I don’t know where I found this video… but it is infuriating. (Warning: if watching videos of basic math being done on a whiteboard makes you nauseous, you might want to skip the video)
Of the three “alternate” methods, the only one I think that can have a reasonable argument made for it is the second one. (The one that looks and functions most like the standard algorithm) the “Cluster Problems” and “Lattice” methods are crap, pure and simple. I could go on, but I won’t. Just watch the video and be amazed.
January 28th, 2007 at 4:03 am
This explains why Verizon doesn’t understand the difference between .002 dollars and .002 cents.
January 28th, 2007 at 1:44 pm
Well, I think it’s actually a great idea that they spend 30p on calculators, since there’s so much more to them now. You can teach use without teaching dependence. But everything else in those books was crap.
January 28th, 2007 at 2:05 pm
I disagree… there’s no reason a 4th or 5th grader needs to learn how to use an advanced graphing calculator. Scientific calculators are incredibly easy to learn, can be explained in two pages, and are all you really need until you hit calc. Hell, I took god knows how many credits worth of upper division math and never once used anything more than a scientific calculator.
So, yeah, I agree that they should spend some time in, maybe, pre-calc teaching them how to use advanced calculators… but there’s no reason to teach them that stuff before high school.
January 28th, 2007 at 2:25 pm
I couldn’t believe that. I started watching thinking “damn, 15 minutes worth? I’m not watching this shit! And then sat there spell-bound going what the FUCK!
I mean, I didn’t really do math (homeschooling rocks!) and my first calculator was the little one on the computer, but even I learned the standard algorithm!
It just goes to show that language and writing skills aren’t all we’re losing with this form of institutionalized education.
January 28th, 2007 at 2:32 pm
Ye gads! I mean really: being able to multiply and divide by hand is super useful! I don’t even OWN a calculator that I can easily use (turning on the computer that buzzes horribly just to add something is pretty annoying, and I hate my graphing calculator (did I give it away or do I still own it?) and my favorite calculator ever was stolen in 2003… so basic math becomes ever so useful — in fact I love the tedium of it! (yeah crazy, I know!)
Please tell me this isn’t coming to a curriculum near me (a.k.a. wishful thinking).
January 28th, 2007 at 5:10 pm
I don’t think it’s possible to argue that any of the alternative methods are any better for division (the cluster method is just guess and check). All of the multiplication methods (including the standard algorithm) are centered around distributive property. The standard and cluster methods are most similar, as the standard method reduces (26×31) into (26×1)+(26×3)x10. For that reason, I think textbooks that teach additional methods that focus on the distributive property and emphasize place value can be a good thing. However, I can’t tell from this video if that is the case in these textbooks.
In addition, this lady is presenting some pretty weak arguments. Thirty pages of stuff for a project: Blasphemous!! If the rest of the book has enough to teach the fundamentals (admittedly in question), then who cares if there’s an project where kids have to dig up real information and use applied math? I have a feeling that the project somehow focuses more on planning routes and calculating expenses than picking sites. She’s largely appealing to BITGODs who feel any difference between modern education and schoolhouses that were uphill and snowy are to blame for the “problem in kids today”(c). For instance, not knowing 6×4 by the fourth grade is not the fault of either of these publishers.
I do agree with the other posters that there’s no excuse for not being able to master arithmetic, and that early calculator dependence is a very bad thing.
January 28th, 2007 at 5:27 pm
If anyone understands the benefit of teaching additional methods, I do… but the key here is that they’re being taught instead of the standard algorithm, and they all (save the second one of the multiplication methods) either obfuscate the distributive property behind a pretty chart or provide no algorithm whatsoever (trying to turn cluster method into an actual algorithm would be painful at best).
I also don’t think her argument about what is being taught instead is weak at all. The authors of the textbook believe it’s a waste of time to teach division (!!) but yet feel it’s not a waste of time to add little projects (or 30 pages worth of calculator reference. for a 5th grader.) to make it interesting. As has been pointed out on other blogs, this is nothing more than pandering to the “kids need to feel involved and interested” at the cost of expressly not teaching them mastery of division because it can be done quickly and accurately with a calculator. The same can be said for every basic mathematical skill, that’s not a reason not to teach it.
The problem with adding methods which demonstrate the distributive property of multiplication is that they haven’t learned that yet, and won’t (most likely) until they get to junior high. At which point it would be perfectly reasonable to show them the new method and say “So why does this work?”
But teaching them these bizarre methods as the “standard” is foolish. They’re more complicated, less obvious, and probably lead to no small number of kids getting frustrated and hating math. As though we need more of that.
January 28th, 2007 at 7:02 pm
The number of pages devoted to projects or calculator work is only relevant to the extent that the additional pages somehow prevent additional meaningful material from entering the textbook. I don’t see this as the case (unless someone can land a copy or pdf of this book for analysis), so I visualize those sections as appendices. I also remember that math classes throughout my education featured a lot of skipping around textbooks, and that may chapters were skipped. While 30 pages of calculator text would be ludicrous as a two-week unit, it might be appropriate as a reference that just exists in the back of the book.
Just to revisit the multiplication debate, there doesn’t seem to be much of a difference between the lattice and standard methods. Both utilize and obfuscate the distributive property, and both take up a similar degree of space. The lattice method simply decomposes to the FOIL method from jr. high algebra: 26×31 becomes (20+6)x(30×1), which is just as complicated as the standard as I described it above. While the lattice could be prone to mistake if the diagonal lines were misdrawn, the standard is prone to mistake as the zero in the second line is often omitted. Both of these are mistakes that are easily corrected with experience. Lattice has the problem of reading numbers around a corner; standard has the problem that the carried numbers become difficult to read if multiple numbers are put over the same digit. The two methods are IDENTICAL, differing only in appearance. The lattice method could perhaps make FOIL, and by extension, algebra, a little easier to understand as well.
I think that the problems being discussed here reflect parental dissatisfaction from having their children learn something differently than they did. And I still agree that the division stuff is crap.
January 28th, 2007 at 7:15 pm
“The lattice method simply decomposes to the FOIL method from jr. high algebra:”
Exactly. Re-read that sentence, it’s important.
These kids are not in junior high. They are in fourth grade. The reason you (and I) notice that it’s the FOIL method, though, is because of our proficiency in math. Go show that to a kid who’s just learning how to do division and ask them why it works. I doubt you’ll find a single kid that can answer it for you.
The more I think about it, the more I like the second method better, the one where you’re basically doing the standard algorithm, but without hiding the fact that you’re multiplying 6×20 instead of multiplying 6×2 in an odd position.
As to the crap in the textbook: textbooks are written to be used cover to cover. That rarely happens in practice, but I don’t think anyone has yet made the mistake of assuming that everything in the textbook is getting used. The point is that the author (not the teacher) believes that there’s not enough time to teach division, but there is enough time to plan a trip to paris. The problem is not what’s in the textbook, per se, but that these are the same people who are writing curriculum for schools and setting standards for what needs to be learned.
And while I don’t think there are many places where too much information is a bad thing, it’s just a waste of money and space to think that a 5th grader needs 30 pages worth of calculator reference. When they won’t even learn graphing until Junior High. The instructions that come with a scientific calculator fit on a sheet of paper the size of a few business cards. What more than that could a 5th grader even use?
January 28th, 2007 at 9:55 pm
The beauty of any of these methods (including the standard) is that a child need not understand why the process works; only that it does work. The fact that the lattice method mirrors an operation that is useful later is a reason for its use, not a reason against. The second and third methods (and first, provided the clusters are defined by digit, i.e. 326=300+20+6) all do exactly the same thing. In fact, I’ve come to like the third method as it seems the easiest to use with larger numbers; 64879*84464 is easier by the lattice method, as you don’t have to erase the carried numbers with each iteration.
As to your next point, I don’t recall that the authors said that there wasn’t enough time to teach division. I do recall them saying that repetitive teaching of long division wasn’t productive given the scarcity of classroom time. I’m not sure how much I agree with that, but I do suspect that many of the problems that are associated with double digit arithmetic stem from insufficient mastery of single digit arithmetic. I’m doubtful that this textbook feels that geography is more important than math.
I dug up some links to the two textbooks in question. I also found a FAQ on calculator use in the TERC book. Although the question is focused on 2nd graders, it may still be appropriate. An author highlights a section called “Beat the calculator”, where students form pairs and one student tried complete problems faster by hand faster than their partner with the calculator.
http://investigations.terc.edu/
http://investigations.terc.edu.....or/q34.cfm
http://www.wrightgroup.com/ind.....L000000004
January 30th, 2007 at 4:03 pm
“The beauty of any of these methods (including the standard) is that a child need not understand why the process works; only that it does work”
-This is INSANE.
If you don’t understand why something works, then you have learned nothing. I know how to plug in a toaster so that electricity flows to the device, that doesn’t make me an electrical engineer.
And when I was in school, we had plenty of time for long division. As far as I know, there are just as many hours in a day now as there were back then.
Finally, on one should use a calculator to do a problem unless they understand what the calculator is doing, how, and why. There is no need to try to beat a calculator. A calculator should be used when you already understand the process, and doing it by hand is no longer fruitful.
January 30th, 2007 at 4:14 pm
**”The beauty of any of these methods (including the standard) is that a child need not understand why the process works; only that it does work”
-This is INSANE.**
I’m going to take a middle ground between these two. I agree with the basic principle that, when teaching basic math, the important concept is not how to do it but why to do it and how the method works.
If one is only going to teach how ot multiply two numbers, I say why waste the time? Just show them a calculator. If you don’t know why you’re doing something or how it works (basically) it’s not terribly helpful. That said, this game can go back to the point that the concepts you’re teaching are incredibly abstract.
That said, I don’t think one needs to learn much abstract math to know how the standard algorithm works. On the other hand, the lattice method may, in fact, be a similar set of operations to the standard method, but it’s obfuscated behind cute lines and novel mechanics. That is the last thing elementary math needs. It needs to be clear why things are being done and, to that end, that’s why I tend to think the second alternate method is just as good (if not better) than the standard.
Finally, on one should use a calculator to do a problem unless they understand what the calculator is doing
I have an economic argument with that. If it is a function that you use very infrequently (for me, for example, double integrals), understanding how to do them by hand is probably far too great a cost for the slight benefit. In the case of basic multiplication, though, I don’t think that’s the case.
January 30th, 2007 at 8:12 pm
Well, you don’t necessarily need to know how to do something by hand to understand the concept behind the math in question. For exapmle, taking the 70th root of 687e, or doing a complex IRR. But you should at least know what’s going on - the theory behind the question.
But what I really meant was that, before you use a calculator in a class, you should have already learned how to do the problem (or at least express it if its impossible by hand) without one. i.e., no skipping steps in teaching.
Obviously, you know that you can’t understand anything beyond double integrals without first udnerstanding them - but I’m not positive everyone understands that you can’t just move past some topic and say “just use a calculator” and then expect a person to be good at the next level, when they don’t really Comprehend the last.
January 30th, 2007 at 8:25 pm
Let me give you a more concrete example: Sue doesn’t know how amortization works, but she can find a calculator to figure out the amortization on a mortgage for her.
Should she burden herself with trying to learn how amortization works if she only needs to do it once? If so, why?
January 30th, 2007 at 9:49 pm
The beginning of my last comment was worded incredibly poorly. I would never (intentionally) suggest that students should simply have faith in whatever formula is handed down to them. My view is very close to Pete’s in the last post: student’s shouldn’t need to know that the lattice method is related to the FOIL method, but they should know that the bottom right corner is single digit multiplication, the next diagonal row is single digit times double digit (230+601), and so on. Thanks for pointing out my error.
I admittedly haven’t gone through all of the evidence on the sites, but TERC at least is an NSF funded program that has shown improved math comprehension and execution relative to traditionally taught same-state samples.
January 30th, 2007 at 9:55 pm
Pete, I only meant in educational setting. No, Sue doesn’t need to understand it if she’s just figuring out her own mortgage. But if she’s a finance major, yeah, she needs to know.
February 2nd, 2007 at 10:27 pm
Ok, not making an argument against learning basic math, but the truth is that our brains aren’t designed to do high order math. We think of math in terms of memorization, not in terms of calculation.
In elementary school we all learn our multiplication tables as the basis on which we develop higher-order multiplication skills. We never learned things like 6×20… we learn that we pull off the zeros at the end of each term, multiply what’s left, and add the zeros back to the final answer, 120.
I don’t know if there is much merit in thinking about 6×20 over 6×2. I’ve used a better method than any of these for years, one that I found in book called the Human Calculator which figures out each digit completely, without the extra addition step (looking at the “Lattice method”, it’s almost identical in nature but without the boxes, lines, and bending around the corner crap). It never looks at things in terms of 6×20… it’s not that I don’t know how to do it, but there’s no use when there is a more direct approach.
But, that’s only good if you have pencil and paper in front of you. I think the “Cluster problem” method is a very practical method when you have to keep stuff in your head (to me, it’s easier to keep a bunch of 6×20 problems in my head than a bunch of 6×3 carry the one problems).
February 2nd, 2007 at 10:34 pm
PS - By “a bunch” of 6×20 problems, I didn’t make much sense. I should have said something like breaking down 26×30 into 20×30 and 5×30 and 1×30 and adding, instead of 0×6, 0×2, next line move over, 6×3 carry the one, 2×3.
And it starts to make alot more sense when you’re talking 3 digits by 3 digits.