As it turns out, our school district is using a controversial math curriculum called Everyday Mathematics, also known as "Reform Math." EM, as Everyday Mathematics is referred to by teachers, was developed by the University of Chicago, and according to their website, it is in use by about three million students nationwide. Here is one example of how simple addition "can" be performed using EM:
An example of what EM calls the "lattice method" for performing multiplication:
What becomes immediately clear is that several extra steps are now necessary to accomplish simple beeline computations. More steps will result in more errors -- only an idiot would claim otherwise. Eventually, EM students are taught four ways to add, five ways to subtract, four ways to multiply, and two ways to divide (traditional long division has been eschewed completely). Rote memorization is de-emphasized, and calculators (as well as estimating) are introduced in grade two.
Here is the basic rationale behind EM, directly from the University of Chicago website:
Research has shown that teaching the standard U.S. algorithms fails with large numbers of children, and that alternative algorithms are often easier for children to understand and learn. For this reason, Everyday Mathematics introduces children to a variety of alternative procedures in addition to the customary algorithms.
Links to or excerpts of said research are not provided -- we are to simply take these statements as fact. EM further claims to "make mathematics accessible to all students" by:
Incorporating individual, partner, and small group activities that make it possible for teachers to provide individualized feedback and assistance.
Encouraging risk-taking by establishing a learning environment that respects multiple problem solving strategies.
This couple is politically conservative, and as a result, the one thing they've got wrong is the idea that liberal parents like this stuff when they don't. Not for the most part.
This passage took me aback:
What's worse, the methods purportedly being used to convince school boards to adopt EM reek suspiciously of Rules for Radicals: *Never thought of these tactics in terms of Saul Alinsky
State that the traditional approach hasn't worked
Disparage testimony from those against the adoption as ideological and politically-motivated arguments
State that the success of any program depends on the teacher
Bring in teachers from affluent school districts as witnesses
Bring in a witness from a university
.Sheesh.
* Barry's article!
The addition method shown is the way that Singapore teaches the student to do mental math. But they also teach the traditional algorithm for pencil & paper work.
ReplyDeleteThe "lattice method" strikes me as a neat parlor trick but I haven't a clue why it works...
I have no objection to alternative algorithms per se. As a programmer, I don't use the same sort algorithm for all data; I pick the one that's best for a given problem based on the strengths and weaknesses of each.
ReplyDeleteIf EM taught kids to use different algorithms in different circumstances based on real algorithmic analysis, I'd support it, as long as they were careful not to spread themselves too thin. Even if all they wanted to do was take one class period and show some alternative algorithms for fun, I'd be fine with it.
But the fact that they aren't using algorithms the way they ought to be used (best tool for each job, practiced to mastery) irks me. I teach my kids different algorithms for paper and exact mental math, with tweaks for mental estimation. I tell them to treat EM algorithms as silly number games that their teachers will forget about a day or two after presenting them, so not to waste any time memorizing them. EM deprecates memorization anyway, so let's start by forgetting EM.
Another problem with the lattice method is that it can't be naturally extended to variables so learning to multiply two polynomials is a greater departure than it is for students schooled in the traditional algorithm.
ReplyDeleteIn addition, the lattice method gets ugly for larger numbers a lot faster.
The education world is dominated by very artsy people. For them, it's not about "the best tool for the job", but more like "the tool that best expresses my personality/mood/needs/etc"
ReplyDeletehttp://mathworld.wolfram.com/LatticeMethod.html
ReplyDelete"Although the process at first glance appears quite different from long multiplication, the lattice method is actually algorithmically equivalent. "
it's actually quite elegant, because place-value carrying works out more efficiently.
now of course it's less transparent -- but is the long multiplication method even that transparent anyway?
I remember in KUMON -- then anyway -- I frequently lost my place for very large factors and would have to backtrack a lot. But then of course, I grew up.
I'm not sure if the process gets uglier for large numbers faster.
'Long multiplication produces the correct answer because multiplication is distributive over addition for the set of real numbers R. '
but do teachers and kids know that?
I mean the reason //why// it worked only hit me two years later after I was first taught it (in first grade).
is the distributive property really that hard for first graders to grasp? For elementary school graders it would be such a fun toy. (Look at how it comes out to the same thing!)
"The education world is dominated by very artsy people. "
ReplyDeleteDisagree.
being artsy is about being innovative and rebellious. and usually comes with problem-solving abilities.
the education world is full of conformism and anti-creativity, and people who do not /solve/ problems, just resolve problems they have already learnt.
sure sure, they say they prize creativity and whatnot, but for some reason when they innovate, they don't design anything fundamentally new.
at TJHSST, Stuyvesant, etc. being artsy is seen as chic. being artsy goes hand in hand with being sciency or mathy.
the people who come up with elegant proofs or a very elegant way of designing an experiment.
Avery–MacLeod–McCarty experiment and the Hersey-Chase experiment -- heck polymerase chain reaction -- artistic, elegant, beautiful.
there is a property inherent a lot of physical systems that is also inherent to aesthetics: emergence.
I encountered the "Math Wars" when I taught high school in the late 1990s. It sent me back to university to improve my credentials so that I could work with people who taught real math. The university was much more focused on mathematics as I understood it to be - I stayed and got my Masters and eventually got a job teaching community college.
ReplyDeleteWe're making a move to use an adaptive software, which I'm fine with because it can individualize the curriculum for the students, which I can't do - BUT, tagging along with the content and the software is a host of "methods."
This is a real problem. Together with this, I now see a new movement called "standards-based grading." As I understand it (and please correct me if I'm wrong) "standards based grading" states that one and only concept should be tested at a time so that if the student gets the problem wrong, we will know which concept they don't understand.
I really see this as a profound misunderstanding of mathematics and math education.
To me, the real value in mathematics has always been the interconnections between the concepts. The real skill in problem-solving is about making decisions about what tools to bring to bear on the problem and in what way.
I'm standing firm and it appears that I'll be able to continue make the important decisions about what happens in the classroom, but it ain't easy!
I don't know how American exams /used/ to be, but the Singapore primary math exam has 3 sections -- Section A, Section B, Section C. Section C is worth 60% of the whole paper.
ReplyDeleteSection A used to be worth 15% of your grade, and the MCQ questions they put in there gave too few points for the amount of work involved -- this included some of Section B (short answers) too.
But now I realised their game. Concept-testing. Section A was really a "debugging" section.
Actually my professors often do the same thing, but often they generously make it 50% of their exam.
Anon 11:29 -- yeah, "standards-based grading" along with rubrics and other assessment joy are coming our way too. It does seem like you have to somehow assign every problem to a single concept, which means you can't ask interesting, multi-layered questions (which are appropriate at the college level).
ReplyDeleteOne trick -- if asked to defend such a question, tell the assessors you are assessing critical thinking. Since no one knows what that means anyway, it usually stops them in their tracks.
This sounds like a cover for equality..certain students can't do multistep problems, so eliminate the multistep problem.
ReplyDeleteLe Rad: [The lattice method] it's actually quite elegant, because place-value carrying works out more efficiently.
ReplyDeleteIn fact, I agree with this, but it's less efficient for the type of problems I would probably do on paper because of the overhead of drawing the lattice. Low digit-count problems are the vast majority in math problems. Of these, the majority are 1x1, which should be done by rote memory. The remainder have too few digits to justify the lattice overhead.
Once the digit count gets large enough that carrying and keeping things in columns becomes unwieldy enough to justify the lattice overhead (say, real science or engineering), it's time to switch to a calculator, spreadsheet, unix command line, or whatever is within arm's reach. I'll train the kids to use these tools (for science, not for math) once they reach algebra.
If I had lived a century ago, I might have used the lattice method for science and engineering calculations but, ironically, in a 21st century filled with calculators, it is the lattice method, not the "traditional" method, that now serves no practical role.
Could someone explain to me how the lattice method shows place value? I see where the digits of the multiplicands is written across the top and those of the multiplier are written on the side. But why does the diagonal adding of the digits on the interior work to solve the problem?
ReplyDeletethe lattice method uses a lattice to carry place-value information rather than leaving the problem-solver to mentally carry the place-value himself.
ReplyDeletethus, you see 8x3 = 24. but because of the position of the lattice, this is really 8x30 = 240. but you realise, because of commutativity, that 80x3 = 8x10x3 = 8x3x10. The lattice method makes use of commutativity and associativity, whereas the long multiplication method doesn't use them so prominantly.
also remember this works nicely with non-whole numbers. much much better than long multiplication when you have to deal with problems like 2.419 cm^3 x 36.56 g/cm^3, and you didn't bring your calculator to lab. Which happens a lot in chemistry.
for some reason, my brain plays nicely with 90x60 = 3600! Intuitively but not with 0.004 x 0.009 = 36 x 10^-5. = 3.6 x 10^-4 I have to start counting zeroes. Horrible stuff.
now of course long mult. should be the default in teaching, and you pull out the lattice method as something based on the long mult. method.
ReplyDeletelook at the way the diagonal lines are drawn. Numbers within the diagonal lines have the same place value.
There are 5 diagonal lines. we see it's derived from an mxn matrix. here place value computations are built into the lattice -- all the problem-solver has to do are the "significant figure" computations.
I think this might work out nicely if you are also working with clock arithmetic and clock multiplication, with a few modifications.
suppose we look at a 88x33 problem. the symmetry is intentional.
ReplyDelete24
240
240
2400
now do it the same way, the lattice way:
you just have to turn the lattice 45 degrees to see it's exactly like the long mult. method, but with a few more lines to store place value more efficiently.
8x30
and 3x80
come out to the same thing, so they are stored in the same diagonal line
the expectation is that say multiplying an m digit number by an digit number will give you an (m+n) digit number most of the time, and sometimes an (m+n-1) digit number.
so you can view any product of an m-digit number by an n-digit number as simply an addition of those two types of numbers. so each cell is divided into two, separated by a diagonal. and the first diagonal is (m+n-1) (that is: 1+1-1 = 1).
each (m+n+1)th diagonal is holding information of the sum of its digits *(m+n-1). But what happens when you multiply an n-digit number by an m-digit number? n+m = m+n, so both types of calculations are included in the same diagonal.
should you spillover, you carry it to the next diagonal.
it's a lot simpler than it sounds. or looks. the diagonals are the equivalents of columns. Because some of the place values end up on the sides rather than the bottom, it looks a little weird, that's all.
btw -- whoever said this doesn't work for variables -- long multiplication doesn't work for variables either!
ReplyDeletebut the lattice method works in helping analyse some pretty abstract problems by induction, especially problems with symmetry. like when an Indian teacher who tries to make you do
1998^1997 = 1997^x
solve for x.
in long multiplication and the lattice multiplication method, you make use of the fact that multiplication is nested addition. and then you add those nested additions.
ReplyDeletein looking at exponent problems like
1998x1997 = 1997^x
you can use the lattice method to help you organise analogously, with a few tweaks. here, you make use of the fact that exponents are products of products.
you will readily see that your calculator, even google calculator, will refuse to do 1998^1997. Only Mathematica is robust enough to computationally solve this problem. Computationally.
Physics Teacher wrote:
ReplyDelete>Another problem with the lattice method is that it can't be naturally extended to variables so learning to multiply two polynomials is a greater departure than it is for students schooled in the traditional algorithm.
No, in fact it extends quite naturally to (single-variable) polynomial multiplication: it may even be faster than the standard approach.
The problem is that it is far from clear to most people (including, it seems a physics teacher (!) ) why why it works in that case.
The paradox here is that the enemies of the standard algorithms claim to be teaching “understanding,” but one of the reasons the standard algorithms have survived is that they have proven fairly easy for ordinary people to understand and implement.
As Crimson Wife implies, it indeed is often easier and faster to do mental addition the non-standard way illustrated in the original post.
And, as Glen says, it is often wise in programming to use non-standard algorithms: indeed, one definition of a good programmer could be someone who is good at inventing (good) new algorithms to carry out a task. Anyone who doubts this should google “fast Fourier transform,” “Grobner bases,” or “bubble sort.” (Bubble sorts are examples of intuitively clear algorithms that are in fact very bad, because they are very slow – I was really shocked some years ago when a comp-sci friend showed me how bad a bubble sort is.)
All this illustrates how really confused the “math wars” have become. The real reason for teaching the “standard algorithms” is not because they are computationally efficient (which often they are not) but because they are pedagogically sound. The standard algorithms are usually the best way to teach a real understanding of what is happening.
Dave
the Singapore Math Olympiad (junior! for 14 to 16 years old) also would ask you to find all solutions to a certain problem that were palindromic.
ReplyDeleteuse of calculators was freely allowed. they usually didn't help you.
I distinctly remember one problem where we had to look at the long multiplication algorithm on the board. What changed everything?
Something like
4x4 * 6y = xy^x (x and y are variables that represent digits, not quantities!)
I have to find the exact problem. but the lattice method might have been useful.
I just remember being at a loss at that time, because I had been taught to solve for variables, not place-values or digits!
ReplyDelete"The problem is that it is far from clear to most people (including, it seems a physics teacher (!) ) why why it works in that case."
ReplyDeleteYes, I was only a physics teacher, as opposed to a "physicist", such as yourself, and so I'm clearly in an inferior intellectual league. And, it is unclear to me how to go from the lattice method to polynomial multiplication. Perhaps you'd be kind enough to actually demonstrate.
When I learned to multiply polynomials in hs it was easy to do because it was a natural extension of the "stacking" algorithm, except that "like" terms were combined instead of those of equal place value.
Perhaps I'm just too stupid to make the connection that you're able to make. But, apparently, so are the vast majority of hs students, even the top of the class, who are completely unable to deal with the simplest algebraic expressions forcing people like me to dumb things down or remove math altogether.
This forum is isn't about the correctness of math algorithms, but the efficacy of teaching them. If you were to spend some time in hs classroom you'd see how effective EM really is.
Place value. Instead of having a tens place or a tenths place or a hundreds place, you have variables.
ReplyDeleteIn fact here it is REALLY useful because xy = yx and yx^2 = x^2y and you can see that if you ever tried to work out a trinomial by hand it gets really ugly and so forth.
and because you can't compare oranges to apples -- there is no carry over!! that is you can get a term 15xy and will remain 15xy and not say 5xy + 10x^2y. Even /more/ convenient.
"being artsy is about being innovative and rebellious. and usually comes with problem-solving abilities."
ReplyDeleteLet me try again. Most people in education come with English or Art degrees. At least they seem over-represented in education schools and in administration. They may not fit your definition of "artsy", but that's often how they fancy themselves. As a teacher I've seen display cases filled with models of molecules that are decorated with all sorts of "fun" colors, and this is the sort of "creativity" that takes center stage in k12.
The physical universe imposes constraints on us, and creativity in science and engineering demands that we understand what those constraints are. Many people in education have no understanding of this and they believe that crafting a mathematical proof is no different from designing a fun new prom dress. Do the contestants on Project Runway need to know much beyond their own moods and preferences? Can we say the same about biochemists and engineers?
I'm probably a little biased here, but my best friend's good friend (both from TJHSST) is studying fashion at RISD (ranked the #1 design school in the US).
ReplyDeletenow when she explains things to ELLE, or sells her clothes in boutiques she doesn't exactly pull out her probability density functions and n log n's, but she doesn't think her time at TJ is a waste (ranked #1 public school in the nation, which you must realise is dominated by math and science).
absolutely -- constraints. take graph theory, or the Seven Bridges of Königsberg problem, proven to have no solution by Euler, or in fact many physics problems, and you shall see that the many strategies involved have become useful for her.
aesthetics has constraint.
the most beautiful thing is designing what can be explored under constraint.
"Do the contestants on Project Runway need to know much beyond their own moods and preferences?"
Project Runway requires pandering to the judges, but yes. You must bear in mind it's TV of course.
balance, solving for optima.
I came out of high school thinking I would major in linguistics. I will probably still do research in linguistics, heavily connected to evolutionary biology or neuroscience, but at that time, my teachers couldn't see why I aggressively pursued linear algebra or higher math.
Ah, but they did not see! And at that time I knew they were intimately connected, even before I came across that PNAS paper using a Markov chain analysis of language evolution, or was exposed to the intense math of evolutionary ecology which I later used to model evolving linguistic systems.
Why would a fashion student at RISD still pursue higher math courses at Brown University? Do you think that they are completely unconnected?
the arts and math are very much connected.
deduction.
ReplyDeleteinduction.
evolution.
revolution.
the process of design. how are parameters similar? how are they different? what is fundamentally different? what fundamentally characterises certain similarities?
you've probably heard of the Taniyama-Shimura theorem -- you know that thing that unites elliptic curves over rational numbers with modular forms and you know, was in turn linked to Fermat's Last Theorem.
in setting out to achieve a proof and understand the relationships at play, the process of design is involved.
and the same mental stack of dominoes that gets trained to prove by induction also is involved in design, and the design of expression.
when cutting up a dress, and sewing pieces on together, you must think of interactions between elements, their pairs, and the sum of those pairs, its distributions.
There is a sense of, "what I feel" or "what others feel" -- but the analysis goes deeper: what exactly characterises that feeling? What are its elements? Is it related to anything? Can it be abstracted, represented? How far do the representations extend into space?
Perhaps, a designer must work with discrete elements, rather than infinitesimal dx's or dy's -- or must she?
Designing an asymmetric catalyst meant to make chiral substances, or designing enzymes, or finding how all the runaway pathways of say, a polymerisation process, or combustion, converges or diverges in product distribution, or designing an experimental technique to MAP the routes involved in a given product spread to caramelisation -- I say, this is art.
Dear Le Radical,
ReplyDeleteI certainly hope for your sake that you never decide to become a k12 teacher. The job openings at places like TJ are few and far between, even for someone with your obvious knowledge. Trust me. I worked for FCPS and everyone looked at job openings at TJ and there never were any.
Regarding our little discussion. You keep getting it backwards. You keep giving me examples of people who are knowledgeable going into art fields. I'm talking about people with art/lit backgrounds making pedagogical decision that affect non-art type fields.
Look at portfolios. An English teacher can collect haikus written by students. But if 100 kids do a hs physics problem correctly then all the solutions will be nearly identical, except for handwriting. There's no way to even be sure whether anyone plagiarized, and there's no point in cluttering up time and space with portfolios in the physical sciences. Yet every teacher in every subject in every class is supposed to keep (nearly) everything ever "created" by every student. Whose idea was this?
Many supporters of portfolios think that when MIT picks out its next freshman class out of tens of thousands it's supposed to have nothing to go on except portfolios. Presumably, these portfolios will be delivered by tractor-trailer and the administrative staff will have to work through lunch for a few days. How many comparison operations must performed to sort 10,000 items? Suppose we partition the problem among various teams. What kind of information must be passed back and forth among the teams to get an accurate sort? How do we determine a comparison operator between arbitrary portfolios created by numerous teachers you've never met?
I'm sure you can answer these questions. But the supporters of portfolios have never contemplated it. Which is exactly my point.
Bloom's Taxonomy: If you're working out a mechanics problem and you apply Newton's 2nd Law, does the word 'apply' necessarily appear on your paper? The education experts seem to think it does because that's how your teacher is supposed to figure out the level at which you're thinking. I went to an "inservice" -- for science teachers no less -- on Bloom's Taxonomy. The session was ALL about the words kids use, or are supposed to use. It didn't dawn on the brass that the language of math/physics isn't necessarily English.
lrg may already know this, but just in case . . . Mark Liberman, a linguistics professor at Penn and a chief proprietor of the linguistics blog Language Log, was once a department head at Bell Labs, when Bell Labs was the stuff of legend. I don't know exactly what he did. But he and Geoff Pullum, and their LL associates, offer penetrating and often funny analysis of the many ways language can be misused to sell dubious theories, and that's highly relevant to the discussions here.
ReplyDeleteMIT accepts portfolios of course.
ReplyDeleteIn the form of recommendations and research abstracts. And math olympiad results -- you really need oodles of creativity to succeed on those. ;-) Competitions, where the results are based on months' worth of work on a robot, or some form of design....
"I'm talking about people with art/lit backgrounds making pedagogical decision that affect non-art type fields."
I think you need both art and science to succeed. As an innovator and researcher anyway.
The realistic thing for me to be as a student in a biochemistry track to be a chemist. Preferably in some private firm, either working on proteins or maybe on food flavors.
I do dream though. Maybe, just maybe, I'll get into Teach for America. I do desire to be in K-12 education reform....because the high school I went to was not at all like TJ. TJ is a dream compared to where I went to.
"I don't know exactly what he did."
So THAT explains why he's such a mathy information theory buff.
In response to Physics Teacher's question:
ReplyDeleteAnd, it is unclear to me how to go from the lattice method to polynomial multiplication. Perhaps you'd be kind enough to actually demonstrate.
Doing this in text only is difficult, because of the nature of the algorithm. But let me describe the method with an example. If you want to multiply, say, (x^3 + 2x^2 - 10x +4) times, say, (3x^2 - 10x + 1), you set up a grid with five columns and three rows. Write the terms of the first polynomial as labels above the columns, and write the terms of the second polynomial as labels for the three rows.
Now in each of the fifteen boxes write the product of the row and column labels.
Notice that, unlike the arithmetic version of the lattice algorithm, there is no diagonal line dividing each square cell into an "upper" and "lower" part. That is because you do not "carry" when multiplying binomials. Nevertheless, like terms do line up along the diagonals of the grid, so adding those cells together provides an efficient way to find the coefficients of the simplified expression.
The whole idea behind the lattice method is to use space to organize the distribution of terms. One needs to multiply each term of the first polynomial times every term of the second polynomial. That can be a lot of combinations -- multiplying a fifth-degree polynomial times a fourth-degree polynomial requires (in general) computing 30 separate products, then gathering the like terms, and ending up with (in general) a ninth-degree polynomial with 10 terms. As the length of the polynomials grows, the likelihood of a student accidentally omitting one or more of those combinations grows too.
Both the lattice method and the stacking method use the spatial arrangement of the terms to ensure that no combinations get missed. In that sense they both play the same role as the FOIL mnemonic: it's just a way to make sure you haven't missed any of the combinations.
What I think this whole discussion illuminates is the distinction between procedure and theory. NEITHER of the two algorithms, no matter how well mastered, confers understanding: it is perfectly commonplace to master the standard stacking algorithm without ever giving a moment's thought to why it works, and exactly the same is true of the lattice algorithm. Indeed a good case can be made that this is precisely why we have pencil-and-paper algorithms: they make it possible to compute complex results without having to constantly stop and think about the meaning of what you are doing. The spatial arrangement of digits on a page allows you to stop worrying about (for example) whether a particular "2" stands for two, or twenty, or two hundred. Put it in the correct column and everything will work out all right.
Should students understand why an algorithm works? Yes, definitely.
If s student does not understand why an algorithm works, should he/she master it nevertheless? Yes, definitely.
Is one algorithm "better" for helping cultivate understanding? Not in my experience.
Does exposure to more than one algorithm help? It depends. Learning a new method for doing something you are already proficient at can sometimes have the effect of making you look more carefully at what you already (think you) know. On the other hand, learning to do the same problem three ways does not magically confer any deeper insight into the structure of our number system.
In response to Physics Teacher's question:
ReplyDeleteAnd, it is unclear to me how to go from the lattice method to polynomial multiplication. Perhaps you'd be kind enough to actually demonstrate.
Doing this in text only is difficult, because of the nature of the algorithm. But let me describe the method with an example. If you want to multiply, say, (x^3 + 2x^2 - 10x +4) times, say, (3x^2 - 10x + 1), you set up a grid with five columns and three rows. Write the terms of the first polynomial as labels above the columns, and write the terms of the second polynomial as labels for the three rows.
Now in each of the fifteen boxes write the product of the row and column labels.
Notice that, unlike the arithmetic version of the lattice algorithm, there is no diagonal line dividing each square cell into an "upper" and "lower" part. That is because you do not "carry" when multiplying binomials. Nevertheless, like terms do line up along the diagonals of the grid, so adding those cells together provides an efficient way to find the coefficients of the simplified expression.
The whole idea behind the lattice method is to use space to organize the distribution of terms. One needs to multiply each term of the first polynomial times every term of the second polynomial. That can be a lot of combinations -- multiplying a fifth-degree polynomial times a fourth-degree polynomial requires (in general) computing 30 separate products, then gathering the like terms, and ending up with (in general) a ninth-degree polynomial with 10 terms. As the length of the polynomials grows, the likelihood of a student accidentally omitting one or more of those combinations grows too.
Both the lattice method and the stacking method use the spatial arrangement of the terms to ensure that no combinations get missed. In that sense they both play the same role as the FOIL mnemonic: it's just a way to make sure you haven't missed any of the combinations. (cont'd...)
What I think this whole discussion illuminates is the distinction between procedure and theory. NEITHER of the two algorithms, no matter how well mastered, confers understanding: it is perfectly commonplace to master the standard stacking algorithm without ever giving a moment's thought to why it works, and exactly the same is true of the lattice algorithm. Indeed a good case can be made that this is precisely why we have pencil-and-paper algorithms: they make it possible to compute complex results without having to constantly stop and think about the meaning of what you are doing. The spatial arrangement of digits on a page allows you to stop worrying about (for example) whether a particular "2" stands for two, or twenty, or two hundred. Put it in the correct column and everything will work out all right.
ReplyDeleteShould students understand why an algorithm works? Yes, definitely.
If s student does not understand why an algorithm works, should he/she master it nevertheless? Yes, definitely.
Is one algorithm "better" for helping cultivate understanding? Not in my experience.
Does exposure to more than one algorithm help? It depends. Learning a new method for doing something you are already proficient at can sometimes have the effect of making you look more carefully at what you already (think you) know. On the other hand, learning to do the same problem three ways does not magically confer any deeper insight into the structure of our number system.
"'The problem is that it is far from clear to most people (including, it seems a physics teacher (!) ) why why it works in that case.'
ReplyDeleteYes, I was only a physics teacher, as opposed to a 'physicist', such as yourself, and so I'm clearly in an inferior intellectual league."
For what it is worth, I read the original quote a bit differently. I read it as something like, "The lattice method is so difficult to understand that even people like physics teachers who are expected to understand difficult stuff have problems with it." With the implied, "and if *those* people, who are pretty good at math, don't get it, how the heck is a non-specialist K-5 elementary school teacher expected to understand it."
Not desparaging (sp?) at all.
-Mark Roulo
Weiss: it also works robustly for any number of variables (y, z etc.), as long as they are consistently arranged!
ReplyDeleteI wonder if that "diagonal space" or something like it can be used to carry information. Do you really need a three dimensional lattice to compute a trinomial for example?
Sorry for the multiple post. Blogger told me my post was too long, so I reposted it broken up into two pieces -- and then all three appeared.
ReplyDeletele radical galoisien: "Consistently arranged" is, alas, a tricky (perhaps impossible?) thing to arrange when more than one variable is involved. Of course the lattice still works as a way of organizing all of the monomial products; the tricky bit is getting like terms to line up along a diagonal. If the terms of the two polynomials have two variables that vary in degree independently, I don't see any way to make the like terms line up along a diagonal.
it's rather arbitrary. I already did it. multiplying an x^2 term by x^2 term is no different than multiplying an x^2 term than a y^2 term. You can't group the x^4 terms with the x^2 terms anymore you can group the x^2*y^2 terms.
ReplyDeleteTo figure out what the coefficients are, simply check out what the factors are for a single cell in the diagonal.
suppose you're multiplying (x^3 + 6x^2 + 4) * (7x^2 + 8x).
ReplyDeleteif you called x^3 "y" there would be no difference, simply that 7yx^2 would be the first term rather than 7x^5, etc.
OK, actually it would affect the grouping slightly. But I see no reason why multivariables can't be used.
ReplyDeletePhysics Teacher wrote to me:
ReplyDelete>Yes, I was only a physics teacher, as opposed to a "physicist", such as yourself, and so I'm clearly in an inferior intellectual league. And, it is unclear to me how to go from the lattice method to polynomial multiplication. Perhaps you'd be kind enough to actually demonstrate.
Ah, I think you mistook my point. My point was that, presumably, you are indeed much better than the average person at seeing such connections, so if you cannot see it, it is unreasonable to think that most ordinary people will see it.
Our place-value system is basically a polynomial representation of numbers. Pretty much anything that words for polynomials will work for our place-value system and vice versa, with the twist that the place-value system has the “carrying” problem, which polynomials do not have.
So, do the lattice thing with, say, 3x + 2 times 2x + 1, doing this just as you would do 32 times 21 on the lattice. Works exactly the same way.
You also wrote:
>When I learned to multiply polynomials in hs it was easy to do because it was a natural extension of the "stacking" algorithm, except that "like" terms were combined instead of those of equal place value.
Of course, so did I, and I agree that this is the pedagogically sound way to teach it.
On the other hand, dozens of times, I have had to implement polynomial multiplication in software (I find it easier to just recode each time rather than search for my old code), and something resembling the lattice algorithm is in fact easier in software than the standard algorithm: You take advantage of the math fact that the coefficient of the mth term of the product is sigma with respect to i of a(m-i) times b(i). This is essentially a convolution, and that connection is useful. There are even more computationally efficient, but more complicated ways of doing this – I had to work through a lot of this stuff when I was co-designer of a Galois-field arithmetic processing unit in hardware.
Yes, I know more about this than you and most other people just because I happen to have had the need to teach it to myself for my work.
You also said:
>This forum is isn't about the correctness of math algorithms, but the efficacy of teaching them. If you were to spend some time in hs classroom you'd see how effective EM really is.
Didn’t I make precisely the same point in my earlier post? To wit:
>All this illustrates how really confused the “math wars” have become. The real reason for teaching the “standard algorithms” is not because they are computationally efficient (which often they are not) but because they are pedagogically sound. The standard algorithms are usually the best way to teach a real understanding of what is happening.
Yes, I really do know more about this than you because of the work projects I happen to have had. But I strongly agreed with you that the standard algorithms, not the lattice approach, are pedagogically sound!
The fact that even a physics teacher finds the lattice approach hard to grasp shows why the lattice approach is pedagogically unsound. I apologize if you did not see that that was my point: I was not trying to attack you personally but to point out that if you do not get it, then surely the average students will not get it.
Incidentally, I too found it hard to see why the lattice approach worked the first time I saw it, even though it is similar to the approach I had already used for multiplying polynomials, and I am pretty sure that I would have found it very difficult to understand why it worked when I was a grade-school student.
Algorithmically, the lattice approach is mildly interesting. Pedagogically, it is a disaster.
We agree, yes?
All the best,
Dave
A note about the history here: the lattice method was how multiplication was taught up to about 1600; the algorithm we view as standard was adopted about 1600 because it was easier and cheaper to typeset.
ReplyDeleteThe history of algorithms can be interesting, but is seldom of anything MORE than historical interest. "Progressive" folks often bring up the age of lattice multiplication as if that might placate us simple-minded "anti-progress" types. "See, it's old-fashioned, just like your way of thinking, so you don't have to reflexively oppose it anymore!" (I'm commenting on this point in general, not replying specifically to Anonymous, who may simply be bringing up an interesting item of history.)
ReplyDeleteBut many of us have formal training in algorithmic analysis and are always in search of a better algorithm, and we just don't find lattice multiplication efficient for either learning or doing small-digit-count, paper-and-pencil multiplication. Regardless of its age.
But many of us have formal training in algorithmic analysis and are always in search of a better algorithm, and we just don't find lattice multiplication efficient for either learning or doing small-digit-count, paper-and-pencil multiplication. Regardless of its age.
ReplyDeleteHow do you feel about long digit count problems? (To sweeten the pot, I'll throw in pre-printed "lattice paper" so you don't have to draw the grid lines and diagonals.) Is there a point at which the lattice method catches up with or passes the stacking method in terms of its computational efficiency?
And I think the key part of Anonymous's comment had to do with why stacking replaced lattice: Not because of algorithmic efficiency but rather because of typesetting restrictions.
Michael Weiss wrote:
ReplyDelete> How do you feel about long digit count problems? (To sweeten the pot, I'll throw in pre-printed "lattice paper" so you don't have to draw the grid lines and diagonals.)
Actually, there is a very good algorithm which is similar to the lattice method, but which avoids drawing the lattice. For example, to do 23 x 76, you note that the one’s place is 8 from the 3x6 and carry the 1. Then you note that the tens' place is 2x6 + 3x7 plus the carried one, giving a 4 and carry the 3. For the hundreds' place, you get 2x7 plus the carried 3 or 17.
Final answer: 1748.
Doing this on paper takes less space and tends to be faster than the standard algorithm. The accumulated products get rather large, though, for, say, five-digit numbers.
This method was often taught as an alternative to the standard algorithm (after kids had already mastered the standard algorithm) at least as late as the 1960s: I learned it in high school around 1970 (though not in class).
It is basically the same way I multiply polynomials in software.
The reason this was not traditionally taught before the standard algorithm was that there was a broad (and, in my judgment, correct) consensus that most people find the standard algorithm easier to understand.
If I get a chance, I’ll post an algorithm that really is weird, but that is actually much more efficient and often used in digital-systems applications: curiously, not even all DSP engineers know about it, even though it is widely used.
Dave
With large digit count problems, I do what every scientist/engineer does and estimate in my head, or use scientific notation plus stacked multiplication to reduce it to fewer digits and get a more precise estimate on paper or use one of the ubiquitous computing machines that fill my pockets, lap, backpack, etc.
ReplyDeleteThis is because I'm talking about best tools for real-life jobs. For mental work, which is a very significant real-life use case, I seldom use the standard stacking algorithm, and certainly not lattice, but use a variety of alternative algorithms, including the one Dave mentions.
HOWEVER, as I've mentioned before, if I were living a hundred years ago and had to carefully maintain many digits of precision for real work, and had no access to machine calculators, I would be much more interested in lattice method for large problems. I don't know where (if anywhere) the breakeven would be, so I would gladly take you up on your offer of pre-printed lattice paper and run some experiments against various stacked algorithms (including using pre-printed stacked grid paper), judging all for speed, accuracy, and effort. If you have a lot of work to do, it pays to invest some time in choosing, and learning, the right tool. Lattice might win by, for example, reducing the likelihood of carrying errors in large problems--or not. I don't know which would have won back then, but I would guess that any case where lattice was the clear winner has probably been eliminated by machines today. (Long polynomial algebra might still be an exception until Pocket Mathematica is ubiquitous.)
And as for *why* stacking replaced lattice, I'm afraid I'm skeptical of folk historical claims. I don't doubt that some printer made such a decision; I doubt that this explains the outcome. When you consider the many scientists and engineers over the centuries who, before machine calculators, spent countless thousands of hours of their lives toiling away on vast calculations, it's hard to imagine that the convenience of book printers was the highest consideration. Those same book printers managed to produce geometry books containing more than "ASCII-art" diagrams, so they could easily have thrown in a diagram or two illustrating lattice method examples, followed by pages of problems laid out the ordinary way, vertically or horizontally. You don't need to print a lattice to ask: 234 x 456 = ?. Printers who made the small extra effort to teach what "industry" wanted would have sold books. If that had been lattice method, printers would have been creating a market for reams of pre-printed lattice paper.
I strongly suspect that lattice faded away as the default algorithm because of the inconvenience to users, not printers.
This is a little off-topic for this thread, but...
ReplyDeleteMy school is in our 2nd year of implementing EM... and I now affectionately refer to it as BM.
Someone should write a guide for EM teachers who want to teach real math while appearing obedient. If EM can pretend to teach the California math standards, then California teachers could pretend to teach EM with some guidance on how to reverse the spin.
ReplyDeleteOne could teach EM reasonably well except for the fact that by third or fourth grade, everyone is all over the place in terms of mastery. My take on the lattice technique is that it's fine. Just master the damn thing. Master something! Do more "math boxes" if you have to. Ensure that learning gets done. All of the talk of pedagogy and understanding works as great cover for low expectations. Schools love EM. Do all of the full inclusion you want, trust the spiral, and then in 7th grade, blame the kids, or parents, or society. Just look at all of the kids who do well, but never ask what parental or tutoring help they get.
ReplyDelete