Learning Math

I am doing a "review of literature" for my very last paper for my MA, which is supposed to be about teaching math to gifted students - or those who have learning difficulties, like Dyscalculia, which I'd never heard about before.
I am getting very tired of review of literature, because almost every promising article or book I look at turns out to have a lot of quotes from other people. So do I have to track down the original, or is it safe to quote the reviewer?

At the same time, I am applying for a job, the reward for all this hard work, and I've already had 3 interviews, which is encouraging. Two of these went well (although I haven't heard back from them yet.)

The third was with the principal of the high school. He asked me how I would teach his 9th and 10th graders Algebra I so they got it (I'm sort of assuming that most have been there at least once before!) so I told him that I've become very interested in Reasoning and Sense-making, which the NCTM is focusing on in many ways, including a summer institute in Orlando I will be attending when this class is done. The principal raised his eye-brows at those words. He seems to believe that kids learn best with the good old-fashioned "drill and kill" that got me dismissed from my student teaching position (when I wouldn't go along with it!) As expected, I was not called back to that school.

One of the articles I've been looking at is How Students Learn: Mathematics in the Classroom from the National Academies Press. I was delighted to read this quote from another source, which corroborates my thinking:

A recent report of the National Research Council, Adding It Up, reviews a broad research base on the teaching and learning of elementary school mathematics. The report argues for an instructional goal of “mathematical proficiency,” a much broader outcome than mastery of procedures. The report argues that five intertwining strands constitute mathematical proficiency:

1. Conceptual understanding—comprehension of mathematical concepts, operations, and relations
2. Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
3. Strategic competence—ability to formulate, represent, and solve mathematical problems
4. Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
5. Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

Note that only one of these mentions "procedures," while the others are about concepts, strategies, adaptive reasoning, and love of math. Not a word about "drill!"

The thing is, a lot of people think that this kind of mathematical thinking is only appropriate for the "gifted" students. The slow ones need drill and kill, evidently, which obviously does kill. These are the students who try and try and try again and don't succeed. Shouldn't we teach them what it's all about, since they don't "get it" through drill alone?

Studies (sorry, I'm not going to look for sources) have proved that students who have been taught to think do better on even multiple choice standardized tests, than students who have memorized all the steps of a procedure. Another quote from the book - observed by John Holt - tells the whole story:

One boy, quite a good student, was working on the problem, “If you have 6 jugs, and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade will you need?” His answer was 18 pints. I said, “How much in each jug?” “Two-thirds of a pint.” I said, “Is that more or less that a pint?” “Less.” I said, “How many jugs are there?” “Six.” I said, “But that [the answer of 18 pints] doesn’t make any sense.” He shrugged his shoulders and said, “Well, that’s the way the system worked out.” Holt argues: “He has long since quit expecting school to make sense. They tell you these facts and rules, and your job is to put them down on paper the way they tell you. Never mind whether they mean anything or not.”

I've been reading that around 50% of gifted students drop out of high school - some figure out other ways to get to college and achieve their potential, others sell hamburgers, or get doped out. We are boring the gifted students with drill and kill, and we aren't helping the weak ones either. Isn't it time for a change?

Connecting with previous knowledge

One of the major techniques we learn as teacher candidates is to connect new learning to previous knowledge. This is a fantastic example of that!

What are you capable of becoming?

An article (in Reaching New Horizons: Gifted and Talented Education for Culturally and Linguistically Diverse Students) I am reading for my very last Education class before I get my Preliminary Credential starts with a wonderful quote attributed to the German poet Goethe. I majored in German many years ago, and don't recall it (maybe its being in English, not German, makes it unfamiliar,) but it tells a lot about how I want to teach:

If I accept you as you are, I will make you worse;
however, if I treat you as though you are
what you are capable of becoming,
I help you become that.

Wenn wir, sagtest du, die Menschen nur nehmen, wie sie sind,
so machen wir sie schlechter.
Wenn wir sie behandeln, als wären sie, was sie sein sollten,
so bringen wir sie dahin, wohin sie zu bringen sind.

- Johann Wolfgang von Goethe
Wilhelm Meisters Lehrjahre

There's a lot of talk in schools about giving students equal opportunity, which no one really knows what entails.
Does it mean equal teacher time, exactly the same books, problems, lectures, papers to write? Does it mean equal opportunity to succeed (or fail?)
On the other hand, what does Goethe mean by accepting "you as you are?" Can we teachers really know who our students are? Maybe he means "as you appear to me," instead.
My final class is about teaching students who are either gifted or have learning difficulties (which could be the same person.) Yesterday we talked out how students who are bilingual often have been tested to be more creative (and more "gifted?") than their mono-lingual peers. There can be many reasons for this, of course. A creative person may have left his country for another to be able to be creative, for example, so bilingualism is a result of his or her creativity. There have also been brain scans that show that bilinguals use their brain differently than others as well, which is one reason some middle class parents are enrolling their children in bilingual classes with students who are learning English as their second language, which appears to be advantageous for all of them.
An article we read yesterday compared the characteristics of creative people with those of bilinguals, which included:

• Risk taking
• Willingness to confront antagonism, ability to freely reject external limits and  rules, and propensity for self-organization.
• Perseverance, total absorbtion, focus, discipline, commitment
• Curiosity, inquisitiveness
• Openness to new experiences, deep emotions and drowth
• High intrinsic motivation

However, often we fail to recognize creativity in language learners, because it "doesn't come through" in their second language, which we are using to communicate with them.  We are treating them as they are, in Goethe's sense, instead of as they are capable of becoming. (You might be interested in the blog I wrote for another class on language learners a couple of years ago, Negotiated Identity.)
The same can be said for up to half of the gifted students in our classes (which is evidently statistically set at the 5% highest results on some test or other in a school.) Some of the most gifted live in another world, evidently, which does not include paying attention in class - because most of it they have already figured out with their own personal research and experiments. We don't see them as gifted, but rather, difficult.
On the other hand, there is a group of high achievers, who know well what they want to become, and let us know that, so we treat them as if they were actually gifted.
Personal experience, and a comment by a speaker yesterday, indicate that up to 50% of gifted students may drop out of high school and not achieve the potential they would have had if someone had treated them for what they were "capable of becoming."
In our class we are learning strategies that won't take too much of the stressed teacher's time, but can enable the creative and the gifted students not to have to waste time on what they already know, by "compacting the lessons," while challenge them with projects of their own choosing, so they learn the skills that "high achievers" learn - good study habits, the joys of learning, the satisfaction of working hard to do something.
As a high school student, I got easy A's, even in "honors" classes - we didn't have AP back then, but when I got to college I discovered that I didn't have a clue how to really study, so I saw those A's turn into B's and C's, my confidence turned into feelings of inferiority, I avoided some challenges because I was afraid of those C's, and stopped being a risk-taker, which has taken me years to recover.
I want all of my students to succeed, even the creative and gifted ones!

Looking - again

Our school year ended early, which has given me time for all those things I didn't have time for before: an on-line statistics class, my last class at CGU - and our first trip back to Denmark in 7 years, which had to be squeezed in between 2 CGU Saturday sessions.

Even though I probably could have continued with a part-time position where I was, I decided that the commute was impossible. After all I'm teaching now instead of retiring. I want to enjoy teaching, not wasting all my time sitting in my car!

Because of the last position, I will be able to qualify in Math and Science. In fact I'm really looking for jobs in continuation high schools, or other alternative schools, where there is a better chance of working closely with the students. I have always felt my greatest triumphs as a teacher when a student has regained his/her curiosity about school. Some of those kids are in fact quite gifted, but got left out somewhere, and gave up.

One of my special tutoring students told me he did great in math until 8th grade, when he began running with the wrong crowd. Then he discovered he'd missed out on so much that he couldn't find his way. He was very disappointed that I wouldn't be returning, since he said that I was the only teacher who had figured out how to help him learn.

My science classes were the really exciting part of this past semester, since I had not taken the official science pedagogy courses at CGU - just summer extension courses at UC Riverside. It was fun figuring out how to present the standards material to the students so they enjoyed it and wanted to learn. I had a lot of help from fellow NSTA email list participants, who suggested books, experiments, provided PowerPoints, etc.

Right now I of course don't know what I'll be teaching in the Fall. It's hard to prepare for the unknown. But I continue to read and take courses, above and beyond CGU. I hope it will be a position with both math and science - and students who really need me to guide their learning!