I have seen similar comparisons before, and they are flawed. The "traditional" algorithm is NOT "what we were all taught to do", because it omits the unwritten part of the algorithm. For example: In the second block of the algorithm, a student is supposed to look at 616, and think "How many times does 82 go into 616"? And then they are supposed to write down a 7. But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized. So what the "traditional" algorithm tells you to do is to try a number that is close, and if it comes out too big or too small you try a different number. That means a lot of side calculations, scribbled in the margin or done in your head. And none of that recorded in the "official" solution.
The Everyday Math curriculum looks cluttered and calculated principally because all of those side calculations get written down on the side. It looks like more work, but in fact it is nearly identical. The main "long division" chart down the left side of the page is nearly identical to the "traditional" form, with the exception that the partial quotients get written down the side and then summed, rather than written on the top one digit at a time -- a superior method, in my opinion, because it takes away the need to "find the 7" (see previous paragraph). If you look carefully you will see that the solution shows, in that step, a student finding the 7 incrementally, as a 5 followed by a 2. The result is one step longer but is more methodical and requires less guessing.
All the stuff on the side is just the intermediate calculation -- the stuff that the traditional algorithm does not provide a space for at all. You will see that they have tried to represent "diversity of algorithms" by using different forms for the 82*40, 82*20, and 82*5 calculations, and for the subtraction that brings down each block from the previous one. Again notice that the traditional algorithm also does not leave space for borrowing/regrouping/whatever you want to call it.
In short: this is a piece of manipulation. The traditional algorithm looks clean and simple only because it leaves out all of the marginalia and mental parts; the EM version looks complex only because it includes all of those in print. Compare apples to apples, and the algorithms are indistinguishable.
"Compare apples to apples, and the algorithms are indistinguishable."
I would agree with this ... up to a point.
First, I want to say that those who tackle the virtues (or lack thereof) of Everyday Math, based on a comparison of the basic algorithms, are way off base. The algorithms that many of these curricula use are perfectly fine.
The huge difference between the curricula has to do with mastery. One could argue that there is no need to do hundreds of long division problems any which way you want to do them, but even that isn't the problem. The problem is that curricula like Everyday Math require little to no mastery of ANY basic algorithm. This really struck me when my son was taking Everyday Math. The assumption is that mastery does not add any understanding, just speed. However, if your child is required to be able to be able to determine how many times does 82 goes into 616 in his or her head, then that indicates a good development of number sense.
The traditional long division algorithm is something I want schools to teach my son just for a practical matter. They didn't do it and I'm still struggling with it now. Just last week he was stuck on how to calculate 365 divided by 7 in his head. I did it in my head very quickly and so did his science teacher standing next to him. We both shook our heads.
The traditional long division algorithm expects more from a student. It's not just rote and it's not just drill and kill. That's just low expectation talk. That's the real difference: low expectations.
But let's not stop with just the real basic algorithms. How about fractions, decimals, and percents? I don't care which algorithm they use, but I want to see mastery, not some excuse that conceptual understanding is more important or that they will see it again in the next spiral (circle) and get another crack at figuring it out.
I also don't want to see an explanation of fractions stop with a graphical representation of a piece of pie. I want my son to have mastery and understanding of how to divide fractions in a way that leads to more abstract rational expressions.
The argument is NOT about which basic algorithm is better. It has to do with the proper role of mastery of basic skills. Everyday math can throw all they want into their workbooks, but that doesn't mean that they have any way to ensure that kids either cover all of the material or master any of it in a timely fashion.
I completely agree with you re: mastery and high expectations, and I am glad you agree that the problem is not with the algorithms. Unfortunately an awful lot of the heat I see around EM seems to get focused on the unfamilarity of the algorithms, which (in my opinion) misses the point. There is nothing wrong with the lattice method for multiplying -- nothing. There is also nothing particulary "better" about it. It is just as useful, and just as easy to screw up, as the traditional stacking approach.
It's a shame they didn't teach your son long division. It sounds like they didn't teach him any division algorithm, which I think is the more salient point here. The one on display in that video works perfectly well for dividing 365/5. I don't think that the "long division alorithm" expects more from students than does the "EM algorithm" (or whatever it's called). It's the teacher and the curriculum that place expectations on students. The fact is, that students who are not expected to learn Algorithm X will not learn Algorithm X, and that's true for all values of X.
But when I see videos like the one Catherine posted, or the ubiquitous "inconvenient truth" video, I get annoyed. It seems to me like the creators of the video aren't actually thinking about what the two algorithms are, and instead are just reacting to the unfamilarity of it (while simultaneously painting a deceptive and inaccurate comparison). That just weakens their argument entirely and, frankly, makes them look like they don't really understand math all that well themselves.
As a parent (of 5 kids) I agree completely that parents need a vote and a veto. And as a former high school math teacher I would add teachers to that list -- let them make decisions about what curricula to use, and hold them accountable for the consequences of those decisions. Professionalism demands no less. But manipulation that distorts the issues and appeals to fear rather than to facts doesn't help matters.
Finally, I take exception to the claim that "real mathematicians" reject the curriculum, and to the false opposition between "mathematicians" and "ed schools". There are a hell of a lot of real mathematicians out there who disagree with each other on this. Some of them work in close collaboration with people in ed schools. And there are people in ed schools who differ, too. Academics are not a undifferentiated block, and it's demeaning to our (*) profession to treat us as if we don't have individual opinions.
Seriously, I love this blog (I have lurked here daily for more than a year, and it's the best place for math ed commentary that I have found), but some of the good-vs-evil typology on here drives me nuts.
(*) Oh, you want my bona fides? B.S. in Mathematics, Physics, and English; M.Sc. in Theoretical Physics (Cambridge University); M.S, Doctoral Cand. in Pure Mathematics; currently completing a joint doctorate in Mathematics and Education (note the "and" there, this is cross-sponsored by two departments). Five years experience as a secondary school teacher, 11.5 years experience as a parent of 5 children. And I'm pretty sure that there's nothing in that list that entitles you to draw any inferences re: how I feel about Everyday Mathematics.
I think there likely is one significant problem with the lattice method, which has to do with handwriting.
I'm told that because children aren't taught handwriting to mastery any more, a lot of kids have problems lining up the digits -- which I believe, after seeing my own son have plenty of problems lining up digits vertically.
Once kids have to start doing calculations in the margins of unlined test papers - which our middle schools kids are required to do - that's going to be a problem, too.
I don't know if you guys remember this, but back when C. was in middle school I asked for extra paper for tests. C.'s handwriting was horrific & he basically couldn't write tiny little numerals AND line them up properly in test margins.
The teacher refused, saying they wouldn't have extra paper on the state tests.
Talk about teaching to the test.
These things matter, btw -- at least, that's what the precision teaching people find.
As a parent (of 5 kids) I agree completely that parents need a vote and a veto. And as a former high school math teacher I would add teachers to that list -- let them make decisions about what curricula to use, and hold them accountable for the consequences of those decisions.
absolutely
from where I sit, administrators churn programs and pedagogies; it's horrible
every five seconds teachers here have to get professionally developed & change everything about what they've been doing
One of our board members here is now pushing for Singapore Math & hitting a brick wall. This is a board member!
I know for a fact that one other board member has significant frustrations with the math curriculum & if I had to bet I'd wager one more would throw in his lot with him. That would make 3 out of 5. But the administration is unbending.
At the last meeting, the asst superintendent for curriculum/instruction/technology, when pressed on the question, repeatedly used the word "we" in explaining how & why they would or would not think about adopting Singapore Math, which they're going to take their time doing.
"We have to see whether our students are where we want them."
"We'll assess the curricula to see whether it meets our needs."
"We need to see whether Trailblazers is doing what we want it to."
That word -- 'we' -- essentially refers to two people: the superintendent & the assistant superintendent. Throw in a couple of building principals & you're up to 4.
None of these women has studied math; the assistant super told me she hadn't taken a math course since high school. (The board member who is pressing the issue majored in math at Dartmouth.)
I suppose there are teachers who agree, but we don't hear from them.
Parents, school board members, mathematicians, & taxpayers are have no involvement in "we."
And taxpayers are funding the district to the tune of $27K per pupil.
Finally, I take exception to the claim that "real mathematicians" reject the curriculum, and to the false opposition between "mathematicians" and "ed schools". There are a hell of a lot of real mathematicians out there who disagree with each other on this.
You can find real mathematicians here and there who approve the curricula, but they are vastly outnumbered by those who don't.
Again, as a parent, when I see a consensus amongst mathematicians that Math Trailblazers or TERC or Everyday Math aren't working, those are the people from whom I take my cue.
I'm told that because children aren't taught handwriting to mastery any more, a lot of kids have problems lining up the digits -- which I believe, after seeing my own son have plenty of problems lining up digits vertically.
That's really interesting. Do they have the same problem with "stacking" algorithms? Those also require you to line up the digits. I'm not sure how lattice is any worse in that respect.
I currently tutor a 10th grader in a Geometry clas who only knows the lattice method, and he is constantly making errors -- drawing the diagonal lines the wrong direction (which ruins the whole algorithm), forgetting to carry, etc. But I'm 99% certain he would make analogous errors (forgetting to put "spacer zeros", forgetting to carry, etc.) with stacking.
The more significant problem is that he lacks any notion of when to multiply, or use an algorithm. Ask him how much sugar is in 8 bags if each bag has 1/4 of a pound of sugar, and he has no idea what to do. Ask him what 8 * (1/4) is and he stares blankly. Have him write down (8/1) * (1/4) and he can produce the answer immediately. It is a classic case of "procedural competency" that masks an utter lack of conceptual understanding. Kids like that (and I have seen many) make me sympathetic to "understanding first" as a slogan. The problem is that I don't think there is any curriculum on earth that teaches understanding. Everyday Math doesn't seem to do it any better, or worse, in my experience.
(addendum to my previous comment) Oh, and of course that student I was referring to doesn't see any connection at all between the "bags of sugar" problem and the related problem 8 / 4, since the latter (to him) exclusively means "divide eight things into four sets", which doesn't match the setting.
(another addendum) I don't have any experience with instructional coaches, so I can't form an opinion of them.
"It sounds like they didn't teach him any division algorithm, ..."
No, he had Everyday Math, so he was taught, if that's what you call it. He can do 365/7 on paper perfectly fine, but I want him to be able to do it in his head.
"I don't think that the 'long division alorithm' expects more from students than does the 'EM algorithm' (or whatever it's called)."
Partial quotients.
The traditional algorithm requires you to be able to determine how many times 82 goes into 616 in your head. Partial quotients does not.
"It's the teacher and the curriculum that place expectations on students."
That was exactly my point. Everyday Math is extremely weak on expectations. Its spiraling (circling) is all about lower expectations. Any talk of understanding and critical thinking is just a smoke-screen.
"I would add teachers to that list -- let them make decisions about what curricula to use, ..."
They're the ones who chose Everyday Math for my son's school, and parents have no veto or choice. Even though some parents might get caught up on the differences in algorithms, they can sense that something is not right. They can tell that mastery is not being achieved.
"Finally, I take exception to the claim that 'real mathematicians' reject the curriculum, and to the false opposition between "mathematicians" and 'ed schools'."
I could argue this, but it's somewhat of a strawman. There are lots of mathemeticians and engineers (and teachers) who have raised very serious and specific questions about curricula like Everyday Math. These are the issues that have to be dealt with, not whether the players fall into one team or another.
You can't pick and choose your arguing points. Your comments about the videos are well taken, and it's not about teams or sides, but that's not all there is. Many of these things have be discussed in detail at KTM over the years.
"...but some of the good-vs-evil typology on here drives me nuts."
An extreme characterization is easy to defend, but the real issue is about power and control. Schools make decisions about educational assumptions and expectations and take little or no account of parental input, even when it's well researched and presented by content specialists. There is a certain pervasive arrogance going on here. I call it turf and ego, not "evil".
Many YouTube videos are easy to take apart, but that doesn't stop anyone from having a serious discussion of the pros and cons of curricula like Everyday Math on KTM. We've actually had quite a few over the years and I would like to see more.
Saxon teaches conceptually, but doesn't necessarily belabor it. I've taught two years of it afterschooling and took one year (8th grade) for myself just to see how it worked.
We have Trailblazers in our district. Both of my sons got out of taking it because one is special ed and one is accelerated in math. However, the accelerated one would tell me stories of the 5th grade math class working with manipulatives on subraction. They spent two solid weeks on it. Since Trailblazers is similar to EM with its spiral, most of the kids didn't have mastery of a number of 5th grade topics (or 4th either.)
One fifth grade straight A Trailblazer kid I know did not know but about a third of her multiplication tables. Ditto on the short division. She had no idea that a fraction could be divided, and had not factored enough to remember how to handle common denominators. This was an A student. I shudder to think what the C student looks like.
One of the problems I have with the partial quotients algorithm is that it allows students to avoid multiplication and division facts that they haven't memorized. It is not a good idea for a mathematically weak student to head towards algebra struggling with basic multiplication, and yet that's what these curriculums do. This puts middle school teachers in the position of having to shore up basic arithmetic before they can move on to algebra.
As far as parents being unfamiliar with new ways of doing things, I do think that is a legitimate beef. Most parents aren't going to have your background in math. I remember well in 1969 when new math was around how my father became so frustrated at not being able to help me. It wasn't necessary then, and it appears it isn't necessary now. Driving a wedge between parent and child is never a good idea unless you plan to blame the parents later, which I've also seen happen.
I think there likely is one significant problem with the lattice method, which has to do with handwriting.
There are other significant problems as well. Multiplying decimals, for one. And understanding how place value is working. Lattice multiplication is simply a rote method--a dirty word amongst refomers. Lattice multiplication is also inefficient. SteveH is correct about mastery but I doubt he would feel better if his son achieved mastery over the lattice method.
As for Michael's desire to teach "understanding first", that's also the desire of many constructivists who seem to think that procedures keep students from learning the concepts. They are not mutually exclusive. Procedural fluency can and does lead to conceptual understanding. The traditional texts do not simply present multiplication facts devoid of problems they are meant to solve. Word problems such as "a bag of sugar weighs 5 pounds, how many pounds do 6 bags weigh" are not uncommon.
many constructivists who seem to think that procedures keep students from learning the concepts
Oh, I know that's a famous straw man of the anti-proceduralists. I certainly don't think that procedural understanding obstructs conceptualization. My point is just that there are students, a lot of them, who can execute algorithms without a whit of understanding, and working with these kids makes me sympathetic to the "understanding first" camp. I don't know what to blame that on -- I have seen this with kids in "reform" classrooms and in "traditional" classrooms, and I suspect it has little to do with the curriculum they are learning from.
"It is a classic case of 'procedural competency' that masks an utter lack of conceptual understanding."
This student shows very little procedural competency OR understanding.
"... make me sympathetic to 'understanding first' as a slogan."
It's quite a powerful slogan, and it's wrong. There are many levels of understanding. You don't teach understanding (conceptual or otherwise) and then expect algorithms to flow from that. As I've said for years on KTM, understanding and mastery are tightly linked. Curricula like Everyday Math try their best to unlink them. They think that understanding concepts with just a little bit of practice is all that you need. This conveniently fits their dislike of drill and kill and how they believe that there is no one way to solve a problem.
Mastery is not just about speed. It's directly related to understanding.
"Ask him what 8 * (1/4) is and he stares blankly."
This indicates really bad everything. There is no magical understanding that would fix this. It's a good example, however.
What does understanding mean when it comes to the basic identities, like
8 = 8/1.
Many of the popular math curricula are stuck with a pie section or manipulable form of understanding, but fall completely apart when it comes to abstract or procedural understandings.
Students can't possibly understand an algebraic approach to multiplication in fifth grade, but Everyday Math seems to think that something like the Lattice method teaches understanding. Actually, it's just a reflection of their marketing goal to avoid standard algorithms at all cost.
In the second block of the algorithm, a student is supposed to look at 616, and think "How many times does 82 go into 616"? And then they are supposed to write down a 7. But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized. So what the "traditional" algorithm tells you to do is to try a number that is close, and if it comes out too big or too small you try a different number. That means a lot of side calculations, scribbled in the margin or done in your head.
But, I know that 8*7 is 56, so 82*7 would be my first choice. With enough practice, I'm not stabbing about in the dark on the side of the paper, trying to pin down 82*7. 8 is obviously too large, as 8*8 is 64.
But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized.Everyone should have the multiples of 8 memorized. If you've memorized the multiples of 8, the multiples of 80 are trivial. The largest flaw in Everyday Math, in my estimation, is the persistent lack of emphasis on learning multiplication facts.
Our district uses Everyday Math in K-5. It's a poor curriculum, made worse by the fact that the teachers have absolute freedom in how to teach it. The teachers who like math do more, and supplement with their own worksheets, which do help to plug the holes in procedural fluency. The teachers who don't like math do the minimum. Our district also does not allot the minimum amount of time necessary for Everyday Math. Of course, it's an affluent district, with parents who care about education, so the tutors (who are frequently elementary teachers) are happy.
Just offhand, does it make any sense to choose a curriculum which most parents do not understand, in order to teach a basic, necessary skill such as math? I am very glad that we have been through the wringer with our oldest child, so we know what to look out for, and where it is essential that we supplement.
We have found that a cross-pollination with Singapore Math did the trick for our eldest. They do a great deal of exercises in Everyday Math, but it lacks a theoretical framework for the students to gain a deeper understanding. The early books of Singapore Math lay out a logical progression for addition, subtraction, and multiplication, which help to knock things into place. Our child is above average in quantitative skills, though. I don't know if it would work as well with other children.
"...sympathetic to the 'understanding first' camp."
But this is only the hypothesis. Where is the proof, or at least some examples? You can't look back at poorly taught traditional math classes and compare them to magically well taught reform classes.
I don't know what understanding first really means. For my son's Everyday Math classes, it meant "don't worry about mastery". By definition. This had nothing to do with bad teaching.
You can't look back at poorly taught traditional math classes and compare them to magically well taught reform classes
This, I think, is the heart of it. I am pretty much convinced that teaching reform classes well is harder than teaching traditional classes well. I also suspect that a well-taught reform class is marginally better than a well-taught traditional class, but both are sufficiently uncommon (and the former even more so) that the difference doesn't amount to much when you go to scale.
I think it's likely that a really good teacher can use any curriculum to teach well (for some value of "use"). I also think it's likely that a really bad teacher cannot use any curriculum to teach well.
But what about the margin? Most teachers are neither really good nor really bad*; is there a difference for these teachers?
From the available evidence, I think the answer has to be "yes", and that EM is not the beneficiary of that answer. Scattershot curricula do little or none of the work for the teacher, while more-structured curricula (like Saxon and Singapore) allow even a borderline ignoramus to have some positive effect on mathematical knowledge.
This turns out to be crucial**.
* My assertion. The population size is large enough, and drawn from an average-enough group, that a gaussian should work pretty well, so I think it's a safe assertion, but I offer it without proof.
** Yes, I would like a saucer of milk with that. 8-/
"I also suspect that a well-taught reform class is marginally better than a well-taught traditional class,..."
I don't even know what a well-taught reform class would look like, but I wouldn't use the word taught. The teacher is the guide on the side. Even the videos that purport to show how this is done correctly are awful. Remember the video about the teacher trying to lead a student or two to discover an answer. Other kids were hanging around waiting to talk to the teacher, and the rest of the kids were fooling around.
The best type of understanding or discovery learning would probably involve a lot of direct instruction, with the real discovery happening individually with homework. It sounds like a well-taught traditional approach.
Have you read Hung-Hsi Wu's work? He is writing math texts for math teachers, both high school and grade school. His website might help you decide about how to fix gaps in understanding fractions in your student.
http://math.berkeley.edu/~wu/
RE: this problem:Ask him what 8 * (1/4) is and he stares blankly. Have him write down (8/1) * (1/4) and he can produce the answer immediately. It is a classic case of "procedural competency" that masks an utter lack of conceptual understanding. Kids like that (and I have seen many) make me sympathetic to "understanding first" as a slogan.
There are a lot of things he doesn't understand that can be taught before "understanding". They are just so well understood by you that you've forgotten them. they must be made explicit to your student.
First: you must drill until automatic that 8 is 8/1 is 8*1. That means you must drill that 8 times 1 is 8, 8 divided by 1 is 8. It means you must drill that 8/1 is 8 divided by 1. It means you must drill that the fraction a/b is the value x such that a = bx. DRILL DRILL DRILL those until they are so automatic that even though he doesn't understand what it MEANS just yet that 8/1 is 8 divided by 1 is x such that a = bx, he does it and knows it. And later the understanding comes.
Then you must drill that 8 * 1/4 is the same as 8/4. Drill drill again. Drill until automatic. So automatic that sometime later, he can have the a-ha that those are the same means both are equal to 2, and all representations are the same.
Of course a curriculum doesn't teach understanding. Curricula should teach truths. The understanding comes from using those truths over and over and over again.
Another related idea to this "can understanding come first?" is this:
when faced with a problem that we don't understand, we can either trust our intuition, or our computation.
If we trust neither, we can't solve the problem.
If we trust our intuition but not our computation, we still can't be sure we solved the problem. The intuition can help guide our computation, but not compute. And in the end, if the computation and intuition disagree, we will abandon our solution but not know what to do to fix it.
If we trust our computation, however, then we can READ THE ANSWER from our computation--even if our intuition said otherwise. (consider any spec rel problem, or even tons of classical mechanics problems.) and we can correct our intuition to match our computation--we can teach ourselves to understand what the problem is telling us.
So mastery of procedural competency has its place: it alone can convince us to fix our logical or rational errors. But we've got to get to the point where there is something CONCRETE we are SURE IS TRUE first.
If procedural competency really truly solid, then it will lead you to correct your misunderstanding--or at least, at some later point in your maturity, allow you to say "huh, look at that. Guess I was wrong." But without it, you're at the mercy of your intuition--and if that's not yet well formed (and how would it be, if your proc. competency is weak?), you will never know when you've gotten the right answer, which answers to trust.
p.s. the video was pointless unless proof were needed that people convinced of the righteousness of their cause don't necessarily make the best case...
"here it is; obviously everybody will draw the same conclusion i did". great. this is what i get for spinning up a YouTube of course... i don't do much of that and'll now maybe do marginally *less*... like TV itself, it's almost always *way too slow* to make its points... if you can read, well then you're much better off *reading*...
I agree with everything SteveH has said so far except: "I don't even know what a well-taught reform class would look like, but I wouldn't use the word taught...Even the videos that purport to show how this is done correctly are awful."
If you want to see some better examples, some of the videos on the Annenberg site have some good examples that would be considered "reform"; I'd trust Marilyn Burns to teach math to my children; and if you ever get a chance to go hear Deborah Ball, you should take it.
In a way, my position is similar to Michael's, only with less patience for the new curricula. I've seen some great stuff from individual teachers, and I think that's what the standards people are trying to bottle, but the curricula that have come out of it are really disappointing. (When your answer to hard math is not "how can I teach it more effectively", but "what can I teach instead" then you've lost my vote).
"...some of the videos on the Annenberg site have some good examples ..."
Are there any that don't require registration?
However, a video is not a curriculum. Which curricula do Marilyn Burns and Deborah Ball recommend? Are they curricula that would prepare a student to get into the College of Engineering at the University of Michigan, or into the School of Education?
"what can I teach instead"
But that's the definition of reform math. Adding a discovery-like presentation to "hard math" doesn't make it reform math, because hard math requires mastery of skills, no matter what the teacher and students do in class. The big problem with a steady diet of group class discovery (even if it's done well) is that takes too much time and you don't cover enough material.
32 comments:
I have seen similar comparisons before, and they are flawed. The "traditional" algorithm is NOT "what we were all taught to do", because it omits the unwritten part of the algorithm. For example: In the second block of the algorithm, a student is supposed to look at 616, and think "How many times does 82 go into 616"? And then they are supposed to write down a 7. But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized. So what the "traditional" algorithm tells you to do is to try a number that is close, and if it comes out too big or too small you try a different number. That means a lot of side calculations, scribbled in the margin or done in your head. And none of that recorded in the "official" solution.
The Everyday Math curriculum looks cluttered and calculated principally because all of those side calculations get written down on the side. It looks like more work, but in fact it is nearly identical. The main "long division" chart down the left side of the page is nearly identical to the "traditional" form, with the exception that the partial quotients get written down the side and then summed, rather than written on the top one digit at a time -- a superior method, in my opinion, because it takes away the need to "find the 7" (see previous paragraph). If you look carefully you will see that the solution shows, in that step, a student finding the 7 incrementally, as a 5 followed by a 2. The result is one step longer but is more methodical and requires less guessing.
All the stuff on the side is just the intermediate calculation -- the stuff that the traditional algorithm does not provide a space for at all. You will see that they have tried to represent "diversity of algorithms" by using different forms for the 82*40, 82*20, and 82*5 calculations, and for the subtraction that brings down each block from the previous one. Again notice that the traditional algorithm also does not leave space for borrowing/regrouping/whatever you want to call it.
In short: this is a piece of manipulation. The traditional algorithm looks clean and simple only because it leaves out all of the marginalia and mental parts; the EM version looks complex only because it includes all of those in print. Compare apples to apples, and the algorithms are indistinguishable.
It is a piece of manipulation & a fine one indeed.
Parents need a vote and a veto.
I am not a mathematician, nor am I particularly well-versed in math.
But when I see real mathematicians rejecting this curriculum, I want school districts to do likewise.
At the same time, I am perfectly willing to fund Everyday Math for those parents who freely choose Everyday Math for their children's education.
Ultimately, this is an issue of power.
On one side: mathematicians & parents.
On the other: ed schools & publishing companies.
It's not a hard call.
"Compare apples to apples, and the algorithms are indistinguishable."
I would agree with this ... up to a point.
First, I want to say that those who tackle the virtues (or lack thereof) of Everyday Math, based on a comparison of the basic algorithms, are way off base. The algorithms that many of these curricula use are perfectly fine.
The huge difference between the curricula has to do with mastery. One could argue that there is no need to do hundreds of long division problems any which way you want to do them, but even that isn't the problem. The problem is that curricula like Everyday Math require little to no mastery of ANY basic algorithm. This really struck me when my son was taking Everyday Math. The assumption is that mastery does not add any understanding, just speed. However, if your child is required to be able to be able to determine how many times does 82 goes into 616 in his or her head, then that indicates a good development of number sense.
The traditional long division algorithm is something I want schools to teach my son just for a practical matter. They didn't do it and I'm still struggling with it now. Just last week he was stuck on how to calculate 365 divided by 7 in his head. I did it in my head very quickly and so did his science teacher standing next to him. We both shook our heads.
The traditional long division algorithm expects more from a student. It's not just rote and it's not just drill and kill. That's just low expectation talk. That's the real difference: low expectations.
But let's not stop with just the real basic algorithms. How about fractions, decimals, and percents? I don't care which algorithm they use, but I want to see mastery, not some excuse that conceptual understanding is more important or that they will see it again in the next spiral (circle) and get another crack at figuring it out.
I also don't want to see an explanation of fractions stop with a graphical representation of a piece of pie. I want my son to have mastery and understanding of how to divide fractions in a way that leads to more abstract rational expressions.
The argument is NOT about which basic algorithm is better. It has to do with the proper role of mastery of basic skills. Everyday math can throw all they want into their workbooks, but that doesn't mean that they have any way to ensure that kids either cover all of the material or master any of it in a timely fashion.
"Ultimately, this is an issue of power."
Yup. That too. Especially when a district has the resources to offer both approaches and chooses not to.
Turf and ego.
SteveH,
I completely agree with you re: mastery and high expectations, and I am glad you agree that the problem is not with the algorithms. Unfortunately an awful lot of the heat I see around EM seems to get focused on the unfamilarity of the algorithms, which (in my opinion) misses the point. There is nothing wrong with the lattice method for multiplying -- nothing. There is also nothing particulary "better" about it. It is just as useful, and just as easy to screw up, as the traditional stacking approach.
It's a shame they didn't teach your son long division. It sounds like they didn't teach him any division algorithm, which I think is the more salient point here. The one on display in that video works perfectly well for dividing 365/5. I don't think that the "long division alorithm" expects more from students than does the "EM algorithm" (or whatever it's called). It's the teacher and the curriculum that place expectations on students. The fact is, that students who are not expected to learn Algorithm X will not learn Algorithm X, and that's true for all values of X.
But when I see videos like the one Catherine posted, or the ubiquitous "inconvenient truth" video, I get annoyed. It seems to me like the creators of the video aren't actually thinking about what the two algorithms are, and instead are just reacting to the unfamilarity of it (while simultaneously painting a deceptive and inaccurate comparison). That just weakens their argument entirely and, frankly, makes them look like they don't really understand math all that well themselves.
As a parent (of 5 kids) I agree completely that parents need a vote and a veto. And as a former high school math teacher I would add teachers to that list -- let them make decisions about what curricula to use, and hold them accountable for the consequences of those decisions. Professionalism demands no less. But manipulation that distorts the issues and appeals to fear rather than to facts doesn't help matters.
Finally, I take exception to the claim that "real mathematicians" reject the curriculum, and to the false opposition between "mathematicians" and "ed schools". There are a hell of a lot of real mathematicians out there who disagree with each other on this. Some of them work in close collaboration with people in ed schools. And there are people in ed schools who differ, too. Academics are not a undifferentiated block, and it's demeaning to our (*) profession to treat us as if we don't have individual opinions.
Seriously, I love this blog (I have lurked here daily for more than a year, and it's the best place for math ed commentary that I have found), but some of the good-vs-evil typology on here drives me nuts.
(*) Oh, you want my bona fides? B.S. in Mathematics, Physics, and English; M.Sc. in Theoretical Physics (Cambridge University); M.S, Doctoral Cand. in Pure Mathematics; currently completing a joint doctorate in Mathematics and Education (note the "and" there, this is cross-sponsored by two departments). Five years experience as a secondary school teacher, 11.5 years experience as a parent of 5 children. And I'm pretty sure that there's nothing in that list that entitles you to draw any inferences re: how I feel about Everyday Mathematics.
Hi Michael!
Thanks for the nice words.
I think there likely is one significant problem with the lattice method, which has to do with handwriting.
I'm told that because children aren't taught handwriting to mastery any more, a lot of kids have problems lining up the digits -- which I believe, after seeing my own son have plenty of problems lining up digits vertically.
Once kids have to start doing calculations in the margins of unlined test papers - which our middle schools kids are required to do - that's going to be a problem, too.
I don't know if you guys remember this, but back when C. was in middle school I asked for extra paper for tests. C.'s handwriting was horrific & he basically couldn't write tiny little numerals AND line them up properly in test margins.
The teacher refused, saying they wouldn't have extra paper on the state tests.
Talk about teaching to the test.
These things matter, btw -- at least, that's what the precision teaching people find.
As a parent (of 5 kids) I agree completely that parents need a vote and a veto. And as a former high school math teacher I would add teachers to that list -- let them make decisions about what curricula to use, and hold them accountable for the consequences of those decisions.
absolutely
from where I sit, administrators churn programs and pedagogies; it's horrible
every five seconds teachers here have to get professionally developed & change everything about what they've been doing
do you have thoughts on instructional coaches?
One of our board members here is now pushing for Singapore Math & hitting a brick wall. This is a board member!
I know for a fact that one other board member has significant frustrations with the math curriculum & if I had to bet I'd wager one more would throw in his lot with him. That would make 3 out of 5. But the administration is unbending.
At the last meeting, the asst superintendent for curriculum/instruction/technology, when pressed on the question, repeatedly used the word "we" in explaining how & why they would or would not think about adopting Singapore Math, which they're going to take their time doing.
"We have to see whether our students are where we want them."
"We'll assess the curricula to see whether it meets our needs."
"We need to see whether Trailblazers is doing what we want it to."
That word -- 'we' -- essentially refers to two people: the superintendent & the assistant superintendent. Throw in a couple of building principals & you're up to 4.
None of these women has studied math; the assistant super told me she hadn't taken a math course since high school. (The board member who is pressing the issue majored in math at Dartmouth.)
I suppose there are teachers who agree, but we don't hear from them.
Parents, school board members, mathematicians, & taxpayers are have no involvement in "we."
And taxpayers are funding the district to the tune of $27K per pupil.
Finally, I take exception to the claim that "real mathematicians" reject the curriculum, and to the false opposition between "mathematicians" and "ed schools". There are a hell of a lot of real mathematicians out there who disagree with each other on this.
You can find real mathematicians here and there who approve the curricula, but they are vastly outnumbered by those who don't.
Again, as a parent, when I see a consensus amongst mathematicians that Math Trailblazers or TERC or Everyday Math aren't working, those are the people from whom I take my cue.
The huge difference between the curricula has to do with mastery.
Absolutely.
I've been thinking about this more and more now that the kids I know have entered high school -- and not just with math but with all subjects.
When schools disdain memory, they use curricula and teaching methods that are pure exposure.
No grownup checks to see if the kids transferred the content or procedures they've "understood" in class to long-term memory.
I'm seeing enormous problems & suffering for many kids as a result.
I'm told that because children aren't taught handwriting to mastery any more, a lot of kids have problems lining up the digits -- which I believe, after seeing my own son have plenty of problems lining up digits vertically.
That's really interesting. Do they have the same problem with "stacking" algorithms? Those also require you to line up the digits. I'm not sure how lattice is any worse in that respect.
I currently tutor a 10th grader in a Geometry clas who only knows the lattice method, and he is constantly making errors -- drawing the diagonal lines the wrong direction (which ruins the whole algorithm), forgetting to carry, etc. But I'm 99% certain he would make analogous errors (forgetting to put "spacer zeros", forgetting to carry, etc.) with stacking.
The more significant problem is that he lacks any notion of when to multiply, or use an algorithm. Ask him how much sugar is in 8 bags if each bag has 1/4 of a pound of sugar, and he has no idea what to do. Ask him what 8 * (1/4) is and he stares blankly. Have him write down (8/1) * (1/4) and he can produce the answer immediately. It is a classic case of "procedural competency" that masks an utter lack of conceptual understanding. Kids like that (and I have seen many) make me sympathetic to "understanding first" as a slogan. The problem is that I don't think there is any curriculum on earth that teaches understanding. Everyday Math doesn't seem to do it any better, or worse, in my experience.
(addendum to my previous comment) Oh, and of course that student I was referring to doesn't see any connection at all between the "bags of sugar" problem and the related problem 8 / 4, since the latter (to him) exclusively means "divide eight things into four sets", which doesn't match the setting.
(another addendum)
I don't have any experience with instructional coaches, so I can't form an opinion of them.
"It sounds like they didn't teach him any division algorithm, ..."
No, he had Everyday Math, so he was taught, if that's what you call it. He can do 365/7 on paper perfectly fine, but I want him to be able to do it in his head.
"I don't think that the 'long division alorithm' expects more from students than does the 'EM algorithm' (or whatever it's called)."
Partial quotients.
The traditional algorithm requires you to be able to determine how many times 82 goes into 616 in your head. Partial quotients does not.
"It's the teacher and the curriculum that place expectations on students."
That was exactly my point. Everyday Math is extremely weak on expectations. Its spiraling (circling) is all about lower expectations. Any talk of understanding and critical thinking is just a smoke-screen.
"I would add teachers to that list -- let them make decisions about what curricula to use, ..."
They're the ones who chose Everyday Math for my son's school, and parents have no veto or choice. Even though some parents might get caught up on the differences in algorithms, they can sense that something is not right. They can tell that mastery is not being achieved.
"Finally, I take exception to the claim that 'real mathematicians' reject the curriculum, and to the false opposition between "mathematicians" and 'ed schools'."
I could argue this, but it's somewhat of a strawman. There are lots of mathemeticians and engineers (and teachers) who have raised very serious and specific questions about curricula like Everyday Math. These are the issues that have to be dealt with, not whether the players fall into one team or another.
You can't pick and choose your arguing points. Your comments about the videos are well taken, and it's not about teams or sides, but that's not all there is. Many of these things have be discussed in detail at KTM over the years.
"...but some of the good-vs-evil typology on here drives me nuts."
An extreme characterization is easy to defend, but the real issue is about power and control. Schools make decisions about educational assumptions and expectations and take little or no account of parental input, even when it's well researched and presented by content specialists. There is a certain pervasive arrogance going on here. I call it turf and ego, not "evil".
Many YouTube videos are easy to take apart, but that doesn't stop anyone from having a serious discussion of the pros and cons of curricula like Everyday Math on KTM. We've actually had quite a few over the years and I would like to see more.
Hi Michael,
Saxon teaches conceptually, but doesn't necessarily belabor it. I've taught two years of it afterschooling and took one year (8th grade) for myself just to see how it worked.
We have Trailblazers in our district. Both of my sons got out of taking it because one is special ed and one is accelerated in math. However, the accelerated one would tell me stories of the 5th grade math class working with manipulatives on subraction. They spent two solid weeks on it. Since Trailblazers is similar to EM with its spiral, most of the kids didn't have mastery of a number of 5th grade topics (or 4th either.)
One fifth grade straight A Trailblazer kid I know did not know but about a third of her multiplication tables. Ditto on the short division. She had no idea that a fraction could be divided, and had not factored enough to remember how to handle common denominators. This was an A student. I shudder to think what the C student looks like.
One of the problems I have with the partial quotients algorithm is that it allows students to avoid multiplication and division facts that they haven't memorized. It is not a good idea for a mathematically weak student to head towards algebra struggling with basic multiplication, and yet that's what these curriculums do. This puts middle school teachers in the position of having to shore up basic arithmetic before they can move on to algebra.
As far as parents being unfamiliar with new ways of doing things, I do think that is a legitimate beef. Most parents aren't going to have your background in math. I remember well in 1969 when new math was around how my father became so frustrated at not being able to help me. It wasn't necessary then, and it appears it isn't necessary now. Driving a wedge between parent and child is never a good idea unless you plan to blame the parents later, which I've also seen happen.
SusanS
I think there likely is one significant problem with the lattice method, which has to do with handwriting.
There are other significant problems as well. Multiplying decimals, for one. And understanding how place value is working. Lattice multiplication is simply a rote method--a dirty word amongst refomers. Lattice multiplication is also inefficient. SteveH is correct about mastery but I doubt he would feel better if his son achieved mastery over the lattice method.
As for Michael's desire to teach "understanding first", that's also the desire of many constructivists who seem to think that procedures keep students from learning the concepts. They are not mutually exclusive. Procedural fluency can and does lead to conceptual understanding. The traditional texts do not simply present multiplication facts devoid of problems they are meant to solve. Word problems such as "a bag of sugar weighs 5 pounds, how many pounds do 6 bags weigh" are not uncommon.
many constructivists who seem to think that procedures keep students from learning the concepts
Oh, I know that's a famous straw man of the anti-proceduralists. I certainly don't think that procedural understanding obstructs conceptualization. My point is just that there are students, a lot of them, who can execute algorithms without a whit of understanding, and working with these kids makes me sympathetic to the "understanding first" camp. I don't know what to blame that on -- I have seen this with kids in "reform" classrooms and in "traditional" classrooms, and I suspect it has little to do with the curriculum they are learning from.
"It is a classic case of 'procedural competency' that masks an utter lack of conceptual understanding."
This student shows very little procedural competency OR understanding.
"... make me sympathetic to 'understanding first' as a slogan."
It's quite a powerful slogan, and it's wrong. There are many levels of understanding. You don't teach understanding (conceptual or otherwise) and then expect algorithms to flow from that. As I've said for years on KTM, understanding and mastery are tightly linked. Curricula like Everyday Math try their best to unlink them. They think that understanding concepts with just a little bit of practice is all that you need. This conveniently fits their dislike of drill and kill and how they believe that there is no one way to solve a problem.
Mastery is not just about speed. It's directly related to understanding.
"Ask him what 8 * (1/4) is and he stares blankly."
This indicates really bad everything. There is no magical understanding that would fix this. It's a good example, however.
What does understanding mean when it comes to the basic identities, like
8 = 8/1.
Many of the popular math curricula are stuck with a pie section or manipulable form of understanding, but fall completely apart when it comes to abstract or procedural understandings.
Students can't possibly understand an algebraic approach to multiplication in fifth grade, but Everyday Math seems to think that something like the Lattice method teaches understanding.
Actually, it's just a reflection of their marketing goal to avoid standard algorithms at all cost.
In the second block of the algorithm, a student is supposed to look at 616, and think "How many times does 82 go into 616"? And then they are supposed to write down a 7. But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized. So what the "traditional" algorithm tells you to do is to try a number that is close, and if it comes out too big or too small you try a different number. That means a lot of side calculations, scribbled in the margin or done in your head.
But, I know that 8*7 is 56, so 82*7 would be my first choice. With enough practice, I'm not stabbing about in the dark on the side of the paper, trying to pin down 82*7. 8 is obviously too large, as 8*8 is 64.
But how do they know it goes in 7 times, and not 6, or 8 times? Certainly nobody has the multiples of 82 memorized.Everyone should have the multiples of 8 memorized. If you've memorized the multiples of 8, the multiples of 80 are trivial. The largest flaw in Everyday Math, in my estimation, is the persistent lack of emphasis on learning multiplication facts.
Our district uses Everyday Math in K-5. It's a poor curriculum, made worse by the fact that the teachers have absolute freedom in how to teach it. The teachers who like math do more, and supplement with their own worksheets, which do help to plug the holes in procedural fluency. The teachers who don't like math do the minimum. Our district also does not allot the minimum amount of time necessary for Everyday Math. Of course, it's an affluent district, with parents who care about education, so the tutors (who are frequently elementary teachers) are happy.
Just offhand, does it make any sense to choose a curriculum which most parents do not understand, in order to teach a basic, necessary skill such as math? I am very glad that we have been through the wringer with our oldest child, so we know what to look out for, and where it is essential that we supplement.
We have found that a cross-pollination with Singapore Math did the trick for our eldest. They do a great deal of exercises in Everyday Math, but it lacks a theoretical framework for the students to gain a deeper understanding. The early books of Singapore Math lay out a logical progression for addition, subtraction, and multiplication, which help to knock things into place. Our child is above average in quantitative skills, though. I don't know if it would work as well with other children.
"...sympathetic to the 'understanding first' camp."
But this is only the hypothesis. Where is the proof, or at least some examples? You can't look back at poorly taught traditional math classes and compare them to magically well taught reform classes.
I don't know what understanding first really means. For my son's Everyday Math classes, it meant "don't worry about mastery". By definition. This had nothing to do with bad teaching.
Do they have the same problem with "stacking" algorithms? Those also require you to line up the digits.
Yes.
The solution for that is to rotate the notebook paper by 90 degrees.
I don't know what understanding first really means. For my son's Everyday Math classes, it meant "don't worry about mastery
My friend K. came up with a perfect explanation of understanding without much knowledge.
She said what they're shooting for is: "That makes sense."
"That makes sense" is better than nothing, but it's superficial.
You can't look back at poorly taught traditional math classes and compare them to magically well taught reform classes
This, I think, is the heart of it. I am pretty much convinced that teaching reform classes well is harder than teaching traditional classes well. I also suspect that a well-taught reform class is marginally better than a well-taught traditional class, but both are sufficiently uncommon (and the former even more so) that the difference doesn't amount to much when you go to scale.
I think it's likely that a really good teacher can use any curriculum to teach well (for some value of "use"). I also think it's likely that a really bad teacher cannot use any curriculum to teach well.
But what about the margin? Most teachers are neither really good nor really bad*; is there a difference for these teachers?
From the available evidence, I think the answer has to be "yes", and that EM is not the beneficiary of that answer. Scattershot curricula do little or none of the work for the teacher, while more-structured curricula (like Saxon and Singapore) allow even a borderline ignoramus to have some positive effect on mathematical knowledge.
This turns out to be crucial**.
* My assertion. The population size is large enough, and drawn from an average-enough group, that a gaussian should work pretty well, so I think it's a safe assertion, but I offer it without proof.
** Yes, I would like a saucer of milk with that. 8-/
"I also suspect that a well-taught reform class is marginally better than a well-taught traditional class,..."
I don't even know what a well-taught reform class would look like, but I wouldn't use the word taught. The teacher is the guide on the side. Even the videos that purport to show how this is done correctly are awful. Remember the video about the teacher trying to lead a student or two to discover an answer. Other kids were hanging around waiting to talk to the teacher, and the rest of the kids were fooling around.
The best type of understanding or discovery learning would probably involve a lot of direct instruction, with the real discovery happening individually with homework. It sounds like a well-taught traditional approach.
Mr. Weiss,
Have you read Hung-Hsi Wu's work? He is writing math texts for math teachers, both high school and grade school. His website might help you decide about how to fix gaps in understanding fractions in your student.
http://math.berkeley.edu/~wu/
RE: this problem:Ask him what 8 * (1/4) is and he stares blankly. Have him write down (8/1) * (1/4) and he can produce the answer immediately. It is a classic case of "procedural competency" that masks an utter lack of conceptual understanding. Kids like that (and I have seen many) make me sympathetic to "understanding first" as a slogan.
There are a lot of things he doesn't understand that can be taught before "understanding". They are just so well understood by you that you've forgotten them. they must be made explicit to your student.
First: you must drill until automatic that 8 is 8/1 is 8*1. That means you must drill that 8 times 1 is 8, 8 divided by 1 is 8. It means you must drill that 8/1 is 8 divided by 1. It means you must drill that the fraction a/b is the value x such that a = bx. DRILL DRILL DRILL those until they are so automatic that even though he doesn't understand what it MEANS just yet that 8/1 is 8 divided by 1 is x such that a = bx, he does it and knows it. And later the understanding comes.
Then you must drill that 8 * 1/4 is the same as 8/4. Drill drill again. Drill until automatic. So automatic that sometime later, he can have the a-ha that those are the same means both are equal to 2, and all representations are the same.
Of course a curriculum doesn't teach understanding. Curricula should teach truths. The understanding comes from using those truths over and over and over again.
Another related idea to this "can understanding come first?" is this:
when faced with a problem that we don't understand, we can either trust our intuition, or our computation.
If we trust neither, we can't solve the problem.
If we trust our intuition but not our computation, we still can't be sure we solved the problem. The intuition can help guide our computation, but not compute. And in the end, if the computation and intuition disagree, we will abandon our solution but not know what to do to fix it.
If we trust our computation, however, then we can READ THE ANSWER from our computation--even if our intuition said otherwise. (consider any spec rel problem, or even tons of classical mechanics problems.) and we can correct our intuition to match our computation--we can teach ourselves to understand what the problem is telling us.
So mastery of procedural competency has its place: it alone can convince us to fix our logical or rational errors. But we've got to get to the point where there is something CONCRETE we are SURE IS TRUE first.
If procedural competency really truly solid, then it will lead you to correct your misunderstanding--or at least, at some later point in your maturity, allow you to say "huh, look at that. Guess I was wrong." But without it, you're at the mercy of your intuition--and if that's not yet well formed (and how would it be, if your proc. competency is weak?), you will never know when you've gotten the right answer, which answers to trust.
welcome to michael weiss.
lots of provocative comments here.
thanks for delurking.
p.s. the video was pointless
unless proof were needed
that people convinced of
the righteousness of their cause
don't necessarily make the best case...
"here it is; obviously everybody
will draw the same conclusion i did".
great. this is what i get for spinning
up a YouTube of course... i don't do
much of that and'll now maybe do
marginally *less*... like TV itself,
it's almost always *way too slow*
to make its points... if you can read,
well then you're much better off *reading*...
"DRILL DRILL DRILL"
This should be the motto of KTM.
I agree with everything SteveH has said so far except:
"I don't even know what a well-taught reform class would look like, but I wouldn't use the word taught...Even the videos that purport to show how this is done correctly are awful."
If you want to see some better examples, some of the videos on the Annenberg site have some good examples that would be considered "reform"; I'd trust Marilyn Burns to teach math to my children; and if you ever get a chance to go hear Deborah Ball, you should take it.
In a way, my position is similar to Michael's, only with less patience for the new curricula. I've seen some great stuff from individual teachers, and I think that's what the standards people are trying to bottle, but the curricula that have come out of it are really disappointing. (When your answer to hard math is not "how can I teach it more effectively", but "what can I teach instead" then you've lost my vote).
"...some of the videos on the Annenberg site have some good examples ..."
Are there any that don't require registration?
However, a video is not a curriculum. Which curricula do Marilyn Burns and Deborah Ball recommend? Are they curricula that would prepare a student to get into the College of Engineering at the University of Michigan, or into the School of Education?
"what can I teach instead"
But that's the definition of reform math. Adding a discovery-like presentation to "hard math" doesn't make it reform math, because hard math requires mastery of skills, no matter what the teacher and students do in class. The big problem with a steady diet of group class discovery (even if it's done well) is that takes too much time and you don't cover enough material.
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