I do the cluster method but that's only in my head really, it's pretty inconvenient and superfluous in paper in my opinion. I prefer they keep teaching the standard methods, they're definitely clear cut and would help students later on, when they need to take higher level math in high school. But that's just my opinion...
1)the so called horrible method you are using is called the distributive property and is taught extensively to early elementary students in China; traditional US students get taught this in 6/7th grade
2)students in china are taught multiple ways to solve a problem; traditional US is taught 1
3)China is conceptually learning focused; the US learning has been procedure based (memorize the algorithm) Seems like your "best" methods are the problem.
@brittymathgeek : 1) No, the horrible method is called "Terc investigations". The distributive property applies to the standard method just as it does to the Terc method. The difference is that the std method gives you a fixed algorithm.
2) reference?
3) How many Nobel prizes does China have compared to USA? Now compare population sizes. The US's problem is not that there is no learning being imparted. Its *who* gets taught by *whom*.
You do of course know that you are doing the exact same thing in all three methods, you're just writing them a bit differently. So the speed difference is just your particular skill at each method. I'd say that clearly the merc (is that it right?) method is superior, because it makes you understand what's going on. I'm a mathematician and when I do multiplication manually I use the merc method. If "fast" is your definition of "good at math" then use a calculator.
@MacLaurin83 : I have a post graduate degree in mathematics, so the speed test is not a question of understanding or familiarity.
The terc method is not a way of conveying understanding, because its not being checked against anything. If you make an error you are lost. The kind of understanding Terc is trying to promote should 1) be taught as part of algebra, where you can foster a better understanding and 2) could be applied to the standard method if its that important.
@websnarf What do you mean by "its not being checked against anything?" I cannot see that the other algorithms are checked against anything in particular. Neither are they self correcting. And why should the understanding of arithmetics be allowed only when doing algebra? Arithmetics is not being taught to kids as a special case of a field or wathever, but is (generally) based on intuition. You could ofc prove all methods, but then why not USE the proof AS our method?
@MacLaurin83 : The standard algorithm can be easily checked by literally anyone, including the person who first did it, by simply rerunning the algorithm. The extra numbers and where they are placed are unique and have to be there for the multiplication to be computed correctly. The terc method is not an algorithm at all, but just a continuous arbitrary regrouping of the multiplication into different equivalences. There are many ways to do the same multiplication using terc.
@MacLaurin83 : Arithmetic is not intuitive -- it must be taught. Please look up the Piraha, or any other hunter-gather group for an examples. Arithmetic only becomes self-reinforcing once algebra is taught. This isn't a matter of proof, all the methods work -- that's not the point. The point is what it takes to give children a mastery of arithmetic which is required *before* you proceed to algebra.
@websnarf I dont think the speed test is a measure of your understanding, but I DO think its a measure of your "training." If you always use one method, then ofc it will be faster. I do the terc method pretty fast let me tell you. And as for polynomial division you could ofcourse use any algorithm for division, not just long. It would have to be pretty hefty polynomials for me to use it. And you could just prove (1-x)(1+x+...)=1 by expanding, which feels simpler to me and does not use division
@MacLaurin83 : My abilities are way beyond the technicalities of any of the methods. So my personal training is a complete non-factor in this test. Like most people, I use a calculator (one that I built myself, but that's another story). You actually cannot prove that an infinite expression is a particular value because you can manipulate it. The long division construction is one way of proving that the result is convergent whenever |x|<1, which manipulation doesn't do.
@websnarf As are my abilities. Yet Im faster with the terc method. Your training is not doing math. It consists in physically being able to draw certain lines faster, not having to spend that split second deciding whether to use an equivalence arrow or not, etc. For the series: In this case we have absolute convergence, so we can rearrange w/o changing the value. We do not prove convergence ofc, but if thats the point we could still use any algorithm. Theres nothing special with long.
@MacLaurin83 : If you are really faster, then prove it. I don't see how its possible, you write many times more things, and group into more ultimate operations. This video has been up for years, and nobody has dared respond to it with an actual video showing anything different.
Series: You don't *KNOW* that you have convergence, unless you use a method like long division.
The lattice method was actually 17th century Europeans first attempts at multiplying with arabic/indian numbers. Over the years we have perfected multiplication and now all of a sudden we want to go back to the inefficient first attempts.
The real inconvenient truth about mathematics education right now is the school systems just passing kids on from one grade to another without caring if they actually are learning anything. Thats why lattice is good because kids don't have to understand it.
Everyday Math is horrible. If your children use Everyday Math at school, like my daughter does, I would suggest buying a book for home use, so that you can explain how to do the math to your children before they do it in class. Also, I would suggest getting a Singapore math, or Saxon math book to teach them how to do math in an organized way.
Thank you for that. I have taught out of Everyday Math, and it is truly frightening. I now teach 8th grade math and it is increasingly more difficult to help students who have risen through elementary out of that program. I have algebra students who will legitimately use lattice to multiply mid-way through a problem. Oh my. Thank you again!
Great video on the utter foolishness of the inquiry based programs which fail to teach computational fluency and claim to teach concepts. I've been teaching HS mathematics for the last 20 years and have seen a steady decline in preparation for higher math as a result of "reform mathematics" . The other day I was multiplying fractions on the board and cross-cancelled, my pretty sharp 10th graders had never seen that before! How can kids divide polynomials if they have never seen long division?
I believe I can explain the benefit of those two methods. The benefit appears to be driving the children, along with their parents, insane. What this accomplishes, I'm unsure of.
I timed myself using a calculator. It took only 3 seconds. That obviously is superior to any method that you used. Thus, based on your reasoning, using a calculator should be the only method taught in school.
Explain to me how you checked your work on the calculator. How do you know you pressed the right key combinations for example? How could anyone check your work?
I tested it on a calculator that displays the calculations. Thus, we know that it is right. How do you know that following the traditional algorithm always works? DId you check it on a calculator?
Anyone else familiar with the traditional method (which includes any educated adult) can check the work, and in fact point out the exact mistakes if any are found.
If there is a bug in your calculator's microcode and it tells you something wrong then what are you going to do?
I am unaware, for example, of any calculator that explains how it applies rounding to its calculations. If I want to double a 200 digit number I have no problem doing that by hand, but many calculators can't do it.
It's interesting that you failed to explain how you know that the traditional algorithm always works. Why avoid answering this question? Isn't this an important part of the mathematics?
The traditional algorithm is nothing more than a deterministic procedure that relies on a*(b+c) + a*b + a*c. Understanding that is available to students after they learn algebra.
That's not what is meant be checking. Checking is for making sure you applied the procedure correctly. If you write out the standard method, then *anyone* who knows that standard method (i.e., *ANYONE* who has gone through a public school) could check the work. If you typo on the calculator, there is no way to check.
Note that checking by using the same procedure does not support why this algorithm works. Students should understand why the algorithms they use work.
What's your problem? Using the procedure and understanding why it works are two *different* things. Obviously the first has to precede the second. If you tie the two together you are just raising the bar so that failure to understand has the side effect of failure to apply.
Besides, knowing *HOW* to prove why it works is *FAR* more important than the mundane act of proving it. And that's the approach that most curricula take.
Is checking your work new criteria that you did not introduce in the video. I'd like to know more about your criteria for teaching algorithms to students. Suppose a particular procedure is efficient but takes a month for most students to learn. Would this matter if another algorithm was less efficient, but students could learn in a week?
This is a response to another video. I only focussed on speed, because the original video made the claim about speed but didn't demonstrate it.
Teaching students elementary arithmetic for the first time automatically includes a requirement about checking your work.
If you could demonstrate an alternative algorithm that was as accurate, as checkable and easier to learn, then you could argue that speed was negotiable. But none of the alternate proposals meet that criteria.
Perhaps you could try some other division algorithms to see how they work for the identity that you refer to. Of course, you don't know any, so you don't know how well they would work.
How do you know whether or not I know of any other algorithms for division? The alternative ones I know are very advanced (meant for machines with fast/large multipliers) and certainly would be no faster, so I ignored them.
Brilliant. Truly compelling and basically common sense. Thanks for all your maths videos, I really do appreciate them. If you don't mind me asking, what is your IQ or what qualifications do you have? PhD in Mathematics? Physics? Thanks.
I understand the bone you have to pick with the system, but the method isn't the problem - many parents offload 100% of the responsibility of raising their child to the school system - and our taxes pay just as much for it per head as parents pay to opt to send their child to private school. (not counting their own taxes).
.. would it not be better to encourage them to think about maths, instead of just apply a formula and lack any further understanding of the methods.
I concede that on pen and paper, the standard algorithm beats almost any method, certainly topping the cluster method for space, and the difference in speed was indeed demonstrated in the video; but mentally, cluster beats all.
Arithmetic is to math as literacy is to poetry. Arithmetic is NOT mathematics.
They are two different things. You want students to think about algebra, calculus and so on. Not calculations.
Methods like the cluster method might have an algebraic justification, but that doesn't make it a worth while thing to think about. Thinking is for algebra, not calculations.
Unfortunately, arithmetic is as far as many pupils ever really get with mathematics and others even view it as mathematics itself, which as you have correctly pointed out, is a limited viewpoint.
In light of this, would you not prefer that pupils are taught to think about arithmetic as opposed to simply carrying out a formula?
Thinking is for mathematics as a whole, including arithmetic. In fact, the very fact that the standard method exists shows that people have thought about arithmetic.
Many students in algebra think that (x+3)(x+3) =x^2 + 9 (as do many college students after they take algebra). Any benefit to understanding the partial products for (10+3)(10+3)? Perhaps this would help students better understand algebra. Some people think that algebra involves generalization and numbers, but many do not. Your thoughts?
My thoughts are that algebra has to be justified from the ground up. TERC investigations does *NOT* do that, because it gives to solid foundation.
TERC investigations tries to half teach you things that are justified by algebra, rather than waiting until algebra can be taught systematically on top of a reliable arithmetic foundation.
If I learn what (10+3)*(10+3) is it will not help me figure out that (1-x) * (1+x+x^2+...+x^n) = 1-x^(n-1). But full algebra will.
What do you mean by "from the ground up"? What does TERC do? Can you provide a specific example? I still have no idea what you mean by "full algebra." The exponential relationship that you describe is not clearly connected to your argument. What point are you trying to make? Is algebra a set of rules to be obeyed or is there some reasoning that underlie these rules? Does reasoning support arithmetic or did these rules descend from above? Where does math come from?
TERC investigations gives you a non-directed way of trying to get at the answer without knowing if you are right or if you've made a mistake. That means there is no fundamental grounding. With the standard method every step is deterministic, so its checkable by anyone.
The only thing TERC does is introduces some basic operations that get introduced in algebra later on, to try to do basic arithmetic. So you never get grounded, and you learn something that will be ultimately superseded anyways.
You continue to talk in vague generalities without specific examples. Why is the standard method "deterministic"? Is it correct just because it is? There is no mathematical reasoning behind it. I'm assuming you say this because you don't know the mathematical reasoning supporting the standard algorithms. Thus, you implicate your own reasoning as faulty. Do you know that different algorithms for subtraction are taught in different countries? Different countries have different standard algorithms.
There is nothing vague about the standard method. Its deterministic by definition. My claim is not about intrinsic correctness, its that it can be *CHECKED* by anyone who knows the method.
I am an expert mathematician who has a web page detailing a personal rederivation of a by-hand square root method. I don't appreciate your attempts at guessing what my level of understanding of these methods are.
The Singapore method is standard in most countries.
Great reply websnarf! No one ever thinks when they calculate. For example, if you took 998+ 1237, I'm sure you use the traditional algorithm rather than think.
When it suits me, I use full algebra. If not I do it the traditional way. That's because cluster, TERC, and other methods are a waste of time -- they offer nothing over algebra, offer less reliability, and are slower than the traditional method on average.
For these methods to be worthwhile, they have to demonstrate that they offer something that traditional + algebra does not.
I really have no idea what you're talking about. What is "full algebra"? What I'm saying is that the traditional algorithm is not the most efficient procedure for all calculations. Many times alternative procedures are much more efficient. See my example above. Thinking and reasoning would be beneficial to determine which procedure is the most efficient, whether an estimate is sufficient, ore whether technology is more efficient.
What happens when pupils want to multiply two three digit numbers? The standard algorithm isn't intuitive enough for that without further teaching, whilst the cluster method is, once learned, intuitive enough to be put to use with any digit numbers: It makes people think about maths, and gives them further understanding.
Admittedly, this does not apply as effectively to less able students, as they often wish to just have a formula and apply it every time, but for more able students..
What happens is that a student who tries to do it with the cluster method does it in their head, they get it wrong some percentage of the time, and they have no way of knowing what they did wrong. So they cannot have their work checked and they cannot learn to avoid future mistakes. If they do it with a pencil and paper then they require someone with good skill, usually with the skill of algebra to check it to find mistakes.
With the standard method everyone who knows it can check the work.
I agree. I'll further argue that the student (depending on age and ability) will get it wrong a larger percentage of time than some of the pro-constructivist math experts are willing to admit. I base that on what I've seen with my own two eyes in the K- 5 age grouping.
I don't think you've even remotely considered the possibility that one of the millions of pupils currently being taught maths in primary school may at one time in their life be without pen, paper and calculator and thus without the means to perform the standard algorithm.
The teaching methods currently employed by primary schools for arithmetic seem to me to encourage children to think about maths, and not just perform the same operations for finding the product of all two digit numbers (cont.)
You're right I didn't think about that scenario, because that scenario is irrelevant. There is no student in the world who is just learning arithmetic whose calculations I would trust if they were doing it in their head, no matter how good the supposed method was for teaching them.
Mental arithmetic tricks by someone who does not know algebra or concepts of estimation is a totally irrelevant skill.
I think the standard algorithm is very difficult to do reliably in your head, which leaves you in a very difficult position if you ever wish to do a calculation without pen and paper. Further to this, I absolutely disagree with your point that to do a calculation in your head is worthless if it's slower than a calculator. There are a thousand and one different situations where you would wish to do this and not even have access to a calculator, making the fact that it may be slower irrelevant.
Utter nonsense. The only time when doing it in your head is worth while, is when you need a fast estimate, which is an entirely different skill (it is aided by, and hence done after *ALGEBRA*). Otherwise using some reliable method will always be preferable. None of these alternates are reliable or even useful in comparison to the earlier methods.
Rock on WebSnarf! My second grader does math great. Now she is not doing well in school because she transfered schools. She learned traditional math from us and her old school. Now she is doing bad (while getting the right answers) because she does traditional math. She is spending her second half of her school year unlearning (in school) traditional math. We are homeschooling her in traditional math. Thanks for your video!
Yes but what you have to realize is that those are *TWO DISADVANTAGES* not a trade off. Being able to do things in your head is worthless if its slower than a calculator and still not as good if you want to check your work afterward.
Math sooner or later must become a subject you do with a paper and pencil. What exactly are you planning to gain by delaying this?
7 year olds should start with understanding the concepts and bag memorizing formulas or algorithms. It's better to understand the problem than to even know how to solve it.
Understanding is meaningless without a foundation. The Nazis "understood" how to deal with jews because their foundations did not include realizing that jews are equally human.
If I understand different ways of how an arithmetic problem breaks down, how does it help if I don't know if I am right, or how I might check my work?
Where did you get that 20 from in the above? Is it 2*(260-250) which you didn't write down? If your arithmetic gets longer or more complicated it just gets worse.
Which is a way of encoding the identity (a + b)*(c + d) = a*c + a*d + b*c + b*d via some "understanding". Which is interesting, but worthless to a student not yet ready for algebra.
where is the evidence that the so-called standard algorithm is better by any measure? Your reasoning (that it's faster) is only one measure and only measured on you. So many people make claims that are based on their own personal opinion. I don't see the evidence being demonstrated here.
Investigations is a FAILURE. For anyone who is a parent and watched how their grade school kids were taught this crap, you will understand the failure. The net/net is the foundation needs to be formed with basics. If nebulous concepts need to be taught to learn why 2x3 is 6, we're putting the cart ahead of the horse for grade schoolers.
While watching this, I was certain this person was making a case AGAINST the original video condemming teaching mathematics so children actually understand what they are doing when they multiply, divide, etc. This person's explanation about why we need to have students master the division algorithm is one of the most ridiculous things I have watched. I am actually speechless!! Has this person stepped out of their office and looked at the world recently? Unbelievable!!
Argument from personal incredulity. You failed to consider that I don't consider anything you think to have any relevance or value to me.
There is no quick or easy division algorithm; the TERC method does not help in any way. Better that there is one standard one that works and is checkable.
Estimate. If someone has a decent understanding of the concepts (not the necessarily the particular algorithm you're promoting) then they would see if the answer was off. How often do you do multiplication this way? I mean, I do it in my head all the time--but just estimating. I say whatever works to teach kids the concepts. Specific techniques have very limited value. Strawman because you're purposely making the other techniques look complex.
Good estimation requires techniques beyond elementary school and would add time to the computation. It would also fail the find low digit errors in large multiplications.
The lattice method is really useful when multiplying large numbers. That's when it comes into its own. Also when using lined paper drawing the lattice is very quick. Try multiplying two 7 digit numbers on lined paper and you'll see the benefits of the lattice method. Plus when using the lattice method it's much easier to trace any errors you've made.
No, the lattice method increases the amount of extra work, and therefore for larger numbers just continues to slow you down to a worse degree.
If these people were serious about showing better methods, they would show the Karatsuba method, which is known to be far superior to all these methods (including the standard method).
Karatsuba requires an *understanding* of algebra in the same way that the TERC method does.
Makes you wonder why American children are so far behind the rest of the world. Using the old way, we have had people innovate and create so much over the past 70 years. Now, we don't need real education, we can all be on reallity tv shows! SUHWEET!
Yes, I can... Now try the same thing with (2x+6)(3x+1). Compare the two methods. Use a geometric model to multiply these binomials and compare to the lattice method. If students learn the reasoning behind multiplication algorithms, then they just might be able to connect them to algebra and other higher mathematics. Doing things faster, doesn't make them better. We need more students to understand mathematics. Some students follow traditional algorithms, many more do not. What about them?
Nonsense. At best, all you have done is skipped arithmetic, moved onto algebra and decided that's good enough. But the lattice method is nothing more than a slow trick. The standard method is equally justified algebraically.
*Understanding* has be based *on top* of a foundation. The lattice method and these other nonsensical tricks are *NOT* a foundation.
Thank you for your input! A discussion on mathematics is always a pleasure. My students have much trouble understanding the distributive property and consequently multiplication of polynomials. They don't understand because the traditional algorithm lays little to no foundation on which to build. Other methods, although "slower" are only so because they are unfamiliar. The standard method is elegant and beautiful, but poorly taught.
There is a simple way of explaining (a+b)*(c+d) but imagining c and d as the cost of two items and a and b two different quantities. So its a bag and a basket of apples and oranges, and you can explain it as a*(c+d)+b*(c+d) or (a+b)*c+(a+b)*d by simply arranging the items physically with some demonstration. That will stick in a student's mind in a way that will never leave them; its a direct appeal to their intuition, rather than some tricked up pattern.
Understanding is what you do *ON TOP* of a foundation. One cannot speak without words, and one cannot understand with a basic palette of facts (including arithmetic.) There is no great structure to learn about arithmetic beyond how to correctly compute with it, and thus nothing to understand about it.
What great *understanding* do you suppose is encoded in arithmetic that I somehow missed?
Numbers don't mean anything unless the child understands conservation of quantity. Many young children can tell you 3 + 2 = 5, but they don't recognize that 3 is conserved when objects are rearranged. How about cardinality? Without an understanding of this idea, facts mean nothing. I've seen many kids who can count, but cannot apply these ideas. Without them, their counting skills are worthless. They can't apply their reasoning. Many ideas underlie arithmetic and counting.
Counting precedes arithmetic which precedes algebra. Nowhere in there is the TERC method or the matrix grid method or any other nonsense. Those methods appeal either to algebra which is getting ahead of the game or artificial nonsense that is useless regardless.
Of course you've never looked at the curriculum have you? If you would, you would see that these ideas are embedded in the curriculum. This is not algebra, but algebraic reasoning that children actually do. You are talking from a nonsensical viewpoint in ignorance, but pretending that you've actually looked at the materials. What a joke!
More straw man. I googled for TERC investigations and found it to be exactly what M.J. McDermott described in the video I responded to.
An example of their stupidity: They consider solving 2-digit multiplication to be a unique benchmark of computational fluency. There is no difference between 2 digit and 10 digit multiplication!
They then ask for students to explain why (a/2)*(2*b) = a*b without teaching fractions.
The harm being perpetrated on the younger generation is infuriating.
Wow, now you've actually looked at some example of the curriculum to draw conclusions! Amazing! Actually they use arrays to explain why (a/2)*(2*b) = a*b. You don't need fractions if a and b are whole numbers and a or b is even. Have you ever taught math before? What do you mean there is no difference between 2 digit and 10 digit multiplication? Other countries build on this benchmark. Of course, let's not learn from them. Let's keep doing what we're doing. Our ignorance will rule the world!
I am not a teacher, so I don't know how to deal with children behavioral problems but I do understand arithmetic and remember how I learned it. I learned 2 digit simultaneously with any multi-digit multiplication. There was no distinction in teaching it because there is no distinction in the actual math of it or the algorithm.
Other westernized countries teach the standard method and they perform better than American schools that don't.
An array can easily be shifted so that you can change 6 x 4 to 3 x 8, noting that the are the same thing. The 2-digit by 2-digit multiplication is important because it is generalizable, thus it is a critical benchmark. Asian countries (high performing) apply this as a primary focus before extending it.
Other westernized countries (high performing) actually talk about mathematical ideas. We don't in the US. One researcher said, after viewing math instruction in the US "I don't see the math"
Investigations introduces fraction work in 3rd grade prior to this work. They also introduce operations with positive and negative numbers in 3rd grade. When did you learn to add positive and negative numbers? I bet not in 3rd grade.
This curriculum is based on research about how students learn. You are basing your conclusions based on your personal opinion that is not grounded in empirical work. Thus, your reasoning cannot be supported. You speak from a position of ignorance.
Why do you bet that I didn't learn negative numbers in 3rd grade? I was a highly accelerated student; trying to peg to my rate learning is besides the point (and more straw man).
There *ARE* empirical studies that show that Washington's students (where TERC is being used) are failing relative to other states and the US failing relative to Europe.
I have seen studies in child learning and they are a joke. We don't understand it well enough to run experiments like this.
Negative numbers are not in any traditional textbook until 6th grade at the earliest. I don't know anything about the situation in Washington, but you can bet the problem is more complex than just the curriculum. Many variable are involved.
I don't know what studies you're referring to. How about looking at a synthesis of reserch such as "How Students Learn" by the National Research Council. The TERC materials employ the recommendations and research in this document. Or look at "Adding it Up"
First of all,I agree with you on your basic point. However, disagree with "Timing" the math problems because it merely diverts the focus on the issue. 3rdly, audio was bad for folks using earbuds! ouch! 4th: Math in general : Math is like a language. You learn to communicate w/numbers instead of words. My college texts dont cover the 'new' stuff, but i'm may not mind OCCASIONALLY using it. - I was taught the old school methods..it doesn't seem much diff. than learning algebra,Cal, etc to me...
You are infuriating. First off, I just finished my bachelors in theoretical math and am entering a doctoral program. The proof I learned of the geometric series is much more elegant and actually shows you what is going on. What you showed illuminates nothing if you only know the algorithm and not why the algorithm works. Second, I think the "new" division is beautiful. Not only does it show what is going on, for me it is faster. It takes forever to try to figure out how many times big numbers...
My son is in summer school because he didn't pass the Math portion of the CRCT. Over 60% of Georgia students this year failed the Math/CRCT. When I sat in on the summer school classes yesterday...I found this cluster method is the way they are teaching the children. I found it terribly confusing. I'm not sure what to do, I think I should teach him at home, the "old method" of multiplication and division. I've been told that will make it more difficult for him to pass the CRCT. HELP!
My son is in summer school because he didn't pass the Math portion of the CRCT. Over 60% of Georgia students this year failed the Math/CRCT. When I sat in on the summer school classes yesterday...I found this cluster method is the way they are teaching the children. I found it terribly confusing. I'm not sure what to do, I think I should teach him at home, the "old method" of multiplication and division. I've been told that will make it more difficult for him to pass the CRCT. HELP!
one thing to remember is that most schools now just try to get kids thu school....
pass the tests then dont worry about them...
tonight i was trying to do an exponential algebra problem for some computer stuff.....i couldnt believe that i couldnt do it....thats why i started watching math vids...
im really ashamed of myself tonight...lol
but i can understand the methods of why they are teaching the lattice crap and all that...
kids pick it up fast and get results on their tests.
I'd have to dock points, considering that timing yourself in methods you aren't skilled with will bias the result, but I'm 100% with you on the argument. I was failed in math (several bad teachers who disliked me) and had to do adult education (90% average on my own). I noticed I couldn't divide on paper anymore, I went out of my way to learn it again, and I still use it in my everyday life despite having a calculator near me and hating math in general.
Do the 13 extra seconds really make those other ways of solving a problem, unimportant and useless? I understand long division as you used it, yet don't see the point and fascination you have with this identity, or your connection with the identity and grade 4 math. Youngins need to understand before they can memorize becuase otherwise they will act without knowing why. Parents get upset because they don't know how to answer the question of why. you know nothin about outsourcing of American jobs
I assure you, I am under severe pressure with respect to outsourcing (I am competing directly with people making one fifth my salary). Those people all learned arithmetic the way I did and recommend. And the ones competing for my job all know the identity I showed very well.
Parents will get more upset when their child gets an answer wrong and their work is impossible to check. So the kids will get bad grades without any obvious way to correct the situation.
using a stopwatch to see which method is better??? the faster the better? Partial products is taught by 3rd grade....an analogy i would use is..eating a pizza..do you eat it as a whole? or bite by bite? so with double digits...or the like...the thing is...if Everyday Math's partial: sums, differences, products, quotients (yes partial + - * %) is taught in a school district, it takes a couple of years to see the result... it takes at least 3-4 yrs...when the 3rd graders are in middle school...
Are you kidding, did you not see the Gram-Schmidt Identity proof? When your high school math teacher asks you to prove that, how are you going to do it without long division? In fact how will you proof our handy dandy power rule for natural numbers without long division?
Great video & proof of the fact that these new methods alone are dangerous to our youth's mathematical development.
My graduating class (I'm a senior in hs now) was the LAST class to learn the conventional methods (they switched to Everyday Math the next year)... I have a brother who is 4 years younger than me. Last year, he asked me for some help with multiplication and I was happy to help, but got quite the shock when he told me that "we don't do it that way!"
Even with that long division method for generating that infinite series, to a beginning student, it is just utterly confusing. I can remember many concepts from math classes that were terribly confusing that required years of rethinking and reprocessing to understand. You can't expect students to just 'get it' because it's right.
I just turned the volume up and I was fine. I definitely agree with this guy though. A day doesn't go by in my calc bc class that my teacher does complain about how his lower level classes can't do simply math like they used to be able to.
I remember in elementary school we had these "mad minutes" where you were given 1 minute to solve as many arithmetic problems as you could. If you were good, you could get all 15-20 or so. However, with these methods it would be hard to solve more than 4.
Of course long division is important. But if you search over web, you will find much more efficient way, how to manipulate with numbers. The mentioned math example (26 x 31) can be computed in mind during max. 5 seconds:-)
Anther good reason for the classic and international algorithm. Thanks. It would be
very nice if you had the opportunity of remaking the video as the quality of the video might prevent non mathematians of understanding your very clear point of view.
i can see your point about teaching children easier ways. And I understand that saying the ideas behind the old methods being important in universities may not matter to some. but i am a college sophomore who currently tutors middle and high school students, and some of these students only know these new methods. The ones who were never shown the old ways are almost always the ones who have trouble with basic geometry and algebra, which we use in our daily lives, whether we realize it or not.
So what if a child has to learn a different way in order to complete the problem? Children are absolutely brilliant at finding new, and sometimes better, ways to solve problems. We are preventing more students from becoming excited and loving math by squashing their creativity. Yes, teaching the standard way is important, but it isn't the only way.
I bet you could show how much faster you can multiply than use an abacus too. But that doesn't mean that your method is faster than the guy who grew up using an abacus.
Kids need number sense more than speed. I have never seen the lattice methosd taught in isolation.
i can cluster in my head. i can also do the trad method in my head, but for some reason clustering is easier. on paper the trad method is better because it's just an algorithm. as for the lattice method i don't know why that's being taught other than it makes multiplication kinda fun and visual. i like all three in the end.
And a calculator (non-human) is faster than any by-hand calculation. So what? The fastest way to compute a function does not necessarily provide the best way to understand the function computed, nor the algorithm that computes it.
What really was the point of the Gram-Schmidt identity digression? This video was absolutely *retarded*.
This video is a response to a video that made claims about the speed of the various methods but without demonstrating this speed difference, so I put it to the test.
The identity is an important one for learning about geometric series. If, as a student, you fail to learn the standard division method, you will not be able to use it to see why this identity works, or why the |x|<1 condition is important. This undermines the claim that the other methods are better at helping the student think.
It can be explained without appeal to the "standard" algorithm for division, viz. long division. So your point is moot even if we ignore the fact that anybody who would learn such an identity would already know long division. Learning elementary arithmetic (even over rationals) doesn't require an ability to understand the most common way of explaining a Gram-Schmidt identity.
You are losing the thread entirely. The methods described in the video I am responding to are suggesting *NOT* learning the standard method for long division, and therefore students will not have a clue as to what I am doing in my demonstration of the identity.
Other explanations for this identity (I misidentified it; its not Gram-Schmidt) tend to be hard to motivate and don't naturally lead to a proof like the long division method.
Knowing long division is not a prerequisite for proving whatever identity you showed. There's very nice combinational proofs of certain theorems, but if one isn't a pro at combinatorial theory, they can still understand different proofs of the same theorem.
Worse (for you), long division is teachable to retarded mongrels. So if somebody was learning a proof of the identity you showed, they could easily learn long division in a couple of minutes and figure out the proof you showed.
And a human who understands an algorithm and has practiced using it can do many problems faster in their head than you can punch it into a calculator, I do it at the grocery store every week.
Teaching a student to solve a problem by inventing a new method every time is re-inventing the wheel. Knowledge builds on previous knowledge, that is how progress is made.
p.s. I have an engineering degree with a minor in Mathematics.
Let me point out that websnarf was making an extremely good point, although it may have been confusing for some nontechnical types. The notion of a geometric series is one that is typically learned in a high school curriculum, yet it is difficult to motivate it intuitively without the method he just described. More generally, I challenge anyone to show how multiplication/division of polynomials can be done using these new-age techniques.
-3(x+y=7) = -3x-3y=-21 y=3 and plugging in 3 as y in either equations would equal
3x+12= 24 -12 -12
1/3*3x=12*1/3
x=4
which would mean you bought three deluxe and four common ones but thinking about it using the cluster method but a little modified to work with algebra is much faster for this type of equation
ok we all can do math but how deep can u go mental with math its endless i thought i was pritty good in math but some video shows some math is going crazier and crazier sometimes i wonder if i can learn that caus im not very high educated bud i have talent for math but im not using it fully but i can understand it that sais enough somebody maybe can teach me
I think the texts books she's talking about basically assume most of the kids in the schools where these 'cluster problem' methods are taught won't be going to University. Quite frankly, the cluster methods seems like a much better way to learn for everyday use. Old methods are usefull and sometimes nessisary building blocks for higher math. If you're never going to study or use higher math, then what's the point, basically. It's sad, but true.
I did my study in Iraq, we were never allowed to use calculators till university, I did my Computer science and Math degree in Baghdad, came to Australia 10 years ago to be shocked how people cant do simple math, you can see it very clear with trade people, cash register people
Inconvenient Truth - Who is responsible for insuring that the government acts within the bonds of the Constitution? Answer = It is the responsibility of citizenship. How can citizens do that? The power of citizenship has widely been known from the example of Paul's Roman citizenship, which sent the centurion scurrying to his higher-ups with the message: "Take heed what thou dost; for this man is a Roman.". Citizenship is a noun.
Totally agree with you on the point on infinite series. There are so many alternate ways to solve a basic math problem. Using grids, intersecting lines etc.. .
All of which teach nothing. None of the fundamental algorithms or methods which are used to describe our more advanced mathematics is taught. Total shame.
Your logic is great if all we needed math for is figuring out gas mileage and what the tip should be if you don't have a calculator. The TERC method is one that teaches through understanding why math works, not just by getting the right answer without consideration for why. Any advancement in math or science requires understanding that the traditional method doesn't provide.
Students in TERC/Investigations districts without outside assistance are floundering. In Seattle only 22% of recent high school graduates can place into a college level math class at Seattle Central CC; a full 50% can not place above the equivalent of high school math one...USA is the worst in math of English speaking nations tested and must be really proud of it. Pick up some Singapore Math books and see why our ass is getting kicked. Even Canada thrashes USA; Canada has a clue we need one.
take what you did and come up with a geometric series? They couldn't. To understand what is happening you have to arrive at the series first through logic, and then use calculus to show that the series = 1/1-x. To do it in the reverse is not possible.
Your argument is totally opposite, and doesn't support your position. Using algebraic long division *REMOVES* the mystery of where this identity comes from. It will match the student's intuition when they understand the idea of the diminishing remainder.
Then proof will reduce to focusing on the remainder.
Once this is done, the student is just one step away from proving why the division algorithm works in general.
The alternative methods will leave students lost to all of this.
What is worse is the horrible second part of your video where you claim that long division proves 1/1-x = 1+x+x^2..., which it doesn't. Then you claim that this is how you begin geometric series, which is absurd. How could anybody
What are you talking about? If you *show* this to a student who is familiar with long division by the standard method they will have a direct reference for this identity.
There is so much wrong here. Applying an algorithm to quickly solve a problem is far different than understanding. Russian Peasants could multiply any natural numbers using Peasant Multiplication, but no one would argue that this meant that they understood the binary behind why it works.
You don't *understand* arithmetic, you just *calculate* it. *Understanding* is for algebra, not arithmetic. There is simply no advantage for omitting the algorithm in favor of partial understanding. The algorithms let you *check* your understanding.
Checking understanding against understanding doesn't work if you don't have something solid that you are sure of that you can fall back on.
While watching that video it didn't occur to me I need long division for algebra factorisation. I suddenly remembered that I had learned that in the 2 years before college.
I think that's the biggest worry, that you need the basic methods for algebra. The new methods are tooled at kids but don't form a good foundation for scientific reasoning.
Great to know the GSI is connected to long-division, the last time I ever used long-division was when I was 8 years old. It seems to be pretty common that teachers skip such things. So I know long-division but noone ever mentioned the connection.
all the methods use multiplication facts. for lattice u just put them in a box and make it alot longer. the standard method is the fastest and easiest to teach. it also relys on multipplication facts. i learned how to do my multipliction facts in 3rd grade. if you know multiplication facts, u can do these problems any way.
the cluster method involves a different way of thinking which the old method does not. its is great for opened problems. that is why you would teach the other methods.
but again, to completely disregard the old method of doing math, and to put such a huge emphasis on calculators...American students are gonna get spanked on international tests.
I really don't like the cluster method. We got something similar with factorising one number and doing smaller multiplication twice. It's the same idea and a more straightforward way, to me.
I'll admit the new methods are more mental, better without paper. But much slower.
its is better to teach a student to do 1 problem in multiple ways, than to teach a student several problems in one way. but why american math is phasing out the old methods is beyond me. i'm going to be a math teacher in Canada, and i do intend to teach the other methods because of the variety of students in a class. NOT everybody will like the old ways, HOWEVER to NOT teach it is wrong.
Well, the proof is cool and all, but when you dont know long devition you cant polynom devition at all. You can always solve polynom devition with cluster problems but in complex rational functions it get really ugly. Long devition just solves it like its any other devition problem.
I agree with you. The reason I feel this is becoming the case is because people feel math is hard and therefore dont want to put any effort in. Hence, the lowering standards of american math education and worsening the student's knowledge. Im still in college as an applied math major, senior, and I realize this exists when I see how some students I tutor examine problems.
The best method depends on the situation. Clustering works for quickly doing math in your head, and it helps to prepare for the concepts of algebra. The traditional method requires less concentration and abstract thinking, provided you have a pencil handy. BTW, the "clustering" technique took me about 4-5 seconds in my head. You just need to become more familiar with manipulating numbers. Keep practicing; you'll get better.
The traditional methods should act as a base for all higher mathematics though other techniques can and should be used if they are faster but not at the expense of the taditional method. by the way there is a faster method of multiplicating the numbers at the start of the video which you can do in less then 10 seconds.
i agree that the good old method is better and it should be taught before the other ones. Nevertheless the lattice method is useful too, as well as the other one. btw you can easily prove the gram identity by using induction and limits.
I'm not really concerned about what method is faster, time wise. I'm not against learning alternative methods either. I am against deleting the standard algorithm from a curriculum, though.
The Terc curriculum actually stresses a lot of tecniques that can't be used universally, like landmark numbers...that concerns me.
I do the cluster method but that's only in my head really, it's pretty inconvenient and superfluous in paper in my opinion. I prefer they keep teaching the standard methods, they're definitely clear cut and would help students later on, when they need to take higher level math in high school. But that's just my opinion...
cuantumk 10 months ago
China is great because:
1)the so called horrible method you are using is called the distributive property and is taught extensively to early elementary students in China; traditional US students get taught this in 6/7th grade
2)students in china are taught multiple ways to solve a problem; traditional US is taught 1
3)China is conceptually learning focused; the US learning has been procedure based (memorize the algorithm) Seems like your "best" methods are the problem.
(Liping Ma...google her)
brittymathgeek 10 months ago
@brittymathgeek : 1) No, the horrible method is called "Terc investigations". The distributive property applies to the standard method just as it does to the Terc method. The difference is that the std method gives you a fixed algorithm.
2) reference?
3) How many Nobel prizes does China have compared to USA? Now compare population sizes. The US's problem is not that there is no learning being imparted. Its *who* gets taught by *whom*.
Liping Ma does not support the Terc method.
websnarf 10 months ago
You do of course know that you are doing the exact same thing in all three methods, you're just writing them a bit differently. So the speed difference is just your particular skill at each method. I'd say that clearly the merc (is that it right?) method is superior, because it makes you understand what's going on. I'm a mathematician and when I do multiplication manually I use the merc method. If "fast" is your definition of "good at math" then use a calculator.
MacLaurin83 11 months ago
@MacLaurin83 : I have a post graduate degree in mathematics, so the speed test is not a question of understanding or familiarity.
The terc method is not a way of conveying understanding, because its not being checked against anything. If you make an error you are lost. The kind of understanding Terc is trying to promote should 1) be taught as part of algebra, where you can foster a better understanding and 2) could be applied to the standard method if its that important.
websnarf 11 months ago
@websnarf What do you mean by "its not being checked against anything?" I cannot see that the other algorithms are checked against anything in particular. Neither are they self correcting. And why should the understanding of arithmetics be allowed only when doing algebra? Arithmetics is not being taught to kids as a special case of a field or wathever, but is (generally) based on intuition. You could ofc prove all methods, but then why not USE the proof AS our method?
MacLaurin83 11 months ago
@MacLaurin83 : The standard algorithm can be easily checked by literally anyone, including the person who first did it, by simply rerunning the algorithm. The extra numbers and where they are placed are unique and have to be there for the multiplication to be computed correctly. The terc method is not an algorithm at all, but just a continuous arbitrary regrouping of the multiplication into different equivalences. There are many ways to do the same multiplication using terc.
websnarf 11 months ago
@MacLaurin83 : Arithmetic is not intuitive -- it must be taught. Please look up the Piraha, or any other hunter-gather group for an examples. Arithmetic only becomes self-reinforcing once algebra is taught. This isn't a matter of proof, all the methods work -- that's not the point. The point is what it takes to give children a mastery of arithmetic which is required *before* you proceed to algebra.
websnarf 11 months ago
@websnarf I dont think the speed test is a measure of your understanding, but I DO think its a measure of your "training." If you always use one method, then ofc it will be faster. I do the terc method pretty fast let me tell you. And as for polynomial division you could ofcourse use any algorithm for division, not just long. It would have to be pretty hefty polynomials for me to use it. And you could just prove (1-x)(1+x+...)=1 by expanding, which feels simpler to me and does not use division
MacLaurin83 11 months ago
@MacLaurin83 : My abilities are way beyond the technicalities of any of the methods. So my personal training is a complete non-factor in this test. Like most people, I use a calculator (one that I built myself, but that's another story). You actually cannot prove that an infinite expression is a particular value because you can manipulate it. The long division construction is one way of proving that the result is convergent whenever |x|<1, which manipulation doesn't do.
websnarf 11 months ago
@websnarf As are my abilities. Yet Im faster with the terc method. Your training is not doing math. It consists in physically being able to draw certain lines faster, not having to spend that split second deciding whether to use an equivalence arrow or not, etc. For the series: In this case we have absolute convergence, so we can rearrange w/o changing the value. We do not prove convergence ofc, but if thats the point we could still use any algorithm. Theres nothing special with long.
MacLaurin83 11 months ago
@MacLaurin83 : If you are really faster, then prove it. I don't see how its possible, you write many times more things, and group into more ultimate operations. This video has been up for years, and nobody has dared respond to it with an actual video showing anything different.
Series: You don't *KNOW* that you have convergence, unless you use a method like long division.
websnarf 11 months ago
The lattice method was actually 17th century Europeans first attempts at multiplying with arabic/indian numbers. Over the years we have perfected multiplication and now all of a sudden we want to go back to the inefficient first attempts.
The real inconvenient truth about mathematics education right now is the school systems just passing kids on from one grade to another without caring if they actually are learning anything. Thats why lattice is good because kids don't have to understand it.
edude2480 11 months ago
Everyday Math is horrible. If your children use Everyday Math at school, like my daughter does, I would suggest buying a book for home use, so that you can explain how to do the math to your children before they do it in class. Also, I would suggest getting a Singapore math, or Saxon math book to teach them how to do math in an organized way.
bjalder26 1 year ago
Thank you for that. I have taught out of Everyday Math, and it is truly frightening. I now teach 8th grade math and it is increasingly more difficult to help students who have risen through elementary out of that program. I have algebra students who will legitimately use lattice to multiply mid-way through a problem. Oh my. Thank you again!
mattybohan4 1 year ago
Great video on the utter foolishness of the inquiry based programs which fail to teach computational fluency and claim to teach concepts. I've been teaching HS mathematics for the last 20 years and have seen a steady decline in preparation for higher math as a result of "reform mathematics" . The other day I was multiplying fractions on the board and cross-cancelled, my pretty sharp 10th graders had never seen that before! How can kids divide polynomials if they have never seen long division?
jarg07 1 year ago
I believe I can explain the benefit of those two methods. The benefit appears to be driving the children, along with their parents, insane. What this accomplishes, I'm unsure of.
SlayerZaraki 1 year ago
Just teach the Mclaurin Series and function expansion to 3rd graders lol
gvsfgdf 1 year ago
High school education is failing America. Not just in math, but across the board.
thisnameisuniq 2 years ago
I timed myself using a calculator. It took only 3 seconds. That obviously is superior to any method that you used. Thus, based on your reasoning, using a calculator should be the only method taught in school.
sleeper2345 2 years ago
Explain to me how you checked your work on the calculator. How do you know you pressed the right key combinations for example? How could anyone check your work?
websnarf 2 years ago
I tested it on a calculator that displays the calculations. Thus, we know that it is right. How do you know that following the traditional algorithm always works? DId you check it on a calculator?
sleeper2345 2 years ago
Anyone else familiar with the traditional method (which includes any educated adult) can check the work, and in fact point out the exact mistakes if any are found.
If there is a bug in your calculator's microcode and it tells you something wrong then what are you going to do?
I am unaware, for example, of any calculator that explains how it applies rounding to its calculations. If I want to double a 200 digit number I have no problem doing that by hand, but many calculators can't do it.
websnarf 2 years ago
It's interesting that you failed to explain how you know that the traditional algorithm always works. Why avoid answering this question? Isn't this an important part of the mathematics?
sleeper2345 2 years ago
The traditional algorithm is nothing more than a deterministic procedure that relies on a*(b+c) + a*b + a*c. Understanding that is available to students after they learn algebra.
That's not what is meant be checking. Checking is for making sure you applied the procedure correctly. If you write out the standard method, then *anyone* who knows that standard method (i.e., *ANYONE* who has gone through a public school) could check the work. If you typo on the calculator, there is no way to check.
websnarf 2 years ago
Note that checking by using the same procedure does not support why this algorithm works. Students should understand why the algorithms they use work.
sleeper2345 2 years ago
What's your problem? Using the procedure and understanding why it works are two *different* things. Obviously the first has to precede the second. If you tie the two together you are just raising the bar so that failure to understand has the side effect of failure to apply.
Besides, knowing *HOW* to prove why it works is *FAR* more important than the mundane act of proving it. And that's the approach that most curricula take.
websnarf 2 years ago
Is checking your work new criteria that you did not introduce in the video. I'd like to know more about your criteria for teaching algorithms to students. Suppose a particular procedure is efficient but takes a month for most students to learn. Would this matter if another algorithm was less efficient, but students could learn in a week?
sleeper2345 2 years ago
This is a response to another video. I only focussed on speed, because the original video made the claim about speed but didn't demonstrate it.
Teaching students elementary arithmetic for the first time automatically includes a requirement about checking your work.
If you could demonstrate an alternative algorithm that was as accurate, as checkable and easier to learn, then you could argue that speed was negotiable. But none of the alternate proposals meet that criteria.
websnarf 2 years ago
Perhaps you could try some other division algorithms to see how they work for the identity that you refer to. Of course, you don't know any, so you don't know how well they would work.
sleeper2345 2 years ago
How do you know whether or not I know of any other algorithms for division? The alternative ones I know are very advanced (meant for machines with fast/large multipliers) and certainly would be no faster, so I ignored them.
websnarf 2 years ago
Brilliant. Truly compelling and basically common sense. Thanks for all your maths videos, I really do appreciate them. If you don't mind me asking, what is your IQ or what qualifications do you have? PhD in Mathematics? Physics? Thanks.
Oneill9293 2 years ago
I understand the bone you have to pick with the system, but the method isn't the problem - many parents offload 100% of the responsibility of raising their child to the school system - and our taxes pay just as much for it per head as parents pay to opt to send their child to private school. (not counting their own taxes).
Saiphes 2 years ago
.. would it not be better to encourage them to think about maths, instead of just apply a formula and lack any further understanding of the methods.
I concede that on pen and paper, the standard algorithm beats almost any method, certainly topping the cluster method for space, and the difference in speed was indeed demonstrated in the video; but mentally, cluster beats all.
cgnicholls92 2 years ago
Arithmetic is to math as literacy is to poetry. Arithmetic is NOT mathematics.
They are two different things. You want students to think about algebra, calculus and so on. Not calculations.
Methods like the cluster method might have an algebraic justification, but that doesn't make it a worth while thing to think about. Thinking is for algebra, not calculations.
websnarf 2 years ago
Unfortunately, arithmetic is as far as many pupils ever really get with mathematics and others even view it as mathematics itself, which as you have correctly pointed out, is a limited viewpoint.
In light of this, would you not prefer that pupils are taught to think about arithmetic as opposed to simply carrying out a formula?
Thinking is for mathematics as a whole, including arithmetic. In fact, the very fact that the standard method exists shows that people have thought about arithmetic.
cgnicholls92 2 years ago
Many students in algebra think that (x+3)(x+3) =x^2 + 9 (as do many college students after they take algebra). Any benefit to understanding the partial products for (10+3)(10+3)? Perhaps this would help students better understand algebra. Some people think that algebra involves generalization and numbers, but many do not. Your thoughts?
sleeper2345 2 years ago
My thoughts are that algebra has to be justified from the ground up. TERC investigations does *NOT* do that, because it gives to solid foundation.
TERC investigations tries to half teach you things that are justified by algebra, rather than waiting until algebra can be taught systematically on top of a reliable arithmetic foundation.
If I learn what (10+3)*(10+3) is it will not help me figure out that (1-x) * (1+x+x^2+...+x^n) = 1-x^(n-1). But full algebra will.
websnarf 2 years ago
What do you mean by "from the ground up"? What does TERC do? Can you provide a specific example? I still have no idea what you mean by "full algebra." The exponential relationship that you describe is not clearly connected to your argument. What point are you trying to make? Is algebra a set of rules to be obeyed or is there some reasoning that underlie these rules? Does reasoning support arithmetic or did these rules descend from above? Where does math come from?
sleeper2345 2 years ago
TERC investigations gives you a non-directed way of trying to get at the answer without knowing if you are right or if you've made a mistake. That means there is no fundamental grounding. With the standard method every step is deterministic, so its checkable by anyone.
The only thing TERC does is introduces some basic operations that get introduced in algebra later on, to try to do basic arithmetic. So you never get grounded, and you learn something that will be ultimately superseded anyways.
websnarf 2 years ago
You continue to talk in vague generalities without specific examples. Why is the standard method "deterministic"? Is it correct just because it is? There is no mathematical reasoning behind it. I'm assuming you say this because you don't know the mathematical reasoning supporting the standard algorithms. Thus, you implicate your own reasoning as faulty. Do you know that different algorithms for subtraction are taught in different countries? Different countries have different standard algorithms.
sleeper2345 2 years ago
There is nothing vague about the standard method. Its deterministic by definition. My claim is not about intrinsic correctness, its that it can be *CHECKED* by anyone who knows the method.
I am an expert mathematician who has a web page detailing a personal rederivation of a by-hand square root method. I don't appreciate your attempts at guessing what my level of understanding of these methods are.
The Singapore method is standard in most countries.
websnarf 2 years ago
Great reply websnarf! No one ever thinks when they calculate. For example, if you took 998+ 1237, I'm sure you use the traditional algorithm rather than think.
sleeper2345 2 years ago
When it suits me, I use full algebra. If not I do it the traditional way. That's because cluster, TERC, and other methods are a waste of time -- they offer nothing over algebra, offer less reliability, and are slower than the traditional method on average.
For these methods to be worthwhile, they have to demonstrate that they offer something that traditional + algebra does not.
websnarf 2 years ago
I really have no idea what you're talking about. What is "full algebra"? What I'm saying is that the traditional algorithm is not the most efficient procedure for all calculations. Many times alternative procedures are much more efficient. See my example above. Thinking and reasoning would be beneficial to determine which procedure is the most efficient, whether an estimate is sufficient, ore whether technology is more efficient.
sleeper2345 2 years ago
Ok, you are making unsubstantiated claims. The alternatives are slower, more error prone, are uncheckable, and remove understanding from the student.
websnarf 2 years ago
What happens when pupils want to multiply two three digit numbers? The standard algorithm isn't intuitive enough for that without further teaching, whilst the cluster method is, once learned, intuitive enough to be put to use with any digit numbers: It makes people think about maths, and gives them further understanding.
Admittedly, this does not apply as effectively to less able students, as they often wish to just have a formula and apply it every time, but for more able students..
cgnicholls92 2 years ago
What happens is that a student who tries to do it with the cluster method does it in their head, they get it wrong some percentage of the time, and they have no way of knowing what they did wrong. So they cannot have their work checked and they cannot learn to avoid future mistakes. If they do it with a pencil and paper then they require someone with good skill, usually with the skill of algebra to check it to find mistakes.
With the standard method everyone who knows it can check the work.
websnarf 2 years ago
I agree. I'll further argue that the student (depending on age and ability) will get it wrong a larger percentage of time than some of the pro-constructivist math experts are willing to admit. I base that on what I've seen with my own two eyes in the K- 5 age grouping.
besheba 2 years ago
I don't think you've even remotely considered the possibility that one of the millions of pupils currently being taught maths in primary school may at one time in their life be without pen, paper and calculator and thus without the means to perform the standard algorithm.
The teaching methods currently employed by primary schools for arithmetic seem to me to encourage children to think about maths, and not just perform the same operations for finding the product of all two digit numbers (cont.)
cgnicholls92 2 years ago
You're right I didn't think about that scenario, because that scenario is irrelevant. There is no student in the world who is just learning arithmetic whose calculations I would trust if they were doing it in their head, no matter how good the supposed method was for teaching them.
Mental arithmetic tricks by someone who does not know algebra or concepts of estimation is a totally irrelevant skill.
websnarf 2 years ago
I think the standard algorithm is very difficult to do reliably in your head, which leaves you in a very difficult position if you ever wish to do a calculation without pen and paper. Further to this, I absolutely disagree with your point that to do a calculation in your head is worthless if it's slower than a calculator. There are a thousand and one different situations where you would wish to do this and not even have access to a calculator, making the fact that it may be slower irrelevant.
cgnicholls92 2 years ago
Utter nonsense. The only time when doing it in your head is worth while, is when you need a fast estimate, which is an entirely different skill (it is aided by, and hence done after *ALGEBRA*). Otherwise using some reliable method will always be preferable. None of these alternates are reliable or even useful in comparison to the earlier methods.
websnarf 2 years ago
...and you wonder why we're so behind in math than the rest of the world?
bubblesnuggles 2 years ago
Rock on WebSnarf! My second grader does math great. Now she is not doing well in school because she transfered schools. She learned traditional math from us and her old school. Now she is doing bad (while getting the right answers) because she does traditional math. She is spending her second half of her school year unlearning (in school) traditional math. We are homeschooling her in traditional math. Thanks for your video!
heywood12xu 3 years ago
Its all about teaching people how to do math by head with the MERC method. Not faster.
ruiruivo83 3 years ago
Yes but what you have to realize is that those are *TWO DISADVANTAGES* not a trade off. Being able to do things in your head is worthless if its slower than a calculator and still not as good if you want to check your work afterward.
Math sooner or later must become a subject you do with a paper and pencil. What exactly are you planning to gain by delaying this?
websnarf 3 years ago
"Being able to do things in your head is worthless if its slower than a calculator..."
I understand your point but i do not agree with you on this one.
The more you can calcule or exercise your mind the faster you´ll do an exercise by head.
And this will result in a faster and better way to solve your day by day problems.
ruiruivo83 3 years ago
Fast and error prone? Writing was invented for a reason.
websnarf 3 years ago
FWIW I got the answer way faster using this approach:
30 x 26... is 260 three times... 250+250 is 500... add 20... 260 more is 780. add 31... done
Whatever works I say.
phillipk 3 years ago
And are you a 7 year old, and what if you messed that up somehow? I don't see you showing all your work there in a way that can be easily checked.
websnarf 3 years ago
7 year olds should start with understanding the concepts and bag memorizing formulas or algorithms. It's better to understand the problem than to even know how to solve it.
phillipk 3 years ago
Understanding is meaningless without a foundation. The Nazis "understood" how to deal with jews because their foundations did not include realizing that jews are equally human.
If I understand different ways of how an arithmetic problem breaks down, how does it help if I don't know if I am right, or how I might check my work?
Where did you get that 20 from in the above? Is it 2*(260-250) which you didn't write down? If your arithmetic gets longer or more complicated it just gets worse.
websnarf 3 years ago
I did this--but very fast--in my head:
31x26
30x26 (hold onto 26)
3x 260
250 two times is 500... plus the 2 10s I left out...
500+20+260 is 780
take that last 26 and you have 806.
phillipk 3 years ago
Or, more reliably:
31 x 26 = (30 + 1)*(25 + 1) = 30 * 25 + 30 + 25 + 1 = 750 + 55 + 1 = 806.
Which is a way of encoding the identity (a + b)*(c + d) = a*c + a*d + b*c + b*d via some "understanding". Which is interesting, but worthless to a student not yet ready for algebra.
websnarf 3 years ago
where is the evidence that the so-called standard algorithm is better by any measure? Your reasoning (that it's faster) is only one measure and only measured on you. So many people make claims that are based on their own personal opinion. I don't see the evidence being demonstrated here.
phillipk 3 years ago
Investigations is a FAILURE. For anyone who is a parent and watched how their grade school kids were taught this crap, you will understand the failure. The net/net is the foundation needs to be formed with basics. If nebulous concepts need to be taught to learn why 2x3 is 6, we're putting the cart ahead of the horse for grade schoolers.
TheBGH 3 years ago
learning how to add, subtract, multiply, and divide is not the end. It is only a mean to an end. The end is application. It is problem solving.
jedeus 3 years ago
Sure, walking is what you do before you learn to run. Does that mean you skip walking and just go straight to running?
websnarf 3 years ago
While watching this, I was certain this person was making a case AGAINST the original video condemming teaching mathematics so children actually understand what they are doing when they multiply, divide, etc. This person's explanation about why we need to have students master the division algorithm is one of the most ridiculous things I have watched. I am actually speechless!! Has this person stepped out of their office and looked at the world recently? Unbelievable!!
saking58 3 years ago
Argument from personal incredulity. You failed to consider that I don't consider anything you think to have any relevance or value to me.
There is no quick or easy division algorithm; the TERC method does not help in any way. Better that there is one standard one that works and is checkable.
websnarf 3 years ago
gee, since you're timing it, why didn't you check how fast a calculator takes?
phillipk 3 years ago
I could. Now how would you *CHECK* that I didn't press the wrong buttons on my calculator?
websnarf 3 years ago
Estimate. If someone has a decent understanding of the concepts (not the necessarily the particular algorithm you're promoting) then they would see if the answer was off. How often do you do multiplication this way? I mean, I do it in my head all the time--but just estimating. I say whatever works to teach kids the concepts. Specific techniques have very limited value. Strawman because you're purposely making the other techniques look complex.
phillipk 3 years ago
Good estimation requires techniques beyond elementary school and would add time to the computation. It would also fail the find low digit errors in large multiplications.
websnarf 3 years ago
Very good video sir, thanks.
nick11927 3 years ago
The lattice method is really useful when multiplying large numbers. That's when it comes into its own. Also when using lined paper drawing the lattice is very quick. Try multiplying two 7 digit numbers on lined paper and you'll see the benefits of the lattice method. Plus when using the lattice method it's much easier to trace any errors you've made.
mousegeek 3 years ago
No, the lattice method increases the amount of extra work, and therefore for larger numbers just continues to slow you down to a worse degree.
If these people were serious about showing better methods, they would show the Karatsuba method, which is known to be far superior to all these methods (including the standard method).
Karatsuba requires an *understanding* of algebra in the same way that the TERC method does.
websnarf 3 years ago
Makes you wonder why American children are so far behind the rest of the world. Using the old way, we have had people innovate and create so much over the past 70 years. Now, we don't need real education, we can all be on reallity tv shows! SUHWEET!
itsmiketonight 3 years ago
It seems the cluster method would help kids understand algebra more easily. That box method is just stupid though
Ben0087 3 years ago
The cluster method is totally unstructured, and it at best teaches them algebra *BEFORE* teaching them arithmetic.
websnarf 3 years ago
Yes, I can... Now try the same thing with (2x+6)(3x+1). Compare the two methods. Use a geometric model to multiply these binomials and compare to the lattice method. If students learn the reasoning behind multiplication algorithms, then they just might be able to connect them to algebra and other higher mathematics. Doing things faster, doesn't make them better. We need more students to understand mathematics. Some students follow traditional algorithms, many more do not. What about them?
MrRodriguez314 3 years ago
Nonsense. At best, all you have done is skipped arithmetic, moved onto algebra and decided that's good enough. But the lattice method is nothing more than a slow trick. The standard method is equally justified algebraically.
*Understanding* has be based *on top* of a foundation. The lattice method and these other nonsensical tricks are *NOT* a foundation.
websnarf 3 years ago
Thank you for your input! A discussion on mathematics is always a pleasure. My students have much trouble understanding the distributive property and consequently multiplication of polynomials. They don't understand because the traditional algorithm lays little to no foundation on which to build. Other methods, although "slower" are only so because they are unfamiliar. The standard method is elegant and beautiful, but poorly taught.
MrRodriguez314 3 years ago
There is a simple way of explaining (a+b)*(c+d) but imagining c and d as the cost of two items and a and b two different quantities. So its a bag and a basket of apples and oranges, and you can explain it as a*(c+d)+b*(c+d) or (a+b)*c+(a+b)*d by simply arranging the items physically with some demonstration. That will stick in a student's mind in a way that will never leave them; its a direct appeal to their intuition, rather than some tricked up pattern.
websnarf 3 years ago
Gee, and I thought understanding was the foundation. I guess the rules without reasons are the foundation. Thanks for clearing that up.
sleeper2345 3 years ago
Understanding is what you do *ON TOP* of a foundation. One cannot speak without words, and one cannot understand with a basic palette of facts (including arithmetic.) There is no great structure to learn about arithmetic beyond how to correctly compute with it, and thus nothing to understand about it.
What great *understanding* do you suppose is encoded in arithmetic that I somehow missed?
websnarf 3 years ago
Numbers don't mean anything unless the child understands conservation of quantity. Many young children can tell you 3 + 2 = 5, but they don't recognize that 3 is conserved when objects are rearranged. How about cardinality? Without an understanding of this idea, facts mean nothing. I've seen many kids who can count, but cannot apply these ideas. Without them, their counting skills are worthless. They can't apply their reasoning. Many ideas underlie arithmetic and counting.
sleeper2345 3 years ago
Straw man much?
Counting precedes arithmetic which precedes algebra. Nowhere in there is the TERC method or the matrix grid method or any other nonsense. Those methods appeal either to algebra which is getting ahead of the game or artificial nonsense that is useless regardless.
websnarf 3 years ago
Of course you've never looked at the curriculum have you? If you would, you would see that these ideas are embedded in the curriculum. This is not algebra, but algebraic reasoning that children actually do. You are talking from a nonsensical viewpoint in ignorance, but pretending that you've actually looked at the materials. What a joke!
sleeper2345 3 years ago
More straw man. I googled for TERC investigations and found it to be exactly what M.J. McDermott described in the video I responded to.
An example of their stupidity: They consider solving 2-digit multiplication to be a unique benchmark of computational fluency. There is no difference between 2 digit and 10 digit multiplication!
They then ask for students to explain why (a/2)*(2*b) = a*b without teaching fractions.
The harm being perpetrated on the younger generation is infuriating.
websnarf 3 years ago
Wow, now you've actually looked at some example of the curriculum to draw conclusions! Amazing! Actually they use arrays to explain why (a/2)*(2*b) = a*b. You don't need fractions if a and b are whole numbers and a or b is even. Have you ever taught math before? What do you mean there is no difference between 2 digit and 10 digit multiplication? Other countries build on this benchmark. Of course, let's not learn from them. Let's keep doing what we're doing. Our ignorance will rule the world!
sleeper2345 3 years ago
How do arrays help a fractions problem?
I am not a teacher, so I don't know how to deal with children behavioral problems but I do understand arithmetic and remember how I learned it. I learned 2 digit simultaneously with any multi-digit multiplication. There was no distinction in teaching it because there is no distinction in the actual math of it or the algorithm.
Other westernized countries teach the standard method and they perform better than American schools that don't.
websnarf 3 years ago
An array can easily be shifted so that you can change 6 x 4 to 3 x 8, noting that the are the same thing. The 2-digit by 2-digit multiplication is important because it is generalizable, thus it is a critical benchmark. Asian countries (high performing) apply this as a primary focus before extending it.
Other westernized countries (high performing) actually talk about mathematical ideas. We don't in the US. One researcher said, after viewing math instruction in the US "I don't see the math"
sleeper2345 3 years ago
Investigations introduces fraction work in 3rd grade prior to this work. They also introduce operations with positive and negative numbers in 3rd grade. When did you learn to add positive and negative numbers? I bet not in 3rd grade.
This curriculum is based on research about how students learn. You are basing your conclusions based on your personal opinion that is not grounded in empirical work. Thus, your reasoning cannot be supported. You speak from a position of ignorance.
sleeper2345 3 years ago
Why do you bet that I didn't learn negative numbers in 3rd grade? I was a highly accelerated student; trying to peg to my rate learning is besides the point (and more straw man).
There *ARE* empirical studies that show that Washington's students (where TERC is being used) are failing relative to other states and the US failing relative to Europe.
I have seen studies in child learning and they are a joke. We don't understand it well enough to run experiments like this.
websnarf 3 years ago
Negative numbers are not in any traditional textbook until 6th grade at the earliest. I don't know anything about the situation in Washington, but you can bet the problem is more complex than just the curriculum. Many variable are involved.
I don't know what studies you're referring to. How about looking at a synthesis of reserch such as "How Students Learn" by the National Research Council. The TERC materials employ the recommendations and research in this document. Or look at "Adding it Up"
sleeper2345 3 years ago
First of all,I agree with you on your basic point. However, disagree with "Timing" the math problems because it merely diverts the focus on the issue. 3rdly, audio was bad for folks using earbuds! ouch! 4th: Math in general : Math is like a language. You learn to communicate w/numbers instead of words. My college texts dont cover the 'new' stuff, but i'm may not mind OCCASIONALLY using it. - I was taught the old school methods..it doesn't seem much diff. than learning algebra,Cal, etc to me...
periwink 3 years ago
You are infuriating. First off, I just finished my bachelors in theoretical math and am entering a doctoral program. The proof I learned of the geometric series is much more elegant and actually shows you what is going on. What you showed illuminates nothing if you only know the algorithm and not why the algorithm works. Second, I think the "new" division is beautiful. Not only does it show what is going on, for me it is faster. It takes forever to try to figure out how many times big numbers...
NSquarticsurface 3 years ago
My son is in summer school because he didn't pass the Math portion of the CRCT. Over 60% of Georgia students this year failed the Math/CRCT. When I sat in on the summer school classes yesterday...I found this cluster method is the way they are teaching the children. I found it terribly confusing. I'm not sure what to do, I think I should teach him at home, the "old method" of multiplication and division. I've been told that will make it more difficult for him to pass the CRCT. HELP!
konefka 3 years ago
My son is in summer school because he didn't pass the Math portion of the CRCT. Over 60% of Georgia students this year failed the Math/CRCT. When I sat in on the summer school classes yesterday...I found this cluster method is the way they are teaching the children. I found it terribly confusing. I'm not sure what to do, I think I should teach him at home, the "old method" of multiplication and division. I've been told that will make it more difficult for him to pass the CRCT. HELP!
konefka 3 years ago
I was wondering how you could be enraged. I'm glad I watched your video. You proved it in less than 7 minutes. Excellent!
Calliber50 3 years ago
one thing to remember is that most schools now just try to get kids thu school....
pass the tests then dont worry about them...
tonight i was trying to do an exponential algebra problem for some computer stuff.....i couldnt believe that i couldnt do it....thats why i started watching math vids...
im really ashamed of myself tonight...lol
but i can understand the methods of why they are teaching the lattice crap and all that...
kids pick it up fast and get results on their tests.
after that...
yocyberness 3 years ago
I'd have to dock points, considering that timing yourself in methods you aren't skilled with will bias the result, but I'm 100% with you on the argument. I was failed in math (several bad teachers who disliked me) and had to do adult education (90% average on my own). I noticed I couldn't divide on paper anymore, I went out of my way to learn it again, and I still use it in my everyday life despite having a calculator near me and hating math in general.
EdwardHowton 3 years ago
Do the 13 extra seconds really make those other ways of solving a problem, unimportant and useless? I understand long division as you used it, yet don't see the point and fascination you have with this identity, or your connection with the identity and grade 4 math. Youngins need to understand before they can memorize becuase otherwise they will act without knowing why. Parents get upset because they don't know how to answer the question of why. you know nothin about outsourcing of American jobs
nattirbee 3 years ago
I assure you, I am under severe pressure with respect to outsourcing (I am competing directly with people making one fifth my salary). Those people all learned arithmetic the way I did and recommend. And the ones competing for my job all know the identity I showed very well.
Parents will get more upset when their child gets an answer wrong and their work is impossible to check. So the kids will get bad grades without any obvious way to correct the situation.
websnarf 3 years ago
How I do 26*31 in my head is 26*30+26 so that is 780+26=806
dudeman209 3 years ago
And how does someone check your work?
websnarf 3 years ago
I use it for quick mental math. I can't use the actual algorithm if I don't have a piece of paper in front of me.
dudeman209 3 years ago
deliberatedumbingdown dot com
esm606 3 years ago
using a stopwatch to see which method is better??? the faster the better? Partial products is taught by 3rd grade....an analogy i would use is..eating a pizza..do you eat it as a whole? or bite by bite? so with double digits...or the like...the thing is...if Everyday Math's partial: sums, differences, products, quotients (yes partial + - * %) is taught in a school district, it takes a couple of years to see the result... it takes at least 3-4 yrs...when the 3rd graders are in middle school...
pinasks 3 years ago
Are you kidding, did you not see the Gram-Schmidt Identity proof? When your high school math teacher asks you to prove that, how are you going to do it without long division? In fact how will you proof our handy dandy power rule for natural numbers without long division?
MobiusCoin 3 years ago
Great video & proof of the fact that these new methods alone are dangerous to our youth's mathematical development.
My graduating class (I'm a senior in hs now) was the LAST class to learn the conventional methods (they switched to Everyday Math the next year)... I have a brother who is 4 years younger than me. Last year, he asked me for some help with multiplication and I was happy to help, but got quite the shock when he told me that "we don't do it that way!"
datadyne007 3 years ago 2
Even with that long division method for generating that infinite series, to a beginning student, it is just utterly confusing. I can remember many concepts from math classes that were terribly confusing that required years of rethinking and reprocessing to understand. You can't expect students to just 'get it' because it's right.
FreemanGordan 3 years ago
I just turned the volume up and I was fine. I definitely agree with this guy though. A day doesn't go by in my calc bc class that my teacher does complain about how his lower level classes can't do simply math like they used to be able to.
I remember in elementary school we had these "mad minutes" where you were given 1 minute to solve as many arithmetic problems as you could. If you were good, you could get all 15-20 or so. However, with these methods it would be hard to solve more than 4.
cialis4you 3 years ago
Of course long division is important. But if you search over web, you will find much more efficient way, how to manipulate with numbers. The mentioned math example (26 x 31) can be computed in mind during max. 5 seconds:-)
heczkojiri 3 years ago
And how long does it take to check the work that happened in your mind?
websnarf 3 years ago
Anther good reason for the classic and international algorithm. Thanks. It would be
very nice if you had the opportunity of remaking the video as the quality of the video might prevent non mathematians of understanding your very clear point of view.
faprama 3 years ago
You should remake the video. The audio recording is terrible, and I can't hear half of the words you are saying.
achan1058 3 years ago
i can see your point about teaching children easier ways. And I understand that saying the ideas behind the old methods being important in universities may not matter to some. but i am a college sophomore who currently tutors middle and high school students, and some of these students only know these new methods. The ones who were never shown the old ways are almost always the ones who have trouble with basic geometry and algebra, which we use in our daily lives, whether we realize it or not.
serv1ngHim 3 years ago
So what if a child has to learn a different way in order to complete the problem? Children are absolutely brilliant at finding new, and sometimes better, ways to solve problems. We are preventing more students from becoming excited and loving math by squashing their creativity. Yes, teaching the standard way is important, but it isn't the only way.
redhead133 3 years ago
Whats your point? only thing concluded from this video is you probably still live with your mom and are virgin. Waste of 7:20 of life.
803brando 3 years ago
theres some proffessior from PA who's putting forward the idea that fractions shouldnt be taught until after calculus
Jovian84 3 years ago
I bet you could show how much faster you can multiply than use an abacus too. But that doesn't mean that your method is faster than the guy who grew up using an abacus.
Kids need number sense more than speed. I have never seen the lattice methosd taught in isolation.
eileenes1973 3 years ago
i can cluster in my head. i can also do the trad method in my head, but for some reason clustering is easier. on paper the trad method is better because it's just an algorithm. as for the lattice method i don't know why that's being taught other than it makes multiplication kinda fun and visual. i like all three in the end.
bytetopia 3 years ago
And a calculator (non-human) is faster than any by-hand calculation. So what? The fastest way to compute a function does not necessarily provide the best way to understand the function computed, nor the algorithm that computes it.
What really was the point of the Gram-Schmidt identity digression? This video was absolutely *retarded*.
nortexoid 4 years ago
This video is a response to a video that made claims about the speed of the various methods but without demonstrating this speed difference, so I put it to the test.
The identity is an important one for learning about geometric series. If, as a student, you fail to learn the standard division method, you will not be able to use it to see why this identity works, or why the |x|<1 condition is important. This undermines the claim that the other methods are better at helping the student think.
websnarf 4 years ago
It can be explained without appeal to the "standard" algorithm for division, viz. long division. So your point is moot even if we ignore the fact that anybody who would learn such an identity would already know long division. Learning elementary arithmetic (even over rationals) doesn't require an ability to understand the most common way of explaining a Gram-Schmidt identity.
nortexoid 4 years ago
You are losing the thread entirely. The methods described in the video I am responding to are suggesting *NOT* learning the standard method for long division, and therefore students will not have a clue as to what I am doing in my demonstration of the identity.
Other explanations for this identity (I misidentified it; its not Gram-Schmidt) tend to be hard to motivate and don't naturally lead to a proof like the long division method.
websnarf 4 years ago
Knowing long division is not a prerequisite for proving whatever identity you showed. There's very nice combinational proofs of certain theorems, but if one isn't a pro at combinatorial theory, they can still understand different proofs of the same theorem.
Worse (for you), long division is teachable to retarded mongrels. So if somebody was learning a proof of the identity you showed, they could easily learn long division in a couple of minutes and figure out the proof you showed.
nortexoid 4 years ago
well you could use the new math method to show the Gram Schmidt Itendity also. but that doesn't mean that the new math method isn't retarded.
oogrooq 3 years ago
And a human who understands an algorithm and has practiced using it can do many problems faster in their head than you can punch it into a calculator, I do it at the grocery store every week.
Teaching a student to solve a problem by inventing a new method every time is re-inventing the wheel. Knowledge builds on previous knowledge, that is how progress is made.
p.s. I have an engineering degree with a minor in Mathematics.
spkrman6 3 years ago
Let me point out that websnarf was making an extremely good point, although it may have been confusing for some nontechnical types. The notion of a geometric series is one that is typically learned in a high school curriculum, yet it is difficult to motivate it intuitively without the method he just described. More generally, I challenge anyone to show how multiplication/division of polynomials can be done using these new-age techniques.
balsam2 3 years ago
and to answer that you would have to do this
3x+4y=24 3x+4y=24
-3(x+y=7) = -3x-3y=-21 y=3 and plugging in 3 as y in either equations would equal
3x+12= 24 -12 -12
1/3*3x=12*1/3
x=4
which would mean you bought three deluxe and four common ones but thinking about it using the cluster method but a little modified to work with algebra is much faster for this type of equation
lilazntown 4 years ago
ok we all can do math but how deep can u go mental with math its endless i thought i was pritty good in math but some video shows some math is going crazier and crazier sometimes i wonder if i can learn that caus im not very high educated bud i have talent for math but im not using it fully but i can understand it that sais enough somebody maybe can teach me
TonyCeylan 4 years ago
I think the texts books she's talking about basically assume most of the kids in the schools where these 'cluster problem' methods are taught won't be going to University. Quite frankly, the cluster methods seems like a much better way to learn for everyday use. Old methods are usefull and sometimes nessisary building blocks for higher math. If you're never going to study or use higher math, then what's the point, basically. It's sad, but true.
hack451 4 years ago
I did my study in Iraq, we were never allowed to use calculators till university, I did my Computer science and Math degree in Baghdad, came to Australia 10 years ago to be shocked how people cant do simple math, you can see it very clear with trade people, cash register people
WillBePilot 4 years ago 2
Inconvenient Truth - Who is responsible for insuring that the government acts within the bonds of the Constitution? Answer = It is the responsibility of citizenship. How can citizens do that? The power of citizenship has widely been known from the example of Paul's Roman citizenship, which sent the centurion scurrying to his higher-ups with the message: "Take heed what thou dost; for this man is a Roman.". Citizenship is a noun.
RoseburgOR 4 years ago
Totally agree with you on the point on infinite series. There are so many alternate ways to solve a basic math problem. Using grids, intersecting lines etc.. .
All of which teach nothing. None of the fundamental algorithms or methods which are used to describe our more advanced mathematics is taught. Total shame.
Bfly99 4 years ago
search for 'Math Warriors' and watch the first one. its a math rapp
justin4eva2008 4 years ago
Your logic is great if all we needed math for is figuring out gas mileage and what the tip should be if you don't have a calculator. The TERC method is one that teaches through understanding why math works, not just by getting the right answer without consideration for why. Any advancement in math or science requires understanding that the traditional method doesn't provide.
austin51289 4 years ago
Math is just a tool for other science, don't take it too seriously.
qwertical 4 years ago
It only is a working tool if it works. These days a lot of what passes for math will not be a tool for science or a tool for learnikng science.
DanaherDempsey 4 years ago 3
Students in TERC/Investigations districts without outside assistance are floundering. In Seattle only 22% of recent high school graduates can place into a college level math class at Seattle Central CC; a full 50% can not place above the equivalent of high school math one...USA is the worst in math of English speaking nations tested and must be really proud of it. Pick up some Singapore Math books and see why our ass is getting kicked. Even Canada thrashes USA; Canada has a clue we need one.
DanaherDempsey 4 years ago 4
take what you did and come up with a geometric series? They couldn't. To understand what is happening you have to arrive at the series first through logic, and then use calculus to show that the series = 1/1-x. To do it in the reverse is not possible.
austin51289 4 years ago
Your argument is totally opposite, and doesn't support your position. Using algebraic long division *REMOVES* the mystery of where this identity comes from. It will match the student's intuition when they understand the idea of the diminishing remainder.
Then proof will reduce to focusing on the remainder.
Once this is done, the student is just one step away from proving why the division algorithm works in general.
The alternative methods will leave students lost to all of this.
websnarf 4 years ago
What is worse is the horrible second part of your video where you claim that long division proves 1/1-x = 1+x+x^2..., which it doesn't. Then you claim that this is how you begin geometric series, which is absurd. How could anybody
austin51289 4 years ago
What are you talking about? If you *show* this to a student who is familiar with long division by the standard method they will have a direct reference for this identity.
websnarf 4 years ago
There is so much wrong here. Applying an algorithm to quickly solve a problem is far different than understanding. Russian Peasants could multiply any natural numbers using Peasant Multiplication, but no one would argue that this meant that they understood the binary behind why it works.
austin51289 4 years ago
You don't *understand* arithmetic, you just *calculate* it. *Understanding* is for algebra, not arithmetic. There is simply no advantage for omitting the algorithm in favor of partial understanding. The algorithms let you *check* your understanding.
Checking understanding against understanding doesn't work if you don't have something solid that you are sure of that you can fall back on.
websnarf 4 years ago
While watching that video it didn't occur to me I need long division for algebra factorisation. I suddenly remembered that I had learned that in the 2 years before college.
I think that's the biggest worry, that you need the basic methods for algebra. The new methods are tooled at kids but don't form a good foundation for scientific reasoning.
AdderSIG 4 years ago 3
some of us has to see what we are doing
Morphelios 4 years ago
Great to know the GSI is connected to long-division, the last time I ever used long-division was when I was 8 years old. It seems to be pretty common that teachers skip such things. So I know long-division but noone ever mentioned the connection.
PS:Dont live in the US, math still fucked up.
LddStyx 4 years ago
I actually mislabeled it. Its not the Graham-Schmidt identity, but rather a closely related one.
websnarf 4 years ago
all the methods use multiplication facts. for lattice u just put them in a box and make it alot longer. the standard method is the fastest and easiest to teach. it also relys on multipplication facts. i learned how to do my multipliction facts in 3rd grade. if you know multiplication facts, u can do these problems any way.
dmeeks11 4 years ago
damn how did i miss this video ?
i like it :D and had the original in my fav's ...
DLSS 4 years ago
MAth education ahve always been sucking, math teacher tells there students how to do it but they don't give them understanding.
I h8 math teachers ... there r the most stupid people on earth ...
TheNaturalStep 4 years ago
Apparently you also hate spelling things completely......
cweeks2 4 years ago
part 2..
the cluster method involves a different way of thinking which the old method does not. its is great for opened problems. that is why you would teach the other methods.
but again, to completely disregard the old method of doing math, and to put such a huge emphasis on calculators...American students are gonna get spanked on international tests.
eugene188 4 years ago
I really don't like the cluster method. We got something similar with factorising one number and doing smaller multiplication twice. It's the same idea and a more straightforward way, to me.
I'll admit the new methods are more mental, better without paper. But much slower.
AdderSIG 4 years ago
its is better to teach a student to do 1 problem in multiple ways, than to teach a student several problems in one way. but why american math is phasing out the old methods is beyond me. i'm going to be a math teacher in Canada, and i do intend to teach the other methods because of the variety of students in a class. NOT everybody will like the old ways, HOWEVER to NOT teach it is wrong.
eugene188 4 years ago
PLEASE next time use a clip-microphone ..
i cant keep my finger on the volume regulator ..
WiAmp 4 years ago
I agree that clustering is great for mental arithmetic, but not at the expense of learning it the traditional way.
montage03 4 years ago
Well, the proof is cool and all, but when you dont know long devition you cant polynom devition at all. You can always solve polynom devition with cluster problems but in complex rational functions it get really ugly. Long devition just solves it like its any other devition problem.
Youngbull01 4 years ago
I agree with you. The reason I feel this is becoming the case is because people feel math is hard and therefore dont want to put any effort in. Hence, the lowering standards of american math education and worsening the student's knowledge. Im still in college as an applied math major, senior, and I realize this exists when I see how some students I tutor examine problems.
fenixreborn 4 years ago
The best method depends on the situation. Clustering works for quickly doing math in your head, and it helps to prepare for the concepts of algebra. The traditional method requires less concentration and abstract thinking, provided you have a pencil handy. BTW, the "clustering" technique took me about 4-5 seconds in my head. You just need to become more familiar with manipulating numbers. Keep practicing; you'll get better.
natlang1 4 years ago
The traditional methods should act as a base for all higher mathematics though other techniques can and should be used if they are faster but not at the expense of the taditional method. by the way there is a faster method of multiplicating the numbers at the start of the video which you can do in less then 10 seconds.
Utubefounder 4 years ago
i agree that the good old method is better and it should be taught before the other ones. Nevertheless the lattice method is useful too, as well as the other one. btw you can easily prove the gram identity by using induction and limits.
draconnis12345 4 years ago
I'm not really concerned about what method is faster, time wise. I'm not against learning alternative methods either. I am against deleting the standard algorithm from a curriculum, though.
The Terc curriculum actually stresses a lot of tecniques that can't be used universally, like landmark numbers...that concerns me.
botanical66 4 years ago
the lattice method is slightly wasteful because
it forces you to record values that you could have used
immediately and then discarded if you carried out
the multiplications in the long-multiplication order.
drawing the thing takes time too.
still interesting though.
mrf00bar 4 years ago