusting but please note I am not suggesting that it should be possible for every PTA to do this. I would love properly funded education.

 

My point is that good implementation requires resources and political will but it's possible.

[Tsuga]

Oh, and adjacent to us, they have even more $. My neighbor did a study. Something like 10% of salaried instruction ("supplemental") was funded by parents. She said it was shocking. The state won't pay. Cities won't pay. People without kids don't want to pay. So parents pay.

 

To the north, the district raises funds by setting up a foundation which PTAs can fundraise "with". So the foundation funds instruction. It's even throughout the district.

[snowbeltmom]

Wow, I have never heard of a PTA funding teacher salaries. Tsuga, how much do you pay per year to the PTA? Are these fees mandatory of every family?

[Tsuga]

There are no fees. It is 100% donations and many employees get corporate matching up to 100% in some cases. There is no public record of membership. Only the PTSA knows and many donations are made anonymously. But they fundraise hard and try to get big money from families that can give. From what I have seen as a non member, member and board member, my kids were treated the same everywhere. Nobody gave us any issues when we did not pay (it was actually an oversight I learned of at year's end).

 

Please note it's not the "base" teachers, but assistants, after school programs like homework helpers, etc. and technically this is all the district, but fundraising emphasizes "freeing up funds for X people in the classroom y hrs / week". So the assistants, math help under the guise of "enrichment", whatever can be done legally to support teachers.

 

I know this is happening in at least five districts, not all wealthy, because I heard about it at a state level meeting. Whoever can raise that much is absolutely raising it for man hours. I know it is happening in Seattle as well because I know people on the PTSA who were raising for addidional English instruction for ESL kids, beyond what the state pays for.

[MarkT]

As so typical with any "standards", this is a problem of implementation.

Of course it is necessary that students can explain how they obtain an answer. mere procedural fluency is insufficient; as a college instructor, I see the fallout of a math instruction that drilled rote procedure without understanding on a daily basis.

And of course some teachers and curricula go completely overboard and require long explanations in specific formats for things that are absolutely trivial.

The problem is usually that the education people who devise these requirements are not the people who are actually using math or who have any first-hand knowledge what mathematicians actually do (they seem to have a rather vague understanding), and that the teachers who are using those materials are themselves often incapable of providing the explanations if they are not provided to them in a teacher manual in a specific format - and thus are incapable of judging whether a particular answer is reasonable and acceptable if it is not a cookie cutter replica of the answer key.

 

The goal of math instruction should be procedural fluency AND conceptual understanding, and one has only really understood something if one is capable of explaining it to another person (all instructors know that, in order to truly understand something, you have to teach it). The problem is not in this requirement, but in the assinine implementations.

Yes they should have knowledgeable folks not directly in the K-12 education field review both the standards and the implementation for clarity.

[snowbeltmom]

Yes they should have folks not directly in the K-12 education field review both the standards and the implementation for clarity.

Didn't the only two educators on the Validation Committee, a mathematician from Stanford and an education guru from MA, refuse to sign off on these standards, yet the standards were approved anyway?

[Serenade]

When you are writing math equations to describe the world it is critical to be able to explain why you chose the variables and values you did and it is a problem in analytics and other firlds when engineers don't do that.

 

I am going to share this with a certain math student who thinks nothing matters except the right answer.   He doesn't seem to believe me when I tell him it is important to be able to explain how he got the answer and that setting up a proper equation is part of that.  This was important even before Common Core.

[Spy Car]

It is absolutely imperative that students understand and can explain their work. Full stop.

 

When one can explain why procedures are valid using mathematical laws and showing ones reasoning, to the point one could teach it to another, one has "mastery" of  a topic. Otherwise not so much.

 

I think the article is another in the long and counter-productive tradition of American anti-intellectualism when it comes to math education. With no other subject is expecting a student to understand what they are doing remotely controversial. It's not "OK" for children to "read" by memorizing the shapes of words in most circles (certainly not on the WTM). We expect well educated students to understand grammar, and not to have their "usage" based on "what sounds right."

 

If there are areas where the expectations of students to "explain" isn't age appropriate, or the teaching has been lacking, then fix those problems. But the backlash (that always comes) when math education extends to actual mastery (as opposed to a subject taught by poll-parrot type memorization) is a battle that needs to be fought, and won.

 

"I got the right answer" is not sufficient. Not close.

 

Bill

[Farrar]

My main complaint is that it's developmentally inappropriate to ask 6 yos to write paragraphs about math. Full stop. Yet the CC makes it almost impossible not to assess young children this way. And invites this type of backlash in doing so. The backlash is a problem because, yes, conceptual understanding is super important.

[Roadrunner]

"I got the right answer" is not sufficient. Not close.

 

Bill

But it is for SM and AoPS. I don't worry.

Yes, Beast and AoPS also ask for explanations, but only for a very small fraction of the total problems. I have seen our local grade 7 math homework. They are just now adding and subtracting negative numbers and spending months on drawing this. Beast taught it in a week to fourth graders. I think there is a way to teach for conceptual understanding and move kids forward without having to write paragraphs and turning a math class into a drawing lesson.

[Spy Car]

My main complaint is that it's developmentally inappropriate to ask 6 yos to write paragraphs about math. Full stop. Yet the CC makes it almost impossible not to assess young children this way. And invites this type of backlash in doing so. The backlash is a problem because, yes, conceptual understanding is super important.

 

So we should aim for fixing the problems without throwing the baby out with the bathwater. I'm all for that.

 

But let's not kid ourselves that there aren't many folks who rabidly oppose excellence in math education, believing (falsely) that grammar stage children are nothing but poll-parrots and that anything beyond treating math as a bunch of math facts to be mememorized is something to oppose. They don't call it the "math wars" for nothing.

 

I've home educated math largely because I know there is no better way to make sure a child understand the material than discussing it one-on-one. The classroom dynamics make that approach difficult. If there are flaws in the implantation of CC expectations, I'm all for fixing them. But giving up on cultivating "understanding?" Not a fight I'm will to lose.

 

Bill

[Spy Car]

But it is for SM and AoPS. I don't worry.

Yes, Beast and AoPS also ask for explanations, but only for a very small fraction of the total problems. I have seen our local grade 7 math homework. They are just now adding and subtracting negative numbers and spending months on drawing this. Beast taught it in a week to fourth graders. I think there is a way to teach for conceptual understanding and move kids forward without having to write paragraphs and turning a math class into a drawing lesson.

 

Really? Because I expect answers very similar to those found in the AoPS Solutions guides. With a detailed series of steps, and an articulation of the mathematical reasoning.

 

Bill

[Farrar]

So we should aim for fixing the problems without throwing the baby out with the bathwater. I'm all for that.

 

But let's not kid reserves that there are many folks who rabidly oppose excellence in math education, believing (falsely) that grammar stage children are nothing but poll-parrots and that anything beyond treating math as a bunch of math facts to be mememorized is something to oppose. They don't call it the "math wars" for nothing.

 

I've home educated math largely because I know there is no better way to make sure a child understand the material than discussing it one-on-one. The classroom dynamics make that approach difficult. If there are flaws in the implantation of CC expectations, I'm all for fixing them. But giving up on cultivating "understanding?" Not a fight I'm will to lose.

 

Bill

 

At no point would I ever say we should stop trying to get kids thinking about the reasons behind the math. I would never say that.

 

However, I think there's zero way to fix this problem without fixing the testing culture. The way this problem came about was at least in part from testing culture - the tests encouraged memorizing the algorithms and drilling to pass the test. So the CC people said, gee, how can we fix this? Good impetus. But their answer was more tests. Because they know that without that, nothing will be taught at all. Except now schools have just found different ways to drill kids on this - by having them practice these canned writing responses over and over and practicing their drawing various arrays and diagrams over and over. It's still memorization, they've just made a more complex algorithm - if they ask that, draw this, if they ask that, write this and plug in the numbers here and here. Really, the only way to fix it is to spend more time with kids and understanding and wait on the testing. To train better teachers who are real professionals. To train them in assessing kids constantly and more informally in the classroom - using manipulatives, listening to them, working in small groups, etc. The places that are consistently producing kids who do have a deeper understanding of math, by and large, do not test them over and over throughout the elementary years in this manner. And I see a connection there. But there's no trust in the teachers and no political cover in lessening the testing - it makes it look like we're "lowering standards."

[Roadrunner]

Really? Because I expect answers very similar to those found in the AoPS Solutions guides. With a detailed series of steps, and an articulation of the mathematical reasoning.

 

Bill

You expect your child to not just solve but also write out in words the solutions basically relicating the manual? We are halfway algebra and with the exception of one writng problem in the class, we do no paragraph writing in math. We do no essay writing or drawing with aops yet we manage to have great conceptual understanding.

[Farrar]

You expect your child to not just solve but also write out in words the solutions basically relicating the manual? We are halfway algebra and with the exception of one writng problem in the class, we do no paragraph writing in math. We do no essay writing or drawing with aops yet we manage to have great conceptual understanding.

 

Jousting Armadillos has been pushing us in this way and it's a struggle. Part of my thinking is definitely being informed by that. JA asks for "Notes to Self" from the student for nearly every section. Basically a paragraph explaining how a concept works. My ds lags a little behind in writing, but no so far behind. His mechanics are weak, but he can sit down and write a pretty good summary after reading a book or can enjoy writing a story. He just finished his first researched report. But this is HARD for him. So, so hard. I feel like it's important so I'm making him do it. And we've been using numberless problems (I did an update of Problems Without Figures, which is on my blog, to help us practice this whole skill of math writing). I do believe in it. But my kid is 11. It feels to me like *now* it's an appropriate skill to be working on. CC asked kids to work on it so young.

 

And, yeah, I felt like ds got a ton out of Beast and understood many great things from it... but I can't imagine him writing anything so coherent and clear as the solutions in the back. That's crazy to me. That's professionally written.

[Spy Car]

You expect your child to not just solve but also write out in words the solutions basically relicating the manual? We are halfway algebra and with the exception of one writng problem in the class, we do no paragraph writing in math. We do no essay writing or drawing with aops yet we manage to have great conceptual understanding.

 

The AoPS Solutions don't involve essay writing. They require a clear representation of the steps and logic used in problem solving. Occasionally here is more need for verbal or written explanations (we alternate between doing things orally and on paper), but never (in my estimation) does the AoPS approach ever say just getting the "right answer" without understanding the math is sufficient.

 

I never treated Primary Mathematics that way either. A correct answer was necessary, but not sufficient. I doubt it was different with you.

 

Bill

[regentrude]

But it is for SM and AoPS. I don't worry.

Yes, Beast and AoPS also ask for explanations, but only for a very small fraction of the total problems.

 

Excuse me, but I disagree. The majority of AoPS problems are of such complexity that it is impossible to just state a correct answer without elaborating on the process that leads to this answer - in other words, without showing work.

The showing of the detailed work IS the explanation; you do not have to write a paragraph or essay, because mathematicians have invented a symbolic language that makes this unnecessary.

 

But unless you have genius children, your kids will fill pages and pages with explanations to how they obtain the answer, because the problems are such that it is impossible to arrive at the answers otherwise.

 

[Farrar]

The AoPS Solutions don't involve essay writing. They require a clear representation of the steps and logic used in problem solving. Occasionally here is more need for verbal or written explanations (we alternate between doing things orally and on paper), but never (in my estimation) does the AoPS approach ever say just getting the "right answer" without understanding the math is sufficient.

 

I never treated Primary Mathematics that way either. A correct answer was necessary, but not sufficient. I doubt it was different with you.

 

Bill

 

I don't have AoPS pre-A, but I worked through a lot of Intro to Algebra for myself. The solutions are wordy. One thing ds has been learning to be good at is expressing the answer in math terms and showing all the work. But he doesn't additionally write out what all the work means or why you do a certain step next the way that the AoPS solution guide does. I think that's one of the problems with that jump between elementary math and algebra. For first graders, all they're doing is simple addition and so forth, so to justify their answer they have to say more. But for algebra students, it doesn't have to become a writing chore. You just slowly go through all the steps and write each one out line by line. No one is asking them to additionally draw out why when they subtracted 2 from both sides of an equation it led to the result it did.

[KSinNS]

Interesting article. I didn't really take it that kids shouldn't understand what they are doing, just that looking for an essay explanation may not be the best way to assess that. Some kids don't write that well. Many kids, especially those with special needs, will have symbolic/mathematical skills ahead of their verbal skills, so to demand an explanation of a strength skill with a much weaker skill  is asking to frustrate the poor soul. I have certainly seen this happen in schools, where a child completely understands a concept but cannot express it in writing, simply because their writing or language is so delayed relative to their math skills. My own kids, I expect, would struggle with this. Because we work one-on-one it's easy for me to say "interesting answer, how did you get there?" and make sure they understand, but asking them to write it would likely not reflect their understanding at all simply because they could not write the explanation clearly. 

 

Obviously, rote memory rather than understanding is a perennial problem, but again, I think their point is that the memorizers will often just memorize the canned written answer, so again, the written paragraph isn't really assessing what we want it to assess. 

 

The thing that gets me though, is that math is a symbolic language in and of itself. You shouldn't need too many words to explain your work. Some here and there, and more as you get to the proof stage and beyond, but for less advanced students, probably not too often. JMHO. 

[Spy Car]

At no point would I ever say we should stop trying to get kids thinking about the reasons behind the math. I would never say that.

 

However, I think there's zero way to fix this problem without fixing the testing culture. The way this problem came about was at least in part from testing culture - the tests encouraged memorizing the algorithms and drilling to pass the test. So the CC people said, gee, how can we fix this? Good impetus. But their answer was more tests. Because they know that without that, nothing will be taught at all. Except now schools have just found different ways to drill kids on this - by having them practice these canned writing responses over and over and practicing their drawing various arrays and diagrams over and over. It's still memorization, they've just made a more complex algorithm - if they ask that, draw this, if they ask that, write this and plug in the numbers here and here. Really, the only way to fix it is to spend more time with kids and understanding and wait on the testing. To train better teachers who are real professionals. To train them in assessing kids constantly and more informally in the classroom - using manipulatives, listening to them, working in small groups, etc. The places that are consistently producing kids who do have a deeper understanding of math, by and large, do not test them over and over throughout the elementary years in this manner. And I see a connection there. But there's no trust in the teachers and no political cover in lessening the testing - it makes it look like we're "lowering standards."

 

We are all colored by our experiences, at least I am. We were in an admittedly fortunate situation in regard to schools and teachers, but drilling kids to pass tests was not the response of the teachers my son had in school. They taught for depth, and did a great job of it in my estimation. And the scores on the CC Smarter Balance tests in my son's outgoing year were just released, and they were phenomenal. 

 

I'm all for improving math education across the board. Better teaches, more professionalism, better ways to discuss mathematical axioms in use, all for it. 100%.

 

Giving up and going backwards? No. 

 

I sure we agree.

 

Bill

[Roadrunner]

Excuse me, but I disagree. The majority of AoPS problems are of such complexity that it is impossible to just state a correct answer without elaborating on the process that leads to this answer - in other words, without showing work.

The showing of the detailed work IS the explanation; you do not have to write a paragraph or essay, because mathematicians have invented a symbolic language that makes this unnecessary.

 

But unless you have genius children, your kids will fill pages and pages with explanations to how they obtain the answer, because the problems are such that it is impossible to arrive at the answers otherwise.

That is my point exactly. There is no need for paragraph writin because solving math problems demonstrate the understanding. Our local PS requires sentence after sentence on everything. It's no longer just a showing solutions but actual writing.

[Farrar]

We are all colored by our experiences, at least I am. We were in an admittedly fortunate situation in regard to schools and teachers, but drilling kids to pass tests was not the response of the teachers my son had in school. They taught for depth, and did a great job of it in my estimation. And the scores on the CC Smarter Balance tests in my son's outgoing year were just released, and they were phenomenal. 

 

I'm all for improving math education across the board. Better teaches, more professionalism, better ways to discuss mathematical axioms in use, all for it. 100%.

 

Giving up and going backwards? No. 

 

I sure we agree.

 

Bill

 

That's not been my experience as a teacher and not what I see with kids around here or kids elsewhere in my life. I think we do disagree if the question is keep the tests like this or not. I don't think it's going backwards to ditch these tests. I think that would be a step forward. I don't know how else you take a decent step forward honestly. Every band aid test fix seems to backfire. There has to be a re-investment in spending time with kids getting them to understanding and to taking whatever time it takes to get kids to understand the concepts. I don't see that happening right now without a big change.

[Roadrunner]

I don't have AoPS pre-A, but I worked through a lot of Intro to Algebra for myself. The solutions are wordy. One thing ds has been learning to be good at is expressing the answer in math terms and showing all the work. But he doesn't additionally write out what all the work means or why you do a certain step next the way that the AoPS solution guide does. I think that's one of the problems with that jump between elementary math and algebra. For first graders, all they're doing is simple addition and so forth, so to justify their answer they have to say more. But for algebra students, it doesn't have to become a writing chore. You just slowly go through all the steps and write each one out line by line. No one is asking them to additionally draw out why when they subtracted 2 from both sides of an equation it led to the result it did.

I should have read your post before writing mine. My English doesn't cooperate, but this is precisely what I was attempting to say.

[Spy Car]

Interesting article. I didn't really take it that kids shouldn't understand what they are doing, just that looking for an essay explanation may not be the best way to assess that. Some kids don't write that well. Many kids, especially those with special needs, will have symbolic/mathematical skills ahead of their verbal skills, so to demand an explanation of a strength skill with a much weaker skill  is asking to frustrate the poor soul. I have certainly seen this happen in schools, where a child completely understands a concept but cannot express it in writing, simply because their writing or language is so delayed relative to their math skills. My own kids, I expect, would struggle with this. Because we work one-on-one it's easy for me to say "interesting answer, how did you get there?" and make sure they understand, but asking them to write it would likely not reflect their understanding at all simply because they could not write the explanation clearly. 

 

Obviously, rote memory rather than understanding is a perennial problem, but again, I think their point is that the memorizers will often just memorize the canned written answer, so again, the written paragraph isn't really assessing what we want it to assess. 

 

The thing that gets me though, is that math is a symbolic language in and of itself. You shouldn't need too many words to explain your work. Some here and there, and more as you get to the proof stage and beyond, but for less advanced students, probably not too often. JMHO. 

 

I believe if students—including those who are not advanced—are not cultivated with the means to express mathematical reasoning in both common English and in symbolic language, then those students are not getting an adequate education. 

 

Bill

[regentrude]

That is my point exactly. There is no need for paragraph writin because solving math problems demonstrate the understanding. Our local PS requires sentence after sentence on everything. It's no longer just a showing solutions but actual writing.

 

The bolded is true only if solving the problem actually requires understanding. That is the generally the case with AoPS - but with many other curricula, it is entirely possible to arrive at the correct answer by performing a rote algorithm without any conceptual understanding. I see students do that every single week; they can manipulate their equations, but do not understand what it is exactly they are doing.

 

ETA: Best example: division of fractions. Most students remember that they have to flip something somewhere. If they are lucky, they remember correctly what to flip and what to do with the flipped thing. But they have no clue WHY they have to do that. They can perform the drilled operation, but they do not comprehend what they do and are evaluate whether their results make sense or correct anything they might have misremembered. They are like trained circus monkeys, performing tricks without thinking. That does not demonstrate mathematical understanding.

[Farrar]

The bolded is true only if solving the problem actually requires understanding. That is the generally the case with AoPS - but with many other curricula, it is entirely possible to arrive at the correct answer by performing a rote algorithm without any conceptual understanding. I see students do that every single week; they can manipulate their equations, but do not understand what it is exactly they are doing.

 

ETA: Best example: division of fractions. Most students remember that they have to flip something somewhere. If they are lucky, they remember correctly what to flip and what to do with the flipped thing. But they have no clue WHY they have to do that. They can perform the drilled operation, but they do not comprehend what they do and are evaluate whether their results make sense or correct anything they might have misremembered. They are like trained circus monkeys, performing tricks without thinking. That does not demonstrate mathematical understanding.

 

But then I would say it would be better to move toward more problems that require real understanding like those from AoPS or Beast than to move toward more writing in math class, especially for younger kids.

 

I think most of us in this thread are against the memorization of algorithms to do math. And probably we've mostly all read that section in Liping Ma's book about the division with fractions. The OP's article seemed to maybe underplay that pitfall. But then the question is what's the best way to do that? Is there any way to change the way we drill for standardized testing and drill for understanding instead? Does it just take a different approach altogether? The problem is especially acute for elementary school. I think by the time students begin to get to middle school and certainly in high school, then there should be some writing involved, but I don't know that I think there should be much as a requirement in elementary school. So how do you use oral language or other means to check and make it meaningful in a big class?

[regentrude]

But then I would say it would be better to move toward more problems that require real understanding like those from AoPS or Beast than to move toward more writing in math class, especially for younger kids.

 

I completely agree - but creating these problems takes expertise, and working with these problems requires expertise on part of the teacher. (I think it would be very interesting if math teachers were asked to solve AoPS problems... my guess is the result would be disturbing)

 

 

 Is there any way to change the way we drill for standardized testing and drill for understanding instead? Does it just take a different approach altogether? The problem is especially acute for elementary school. I think by the time students begin to get to middle school and certainly in high school, then there should be some writing involved, but I don't know that I think there should be much as a requirement in elementary school. So how do you use oral language or other means to check and make it meaningful in a big class?

 

I do not see that there needs to be essay writing involved in middle or high school math courses at all.

Students can be asked to demonstrate understanding by writing their solution in mathematical symbolic short hand. In my school back home (and that is still the case), the geometrical proof is taught in 6th grade. Students are taught to prove relationships, to use "if" and "if-and-only-if" correctly, i.e. apply precise language to distinguish between sufficient and necessary conditions, to use the double arrow to indicate "therefore it follows that", to construct a logical chain of argument without a single complete sentence. We derived immense satisfaction from concluding our proofs with the three letters q.e.d. :-)

None of this requires any essay writing skills.

One problem with math instruction in this country is that such skills that elsewhere are deemed appropriate for middle schoolers are postponed until 10th grade. In the absence of precise mathematical notation, there are only sentences and paragraphs.

 

On an elementary level, understanding can be checked by incorporating good quality word problems. Students can take turns verbally explaining their solutions to the class or by working in groups talking about problems (which, contrary to some pedagogy schools does NOT serve as a substitute for instruction by a competent teacher!). It is, IMO, unnecessary for the teacher to require that every single student presents a narration of every single problem. The entire math class could be structured to emphasize that math is about thinking, not merely about manipulating numbers or drilling facts.

This, however, would require competent math teachers who have a thorough conceptual understanding of math and actually like the subject. Having encountered elementary teachers who were incapable of comprehending the meaning of percentages, this seems to me the biggest obstacle to improvements. As long as students who can't hack their major drop down into the educational track, and as long as there are math teachers who openly admit to be afraid of math (a problem especially in elementary education), schools are doomed to mediocrity.

[JanetC]

I don't disagree that learning to understand the concepts behind math procedures is important. The question is really more about when and how students demonstrate that understanding.

 

Little kids are very concrete thinkers:

 

https://www.yahoo.com/parenting/6-year-olds-hilarious-math-answer-shows-hes-180019539.html

 

It's just an age level thing.

 

Kids normally have writing skills that lag (sometimes significantly) behind the grade level of the sentences they speak or understand verbally. And although I haven't seen a study, I'm guessing that there is a similar lag between the math a kid can do and the math a kid can explain or teach. It makes a lot of sense to me when I think about what I have observed in homeschooling my kids and in helping a few others with their homework.

 

So, faced with the demand from standards writers that kids produce explanations, math textbook writers churned out the algorithm for writing explanations featured in the OP's article and teachers dutifully taught kids to create these things by rote. Just because I agree that this type of work is of dubious educational benefit does not mean that I don't think understanding concepts is important.

 

I have my doubts as to whether mathematical understanding can be demonstrated on a standardized test, particularly on a computer where it's hard to enter equations and draw diagrams. I also question whether standards designers are in touch with the cognitive development of children, and whether test designers can really define and evaluate "correct" explanations versus "incorrect" ones.

[regentrude]

I have my doubts as to whether mathematical understanding can be demonstrated on a standardized test, particularly on a computer where it's hard to enter equations and draw diagrams.

 

It can be demonstrated if the test question is well designed: it simply has to be designed in such a way that it is impossible to solve without conceptual understanding.

ETA: A well designed multiple choice question would even be able to pinpoint exactly which misconception the student has about the problem - by making the incorrect answer choices something that would be obtained if they made the most typical sorts of mistakes. This would also distinguish between simple computational errors and fundamental misunderstandings.

But that requires expertise and insight from the test developers.

 

I also question whether standards designers are in touch with the cognitive development of children, and whether test designers can really define and evaluate "correct" explanations versus "incorrect" ones.

 

To the latter: the test designers may possibly be able to distinguish correct vs incorrect explanations. Many teachers probably won't.

[Farrar]

Regentrude, I think we basically agree. I think the solution here is to invest time and money into better teachers at all levels.

 

However, I'm not sure why you perhaps think the test designers are any better than the teachers. There have been articles about the test scorers and writers that make them sound massively less qualified for the positions they hold than most teachers are.

[regentrude]

However, I'm not sure why you perhaps think the test designers are any better than the teachers. There have been articles about the test scorers and writers that make them sound massively less qualified for the positions they hold than most teachers are.

 

That's why I said "may possibly". I have no idea who designs which tests - some test designers may know less than the teachers do. I do know, however, that some tests are designed in close cooperation with, or entirely by, people with subject expertise; I know physics professors who are involved in constructing the AP and SAT subject tests in physics - so in some instances at least, the test designers have more subject expertise than the teachers.

But I also know that many math teachers do not understand math.

[Farrar]

That's why I said "may possibly". I have no idea who designs which tests - some test designers may know less than the teachers do. I do know, however, that some tests are designed in close cooperation with, or entirely by, people with subject expertise; I know physics professors who are involved in constructing the AP and SAT subject tests in physics - so in some instances at least, the test designers have more subject expertise than the teachers.

But I also know that many math teachers do not understand math.

 

Yes, I have (or had... I haven't taught an AP course or been in a school doing that in a long time) a lot of trust in the creation and scoring of AP tests. I have a lot less in most of these newer tests. In general, AP tests are examples to me of good standardized tests. Clear and specific body of knowledge for the test. Clear method of questions. Clear body of texts from which to study. Yet still rigorous. I wish all standardized tests could meet that sort of simple criteria.

[Roadrunner]

I remember solving in first grade problems like this:

 

X + 5 = 9

x = 9-5

X = 4

 

I didn't have to invent a fairy tale. 

 

This is a very basic example. 

My younger son is working through SM. No writing requirement there either, but I know he is being taught conceptual understanding.

 

Nobody is in favor of mindless drilling. Of course that would be a waste of time. I just feel for kids, often boys, who really struggle with writing aspect. There are better ways. 

[Spy Car]

If X +5 = 9, and I want to isolate the X, I can subtract 5 from both sides of the equation (because if you do the same thing to both sides of an equation it stays balanced).

 

So, X +5 -5 =9 -5 =

 

X = 4

 

Not a fairy tale. Not ultra wordy. 

 

Bill 

 

ETA i might give bonus points for:

 

So, X +5 -5 =9 -5 =

 

X + (5 -5) = 4 + (5 -5) =

 

X + 0 = 4 + 0 =

 

X = 4

 

 

 

 

 

[regentrude]

If X +5 = 9, and I want to isolate the X, I can subtract 5 from both sides of the equation (because if you do the same thing to both sides of an equation it stays balanced).

So, X +5 -5 =9 -5 =

X = 4

Not a fairy tale. Not ultra wordy.

 

Yes, but this is not 1st grade level thinking - which Roadrunner was referring to.

In 1st grade, the concept of a balanced equation has almost certainly not yet been introduced. This would be supposed to be a "puzzle" problem where the student has to apply his knowledge of math fact families.

A 1st graders logic would go more along the lines:

I want the number for which five more is 9.

I know that 4+5 =9.

So the number I am looking for is 4.

[Deee]

In my experience, MEP does this stuff really well. It teaches concepts, not procedures and it starts very early. But it does require a commitment to spending up to an hour on maths everyday, teachers who can really teach rather than just set work, class participation and quite a lot of preparation. If you don't like maths, this is pretty unpalatable.

 

I agree with Regentrude that mathematical symbols are preferable to word-based explanations. Scientists, statisticians and mathematicians use symbols. They are the tools of the trade, they feel more "grown-up," and writing-phobic kids love them!

 

Bill, Australia has been using the "see if it sounds right" method of grammar instruction since the early 1980s. While we are inclined to blame our dodgy grammar on American television, I think our education system may have something to do with it......

D

[Farrar]

The balanced equation idea is taught at the first grade level in MEP, but through workbook pages and manipulatives where kids see little drawings of balances and little mystery sacks. I guess my hope would be that a first grader would be able to say something like what Regentrude wrote there as an explanation. Or be able to show that they get it by moving manipulatives around on a balance or on a pretend balance. I don't think they have the skills to do what Bill showed yet - and any effort to get them to be able to is just teaching another algorithm, one that will take a lot of time that could be better spent in other ways. But my understanding is that a lot of the CC materials would find Bill's explanation insufficient (which, yeah, is maybe a problem with the materials and not with the idea behind them, but still). Kids would need to be able to write a sentence along the lines of what Regentrude said or be able to draw a series of pictures of balances and items. That's what's just mind boggling to me - I really don't think kids can do that - or, rather, I think you can monkey train them to do it but I don't think it actually represents deeper thinking.

[regentrude]

 Kids would need to be able to write a sentence along the lines of what Regentrude said or be able to draw a series of pictures of balances and items. That's what's just mind boggling to me - I really don't think kids can do that - or, rather, I think you can monkey train them to do it but I don't think it actually represents deeper thinking.

 

No, I don't think it does either - because to come up with the correct answer to the problem X+5=9 already requires the student to understand the concepts of what addition means and how a variable functions. If they figure out X=4 "because 4+5=9", there is really nothing else left to say about this problem.

 

It seems that the educators who come up with those nutty requirements do not realize which things are trivial and require no explanation, which problems have the requirement for conceptual understanding automatically built in, and which problems benefit from additional explanations. It reminds me of the "zero tolerance" policies: somebody makes blanket rule and then circumstances are no longer evaluated on an individual basis - because discerning and using common sense is deemed too difficult. Sigh.