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Monday, August 19. 2013
Can't kids just learn their multiplication tables anymore?
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Draw 4 rows of 3 squares, or 3 rows of 4, your choice (multiplication is commutative).
Count them, starting with 11 and going downwards.
You'll exhaust the squares when you get to zero.
The irony about the teacher talking about different approaches etc leading a student to conclude that 3X4=11 is that in most cases, the 3X4=11 answer was achieved by a S.W.A.G. Which to those who are not STEM majors, is a scientific wild-assed guess.
A further irony is that while the teacher may imagine a dialogue with students about explaining approaches etc., in most cases elementary and middle school students have no interest whatsoever in discussing their approaches. In most cases a teacher will have to drag such explanations out of students. In general, students would rather have their teeth pulled instead of explaining how they came up with a given math answer.
Too many adult educators make the assumption that what THEY like to do in learning is also what children like to do. For example, the rote and repetition used in learning to read by the phonics approach is to an adult sensibility, deadly and boring. By contrast, young children like repetition, as their knowledge base is much less than that of an adult. For a child, repetition reinforces what they have learned. For an adult, repetition is boring. How many times does a four or five year old like to have the same story being read? How soon does the adult get tired of reading the same story over and over?
In the last 40 years, math educators have added a smorgasbord of topics to be covered in the lower grades. For an adult with at least a BA in math, which would describe math educators in Colleges of Education [I hope], one more math topic is like reading one more article in the newspaper. The more the merrier, especially since they devote as much time to the new topic as they would to reading another newspaper article- as they have already been exposed to the topic. Which is not the case for the younger students.
Unfortunately, this does not go over as well for the elementary or middle school student. One more topic covered in a cursory, superficial manner - no "drill and kill" here- is one more topic that a student doesn't understand very well- if at all.
Students would be better served by learning fewer math topics in a more thorough manner.
Disclaimer: I taught math for two years.
True story. My kids were required to memorize times tables (which automatically gave them division tables) to the whole number 20. They absorbed it easily and were rather proficient in basic arithmetic by the time they got to second grade.
I was told, point blank, that I was harming my kids by making them learn by discipline rather than allowing them to reach their own system of computation. This low level Bachelors in Education was telling a Phd mathematician that rote learning was incorrect and that only by allowing my kids to learn by "exploration" would they be successful in learning math.
I told the Mrs. that even thou she taught in that same school, we were paying for private education because her colleagues were idiots. Oddly, she agreed. :>)
There is something to be said for "exploration." Unfortunately for exploration-oriented teachers, the best explorations are those which teachers do not script, but when a student spontaneously discovers something. Moreover, when the discovery comes from the student's own endeavors, not from a teacher's prodding, the discovery sticks better.
But in general, most kids just want to be told how to do it. It is faster than discovery/floundering. There is a reason why Direct Instruction has lasted for several thousand years. By and large, it works.
My first 8 years of learning math were run of the mill to me, but I at least got a good grounding in the basic arithmetic operations. In high school I got exposed to New Math [UICSM]. From the beginning, we had to write proofs. In 9th grade, proofs involved the commutative, associative and distributive properties of addition and multiplication. My enjoyment of math increased greatly. At the same time, I recognize that not all students can handle writing proofs- for many or even most it is "math misery," to quote a fellow student.
My thorough knowledge of the times tables, coupled with proofs about distributive properties etc., led me to explorations in estimating. As a result, I can come faster to a ballpark estimate than most people can with an exact answer on a calculator. It's a useful skill. [I can often beat them on an exact answer.]
But it couldn't have been acquired without already knowing the times tables.
There is also a problem in teaching math at the elementary level- for many elementary school teachers, math was their worst subject. While they may not know their times tables very well, they are even less able to explain the mathematical nuances that College of Ed Math educators want elementary school students to handle.
Which reminds me of sitting in math ed courses in my middle age years. We prospective math teachers were told that we were being taught the- let us call it- New Reform method of teaching math. I said to myself, what happened to the New Math I learned when I was in high school? How many iterations of New Reform method of teaching math have we gone through? I suspect a new model ever 5 years- or less. New Improved. Yeah, right.
It is all about process in the same way that policy is all about intent (assuming some of this nefarious shit is really well-intentioned); consequences and objective reality be damned. Welcome to it.
Heh - Oh my here's another example of what is wrong with us as a society.
Which is easier, adding four copies of three or just memorizing four threes are twelve?
This is the crux of the matter. There isn't any replacement or easier method to learn elementary arithmetic other than rote.
Oh boy. Had a rather spotty early education:started K class a year eaerly, and somewhere among the several moves skipped most of second grade and never saw third. HOWEVER, I did do first grade all the way through and most of second. During those years, I was required to learn by memory the times tables through 12. Not just memorize--but be called to the front of the class to answer any question the teacher through at you and two students also. A hell of lot better than feeling sorry for myself and becoming a victim. I am old now and can still multiply as fast as a hand held, or even this calculator on the computer ! Come on folks -- the best thing you can do for a kid is to convince them YOU CAN--not provide excuses for not being able to do something.
I'm not sure how demonstrating how you got to a wrong answer, at least with finding your error, helps with learning maths. However, rote memorization is also a terrible way to learn maths.
What is needed is a combination of both, rote memorization with intelligent explanation....that invokes thought. The problem is, most teachers, especially at the elementary and middle levels, know little about maths other than the monkey math (monkey see, monkey do, monkey gains no deeper understanding).
It doesn't help that we break up the great adventure of math into years of travails. Nor that we do not teach to competency but rather adequacy. Maths is a construction and far to often the foundation or intermediate layers are deemed 80% adequate so let's move on. And forget those who may only be 70% adequate. On the other hand, those who understand must mill about smartly in eternal boredom till a plurality is deemed sufficiently confused to move on.
Salman Khan contrasted maths education with learning to ride a bicycle. No one deems a kid ready to ride a bike at 80% proficiency so now let's move on to BMX riding.
But we now have the technology, or at least Khan Academy does. And it's free. Group progress based on age and time in grade is a stupid way to teach maths and leads to far to many kids being left behind and quite a few losing interest due to the boring pace. Maths education should utilize the flipped classroom with kids working, hitting their rough spots and as long as they are working at and progressing, age competency doesn't matter.
BTW, group progress based on age and time in grade is also a stupid way to teach any foreign language, but, one that demands the adoption of a new and formal way of thinking such as maths, it is really ignorant.
"Thinking leads man to knowledge. He may see and hear, and read and learn whatever he pleases, and as much as he pleases; he will never know anything of it, except that which he has thought over, that which by thinking he has made the property of his own mind. Is it then saying too much if I say that man, by thinking only, becomes truly man? Take away thought from man's life, and what remains?"
- Johann Pestalozzi
I forgot to include this quote on how one truly learns something. The problem with so much maths instruction is that it is heavy on memoriter and weak on presenting the reasoning from a variety of perspectives to provoke thought. The student learns to depend on brute mental exertion, until they reach their limit, then falls behind.
The way pupils study, depends on what is emphasized. The methods that are best to develop a sound knowledge of geography in pupils, will, as a rule, be the best to teach them how to study geography. The reason that mechanical memorizing is the main part of study in the elementary school, high school and university, is that reproduction is the primary thing required. If boys and girls find that the teachers' questions ask for a reproduction of the text, they will memorize before thinking and without thinking. If, however, there is a thought question, it will cause them to organize and analyze the subject matter of the book, and then mechanical memorizing can not occupy such a prominent part.
So as we see, there is merit in the "how arrived" question but not to the exclusion of the "learning your times table" method. The problem is that the desultory teacher will come to teach one way over the other with damaging results. History as a guide would indicate that if necessary the memorization method at least give the student capability in a rote manner whereas all the exposition on method in the world is of little value if one can't arrive at the correct answer in a prompt fashion.