I know that this may be frustrating to watch from the outside, as teacher, I find it very exciting to witness this type of process. It’s time consuming, yes. But, once you get past that, you have a little girl here that is able to make sense of numbers, talk about what she is doing, and understand where and why she has made a mistake.

This is good stuff. Out of context, it may seem ridiculous, but in the context of a high quality math program, this is top-notch stuff.

If you say so Stephen, but it really does impact a good percentage of children that they will think math is hard at the early stages, and really confuses them over number operations and number sense. That is the psychological part, and than the children must have the ability to draw shapes that actually resembles squares, in some orderly fashion. How many children can actually draw shapes, since that is another area that is no longer practice in any great detail in the primary grades?  I have no idea, but I do have an inkling with my youngest and her dyslexia problems.

Although it was not confirm until my youngest was at the end of grade 3, she often question the silly notion of drawing pictures, when it was much easier to write down the numbers in the new name that they are calling addition, stacking. Mind you when the girl in the video did it by stacking, she did not include the operation sign, but she did it with the drawing of squares. As it was told by my youngest so many years ago that this is how it is done. My youngest grew to hate math, a far more hatred than reading.And by the end of grade 3, she seen herself as being stupid, and the psychological educational assessment, pretty well confirm it, she was at grade 1 level in math and writing, but the bright spot was in reading which was barely at the grade 3 level.

I found that hard to believe that her reading level was at a grade 3 level, when she always had much difficulty reading the math text book, that for all purposes was heavy in language, and less on numbers and number operations. It never did make sense to me, what can be told in one page using numbers, with a few words, is now being done using 10 pages of words, using far less numbers.

Conceptual strategies, or as my youngest sneers saying the words, ‘picture, words and numbers’, is a recipe for weak foundational math skills, that leads to lower achievement in advance math. Since grade one, I have been at war with the math curriculum, that where once my youngest once knew how to do simple addition and subtraction by the age of 3, to where she could no longer add, but she could subtract. Mind you that is if she did it their way, using pictures, words and numbers.  By grade 4, I rolled up my sleeves and taught her using the correct math terms and laws. Terms such as addition, subtraction, a + b = b + a,  important math laws such as the associative law of addition, and other important concepts. Lots of practice without using pictures, manipulatives, and conceptual understanding emerged, to where my youngest is an excellent math student.

Before I undertook the teaching of math for my youngest, I contacted private tutors and a math professor who was in the news at that time, criticizing the math curriculum. I asked rather a simple question, what are the core elements that are essential to do advance math with ease. I took their advice to heart, and was comforting that they thought in the same lines as I did, that all children are quite capable of mastering arithmetic, including basic algebra and geometry. But as it is now with the current math curriculum, teaching methods, the dumbing down, and many other missing elements, very few children are getting a firm foundation in math, in order to do advance math with ease.

On Wikipedia, Advocates of conceptual understanding, “A considerable body of research by mathematics educators has generally supported reform mathematics and has shown that children who focus on developing a deep conceptual understanding (rather than spending most of their time drilling algorithms) develop both fluency in calculations and conceptual understanding.[4] Advocates explain failures not because the method is at fault, but because these educational methods require a great deal of expertise and have not always been implemented well in actual classrooms.”

The critics, “Those who disagree with the inquiry-based philosophy maintain that students must first develop computational skills before they can understand concepts of mathematics. These skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method.]”
http://en.wikipedia.org/wiki/Math_wars

My youngest who is in grade 10, came home the other day, saying she is so bored in math. I smile, because she is finding grade 10 academic math relatively easy, because she does have a firm foundation in math. Not bad for a child, who was at a grade 1 level in math, by the end of grade 3, and the ominous predictions by the so-called experts that at the best she would only be a C minus student in math.

Thanks for posting this video.  I just posted my thoughts about this on my blog, if you are interested:

http://www.ahypatia.wordpress.com.

Hi Nancy,

I have come to appreciate your perspective on this very much. The quote on expertise is an important one.

I remember clearly the day that it was announced in our district that we were going to be purchasing a new math textbook. Millions of dollars were at stake. At that time, the National Council of Teachers of Mathematics (NCTM) in the U.S. were advocating for a whole new approach to teaching mathematics. And the new textbooks reflected that approach.

I was very concerned that the program would just be placed in classrooms with very little initial support. I remember sitting in our associate director of education’s office and pleading with him to put the money into building a sense of expertise among elementary teachers first, before introducing the new textbooks.

You’re absolutely right about the reading level of the program texts!

Those books are no longer being used and, for many students, the program was never really implemented in the way that it could have been.

It’s an interesting thing. It’s OK for a teacher to say admit that they really weren’t that good at math. We accept that. We wouldn’t accept a teacher saying, “I’m really not that good at reading or writing.”

But, in elementary schools, there seems to be a sense that everyone can teach it. Well, not without a firm sense of what they are doing and why.

I don’t want to go on too long here, but I will say that my opinion has changed in the last several years. At one time, I thought that memorization of number facts was not necessary, and that kids would eventually come to know them. Now, after teaching grade 8 math for many years, I believe that deep conceptual understanding must be accompanied by a facility with facts and computation skills.

But, just being to handle the algorithms does not guarantee an understanding of what students are actually doing when they add, subtract, multiply and divide.

Thanks again for the video, Malkin!

I apologize for the omission of some words above…I was rather over-caffeinated this morning, trying to get too many things done too quickly!

stephen

“But, in elementary schools, there seems to be a sense that everyone can teach it. Well, not without a firm sense of what they are doing and why.”

I learn to appreciate it as I went along, getting into more difficult concepts such as fractions. I had to understand it first, before I could proceed with methods. I had to be prepared for the many questions that my youngest would ask. According to the research, dyslexics have to know the whole concept, not just the small part that is being taught, in order for understanding to occur. Many of a night, we spent hours on math concepts that were actually at 2 to 3 levels above. One night, it was three hours exploring the concepts of squared numbers. By the end of the 3 hours, not only did she know the rather simple but essential foundation knowledge for squares, she also had a firm base of knowledge on everything else on squared numbers for future grades. It has been stated in many different places, including the experts, that if every student in a classroom was taught as if all of them were dyslexics,  achievement would be no problem at all.

But it would mean getting rid of the progressive math and language curriculum and teaching methods, and the requirement of teachers to have full understanding of knowledge that is being conveyed.

It rings true to me, because under the current math curriculum and teaching methods, it is a very difficult task to teach conceptual understanding first. And note, when I started to teach my youngest at home, the major complaint from the school was that I was teaching her things that she did not need to know, and my way did not lead to understanding but confusion. The only confusion she had, was why it was necessary to used the new fangled methods, when putting down the numbers was so easy. Or the other problem she had with word problems, and how we came up with a strategy that circled the numbers and underlined the small words that would indicate the type of operations. That they did not like, because it only leads to putting down the numbers once again. Apparently she was still not learning to conceptualized the math problem. I beg to differ, since the current methods in teaching math at the elementary level, are very difficult to teach, when parents of high levels of education, cannot even understand why are they doing it this way?

My point of view has been expressed very well by Hypathia.

I am an engineer; I did a lot of math in high school and university. There is no way anybody can make sense of not advanced math but just regular arithmetic and algebra having been taught that way.

Stephen, I’m really curious which way you will chose for your children to be taught. I hope you will continue describing your experiences on your blog.

I really liked the last point in the video.
It is absolutely true. The textbooks and workbooks for let’s say grade eight use very, very small and simple numbers.
As soon as you ask even a good student a simple - conceptually - problem with bigger numbers they freeze like a dear in the headlights.
For example
2/ 999 + 1/2 would scare the hell out of 75% of grade 8 students.
Let’s not even talk about something horrific like
0.3x - 2 = 4

Actually, take a look at the Canadian math contest.
They are organized as follows:
- about 1/3 very, very easy problems that can even be guessed without a method
- about 1/3 standard computation problems with easy numbers
- about 1/3 so-called reasoning problems that are almost never taught explicitly requiring the student to explore all the posibilities in an organized way.
Problems of the type in how many ways can a student pay 1.35 CAN using only quarters, dimes, 5-cent coins and pennies and the total number of coins has to be less than 15 for example.

These are the types of problems that John Mighton describes in one of the last chapters of “The Myth of Ability”. They are always posed but in my experience almost never taught.

I’m at a loss why this type of problems is considered so important in elementary years and if they are, why the method of solving them is not in the textbook.

Speaking of math, my son has been using the Khan academy math site for almost 3 months now.

Excellent! The practicing of skills to mastery is built into the progression of modules and the review questions, the way the problems are grouped into modules makes sense and the videos that explain things are very good.

My son was able to learn new math concepts by watching the video for the concept two or three times. Very, very good!

Thank you Malkin for suggesting the site.

Europe, I have to say that mastery of math skills, or the arithmetic skills, sure is making light of my youngest math work for grade 10. She is the only one that is two weeks ahead of her class, and the only difference between my child and the other students, , she has mastery and a solid foundation in good old fashion arithmetic. Those long nights of working on fractions, are paying off.  I no longer teach her anything, and in fact she is showing me things I have long forgotten, and she too goes to the Khan site if she is having trouble.

I had no choice but to re-teach or tutored my child, because she was going no where too fast with the current math curriculum. When she comes through the door, telling me about her day, the first thing is the pride in her voice telling me about the math class, than one of the sciences. She may be still weak in written language, but she is a perfect example of what mastery can do for any child, at any ability or skill.