Can our students all be geniouses?

I discovered this great video on education on YouTube while looking for the cartoon that follows. I've seen others by this group RSA. It's a look at education - why, for whom, etc. - and creativity. Are we educating creativity out of our students?



Here's the cartoon I saw on Facebook. I can't figure out what the original is.

 

We're theoretically supposed to be providing diversified teaching, but it's hard with a classroom filled to the window-sills. But politicians evidently don't want students to be creative.

Why do we have to learn this stuff?

I once had a student who knew exactly how long 5 cm was. A teacher had had each student find 5 centimeters somewhere on their hands. Connecting up to prior knowledge?
2 cm is a good deal more than half an inch, though, more than 3/4 inch. I think they need more practice working with rulers!
Here's another reason to learn math.
American Chopper vs The Metric System:
(another reason to learn the metric system)

Math is Reasoning and Sense Making

When I was a "pupil" in elementary school, we had a subject called "arithmetic." We learned to add, subtract, multiply, do fractions, percentages and convert from inches to rods, and similar activities. Sometime we had word problems. For some reason, this subject interested me. Or maybe it was science that interested me, and science needs math.
In high school we had 2 years of Algebra, Plane, Solid and Analytic Geometry and then Trigonometry, which I looked forward to, because Dad was an engineer and loved to survey things, which involved trig. None of this was called mathematics, as far as I can remember. That was something I would learn in college.
College came, along with math. Suddenly I was expected to understand very abstract thinking, which I had no training for. It took me a week to understand the necessary predecessor of Calculus: Limits, which is hard to comprehend, now that I know what they are. I expect we were given the definition in all its glory,

DEFINITION: The statement has the following precise definition.
Given any real number , there exists another real number so that

if , then .

(Source:

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html)

Now there are all sorts of ways to ensure that limits and other math concepts make sense, for example this new tool in the NCTM Illuminations collection of lesson plans and activities: Illuminations: Interactive Calculus Tool. Why are teachers still befuddling their students with definitions and procedures to be memorized instead of helping them reason their way to a point where math actually can make sense?
There are far more students taking Algebra and Geometry now than when I went to school. I guess we were somehow motivated to learn it because it was required for college, and those who took it were planning to go to college (about 40% of my suburban high school class. Far fewer from the inner city high school.)
Research has shown that all (i.e. most) students can learn Algebra and Geometry, and many schools are now requiring the 4 years of math in high school that we "college prep" students had back in the late 50's. But I don't think they are motivated in the same way we were.
The National Council of Teachers of Mathematics (NCTM) has been on to this for many years (even before I went to school.) In his closing words at the summer Institute on Reasoning and Sense Making, NCTM 's president Michael Shaughnessy offered several quotes about using reasoning in math instruction, that went back to 1830, which he included in the latest edition of the NCTM newsletter Summing Up:as Reasoning and Sense Making—Expanding Our NCTM Initiative. For example

Continued emphasis must be placed on the development of processes and principles in the solution of concrete problems, rather than on the acquisition of mere facility or skill in manipulation. The excessive emphasis now commonly placed on manipulation is one of the many obstacles to intelligent progress.
—MAA, Reorganization of Mathematics in Secondary Education, 1923

and

Students should be encouraged to question, experiment, estimate, explore, and suggest explanations. Problem solving, which is essentially a creative activity, cannot be built exclusively on routines, recipes, and formulas.
—An Agenda for Action, NCTM, 1980, p. 4

Why was I not learning "the development of processes and principles" back in the 50's? Why are math teachers still teaching "routines, recipes and formulas"?

The NCTM is trying once again to get teachers to help students learn to reason with math so that it makes sense, with conferences like the one I attended and several series of books to encourage teachers to go beyond their textbooks. That this is important is obvious when you inspect the standard mass-produced textbooks, which thrive on steps, "recipes and formulas," with a picture added every once in a while to try to have it make sense. For example, Glencoe Mathematics, Algebra I (which I just happen to have on hand) introduces Polynomials this way:

Why It's Important
Operations with polynomials...form the foundation for solving equations that involve polynomials[!] In addition, polynomials are used to model many real-word situations. In Lesson 8-6 you will learn how to find the distance that runners on a curved track should be staggered. (This is accompanied by a picture of a track race.)

Fortunately, 45 states and the DC have adopted the Common CoreState Standards. As you can see, most of the Standards for Mathematical Practice in the CCSS explicitly refer to reasoning and sense-making as part of mathematics instruction:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

We can look forward hopefully to future textbooks that take these to heart, and help teachers facilitate students' reasoning, rather than require that students memorize steps and procedures that won't even help them pass the current state tests! In the meantime, I hope that math teachers use the many resources provided by the NCTM, such as these in the Illuminations site and the Focus in High School Mathematics: Reasoning and Sense Making books.
.

Reasoning and Sense Making in Geometry

I've finally caught my breath after returning from a fantastic 3-day Institute in Orlando about motivating students to learn mathematics through Reasoning and Sense Making (the link goes to a page with handouts from the many presenters.) What a wonderful experience being together with about 700 other math teachers from all over the country (including Puerto Rico and the Virgin Islands - but only one other person from my neck-of-the-woods) who are also concerned about the state of math instruction these days.

Funny thing is, there has been concern about the way mathematics has been taught in this country since way back in 1833, when a guy name Colburn wrote about using the Pelozzi method for teaching arithmetic. He complained that arithmetic was all drill and memorization, not reasoning. Sound familiar? The NCTM president Mike Shaughnessy went through a long list of early quotes, including 1923, 1935 and more recently since the '80s. I hope he'll upload his PowerPoint, because I'd love to have those quotes!

I mostly followed the sessions about Geometry, since I couldn't figure out last winter how to get students interested in doing proofs, which is what I think is the fun part of geometry. It turns out that kids are turned off by having to prove the obvious, when we ask them to prove things like vertical angles as congruent.

Michael Battista, in his presentation, The Role of Proof in Geometry, said that proof in Geometry is a caricature, since we are teaching the form of proof, rather than the content. We start too low to teach the method of proof at a point where it just doesn't make any sense to prove things. Kids experience that as busy work, which just doesn't make sense! We should start where it takes some thinking, reasoning, struggling, to figure out how to get from the given to what is to be proved. When the students have figured it out and can explain it - that is when they will be open to learning how to do formal proofs. The proof is just a final written justification of the work they have already done. No mathematician would start writing the proof before having reasoned his way to a solution. We shouldn't expect our students to do so either. Proof is a personal sense making, he said, where we go from saying what is "true" to "why" it is true. We are explaining our reasoning to others. Students move through 5 levels of geometric understanding, he said (known as the van Hiele levels:)

1. A visual, holistic examination of the shape
2. A description of the parts and their relationships
3. The interrelating properties (like vertical angles and the properties of parallel lines)
4. Conceptual proofs (explaining verbally)
5. Formal proof.

We have been trying to get students at level 2 to write formal proofs. They need to work with geometric shapes a while to get to level 4. At that point we can introduce formal proofs.

Battista has written a lot about using the software Geometer's Sketchpad to help students reason about geometry so that it makes sense, and has contributed to several books published by the NCTM. The man behind Sketchpad, Michael Serra, honored us with a wonderful collection of Investigations in Geometry from his textbook Discovering Geometry. We had a nice break in all the talk working in groups to figure out a variety of geometric problems. I worked with Origamics problems, developed by Kazuo Haga. (I just ordered the book. What a fun way to work with geometric reasoning!)

Jeffery Wanko provided a fun session, Developing Proof Readiness with New Logic Puzzles. He uploaded his materials to the Institute website, so you can enjoy them, too: Presentation (PDF)and Handout (PDF).We started with puzzles and their solutions, which we studied to figure out the rules. Then we did a big one together (this was a whole roomful of Sudoku addicts, of course,) and worked individually and in pairs to solve some smaller ones. He provide several pages of puzzles, which gave me something to do besides reading on the long plane trip back home! He recommended the Japanese puzzle magazine, Nikoli.com. Students can become ready for writing formal proofs through talking about puzzles like these with each other, getting to at least levels 3 and 4 listed above.

All that confirmed my previous experience that geometry is fun. I hope I can inspire my students the same way!

I will try to find time to write more about my experiences at the Institute in another blog. Besides swimming every day,of course, the most valuable part of the Institute was the many discussions with teachers from all over the country with many different school experiences.

Memory and Sense Making

On Wednesday I am off to Orlando to participate in the National Council of Teachers of Mathematics Summer Institute for High School Teachers on Reasoning and Sense Making. I am looking forward to being with a group of teachers who really want their students to understand mathematics. Too often during teacher training I ran across teachers who were more of the "drill and kill" school.

With my experience with myself, my own children (now successful adults,) and the children and young people I have taught, kids don't learn because you force them to memorize something or give them drills to do whatever time and again until it sinks in. Kids learn because they are curious about something and want to find out about it. If they have a reason to learn something that means something to them (and I doubt "to get into college" or "because it's in the standards" are reason enough for most students,) they will want to learn it, and will dig into a topic until it is theirs. They might even ask someone for the answer - or help to find the answer.

I read a short article yesterday about some research that implies that people don't remember as well as they used to because now they can just Google stuff to get answers they don't have to remember. Evidently some people were tested on how well they remembered things (probably a list of unrelated facts) and some were given the opportunity to enter them on a computer. That last group, of course, forgot them immediately. But that doesn't prove the thesis that we remember differently now. The author of the article pointed out that Socrates was just as worried that the new-fangled techniques of writing would ruin people's ability to memorize things - which is probably true, of course. I write things down so that I can go on to investigate other things. In a sense, the written word is an extension of our long-term memory.

During my teacher ed classes I came upon several references comparing the brain to a computer. You know, data comes into short-term memory, but it has to be connected to other information to be transferred to long-term memory. If we just give students facts, or formulas, or steps to solve problems, they may remember them long enough for the unit test, but if they don't have a way to connect those data with something else - something that makes sense to them, and they want to know about - that data we tried to stuff into their heads probably won't be around for the final, or state exams - or life.

I remember a newspaper opinion piece written by a teacher years ago in Denmark, who claimed that a teacher's job is not to fill in the holes in students' brains, but to create the holes in the brains, so that students would go around looking for what they could put into them. Learning, he said, is making holes, not filling them in. Those holes are what students create while they are making sense of their world. And the holes will never get filled. They will be dug deeper, with lots of side channels that connect up with other holes.

This was illustrated beautifully in a very moving film we saw on Saturday, Buck, which is about a guy who spends 9 months out of the year telling people how to train their horses (not break them) at clinics all around the country. Buck likes to say he's not helping people with horse-trouble, he's helping horses with people-trouble.

I kept thinking that he was talking about classroom "management," where teachers are figuring out how to train their students and need help with "student-trouble" while in reality, it's the students (who have to be there, just like the horses had no choice in the matter) who have "teacher-trouble." The movie was about the best movie on education I have seen. I kept wishing I had a notebook, so I could write down all his words of wisdom. So I bought the book that became the movie The Faraway Horses, in hopes that some of those bits of wisdom are stored there.

One of the most telling episodes in the movie was a woman who told about how Buck had changed the way she trained her horse for dressage. Evidently in the bad old days, horses were trained to get into various unnatural positions by harnessing them with torture instruments (there were examples shown in the film.) Finally the horse gave in and did as required to avoid the pain and humiliation of the harness. But the woman had participated in a sheep-herding clinic with Buck, and discovered that all those unusual positions came naturally to a horse when he was using them to herd sheep. The horse found a connection where he needed to be in that position. And then during dressage, he easily moved in the position (probably fondly remembering the weekend herding sheep.)

Are our students being difficult because they don't want to be harnessed to a school desk when it doesn't make sense to them to be there? Are we trying to break them rather than helping them make sense of what we think they should know?

At the NCTM institute, we have each selected a different area to concentrate in, which for me will be Geometry, which I think was my favorite math subject in high school. I taught some Geometry this past year, taking over from another teacher. It was very difficult teaching students to do the proofs of geometry, which is what I liked best, and which is what geometry is all about. I hope that the Institute will help me see how to present geometry so it makes sense to them. Of course it's easy enough to make sense when you're talking about things that can be represented physically, like area and volume, circles and cylinders. But the abstract high-order thinking of proofs seems to have been distracted by low-level memorization of theorems.

I expect to be a better teacher after the Institute - but it is only one of many ways I am trying to make sense of my job as a teacher.

Addendum

While reading this afternoon I happened upon a note that is so pertinent to this, that I am quoting it here:

When reviewing radioactivity for this book, I was reminded that too often in science resources, authors explain what happens without really explaining why it happens. If you can only describe occurrences,then you really don't understand what's going on, and you end up only memorizing what happens. If you can figure out a mechanism for the occurrences, though, then you can build a lasting understanding of what's going on. Even though scientists often can only describe what happens when they first encounter a phenomenon, the ultimate goal is a mechanism for the phenomenon and the resultant understanding. You can compare this to mathematics, in which there are rules to follow. Only when you understand the reasoning behind the rules do you understand math.

William C. Robertson, in More Chemistry Basics, p 109 (my italics)

The wisdom of experience

I've been having a few talks with friends about the difficulties of getting a job now that I have my credential. Although I have had a few interviews, someone else (younger) seems to get the job each time. I have been charitable and figured that the younger person is probably more qualified than I am. Perhaps she majored in a subject that I have "only" learned through enormous amounts of reading, discussion, email exchanges and a few courses. Perhaps she has more science teaching experience. Perhaps she has actual science laboratory work experience. I can't beat that.

But recently one of my young fellow students got a job for which I felt I was the more qualified. I was teaching the subjects this last spring to students very much like the ones at this particular school, and had selected the same chemistry book that is being used there. I read incessantly about science pedagogy and love going to professional development courses.

As I was teaching this spring, I tentatively introduced the "when I was your age, we didn't have calculators/ computers/ know about DNA..." comment to see how the kids reacted. It turns out they loved it. They also loved that I could be teaching and suddenly come up with some example from my past that just fit the topic perfectly. I have close to 50 years of life experience more than my young fellow students that has not been spent knitting (at least not most of the time.) I taught, I started an environmentally based business, using a lot of chemistry, was a technical writer, learning how to explain things clearly. In fact, most of my career has been about motivating people (to learn German grammar, to treat our world respectfully and sustainably, to use some piece of software efficiently...) Some of my other older new-teacher friends have been engineers, lawyers, economists, business owners - all with fantastic stories to tell.

In the really old days, the elder members of a tribe were called upon as teachers of the young, because people recognized their wisdom. Elderly people in some cultures were revered greatly for their wisdom. In others they were considered doddering fools - maybe because they couldn't hear well, or see well, so they couldn't hear the question properly, or negotiate their surroundings agilely - or maybe they were senile (although I doubt they got old enough for Alzheimer's back then, although they might have gotten mercury or lead or antimony poisoning.)

People my age are often of good health and mind, and they aren't going to take time off to have babies or have to pick up a sick child from school. They may have older parents who need some help, or a spouse who needs surgery. But my spouse cooks all the meals when I'm working!

I've been told about a principal who said that he didn't think an "old fogeys" (like me) would hang around very long - like more than 5 years. Statistics show that young people, unfortunately don't either. I figure I'll teach until I don't like it any more, or until my health deteriorates. Who knows how long that will be. (I sure don't like the idea of sitting around knitting and reading books the rest of my life!) Another told a colleague that she was not going to hire any more baby-boomers (for some unknown reason.)

Of course there are a lot of teachers even younger than I am who no longer enjoy teaching and do not renew their skills and content knowledge. Some of them aren't very far out of college, in fact.

I was enticed to teach by an organization called EnCorps Teachers, who are recruiting experienced people to teach science, math and engineering. I have spent 2 1/2 years studying and practicing to become a good teacher, and run up a bill of close to $60,000 at a private school of education. I'm not quitting any time soon! And neither are my other older fellow students. We have a lot to share and we enjoy kids. We want to give a little back.

Summer Reading

I've had lots of time to read this summer (also to knit and to swim.) I thought someone might be interested in the great books I've found.I am a member of a number of email lists which have asked about summer reading ideas, and I jumped at the chance when I read about books that seemed useful.

For new teachers (like me!)

One of the best books I found about classroom management is K. Cushman's Fires in the Bathroom, which is advice by high school students for new well-meaning teachers, who don't always get it right.
 Learning Outside The Lines: Two Ivy League Students with Learning Disabilities and ADHD Give You the Tools for Academic Success and Educational Revolution is a completely different book - written by 2 students with ADHD who finally figured out how to get their life together to graduate from college. It is a real eye-opener for teachers.
Along the same line is one of the books that is waiting for me: Fair Isn't Always Equal, which I am looking forward to reading.
I was taking a class this summer about teaching  students who are gifted and/or have a learning disability. One of the books I read for my paper was When Gifted Kids Don't Have All the Answers: How to Meet Their Social and Emotional Needs. It helped explain a lot about the problems low achieving but smart kids are having in school.
Since I just got myself a Kindle, I decided to read a couple of books on it. The Accidental Teacher by Eric Mandel tells his story of trying to teach English with no credential and little support. Teaching Outside the Box: How to Grab Your Students By Their Brains by LouAnne Johnson.
Unfortunately most of these books are by English teachers, which is often a different kind of teaching. (I guess English teachers like to write more than math teachers!) So I am very happy to have found Coach G's Teaching Tips, since he is a math teacher.

More books!

I've also been reading about chemistry, biology, physics and math, so I guess there will be at least one more summer reading blog coming up.