As a way of showing how these techniques are used, I will take you through the sequence of events which would take an innumerate (fancy way of saying someone who can’t do arithmetic) person to the end of Grade 9 math. By the way, if someone can’t do math, there is a name for the cause. The unseen villain is dyscalculia. Sound familiar? First a problem behavior is named and then the behavior is explained by the label. Making fun of the people who do this is too easy, so I’ll go back to what can be accomplished, even for someone "with" dyscalculia. We’ll call our hero Rick.
Rick had a poor start in math, and by Grade 6, as far as he was concerned, the math teacher was speaking in a combination of Gregorian Chants and hexadecimal. In spite of all the attempts of the social-work oriented people in Rick’s school to enhance his self-esteem and create an environment in which his creativity would flourish, Rick felt incredibly bad in math class. He avoided it when he could and was in a trance when he couldn’t. It is fair to say that Rick hated math. When his mother brought him to me she asked if I had a teaching certificate. I said, "No, I don’t have the same credentials as those who have failed to teach him so I won’t be doing more of the same." Every good story needs irony.
Even if Rick had thought he was good in math or had high self-esteem, that would not reflect in his mathematical behavior. A worldwide study of math students found that Korean students ranked first and students from the U.S. ranked last. Twenty-three percent of the Korean students answered that they were good in math compared to 68 percent of the American students.
When Rick gets in the clutches of a Power Teacher, our teaching argonaut needs to do a number of things: 1). Determine what skills our young lad has. Standardized achievement tests are of little value for this. As you will remember, these kinds of tests are made to compare students. Not much help at all. The Power Teacher will find that Rick doesn’t have many skills after taking math for six years because they’ve been teaching non-behavior at school. 2). Keep Rick motivated or, in behavioral terms, make sure that his behavior is reinforced by reinforcing success and by ensuring that success is possible through the sequence of exercises given him. 3). Ensure that Rick starts his exercises below his level of competence. 4). Ensure that Rick has attained a high level of fluency in each skill before going on to more complex skills. There are many other components of this program, but I’ll content myself with this partial list. Notice that these steps are the opposite of what happens in the average school.
The first test of Rick’s skills (remember we are not testing Rick, his essence will remain unknown to us, we are interested in his mathematical behavior) will show us that he can do perhaps 12 to 15 digits correct per minute in simple arithmetic facts such as 21/3, 9+6, 17-9 and 7X8 with the help of his fingers. He hides his digital teaching aids under the table, but our Power Teacher has seen this before and pretends he doesn’t notice. There is no point in telling him not to use his fingers. He will stop soon enough. At such a slow rate, he makes many mistakes. We do not inquire after his self-esteem or test his creativity. We check how fast he can write digits. This may be holding him back and we may start with the exercise of writing 0 1 2 3 4 5 6 7 8 9 and repeat.
If his speed in digits is OK or we make it OK, we’ll go to the basic math facts. If he can improve on exercises in which facts such as 7+9, 14-9 are mixed, we’ll continue with this exercise. If he doesn’t show early improvement, we’ll move to practice sheets with fewer different facts. Since most mathematics instruction, even on basic facts is inefficient, we may start with 9+0, 3-0 as special (and easy) cases until we establish the component skills necessary for the final exercise sheet. Remember, we are doing two things, teaching skills and maintaining and improving motivation. We plot results on the Standard Behavior Chart and make sure to show Rick when he’s improving although he soon starts to determine this himself. The teacher claps, squeals, shakes Rick’s hand and generally carries on when Rick makes an aim. Rick knows enough math to realize that 32 per minute is faster than 29 per minute. The reinforcement for improving is automatic. We continue through the math curriculum, ensuring that Rick’s behavior is fluent for each activity before moving on to the next. This ensures that we are using the, "teach once, remember always" method rather than the process followed by the spiral curriculum, "teach a little bit, have the student forget, wait a few months and teach a little bit again…." a procedure which is like looking at a reflection in opposite mirrors. It just goes on and on.
Rick gradually begins to understand the language of the math teacher. The teacher says things such as, "Eighty-one divided by 9 is 9." and, glory to Pythagoras, Rick can "understand" this. His math projects show large increases in frequency. Rick starts to get a little cocky about math. "What’s so hard about this? Is this the best you can do? Give me something tougher?" The Power Teacher seizes this opportunity to throw in an easy exercise or two so that Rick reaches his fluency aim in one try. This is quite a powerful reinforcer and I recommend that the school system use it more. The school system won’t, of course, because the student is supposed to be motivated "internally" and fluency is beyond them. The Power Teacher has been using number equations, so when algebra is introduced it holds no mystery. This story, being hypothetical, has a happy ending. Rick catches up in math. I used the Power Teaching techniques to boost Rick’s ability in math from Grade 1 to Grade 6.5 in 50 hours of instruction and then to Grade 9.5 in another 30 hr. The last two tests were given by the school. The story is hypothetical only because the student’s name wasn’t Rick and the total number of math instruction hours was 40, not 80 and Rick’s grades in math, in spite of knowing practically no math to being above grade level, never rose above the minimum required for promotion. These results have been replicated hundreds of times by myself and other people. The results are as real as they are ignored. I still puzzle about what happened to Rick’s dyscalculia, that "brain condition" cured without neurosurgery.
Cheerio and ttfn,
Grant Coulson
November 2, 2009 at 9:16 pm |
It doesn’t surprise me that no one noticed Rick was above grade level in math. He wasprobably never asked to do anything that required above grade level skills. And if hewas actually good at math he probably hated “math journals” and “explaining” his thinking.
November 2, 2009 at 9:38 pm |
I don’t know whether he had a journal, but it wouldn’t surprise me if he did. He did show me a test question where he got 2 out of 4 for the correct answer. The question was a typical “ambush” question something like: “A man’s suit had been on sale for 14 days. Its original price was reduced by 25% on the first day and 25% of its original price on the 10th day. It is now selling for $140. What was its original price.” He didn’t show his work. I have a theory about success and rules for it. When a person is successful, that person can write a summary of rules for success, but that does not mean the rules preceded the success. They were probably distilled from the success.
Cheers Grant