Old Math

I have been taken to task for using the term "Progressive Education" to describe the modern instructional practices. Progressive education is all about methods that work, since the methods I complain about don't work they are, ipso facto, not Progressive.
As my charges might say: whatever. I readily admit I don't know anything about Education, which is the academic study of how to teach. I am, however, starting to learn something about education, which is the practical art of teaching.

Every number can be written as the product of a unique set of prime numbers. For example the number 60 = 2 x 2 x 3 x 5 and that is the only product of prime numbers that equals 60, discounting permutation of course. This is known as the Fundamental Theorem of Arithmetic. Factors are wonderfully useful for finding common divisors and other basic arithmetical operations.

Note the word arithmetic, this is pretty basic stuff. Factors also show up in higher math, for example every polynomial can be factored into smaller polynomials.

We are learning about factors in the 6th grade special ed class. The kids, for the most part, grasp the concept of a factor and prime numbers but in order to seal the deal they need to practice.

Find the factors of 24. They are 1,2,3,4,6,12.

Is 7 a factor of 135? Divide 135 by 7 and you get a remainder of 2, 7 is not a factor of 135.

Here is my problem: These students do not know times tables. Every time you ask them what is 6 x 4 they must count to 24 by 4s. Using their fingers. Rote learning of times tables, when they were young, was not considered proper modern teaching procedure.

Since they can't do multiplication effortlessly then can't do division easily. Since they can't do division they can't quickly do factors. Without factors they can't do fractions. Without fractions they will fail at algebra. etc.

Learning 50 or so facts about numbers (2x2=4, 2x3=6, etc.) unlocks a whole chain of knowledge. Without those facts everything becomes difficult in math.

The lack of factual knowledge prevents children from learning new conceptual ideas.
Argh.

I've been tagged by gracyie at Today's Homework. Might as well get the ball rolling by saying this is one of my favorite blogs.

Jobs I have held:
- Golf Caddy
- Hospital maintenance mechanic
- Precision machinist
- Electronics Engineer

Places I have lived:
- Chester County, Pennsylvania
- Palo Alto, California
- Grenoble, France
- Denver, Colorado

Blogs I enjoy:
- The Education Wonks
- Betsy's Page
- joannjacobs
- Pharyngula

Places I would like to visit:
- The Burgess Shale site in Canada
- The Amazon River
- Angkor Vat
- Great Pyramid of Giza

Music I love:
- Beethoven Symph. #7
- Mozart Concerto for Oboe
- Bach Brandenburg Concertos
- Little Walter

I have been reading Martin Gardner's "The Colossal Book of Mathematics" which is a collection of topics from his very long career as a writer about interesting mathematical oddities. Gardner's books are a wonderful source of material for teachers looking to challenge, perplex or intrigue students with an aptitude for math.

Here is a (paraphrased) paradox from the book:

One Friday the principal at school announces over the PA system that there will be a fire drill next week. It will be at 10 AM but in order to make the drill more authentic it will be a surprise. The principal (like all principals) is completely honest - there will be a fire drill and it will be a surprise.

Can we determine, logically, what day the drill will take place? If Friday of next week comes around and there has been no fire drill then everyone in the school will know there is going to be a drill and thus it will not be a surprise. Thus Friday is out.

On Thursday morning we know, from the previous argument, that the drill will not be held on Friday, therefore it must be held today. It will not be a surprise and thus Thursdays is not the day of the drill.

On Wednesday morning, we know that Friday and Thursday are not the day of the drill so that means today must be the day, which means it won't be a surprise… You get the idea; we can logically show that the fire drill cannot happen on any day of the week.

It is a very simple paradox to explain, yet it is a non-trivial problem, Gardner has three full pages in the bibliography on this one paradox!

I will try it out on the 7th and 8th graders this week.

Here is how students are taught to multiply two numbers in the school district where I live. Consider the problem 12 * 34. We can write this as (10 + 2) * (30 + 4). Using the distributive law twice this can be written as 10 * 30 + 10 * 4 + 2 * 30 + 2 * 4 = 300 + 40 + 60 + 8 = 408. A helpful memory aide is FOIL, multiply the First, Outer, Inner and Last pairs of numbers.

Seems like a reasonable approach, and it can actually help to use this technique for mental arithmetic. If you are familiar with the “old” way of multiplication it goes like this: 12 * 34 = (10 + 2) * 34 = 340 + 68 = 408 (assuming the 34 was written above the 12).

The "old" and "new" methods have been around for centuries, but (until recently) the "old" method has been preferred because, well it works better.

The number of multiplications and additions for the new method scales with the number of digits in the bottom number. With the new number the number of operations scales with the number of digits squared (albeit they are simpler operations). If you are planning to multiply anything large than two digit numbers you want to use the "old" method.

Whatever the logic behind the switch the students in 6th grade do not know how to multiply three digit numbers and they do not know how to divide. (Long division? Are you CRAZY?). The curriculum simply does not provide for those two skills.

There are several root problems. Parents, for the most part, hate math. Elementary teachers, for the most part, hate math and it only takes one to ruin an entire cohort of students for the subject. The progressive education system rejects any kind of rote learning or systematic procedure like times tables or long division.

Net result is students do not know the basic math facts like 6*7, do not have any reliable algorithms to solve problems and get a near constant diet of negative feedback about the subject.

The oldest charter school in this district was started, in part, because parents were justifiably horrified by math education in the other public schools. That school emphasizes traditional math education. It has a large waiting list and ranks at the very top in all rankings of academic performance.

I have been helping out in a sixth grade special ed math class since the school started, originally on account of a girl with big time attention problems but generally walking around helping students individually. Things have been going well, my protégé has made huge improvements and the class was really starting to "click" and I was talking to the teacher about spending less time in the class. Throw that crutch away I say.

And then in one week we picked up two new boys. Both big time victims of SLS (Sh*tty Life Syndrome, closely related to SPS, Sh*tty Parent Syndrome). Emotional and physical abuse, etc. Not an excuse of course but an explanation for distressing behavior: tantrums, lies, refusals; the whole gamut.

Both these kids are quite capable of doing the work, they are ahead of most of the class in knowledge and ability. But they are used to using bad behavior to get attention, which is probably one of the big things that has been missing in their lives.

Needless to say the class has not absorbed TWO bad apples with grace, kids are reverting to where they were at the beginning of the year. I call it The Math Class From H*ll. It will take us weeks to get things back on track and these two boys moving in a positive direction. At which point the parent will probably pick up and move, leaving me with an empty feeling and the boy with a new school to face.