Update:
A poll has now been created in the sidebar! Please express your preference. (Thank you, Jonathan, for making this suggestion).
As if the educational fate of our children could be determined by a poll of our readers...
Hey, you never know.
What's even more incredible is that individual teachers, schools, districts, and states still feel that this is a 'local' decision. After over two decades of this debate, children and their parents still have little to say about these kinds of curricular decisions that will impact on the mathematical futures of another generation. After all, the experts know what is best, right? Well, there are many many experts who all feel sure they know the answer to this question. Trouble is, they do not all draw similar conclusions and spend most of their time defending their choices or their particular agenda or favorite set of materials.
Finally, MathNotations will settle this debate once and for all. The results of this survey (assuming a minimum of 3 responses) will be used to influence national policy for years to come!
Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5 :
(A) Teach only the traditional algorithm and expect mastery
(B) Teach the 'partial products' method to develop understanding of place value and the traditional algorithm; teach the traditional algorithm as a more efficient method and require it; expect mastery
(C) Teach the 'partial products' method to develop understanding of the traditional algorithm; teach the traditional algorithm as a more efficient method; give students a choice of methods; expect mastery of at least one method
(D) Model other methods (e.g., 'lattice method') and encourage students to invent their own method; do not require any particular method or mastery
Now, don't miss this opportunity to be part of an historic decision. Your vote does count...
Saturday, January 26, 2008
43x27? Which Algorithm should be taught? A MathNotations survey will end the debate!
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28 comments:
B+ Multiple distribution. Partial Product. Standard algorithm (and bring standard algorithm to mastery)
E) but close to B)
My vote is:
1) Teach the partial products method on 1x2 and 2x2 problems (that is a two-digit number times a one or two-digit number). Introduce but do not yet expect mastery of the distributive property of multiplication over addition, which is why it works. Expect comprehension (on an age-appropriate level) and mastery.
2) Demonstrate that partial products scales poorly to larger problems, such as 3x2 3x3 and 4x2. Demonstrate the traditional algorithm, showing that it is a shortcut for the partial products method. Expect comprehension, mastery and automaticity.
3) Much later, like pre-algebra or algebra I, teach the lattice, egyptian and whatever other methods float your boat. Make students figure out why they work. At this point, you shouldn't be paying attention to how students multiply when they don't have a calculator handy, so if they want to use an alternative method, that's fine.
Great response, Jonathan! You vote has been duly recorded.
As of this moment:
(A) 0
(B)* 1
(C) 0
(D) 0
(E) 0
I hope I'll be able to keep track of all these. My calculator just broke.
Thanks, Mathmom! I wrote the previous comment before reading your vote. Now I have to recalculate the tally. This must be how the presidential candidates feel this evening!
(A) 0
(B)* 2
(C) 0
(D) 0
(E) 1
Note: B* is 'modified' B. In some cases, votes will be double-counted.
Wouldn't this depend on the level/grade of the student targeted?
For younger students, I assume it is easier to grasp methods, rather than why they work -thus the choice may be A. Older students can understand concepts such as the distributive law better, and may be more amenable to B or C, and some may even appreciate the beauty of the application of the basic law to the computational method, but by this time, they should have mastered the traditional method anyway. More experienced students can be exposed to D, and might be able to actually use some of these methods when the occasion demands it.
Of course, there is method Z, which is just keying the numbers into a calculator - which is what most people seem to use later in life anyway.
TC
Hey folks...
Please feel free to make comments here but please, please vote using the new poll in the sidebar of this blog. I will run the survey through Feb 29!
Tc - you're absolutely right! The poll refers to grades 3-5. I should probably clarify that.
Through February 29, as in the Florida primary date? I hope all the votes will be counted!
Sherman--
Ah, you saw through the method to my madness. I've already conferred with my poll advisor, Chad, and he says it'll all be hangin' fine...
I vote for a modified C: Partial products sets you up nicely for polynomials, and they reinforce place value. I think with a little encouragement--wheich should be provided--most children will move to the more efficient standard algorithm, but if they don't it's not a show stopper, so being proficient at partial products is sufficient, even though it's not particularly efficient.
totally_clueless: my daughter in fourth grade could do partial products (and understood it) up to 4digit by 4 digit. It took maybe 6 months of exposure until she felt comfortable moving to the standard algorithm. I'm not trying to say that the partial products algorithm is easier for all kids (statements about all kids are dangerous) but it is for some.
If you think closely about the standard algorithm, you'll see what my daughter had trouble with. She was OK with the standard algorithm for multiplying by a 1-digit number, but when you're multiplying by a 2-digit number (think 30x24) and you do the tens digit (think 30x4=120) you put down the 20 and carry the 1. 1 stands for 100, but where do you write the 1? over the 10's column! (huh?!) She spent many confused hours trying to figure it out and putting the carry digits and stuff in the wrong places, and getting tangled up. She eventually straightened it out, but it was after spending a lot of time with the expanded/partial products algorithm. All's well that ends well.
Thank you Lsquared for that insightful comment. I don't want to influence the poll so I won't express my feelings at this time.
I hope you also voted in the poll by choosing the option closest to your preference.
I agree with MathMom.Also, show students some interesting 'tricks' about multiplication, and explain them.It helps (a lot).
L^2, I can appreciate the fact that some kids may have problems with the traditional method.
I don't quite recall the way I was taught this - but I would imagine something like this sequence:
(1) 1 digit x 1 digit (mainly by tables)
(2) multi-digit by 1 digit (with carry).
(3) multi-digit by multiple of power of 10. (essentially (2) with trailing zeros added)
(4) putting the thing together in terms of combination of (2) & (3)
I guess this is slightly different from partial products, but it seemed to work in my class.
TC
I've linked to yours from mine:
Multiplication Algorithm Poll
Hope you get lots of responses! It's a good idea!
(I feel we need to teach the partial products first, then present the traditional and require mastery of at least one; I would try to limit the usage of the partial products algo eventually to those students who have lots of trouble with the traditional.)
I'd be a lot happier about answering if I liked the choices better. From my perspective, the best approach when it comes to exploring algorithms is to be sure that students see that most, if not all, successful, sensible algorithms are grounded in the distributive property. Thus, they need to see in one or more ways that 43 x 27 = (40 + 3)(20 + 7) = 40*20 + 40*7 + 3*20 + 3*7. This underlies both the standard algorithm, the lattice method, the area model, and probably many others. What has always concerned me about the traditional algorithm is that because it is designed to reduce additions through using place value (obviously, some sort of compression process is needed to gain speed over simply adding 27 addends of 43 together or, worse, 43 addends of 27), but does so in a way that "loses" information. That is, there is by necessity a hiding of the underlying process that justifies the "moves" one makes in carrying out the algorithm. This hiding for the sake of compression and speed is even worse in long division and leads to even greater confusion and error in carrying out the algorithm.
Once students see the underlying structure that makes multi-digit multiplication work, the method they choose is strictly up to them. Trying to force or intimidate them into using any particular method is pointless and in some cases very counterproductive.
As to whether all of the above should be preceded with a period of experiment, invention, discovery, etc., I'd say that ideally all mathematics work with children should allow time for this. However, good teachers will be able to craft lessons and discussion that help lead students to certain important ideas and understandings. What additional insights will be gained through the preliminary exploration phase is unpredictable and potentially very rich in classrooms where a true mathematical learning community has been fostered. Unfortunately, simply allowing exploration time without paying careful attention to the important ideas that emerge is not broadly useful. Teachers need to be highly attentive, flexible, and creative in how they use this time to move towards the key mathematical ideas that don't come up spontaneously amongst the students.
I can't believe that so far only one other commenter has mentioned the problem with the poll. (Michael Paul Goldenberg, you da' Man! I just checked out your blog, and I will live there for days! )
MathNotations is a great blog, (I've just added it to my blogroll) but I am leery of this poll. I'm sure everyone here understands the logical fallacy of false dichotomies. This poll is a pretty good example.
"Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5:"
Um, here are my options? I think not.
One of the great problems in (at least) American education today is that we're firmly locked, sealed, and vacuum-packed into the box of pedagogical dogma.
standard algorithm, partial products, lattice method, indeed! The myth of the "one best method" is still so rampant in our "developed" nation.
Education is not about inculcation of any algorithm. It is about students gaining insight, knowledge and lasting value. You can't do that with "just shut up and learn this method," just as you can't do it with, "I'll shut up and let you teach yourself." Those are the ultimate false dichotomy in education of our time.
It's too big a subject to tackle in a comment. But you can read more about it at my post at:
The Math Mojo Chronicles
Hotcha!
Brian Foley (a.k.a. Professor Homunculus at http://MathMojo.com )
Michael, Brian--
If one of my objectives in creating this poll was to generate reactions, I've certainly succeeded!
Of course, this poll is simplistic. However, I believe strongly one must distinguish between PEDAGOGY and CONTENT here.
No one ever seems to question that students should know some method for solving quadratic equations. Certainly there are myriad ways to motivate and explain these methods conceptually. But, in the end, students need to have confidence in some method. Many students may not be successful in developing skill with several methods and need to develop proficiency with one. When working with students from other countries, they listen incredulously as I demonstrate, the square root method, the factoring method, completing the square and ultimately the quadratic formula. When questioned about their method(s), they simply reply, "Oh, we use the quadratic formula for almost all of these except, maybe, the '3x^2 = 12' type. I'm not saying that's the best way, I am suggesting there needs to be a clearly defined expectation.
While I thoroughly agree with both of you that elementary students should develop a conceptual understanding of place value and the meaning behind one or more algorithms for multidigit multiplication, the issue of learning some algorithm well should NOT be debated. However much some may detest the Standards Revolution, the vast majority of K-8 teachers I speak to want clear direction as to WHAT students are expected to know and know well.
My original poll had a choice (E) None of the Above or Some Other Algorithm or Combination.
I chose to eliminate this option to require the reader to choose the approach that most closely coincides with his/her preference for what students should learn. OR to simply choose not to vote... Although there is an aspect of pedagogy embedded in the options, the essence is the WHAT not the HOW.
I knew in my heart that this innocent question would strike at the heart of the Math Wars. I've never hidden my 'centrist' position in this debate, a position that means one gets pummeled from both sides but in a nicely balanced way of course!
Again, I appreciate your reasoned responses and, pedagogically speaking, we are not far apart. However, I believe that our nation's teachers need to have clearer direction than has been given thus far by NCTM or most states' standards.
I encouraged replies to this post specifically to give the opportunity to those who didn't like the choices offered or to go beyond a multiple-choice response.
Brian & Mike,
If you've glanced at my blog then you know that teaching for understanding was my raison d'etre in the classroom and my primary impetus for this blog.
However, after my students recognized the need for a procedure (motivation) and understood the ideas underlying a procedure, I expected students to learn some procedure well and be able to implement it accurately.
I take full responsibility for content and semantics of the options in the poll. I didn't spend weeks formulating this. I didn't run it by various organizations. I simply laid it out there. Thus far, the results have been interesting to say the least. I have not yet heard from many stating that the choices are overly biased, ambiguous or ill-defined. But they may well be...
Dave,
I feel your pain. I think you do a great job (much better than I ever could) at what you do, and it's apparent that your intentions are absolutely on target. I also think some sort of poll is a cool thing, even if it is just to stir up thoughts.
Still, the basic assumption that schools should have some standard that they require (other than "the student should understand multiplication and should be able to multiply with mastery, confidence, ease, and enjoyment") is something I find unnecessary, and counterproductive.
I think all students should be able to several methods with mastery. That should be a given. The problem really starts when a student finds a method that is seriously superior to the "standard" (or whatever algorithm is the "method du jour") and are then penalized for using their superior method. That is unavoidable when some desk-jockey in the state capital sets some "standard."
Sure, every child should learn some algorithm well. The problem is when there is an artificially limited set of choices. If a child can multiply better than expected, but doesn't use the method that's expected, ah, there's the rub.
I'm sure the reason that many teachers would like guidelines, is because they know they are going to be judged on something other than if their students can really multiply. If we put more effort into curing PST (Pathological Testing Syndrome), and less time trying to define some unnecessary standard, we'd free teachers to actually teach in more valuable ways (valuable to the students' minds, that is).
The American school system has been around for a very long time, and we still have not come up with "the one best system." It shouldn't take that long to figure out that the search for the "one best system" is a Snark-hunt.
One last comment: When you mention that:
"...the vast majority of K-8 teachers I speak to want clear direction as to WHAT students are expected to know and know well..."
it leads me to believe that somewhere in their professional educations, budding teachers need to be told that we trust them enough to teach, once they become teachers. I mean, simple multiplication of multidigit numbers is a very, very basic skill, like reading.
Each teacher knows, more or less (and without having to test) at what level his/her students should be able to read, and if they can read at that level. How do they do this? They ask the child to read. If the child can't read, well, that's a good sign that the child can't read.
Does a teacher really need a rubrik, a standard, a curriculum, a list of proscribed algorithms, and a state-mandated test to be able to tell if a kid can multiply? If they do, isn't something wrong our expectations of the teacher?
Still, I can't really say that I'm right. And your poll will surely accomplish some decent goal. I don't mean to be picking nits. It's clear that you didn't try to craft "the definitive poll." It's just that I really feel that the fundamental assumption of the question might need to be revisited. Then again, maybe not.
Best of luck,
Brian
Brian,
I appreciate what you're saying and I essentially agree with you. It would be unethical if a student were penalized for correctly using an algorithm that wasn't the 'prescribed' one. The exception would be if the objective were to demonstrate knowledge of a particular algorithm such as completing the square. That's a different situation.
The issue of how many algorithms, what kinds of algorithms, what should be required or not is a very complex issue. The child with stronger reasoning ability can process several methods, may be able to invent their own and can select the method most appropriate for a given problem. Having worked personally with many children who have learning disabilities has helped me to understand better that some children need to be shown straightforward step-by-step procedures without many options. If the concepts are presented in a mode that fits their way of thinking and processing, they can begin to understand the ideas underlying a procedure, but, in the end, I would like them to experience success in accomplishing some procedure consistently well.
I have never been a believer in rigid prescription of method. The options in the poll do reflect a range of learning methods, some far less prescriptive than others. It is far from perfect but it's the best I could do at this time. I suspect i will do a better job on my next poll!
Partial products are excellent. When converting to column multiplication, left to right works well.
e.g. 43,251 x 1038 is handled nicely by multiplying 40,000 x 1000, then below that 3000 x 1000, then 200 x 1000 ...40,000 by 30... Lots of rows to sum, but then kids practice addition. Then after the mathematical concepts here are understood, go to a calculator for highly accurate multiplication results. First use it for answer verification. Then use it as a primary tool.
Don't misunderstand me about calculators. Home-teach kids to master a 20x20 multiplication table. Show them how the 1 to 20 columns and rows are serial additions. Teach them useful techniques like mentally multiplying 23 x 22, by multiplying 23 x 20 and then adding 23 x 2 in their heads.
Teach them "Think different" concepts on multiplication like 31 x 28 being 30 x 30, then adding 30 (31 x 30), then SUBTRACTING 3 x 28 ( 3 x 30 -3x2), in their heads.
But calculators are amazing tools. HPs that do exact operations, and return exact values such as 28/13 and 4 pi, are more powerful than TIs that only give decimal approximations. (HPs give decimal approximations at the users' command.)
Even when a student uses pencil and paper (or purely mental) operations, a calculator is great for answer-checking.
Mark (the last commenter),
Good comments. I must say that all of your Ideas beat the poop out of what's commonly taught.
You can "think much more different" if you don't stop anywhere near there, though. Left to right multiplication can be done with a much more elegant and sensible algorithm that the ones you mentioned (although yours should be taught as well, as a stepping-stone).
Also, it is a travesty that almost no school-children are taught how to memorize numbers instantly, so that they can keep all "partial product" in their head easily.
As far as checking - using estimation and modulo 9 you can learn to check quicker than most kids can use a calculator. And when you get decent at it, the odds of making a mistake are lower than the odds of mistyping numbers into that infernal machine. (Calculators were invented by vampires to suck kids' brains out.)
Yours truly,
Brian Foley (a.k.a. Professor Homunculus at http://MathMojo.com )
Brian,
I don't know much math. I just had two kids who were underperforming in math many years ago, so I felt had to intervene. One was a B student from 5th-10th grade, who never did math homework because he said he got it done at school. His English teachers gave him a lot of homework.
I had a wake-up call when he got a 55 on a practice PSAT in 10th grade. Why did the alarm bell ring? He had scored 480 in October of 8th grade on the SAT, with a year of pre-algebra and a half-year of algebra I. 1.5 more years of algebra and a year of geometry should not have translated into a mere 70 point score rise. Something was seriously amiss. So I taught him a summer of intensive remedial math after 10th grade.
His 11th grade honors precalc teacher—the first teacher he ever encountered who had a University of California mathematics degree— said to me in early second quarter, "Can you help me understand something? I asked last year's teacher who the best students in his class were. He didn't mention your son. But he's head and shoulders above everybody else in my class. H's getting 10's on every problem set, he got a 10 on the first two quizzes and 105 on his first test (w/ extra credit challenge problem). He understands things nobody else does. He writes beautiful solutions. Can you tell me what's going on?"
I explained that earlier, he did problems in his head, often incorrectly, he didn't know how to compose arguments (mathematics is a language) and he was almost never assigned C-category problems, so he viewed math as a trivial waste of time.
Upshot: he got his first A in many years, actually an A+, and his teacher set up two paid alg I tutoring jobs (private school) that worked out well for his tutees. He scored a 710 on the SAT and an 800 on the SAT Level 2c subject exam.
I had a 4th grader who was being just totally crushed. I tried to help with math homework, but the assignments were chaotic. He just finally refused to even try to do his math homework. To see tears in his eyes and protest, "I'm not good in math, Dad!" broke my heart.
I took him home, and found out that he could do not complete a 10 x 10 multiplication table or do any long division. I concocted a non-stressful systematic build-from-a-foundation scheme. We used a variety of manipulatives, including an abacus for computation. (I guess you and I think alike, eh?) I taught him basic algebra using a balance and weights: keep the pans level, and that's an equation analogue.
Upshot: he started calculus at age 16. He too became a math tutor, starting with a University of Chicago summer program in which a 28 year old African-American reentry student in his calculus class approached him, and asked if he could tutor her. So he did.
What happened? With home-schooling he learned that he should ask questions whenever he was confused, and every question he posed to become unconfused was a good question. At Chicago, he didn't know that if he asked questions in class he might be ridiculed as "dumb". When his classmates said to him after class, "I'm glad you asked those questions today, because I didn't understand what he was saying," and his instructor (a grad student) repeatedly said, "That's a very good question," my son felt self-confidence. By his example, other students mustered the courage to start asking questions too, and a one-way-communication lecture course was transformed into a bilateral-communication experience. Along the way, a struggling student saw somebody who she thought might be able and willing to help her overcome her own confusion. (This one also scored an 800 on Level 2c.)
He got into an undergrad research project that led to training at NASA Goddard, and is a having a blast as a grad student at Columbia now (not in math). He knows a lot more math than I do. Actually, all my kids, including my home-schooled youngest, who completed multivariable calculus, differential equations and linear algebra as a college freshman, know a lot more math than I do.
I've been assisting a 9th grader in "Honors" Algebra II. He missed a test problem a few weeks ago: | x + 2 | + | x – 2 | = 4.
There was no problem like this in the book or any explanation of absolute value sums. So I showed him how to do them: Treat the absolute value expressions as independent functions f(x) =y1 = |x + 2| and g(x) = y2 = | x – 2|. Convert these to linear equations: y1a = x+2 for x + 2 > 0 and y1b = -x - 2 for x +2 < 0, and y2a = x – 2 for x – 2 > 0 and y2b = -x + 2 for x - 2 < 0. Now graph them on the Cartesian plane (by hand). Now we add the linear y values together in each interval. For x > 2, x + 2 + x – 2 = 2x, for the interval x < -2 -x – 2 + -x + 2 = -2x, and for the interval -2 < x < 2, | + 2 + -x + 2 = 4, i.e. y abs = 4, a horizontal line. So that is your test problem's solution. Now this is a special case of absolute value sums, because the expressions generate lines with opposite slopes in the middle interval. You try this on |2x +2| + |x – 2| = 4. See how you only get a two-point solution? Now let's try |2x + 2| + |2x – 2| = 4. "Oh, I get it now." Did your teacher show you this? "No."
On HPs, I love them. It started when a friend let me borrow his HP-45 for P Chem. I had a lovely eyesaver-yellow Pickett 320 circular/expanded scale. I could get 5-decimal precision on some calculations. But when I flubbed exponents on tests (10^-13 vs. 10^-12, darn!), the HP gave me the RIGHT answers. Then I learned how to program a desktop 9810 with my own concocted linear and exponential regression statistics programs for biomed research.
The HP49g+ and 50 calculators allow you to input integrals that show on the screen in textbook-format. Then they give you textbook-format symbolic answers if you don't specify any intervals. Those engineers in Palo Alto are some real geniuses. Using their inventions is connecting to their amazing brains.
Mark
Mark,
Wow, what a great example of what a good homeschool parent can do! I hope it's ok that I'm quoting an except from your comment on my blog (with a link to this page) at:
The Math Mojo Chronicles, as an inspiration to some of the parents who struggle with their kids' math education.
I'll be sending my readers to this page, and pointing them down to your comments.
Cool runnins!
Brian
The best math teacher that my son had in public school, the only one that really knew his math rather than just teaching the subject, told them "I have to teach you this, but don't use it." And then he taught them the right way to do it.
The students did better the traditional way. (According to my son) Many wondered why anyone would do the lattice method (most couldn't remember the name) when it seemed like more work to do the same thing.
Procedurally, isn't the partial product method the same as the traditional method. I looked at the everyday mathematics site's example. It's the difference between (a+b)(c+d) and (b+a)(d+c). It does not matter which way you do it.
Of course, I never said, but 43 x 27?
43*3 = 129
129*3 = 387
387 * 3 = 1130 + 21 = 1161
That's what I really did (in my head),
though I paused to consider instead trying:
(35+8)(35-8) = 35^2 - 64
= (70^2)/4 - 64
= 4900/4 - 64
= 1225 - 64
= 1161
But, as I said, I chose not to try, correctly estimating that I would have to hold more intermediate results in my head at once.
Alternates are ok (but the standard algorithm must be learned)
Jonathan
1130 was a typo for 1140
Hey Dave,
I hope you see this comment, as your contest is many months old! We're designing a free tool that helps teach multiplication in each of the three ways your contest targeted - standard, partial products and lattice.
The tools we develop (we're a nonprofit) all engage lots of educators during the design phase - and we'd love it if some of your readers, or you, if you'd like, would join the design team!
There's more on our site, http://www.plml.org/node/73/ - including a signup form, for anyone interested to join!
The result? An awesome, free educational tool for multi-digit math instruction!
Cheers,
--Dave
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