Many sets of state and national mathematics standards have come and gone in the past two decades. The Common Core State Mathematics Standards (CCSMS), which were released in June of 2010, have been adopted by almost all states and will be phased in across the nation in 2014. Will this be another forgettable standards document like the overwhelming majority of the others?

Perhaps. But unlike the others, it will be a travesty if this one is forgotten. The main difference between these standards and most of the others is that the CCSMS are mathematically very sound overall. They could serve -- at long last-- as the foundation for creating proper school mathematics textbooks and dramatically better teacher preparation.

Wow! This should be three separate articles; one discussion on CCSMS versus TCM (as he calls it), one on basic math concepts and examples, and one on his ideas of "mathematical engineering". In it's current form, it's too much.

Its optimistic tone (about CCSMS) is not entirely backed up in the article.

"Unfortunately, textbook developers have yet to accept that the CCSMS are radically different from their predecessors, Most (and possibly all) textbook developers are only slightly revising their texts before declaring them aligned with the CCSMS."

and

"The advent of the CCSMS sends out the signal, for the first time from within the education community, that ESM has no place in the school curriculum. TSM is incompatible with the CCSMS, and now colleges and universities are duty-bound to provide future mathematics teachers with a replacement of TSM. Would that those institutions were aware of their duties!"

The overall tone of the article is wishful thinking. I see no indication that his ideas have any chance of success. We're talking mostly about problems in K-6 math. A better option would be to take math away from general teachers and require math certified teachers in K-6. To wait for some systemic "mathematical engineering" solution to filter through the education community requires even more wishful thinking.

Something can be done right now. Take K-6 math away from general teachers. Define specific grade-level expectations in content, skills, and performance. Do not pass kids along without achieving those standards. Nobody argues with balance, so call their bluff and require schools to define it and explain how they will get the job done. Define the 6th grade (or whenever) test that determines if kids will be allowed into pre-algebra in 7th grade. Show parents how their kids are progressing on this path for each grade.

Don't consolidate skills with rubrics. Don't call fractions number sense. Don't talk about general problem solving or critical thinking. Give specific skill tests and provide raw percent correct scores. Put the results online so that parents can see the results immediately and know how well their child is progressing towards the major tracking branch of pre-algebra in 7th grade.

Given Wu's take of the problems in math education, I'll put my bet on technology and/or bottom up (choice) solutions any day over systemic, top down solutions. There are simple things that can be done right now by any school system. While educators and education wonks are trying to figure out some grand top-down solution, give parents the means to find their own individual solution.

San Francisco is one of the first distrcts to implement the new standards, but this doesn't sound very promising - no practicing, lots of "real life" group work, and trying to create 8-year old "engineers".

Instead of teaching students to add fractions (cross multiply the numerators and denominators) and then solve 15 problems to get the right answer, they'll be asked to apply the concept to a real-life situation. Students would be encouraged to bounce ideas off each other about how to solve the problem.

Bergeson dream is to see 10-year-olds working out how to build a bridge in place of the current method of having a class of kids sitting in rows doing two-digit multiplication problems until their hand hurts.

"I think it's going to be a revolution in mathematics across the country," she said. "They can be a little engineer in the third grade."

I read the sfgate article - yuck! The worst quote: "don't ank the teacher until you ask everyone else in the group."

And my mother (who teaches K-1) already thinks California's math standards for kindergarten are unreasonable. I can't imagine what she'd think of adding quasi-algebra.

"Bergeson dream is to see 10-year-olds working out how to build a bridge in place of the current method of having a class of kids sitting in rows doing two-digit multiplication problems until their hand hurts. 'I think it's going to be a revolution in mathematics across the country,' she said. 'They can be a little engineer in the third grade.'"

That's my favorite. Completely clueless! They interpret the stadards in their own image and STILL mastery of basic skills won't be achieved.

Wu's wishful thinking or dreams of mathematical engineering will have no effect.

The ed school adage of students must struggle first is alive and well. There's a fine line between asking students questions to get them to think, and having them struggle uselessly. And why is getting direct information from a fellow student better than getting the same information from a teacher?

Wu's article is spot on. His description of what happens when kids try to learn addition of fractions by memorizing an algorithm is perfect. The problem, for those of us in computer science, is that students who have memorized their way through math have no hope of creating computational solutions to even the simplest problems. All they can do is continue to wait for the solutions to be spoon-fed, so they can memorize. Eventually, they will need to be able to think for themselves or else they have no hope of succeeding in the field.

And that is precisely why we don't want the student to ask the teacher until they have worked for a while. Anyone who has learned math effectively knows that you can't learn to solve the problems without struggling a bit - if you immediately rush to check the solutions in the back before you have figured out an answer, you will never learn the material. Struggle is important to learning. Sorry, but that's the truth.

Bonnie said: --that is precisely why we don't want the student to ask the teacher until they have worked for a while.

But that has nothing to do with asking another student for help or talking about their problem together.

Do this thought experiment: the elementary grade class is given a problem that requires "thinking". Most have no idea how to solve it. For them, they are being asked to cross the grand canyon by jumping. Maybe a few will attempt it once--"struggling" to come up with some idea, and fall flat. No idea what to do next. Most, having done that once, thank-you-very-much, won't even try to do that again. So they won't be struggling at all. They'll be refusing. Some will then ask another student what to do--if one asks another student who doesn't know, both have gotten nowhere and wasted their class time. If one asks a student who does know, all they will find out is the answer "here's how you do it". That doesn't help anyone to think at all. Even if that student tells you why the answer is true, that doesn't help because recognition isn't the same as being able to produce the right reasoning.

What a student needs to go from "struggle to jump the grand canyon" to "build oneself a bridge" is a teacher who can help them do that.

The best way to do this is as follows:a student starts with no idea what to do. Teacher, *knowing what they DO know*, points them to a simpler but related problem that they can solve, and asks them to think about that and solve it. (If student can't do that, teacher goes back more, to where the bedrock is solid, rinse and repeat.) Then student solves that, and teacher says good and asks leading questions that point to the new thing they don't understand. Teacher breaks down the new thing into small bites, each of which is something they can learn to recognize that they already can solve, or that they need to learn to solve.

Students talking to each other cannot break down tasks appropriately into smaller solveable subtasks, because by defn, students don't know what they don't know, and often don't know what they do know. Teachers must model how that's done for the student.

Now, the above isn't going to happen in most classrooms either--but the blind leading the blind is an utter waste of time and prevents the above from ever happening, since teacher time on task is what's needed to do the above.

One thing I find frustrating in math ed debates is the apparent fallacy of the excluded middle. It's as if when someone says "students won't learn from just rote memorization of equations", the assumption is the person solely believes in inquiry based learning, and if someone says "students learn nothing from inquiry based learning, because no one is guiding their inquiry" that they are assumed to be in favor of rote learning of equations.

Wu gets attacked constantly because of this fallacy. Someone hears him say something that puts Saxon in a bad light and they assume he thinks Everyday Math is a good text; someone hears him say something that puts Everyday Math in a bad light and they assume he loves Saxon.

The "struggling" argument is the same. If you're against "struggling", you're often assumed to be anti "learning to think for themselves". If you're "for struggling", you're often assumed to be direct instruction of concepts.

How about I phrase it as: students need to struggle but NOT IN VAIN. They must be given the tools so struggling is very nearly always successful.

I wonder if Bonnie's experience of seeing students who have "memorized their way through math" is a selection bias born from constructivism.

My guess: since no one can actually teach the understanding of WHY K-8 or K-12 students are being asked to do what they do in math, none of them have any understanding or clue. None of them know how why the math works, but some of them were math-natural enough to grok the procedures or memorize them and at least get the right answers.

Only these students who breezed through K-12 by quickly grokking the procedures, regardless of actual understanding, even attempt to go into engineering, CS, or science fields in college. The ones who didn't memorize their way through gave up on those interests years ago, because math was horribly senseless AND difficult. So yes, the selection bias in math-requiring college majors is for students who memorized. But that's because no other students could even get far enough to be in these majors.

"Struggle is important to learning. Sorry, but that's the truth."

"the truth"?

I don't buy this one bit. It's a nice idea to structure learning so that you create "light bulb" moments, but that's neither necessary or sufficient. However, when you group kids together in an "active learning" environment, maybe one or two get the desired light bulb moment. They directly teach it to the other kids who haven't figured it out. What's the benefit of being directly taught by kids who have no training as teachers?

Is there something special about struggle if you don't end up with the light bulb moment? It's good to learn to work hard and be persistent, but that's not struggle. You can't set up everything to learn as a series of struggles. If you don't struggle, then you haven't really learned the material?

There will be plenty of struggle with 25 question homework sets that gradually move kids from simple questions to more difficult ones, but the goal is hard work, not struggle. If you do those problems in sequence, then there should be plenty of light bulb moments with minimal struggle.

Besides, this isn't what Wu is talking about. This struggle idea came from the sfgate article that represented what Wu is afraid of.

"Unfortunately, textbook developers have yet to accept that the CCSMS are radically different from their predecessors, Most (and possibly all) textbook developers are only slightly revising their texts before declaring them aligned with the CCSMS."

This is true for schools. They will change CCSMS to reflect their own simplistic ideas of learning. In the process, the mathematical rigor that Wu talks about will be lost. When you spend all of your class time tackling few problems in active learning groups, less content will be covered, few will really benefit from light bulb moments, and mastery won't be achieved because they really don't like the "struggle" of homework sets.

This is a key point. Current K-6 educational pedagogy talks a lot about understanding and critical thinking, but their goal is to try to teach this by simulating the process in class using heterogeneous groups with the teachers as the guide-on-the-side. Do they try to accomplish this goal any other way, such as with individual homework? No. Do they try to ensure that kids have mastered content and skills? No.

They try to claim the pedagogical high ground of understanding and critical thinking, but what they do doesn't get the job done. If you look closely, you will see that what they care about is only what goes on in the classroom. The goal is not understanding and critical thinking. The goal is a happy learning environment with mixed ability groups.

Wu's pedantic math discussion does not deal with the reality shown in the sfgate article. We're talking about schools that jabber on and on about understanding, but allow kids to get to fifth grade without knowing the times table.

"... is that students who have memorized their way through math have no hope of creating computational solutions to even the simplest problems."

How do you memorize your way through math? How is it possible to pass exams just with memorization? As soon as you are confronted with a slightly different problem, how can memorization help? You have to define the issue better than this.

You can memorize a particular algorithm for multiplying numbers, but you can use it to solve all sorts of problems. You can understand how a multiplication algorithm works, but will that help you solve more problems that require multiplication? What level of understanding are you talking about; place value, algebraic, number base system, linear space?

You can memorize how to write a binary search algorithm, but you can apply it to solve all sorts of problems. If you are talking about writing a more elaborate search technique, then you need a different kind of understanding.

It's been at least a couple of decades that K-12 have been applying their non-memorization pedagogy. What CS or math students are you talking about? However, since you don't define what memorization means, and you don't define your problem, I can't possibly guess at what's causing it.

Here's one of my problems with Wu. Like so many university-level mathematicians, he seems completely out of touch with the perspectives of young children/mathematical novices. How readily will most 3rd graders absorb statements like "The segment from 0 to 5/6 is 5 copies of the segment from 0 to 1/6"?

Also, Wu says he favors precise definitions over intuition, but the word "copy" seems ill-defined here. Copies aren't necessarily concatenated (or, to use Wu's term, "combined"). And what if a child assumes that part of what is "copied" is the line segment's position (from 0 to 1/6), and not just its length?

Wu's proposal for how to demonstrate to 6th and 7th graders why multiplying two negative numbers yields a positive number using the distributive law (see p. 7) strikes me as even more inaccesible to most of his intended audience. He's right to argue, as he does later on, that this demonstration is more accessible than a general proof would be (and I like the demonstration very much and will try it out on some kids I know), but still... At the very least, Wu should await the results of some sort of pilot study before recommending this to teachers of 6th and 7th graders.

On a different note, I whole-heartedly agree with everything SteveH writes about the Common Core Standards and the problems of wishful thinking, top-down reform, and lack of specifics. In the context of a dominant paradigm, vague standards like these will simply be used by those in power to further augment their "reforms." We already see this happening with reading standards, as I've noted here: http://oilf.blogspot.com/2011/10/reform-reading-in-montgomery-county-md.html and here: http://blog.coreknowledge.org/2011/05/12/common-core-standards-a-cautionary-tale

--Like so many university-level mathematicians, he seems completely out of touch with the perspectives of young children/mathematical novices. How readily will most 3rd graders absorb statements like "The segment from 0 to 5/6 is 5 copies of the segment from 0 to 1/6"?

Katharine, this was not my experience teaching elementary teachers Wu's fractions material. They didn't think Wu was out of touch with what their kids could do. Re: 3rd graders and the above: the issue isn't trying to get 3rd graders to understand the *statement* at all--the way it's written in words is difficult, sure. But that's not the math. And the words used for an AFT paper aren't meant to be the words used for a child.

And Wu isn't suggesting you *start* with the above definition either. You get there, over time. The 3rd grader can readily understand 5/6 from pattern blocks and infix cubes. You start with those manipulatives. With practice, then you move them to the number line, which should not be difficult because they've seen that for whole numbers. So now you tell them about candy bars, and breaking it into 6ths, and then work up to the 5 pieces, each of size 1/6th. Children don't have a problem understanding segments and "segments of equal length" if you are clear from the beginning. For 3rd graders, you don't need the formal definitions, but you DO need the teacher to know the formal definitions so they aren't imparting false information and misleading them at that age. Then, as they progress to grades 5 and 6, you can expect the students themselves to know the precise defn.

--Kathrine said Also, Wu says he favors precise definitions over intuition, but the word "copy" seems ill-defined here. Copies aren't necessarily concatenated (or, to use Wu's term, "combined"). And what if a child assumes that part of what is "copied" is the line segment's position (from 0 to 1/6), and not just its length?

Wu's book is quite precise. After a precise defn is used, he lightens up, and moves back to words like "copies". Again, his book is not for *children*, but for the adult teachers. So they need to learn the proper defn (which is too precise and wordy for an AFT article), and then they need to learn how to adapt that to the classroom.

Similarly, the negative times negative issue is something where your feeling about it comes without doing the prior week of fractions and near week of rational numbers. Teachers would be incredulous on day one with his suggestion, sure, but many don't feel that way on day 9, because they see how they'd have led their students there. Incidentally, the Japanese text for grade 7 teaches it that way, so it certainly is possible to be done at that point with proper background.

I would suggest a look at his book. I think you would find it addresses these issues.

Grant Wiggins is also the author of Understanding by Design, and his reasons for trashing CCSS appear to him to 1) be in conflict with his top-down "just in time" philosophy of learning and 2) implementation of the standards may cut down on his book sales and speaking engagements. Just a guess.

The all seeing eye, symbolizing Lucifer, with wings symbolizing the phoenix rising out of the ashes, a masonic concept that symbolizes the emergence of the New World Order out of chaos. The lightning bolt is an occult symbol as well and could be seen as a symbol for Lucifer falling from Heaven: “And he said unto them, I beheld Satan as lightning fall from heaven.” Luke 10:18. Notice the masonic checkerboard pattern. This pattern is the one used in the floor tiles of masonic lodges and is a symbolic representation of good and evil, or knowledge of, symbolized by other esoteric symbols like the hexagram, ying/yang, moon and star, and the crossing of the swords near the bottom of the picture.

## 19 comments:

Wow! This should be three separate articles; one discussion on CCSMS versus TCM (as he calls it), one on basic math concepts and examples, and one on his ideas of "mathematical engineering". In it's current form, it's too much.

Its optimistic tone (about CCSMS) is not entirely backed up in the article.

"Unfortunately, textbook developers have yet to accept that the CCSMS are radically different from their predecessors, Most (and possibly all) textbook developers are only slightly revising their texts before declaring them aligned with the CCSMS."

and

"The advent of the CCSMS sends out the signal, for the first time from within the education community, that ESM has no place in the school curriculum. TSM is incompatible with the CCSMS, and now colleges and universities are duty-bound to provide future mathematics teachers with a replacement of TSM. Would that those institutions were aware of their duties!"

The overall tone of the article is wishful thinking. I see no indication that his ideas have any chance of success. We're talking mostly about problems in K-6 math. A better option would be to take math away from general teachers and require math certified teachers in K-6. To wait for some systemic "mathematical engineering" solution to filter through the education community requires even more wishful thinking.

Something can be done right now. Take K-6 math away from general teachers. Define specific grade-level expectations in content, skills, and performance. Do not pass kids along without achieving those standards. Nobody argues with balance, so call their bluff and require schools to define it and explain how they will get the job done. Define the 6th grade (or whenever) test that determines if kids will be allowed into pre-algebra in 7th grade. Show parents how their kids are progressing on this path for each grade.

Don't consolidate skills with rubrics. Don't call fractions number sense. Don't talk about general problem solving or critical thinking. Give specific skill tests and provide raw percent correct scores. Put the results online so that parents can see the results immediately and know how well their child is progressing towards the major tracking branch of pre-algebra in 7th grade.

Given Wu's take of the problems in math education, I'll put my bet on technology and/or bottom up (choice) solutions any day over systemic, top down solutions. There are simple things that can be done right now by any school system. While educators and education wonks are trying to figure out some grand top-down solution, give parents the means to find their own individual solution.

San Francisco is one of the first distrcts to implement the new standards, but this doesn't sound very promising - no practicing, lots of "real life" group work, and trying to create 8-year old "engineers".

Instead of teaching students to add fractions (cross multiply the numerators and denominators) and then solve 15 problems to get the right answer, they'll be asked to apply the concept to a real-life situation. Students would be encouraged to bounce ideas off each other about how to solve the problem.

Bergeson dream is to see 10-year-olds working out how to build a bridge in place of the current method of having a class of kids sitting in rows doing two-digit multiplication problems until their hand hurts.

"I think it's going to be a revolution in mathematics across the country," she said. "They can be a little engineer in the third grade."

Read more: http://www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2011/09/26/BA121L7K1D.DTL#ixzz1ZY8qCMJH

I read the sfgate article - yuck! The worst quote: "don't ank the teacher until you ask everyone else in the group."

And my mother (who teaches K-1) already thinks California's math standards for kindergarten are unreasonable. I can't imagine what she'd think of adding quasi-algebra.

"Bergeson dream is to see 10-year-olds working out how to build a bridge in place of the current method of having a class of kids sitting in rows doing two-digit multiplication problems until their hand hurts. 'I think it's going to be a revolution in mathematics across the country,' she said. 'They can be a little engineer in the third grade.'"

That's my favorite. Completely clueless! They interpret the stadards in their own image and STILL mastery of basic skills won't be achieved.

Wu's wishful thinking or dreams of mathematical engineering will have no effect.

Note the misuse of engineering at both ends.

The ed school adage of students must struggle first is alive and well. There's a fine line between asking students questions to get them to think, and having them struggle uselessly. And why is getting direct information from a fellow student better than getting the same information from a teacher?

Wu's article is spot on. His description of what happens when kids try to learn addition of fractions by memorizing an algorithm is perfect. The problem, for those of us in computer science, is that students who have memorized their way through math have no hope of creating computational solutions to even the simplest problems. All they can do is continue to wait for the solutions to be spoon-fed, so they can memorize. Eventually, they will need to be able to think for themselves or else they have no hope of succeeding in the field.

And that is precisely why we don't want the student to ask the teacher until they have worked for a while. Anyone who has learned math effectively knows that you can't learn to solve the problems without struggling a bit - if you immediately rush to check the solutions in the back before you have figured out an answer, you will never learn the material. Struggle is important to learning. Sorry, but that's the truth.

No one's disagreeing with you. You use the term "struggling a bit". That's correct. But struggling a lot is not a good thing.

Bonnie said:

--that is precisely why we don't want the student to ask the teacher until they have worked for a while.

But that has nothing to do with asking another student for help or talking about their problem together.

Do this thought experiment: the elementary grade class is given a problem that requires "thinking". Most have no idea how to solve it. For them, they are being asked to cross the grand canyon by jumping. Maybe a few will attempt it once--"struggling" to come up with some idea, and fall flat. No idea what to do next. Most, having done that once, thank-you-very-much, won't even try to do that again. So they won't be struggling at all. They'll be refusing. Some will then ask another student what to do--if one asks another student who doesn't know, both have gotten nowhere and wasted their class time. If one asks a student who does know, all they will find out is the answer "here's how you do it". That doesn't help anyone to think at all. Even if that student tells you why the answer is true, that doesn't help because recognition isn't the same as being able to produce the right reasoning.

What a student needs to go from "struggle to jump the grand canyon" to "build oneself a bridge" is a teacher who can help them do that.

The best way to do this is as follows:a student starts with no idea what to do. Teacher, *knowing what they DO know*, points them to a simpler but related problem that they can solve, and asks them to think about that and solve it. (If student can't do that, teacher goes back more, to where the bedrock is solid, rinse and repeat.) Then student solves that, and teacher says good and asks leading questions that point to the new thing they don't understand. Teacher breaks down the new thing into small bites, each of which is something they can learn to recognize that they already can solve, or that they need to learn to solve.

Students talking to each other cannot break down tasks appropriately into smaller solveable subtasks, because by defn, students don't know what they don't know, and often don't know what they do know. Teachers must model how that's done for the student.

Now, the above isn't going to happen in most classrooms either--but the blind leading the blind is an utter waste of time and prevents the above from ever happening, since teacher time on task is what's needed to do the above.

One thing I find frustrating in math ed debates is the apparent fallacy of the excluded middle. It's as if when someone says "students won't learn from just rote memorization of equations", the assumption is the person solely believes in inquiry based learning, and if someone says "students learn nothing from inquiry based learning, because no one is guiding their inquiry" that they are assumed to be in favor of rote learning of equations.

Wu gets attacked constantly because of this fallacy. Someone hears him say something that puts Saxon in a bad light and they assume he thinks Everyday Math is a good text; someone hears him say something that puts Everyday Math in a bad light and they assume he loves Saxon.

The "struggling" argument is the same. If you're against "struggling", you're often assumed to be anti "learning to think for themselves". If you're "for struggling", you're often assumed to be direct instruction of concepts.

How about I phrase it as: students need to struggle but NOT IN VAIN. They must be given the tools so struggling is very nearly always successful.

Last comment:

I wonder if Bonnie's experience of seeing students who have "memorized their way through math" is a selection bias born from constructivism.

My guess: since no one can actually teach the understanding of WHY K-8 or K-12 students are being asked to do what they do in math, none of them have any understanding or clue. None of them know how why the math works, but some of them were math-natural enough to grok the procedures or memorize them and at least get the right answers.

Only these students who breezed through K-12 by quickly grokking the procedures, regardless of actual understanding, even attempt to go into engineering, CS, or science fields in college. The ones who didn't memorize their way through gave up on those interests years ago, because math was horribly senseless AND difficult. So yes, the selection bias in math-requiring college majors is for students who memorized. But that's because no other students could even get far enough to be in these majors.

"Struggle is important to learning. Sorry, but that's the truth."

"the truth"?

I don't buy this one bit. It's a nice idea to structure learning so that you create "light bulb" moments, but that's neither necessary or sufficient. However, when you group kids together in an "active learning" environment, maybe one or two get the desired light bulb moment. They directly teach it to the other kids who haven't figured it out. What's the benefit of being directly taught by kids who have no training as teachers?

Is there something special about struggle if you don't end up with the light bulb moment? It's good to learn to work hard and be persistent, but that's not struggle. You can't set up everything to learn as a series of struggles. If you don't struggle, then you haven't really learned the material?

There will be plenty of struggle with 25 question homework sets that gradually move kids from simple questions to more difficult ones, but the goal is hard work, not struggle. If you do those problems in sequence, then there should be plenty of light bulb moments with minimal struggle.

Besides, this isn't what Wu is talking about. This struggle idea came from the sfgate article that represented what Wu is afraid of.

"Unfortunately, textbook developers have yet to accept that the CCSMS are radically different from their predecessors, Most (and possibly all) textbook developers are only slightly revising their texts before declaring them aligned with the CCSMS."

This is true for schools. They will change CCSMS to reflect their own simplistic ideas of learning. In the process, the mathematical rigor that Wu talks about will be lost. When you spend all of your class time tackling few problems in active learning groups, less content will be covered, few will really benefit from light bulb moments, and mastery won't be achieved because they really don't like the "struggle" of homework sets.

This is a key point. Current K-6 educational pedagogy talks a lot about understanding and critical thinking, but their goal is to try to teach this by simulating the process in class using heterogeneous groups with the teachers as the guide-on-the-side. Do they try to accomplish this goal any other way, such as with individual homework? No. Do they try to ensure that kids have mastered content and skills? No.

They try to claim the pedagogical high ground of understanding and critical thinking, but what they do doesn't get the job done. If you look closely, you will see that what they care about is only what goes on in the classroom. The goal is not understanding and critical thinking. The goal is a happy learning environment with mixed ability groups.

Wu's pedantic math discussion does not deal with the reality shown in the sfgate article. We're talking about schools that jabber on and on about understanding, but allow kids to get to fifth grade without knowing the times table.

"... is that students who have memorized their way through math have no hope of creating computational solutions to even the simplest problems."

How do you memorize your way through math? How is it possible to pass exams just with memorization? As soon as you are confronted with a slightly different problem, how can memorization help? You have to define the issue better than this.

You can memorize a particular algorithm for multiplying numbers, but you can use it to solve all sorts of problems. You can understand how a multiplication algorithm works, but will that help you solve more problems that require multiplication? What level of understanding are you talking about; place value, algebraic, number base system, linear space?

You can memorize how to write a binary search algorithm, but you can apply it to solve all sorts of problems. If you are talking about writing a more elaborate search technique, then you need a different kind of understanding.

It's been at least a couple of decades that K-12 have been applying their non-memorization pedagogy. What CS or math students are you talking about? However, since you don't define what memorization means, and you don't define your problem, I can't possibly guess at what's causing it.

Here's one of my problems with Wu. Like so many university-level mathematicians, he seems completely out of touch with the perspectives of young children/mathematical novices. How readily will most 3rd graders

absorb statements like "The segment from 0 to 5/6 is 5 copies of the segment from 0 to 1/6"?

Also, Wu says he favors precise definitions over intuition, but the word "copy" seems ill-defined here. Copies aren't necessarily concatenated (or, to use Wu's term, "combined"). And what if a child assumes that part of what is "copied" is the line segment's position (from 0 to 1/6), and not just its length?

Wu's proposal for how to demonstrate to 6th and 7th graders why multiplying two negative numbers yields a positive number using the distributive law (see p. 7) strikes me as even more inaccesible to most of his intended audience. He's right to argue, as he does later on, that this demonstration is more accessible than a general proof would be (and I like the demonstration very much and will try it out on some kids I know), but still... At the very least, Wu should await the results of some sort of pilot study before recommending this to teachers of 6th and 7th graders.

On a different note, I whole-heartedly agree with everything SteveH writes about the Common Core

Standards and the problems of wishful thinking, top-down reform, and lack of specifics. In the context of a dominant paradigm, vague standards like

these will simply be used by those in power to further augment their

"reforms." We already see this happening with reading standards, as I've noted here: http://oilf.blogspot.com/2011/10/reform-reading-in-montgomery-county-md.html and here: http://blog.coreknowledge.org/2011/05/12/common-core-standards-a-cautionary-tale

--Like so many university-level mathematicians, he seems completely out of touch with the perspectives of young children/mathematical novices. How readily will most 3rd graders

absorb statements like "The segment from 0 to 5/6 is 5 copies of the segment from 0 to 1/6"?

Katharine, this was not my experience teaching elementary teachers Wu's fractions material. They didn't think Wu was out of touch with what their kids could do. Re: 3rd graders and the above: the issue isn't trying to get 3rd graders to understand the *statement* at all--the way it's written in words is difficult, sure. But that's not the math. And the words used for an AFT paper aren't meant to be the words used for a child.

And Wu isn't suggesting you *start* with the above definition either. You get there, over time.

The 3rd grader can readily understand 5/6 from pattern blocks and infix cubes. You start with those manipulatives. With practice, then you move them to the number line, which should not be difficult because they've seen that for whole numbers. So now you tell them about candy bars, and breaking it into 6ths, and then work up to the 5 pieces, each of size 1/6th. Children don't have a problem understanding segments and "segments of equal length" if you are clear from the beginning. For 3rd graders, you don't need the formal definitions, but you DO need the teacher to know the formal definitions so they aren't imparting false information and misleading them at that age. Then, as they progress to grades 5 and 6, you can expect the students themselves to know the precise defn.

--Kathrine said

Also, Wu says he favors precise definitions over intuition, but the word "copy" seems ill-defined here. Copies aren't necessarily concatenated (or, to use Wu's term, "combined"). And what if a child assumes that part of what is "copied" is the line segment's position (from 0 to 1/6), and not just its length?

Wu's book is quite precise. After a precise defn is used, he lightens up, and moves back to words like "copies". Again, his book is not for *children*, but for the adult teachers. So they need to learn the proper defn (which is too precise and wordy for an AFT article), and then they need to learn how to adapt that to the classroom.

Similarly, the negative times negative issue is something where your feeling about it comes without doing the prior week of fractions and near week of rational numbers. Teachers would be incredulous on day one with his suggestion, sure, but many don't feel that way on day 9, because they see how they'd have led their students there. Incidentally, the Japanese text for grade 7 teaches it that way, so it certainly is possible to be done at that point with proper background.

I would suggest a look at his book. I think you would find it addresses these issues.

Grant Wiggins pretty much trashed the standards in the new issue of Education Week.

Grant Wiggins is also the author of Understanding by Design, and his reasons for trashing CCSS appear to him to 1) be in conflict with his top-down "just in time" philosophy of learning and 2) implementation of the standards may cut down on his book sales and speaking engagements. Just a guess.

The all seeing eye, symbolizing Lucifer, with wings symbolizing the phoenix rising out of the ashes, a masonic concept that symbolizes the emergence of the New World Order out of chaos. The lightning bolt is an occult symbol as well and could be seen as a symbol for Lucifer falling from Heaven: “And he said unto them, I beheld Satan as lightning fall from heaven.” Luke 10:18. Notice the masonic checkerboard pattern. This pattern is the one used in the floor tiles of masonic lodges and is a symbolic representation of good and evil, or knowledge of, symbolized by other esoteric symbols like the hexagram, ying/yang, moon and star, and the crossing of the swords near the bottom of the picture.

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