If that is the case I would like to present the school board with it. Problem is, like everything else for every page talking bad, another says good. This is in GD because it is apparently a debate or all schools would be using one way or another. So Dopers, did my son get mathematically screwed by his school?
She is a meterorologist and not a mathematician or educator.
She had to look at her notes to define “algorithm”
She says 806 as eight-oh-six
She says kids don’t learn with TURC with no citation or reference
She is talking about constructivism in math. Constructivism (when properly done) is more effective for learning for understanding and memory than teaching algorithms with no real anchor in real-life applications.
My qualifications? Well I don’t have a degree in meteorology but I do have a bachelor’s in mathematics (interestingly also from the University of Washington), a master’s in mathematics, I have taught math since 1998 in middle-school and high-school to regular students, honors, second-language learners, and special education students. I spent two years as a math coach and I am currently writing my dissertation for my Ed.D. in math education where I researched (you guessed it), the effectiveness of constructivism in algebra.
You are never going to get agreement on how to teach arithmetic to elementary school students.
I’ve looked briefly at the first video in the link you provided. Obviously, the person in the video (why does a meteoroligist think she knows how to teach math?) has an agenda: she’s not comfortable with the method she’s warning against.
But the problem is that, as she very neatly glosses over in her presentation of the “standard algorithm” for teaching multiple digit multiplication, the method is wrong. Oh, it will produce a proper result, but it will not produce proper understanding of what is going on. Thus, when you multiply 1 times the 2 in 26, you end up writing a 2 down, but you really aren’t getting two, you are getting 20.
One of the current theories for how to teach math is that students should be learning not just an algorithm for a particular situation, but should comprehending the underlying process that is being accomplished. There are any number of ways that are being used these days in elementary schools to accomplish that. For example, some students get taught properly to do the problem she demonstrated (26 x 31) by doing (1 x 6) + (1 x 20) + (30 x 6) + (30 x 20). Other students are lead to an understanding of what is going on by using the method against which she is railing in the video, which involves understanding the basic concept behind what the number 26 represents (20 + 5 + 1). Of course, she presents this in the most unfavorable light she can, because she doesn’t like this method. Why? Ultimately, mostly because she’s not been taught to think that way, and it makes her uncomfortable.
One of the current examples given most often for how math as taught in our childhoods should change is the famous algorithm of FOIL: First, Outside, Inside, Last. Used to mutiply two binomials (for example 2x + 4 by 3x - 3), it has you multiply the first terms of each binomial, then the “outside” terms (the ones at the beginning and the end, that is, 2x and 3), then the “inside” terms (the ones next to each other, here 4 and 3x), and lastly, the “last” terms (here 4 and 3). This algorithm, with its convenient mnemonic FOIL, has been taught for decades.
The trouble with the algorithm is that it relies upon writing the problem in a linear fashion (the whole acronym loses meaning if you do the problem vertically), and it doesn’t teach the student what they are really doing (applying the distributive property of multiplication and addition). So, for example, when the student is presented with the first multiplication problem consisting of a binomial and a trinomial, FOIL fails, and the student is left wondering what to do instead. Usually, the student asks the teacher for some other handy algorithm, with a catchy name. Or, the student may stubbornly try to FOIL the problem, without ever really understanding why it isn’t working. But a student who has been properly taught the distributive property and its application in multiplication of polynomials (for example, through the use of algebra tiles) will have absolutely no problem with a trinomial by binomial exercise, or any other combination of polynomials multiplied together.
Yet, I would almost guarantee that the woman in the video, presented with two binomials to multiply, would tell you that you FOIL them, and would be upset if her children were taught some other, “fuzzy” method of understanding the process.
Now, mind you, I’m not willing to say that any method demonstrated in the videos is the one, right method for teaching multiplication or other arithmetic operations to elementary students. But I can guarantee you this: the student taught the “standard” algorithm hasn’t got a clue what is going on, only knows a process for a particular exercise. Against this type of stupidity the gods themselves struggle (in vain).
I am convinced that people comprehend and learn math differently. Therefore this debate is looking for the wrong solution - because there is no single correct solution.
I have ALWAYS solved the above problem this way: (30 X 20) + (6 X 30) + (1 X 20) + (1 X 6). The results give this: 600 + 180 + 20 + 6 =806. In other words, I start from the left with the larger numbers. This allows me to do the entire calculation in my head. It is easier to add smaller numbers to larger numbers. So I end up adding 600 plus 180 = 780 plus 20 = 800 +6 =806. Same thing with adding a series of numbers. I add them starting with the digits on the left (no carrying over! ).
When I did this in grade school I got poor grades in math for not showing my work, but I always had the right answers. I could not understand why the teachers had a problem. I would read a math problem and write the answer. No big deal. Without realizing it, I had a problem “showing my work” because I wasn’t doing it in the classical way. And it was so easy and fast for me, that I didn’t even know at the time that I was doing it “backwards”.
I have worked as an accountant for over 30 years and I do not use a calculator. I do all my math “in my head”. I have had people who were a real whiz on a calculator add a long column of numbers and I could always do it quicker (and correctly).
My point is that math is not the same kind of “learned skill” like riding a bike or learning to cook. We do not all have the same way of perceiving abstract concepts. Ideally we would be able to test for basic differences in learning and have more than one approach to teaching.
I’m no math educator; I have no math educator credentials. And, I’ve only watched part of the top video in the link. My uninformed, possibly inconsistent, not well-thought-out views are as follows:
There’s no right algorithm for any of these things; the very concept of “The One True Algorithm” is silly. But, it’s also silly to deprive students of learning any algorithms at all, telling them instead to only proceed non-algorithmically from undirected application of little tricks like “Well, if we break 26 up into 20, 5, and 1, all of those multiplications will be nice and round and we can handle those differently” (i.e., “cluster problems”). Tricks are nice, but they aren’t everything, and certainly not fundamental.
People should learn the concepts, and some algorithms, and perhaps even link the two to see why the algorithms work, and perhaps even discover some nice tricks on their own. However, one cannot make the perfect the enemy of the good; the most important thing is to know the bare bones of the concept, the second most important thing is to know the bare bones of some reasonable algorithm for very basic calculation, and everything else is tertiary.
Many math educators’ disdain for calculators is overly antagonistic, to the point of ridiculously strident "handbasket-Cassandra"ism. Truth is, there isn’t that much need for most people to be able to multiply arbitrary 4 digit numbers by hand; that would be a fine example of a case where knowing the concept of multiplication (What is it? What is it useful for?) is much more useful than knowing any specific algorithm for it. But then again, there’s not much need for most people to be able to solve quadratic equations or take derivatives or know anything about prime numbers, either, and yet we eventually teach those too, so… I dunno. But knowing how to use a calculator is a much more useful skill than knowing how to carry out pencil-and-paper calculations; not that the two need be exclusive, but the tone of the woman in the video drips with disdain for emphasizing the former more than the latter. It’s a bit like the debates we had over print and cursive writing, I suppose.
The woman in the video repeatedly states that what she calls the standard multiplication algorithm is “more efficient” than others, or “the most efficient”. She gives no metrics by which to back this up; indeed, it strikes me as bullshit (as far as I could tell, all the actual algorithms she discusses are Theta(n^2), which is sub-optimal for huge numbers, but nobody’s multiplying huge numbers by hand anyway. They all seem reasonable to me.). She feels that the standard multiplication algorithm emphasizes how place-value notation works in a way which other algorithms do not, but again, she says nothing to back this up, and I find this assertion dubious.
Oh, I dunno. The more I type, the more aware I am of my lack of expertise to speak on this topic. Still, having typed this out, I suppose I’ll post it.
I work in a store and am constantly amazed that people can’t do simple things like calculate 30% off of $55 or convert 120 meters to yards without a calculator. This is both young and old people. Not sure how to fix this however.
Embarrassingly, I had to look this one up; at least, to get beyond the accuracy of my original approximate answer (in my head, the generally useful enough approximation is 1 meter = 1 yard).
(Any time I convert from metric to US or vice versa, I have to go down to inches or centimeters because the only conversion that’s stuck in my brain is 2.54 cm in 1 inch. If I have to convert something from miles to kilometers, it takes me a while.)
Further reading on that website does compare some methods. Saying the Saxon math is a much better method. Even using graphs and bars. Anecdotal evidence, my son is horrible at simple math in his head, but has had all A’s in the subject since they started grading it. That cannot be good needing a calculator or a sheet of paper for simple math. And speaking of calculators his school has made them mandatory the last few years, we used to get detention for sneaking them in. :smack:
The question I’d ask is, at what age do kids usually understand the distributive property? (That’s the property used in FOIL - it’s the one that says a(b+c) = ab + ac.) My WAG is that the standard algorithm for multiplication of multi-digit numbers can be mastered at a younger age than the age by which kids have a decent grasp of the distributive property.
By the time kids are multiplying binomials by FOIL, they’re usually in eighth grade, give or take, so by then they should have that concept down. But the genius of the standard algorithm is that it relies on two very simple concepts - taking the product of two one-digit numbers, and carrying - to enable kids to do more complicated arithmetic. The distributive property is definitely a higher-level concept than either of those, and I’d personally be a little skeptical of relying on it to teach basic multiplication.
What generally happens is people will ask “How many yards is this” and when I say “add 10%” the blank looks ensue. Not everyone knows metric conversions, but I expect anyone over 10 to be able to multiply by 1.1.
I agree, but only after you learn the basics. Relying on a calculator because you cannot use your noggin is bad, relying on it to quickly compute a more complicated problem or to ensure something is correct is not bad, all IMHO.
I don’t trust myself to do arithmetic without a calculator at all, but I don’t feel that I’m at any disadvantage as a result. Regardless, I do think that doing arithmetic by hand is very helpful in developing an intuition for estimating the outcome of a calculation, and so I am very much in favor of it.
ETA: I think a lot of the anti-calculator sentiment is plain old fogeyism. If there’s really a disadvantage, then let’s be cautious, but let’s not do it the old way just for the sake of doing it the old way.
I ran the 440 in school track (before we sold our soul to metric :)), so that’s how I remember it.
Physics/EE here…since grade school I have always personally relied on mnemonic tricks first when learning a math process, and have had true understanding of the mechanics come later. I continue to do so in grad school. I find that it isn’t until much later, after the initial memorization is complete and I start to functionally use it, that the intuitive Neo “Whoa!” vision of the actual math becomes clear. It sounds like the push with this new approach is to achieve “Whoa!” first, and functional ability second. The risk in this approach is that some students will never achieve “whoa”, even though they are perfectly capable of achieving functional mechanical proficiency.
For example of techniques that saved me countless hours and allowed me to score much higher on exams…
FOIL
Derivative rhyme “Low d high minus high d low, and under them all low squared will go”
“B.A. Barracus, fool!” for remembering the order of which integrand goes on top of the integral symbol, and the order of calculating.
M2I2ACIDS for analyzing curves.
And countless others that I can fall back on without having to reinvent or rederive a fundamental concept.
Is it the best way? For me it is, it’s just the way that my brain is wired and most effectively processes the information (if we define “effective” as “being able to pass mainstream standardized examination formats in a timely manner”)
My math teacher taught me using the standard algorithm. She also taught me why it works. I always use all the methods she described in the video. Except for the wierd crosstable thing, I don’t see a difference in any of the methods. She’s just breaking the numbers down in differently sized chunks.
FWIW, the thing that helped me the most with seeing the “why” was a farmer analogy. He’s plowing a field 26x31 units. If you draw the field and draw lines splitting the field at 20 and 30, you see it makes 4 squares of sizes 20x30, 1x20, 30x6, and 1x6. It then becomes obvious that if you add them up, they equal the whole field. That image stayed in my head and I used it during calculus integration (except now the field’s fence was moving).
In conclusion, any of the methods she mentions work if you just teach the kid why he’s doing what he’s doing, to his satisfaction.
Yes, but the problem is that they don’t teach the kids how to **pick a method **and get good at it. Quite the opposite. Instead they teach them that there are many methods (true) and they cheat the kids out of the necessary practice using one method to make them comfortable with that method (and they don’t know what they’ve missed because they believe that school is supposed to be mystifying).
So the poor kids wind up slaves to having a calculator. Or hating arithmetic which has been made unnecessarily complex so some educrat can show off his or her math education.
I have no objection to really bright kids learning or discovering that there’s more than one way to skin the multiplication cat, but hard as it may be for SDMB members to realize this, SOME of us have to be AVERAGE. Those kids need a simple, reliable method to produce accurate results. They need a practical solution, not one that explores number theory every time they need to multiply 26 X 31.
In everyday life, we need accurate numbers. Not whizbang therories.
Said by one who got all A’s in math back when there was no grading curve–but thinks this fuzzy math is a very bad idea for today’s students.
Setting aside the “cluster problems” non-algorithm, none in the video seemed to me any more describable as “exploring number theory” than the standard algorithm. Just like the standard one, they can all be memorized and carried out very easily, without needing to understand how or why they work. (Frankly, they all actually seemed like only minor regrouping variants on the standard algorithm)
Bingo. Some of the stuff I routinely used calculators for in the classroom when I was teaching involved stuff we used to do with tables and interpolation, like computing values of logarithms and trig functions. There’s NO point in learning to do any of that the old-fashioned way.
And as late of the mid-1990s, I was teaching elementary statistics with a standard normal table, and having to teach the kids how to use it to compute more complicated normal probabilities. I still remember the day when, finally teaching with a TI-83, I put the transparency of the standard normal probability table on the overhead - and, with a big grin, picked it up and sailed it across the room. I had instantly freed up four lectures in this class I’d taught a dozen times before, because I finally didn’t have to teach kids to use a damned table. I got to teach concepts for the first time that I’d never had time to teach in an elementary-stats class. It was great.
There are two goals to teaching math in elementary school. The first goal is the more directly practical one: People need to be able to get answers to arithmetic questions. The quickest, easiest, and most practical way to do that, nowadays, is using a calculator, and it’s pure elitism to say otherwise. The calculator is superior to all of the methods discussed on that page, in that respect. In fact, I wouldn’t mind at all seeing a class (or at least, a portion of one) on how to use a calculator taught at the elementary or high school level.
The second goal to teaching math is to teach the students how to think well. This is a much more elusive goal to pin down, but for purposes of math education, it comes down to getting the students to understand why and how methods work. For these purposes, the methods disparaged in that video are better than the standard algorithm, and the more methods learned, the better. Sure, it’s not immediately apparent why that lattice method always works, for instance, but a student who does figure out why it works will come away with a better understanding of the mathematics behind it, and more importantly, with a better understanding of how to gain understanding of other things.
).
Anecdotal evidence, my son is horrible at simple math in his head, but has had all A’s in the subject since they started grading it. That cannot be good needing a calculator or a sheet of paper for simple math. And speaking of calculators his school has made them mandatory the last few years, we used to get detention for sneaking them in. :smack: