Project Based Learning to Support Math Standards

Since the one job that I've ever been dismissed from - as an Intern for my credentialing through Claremont Graduate University - questioning my ability to teach math using Project Based Learning, I just took a Teacher's Toolkit course about PBL from the UCLA Extension, Education Department. I applied for the job, and was delighted to get it, because I want math to be authentic so that students can see that they really can and will use it in their daily life. I was even promised PD on PBL, but that fizzled out soon after I started around Nov 1. After discovering that the students were drastically behind in learning what they needed of standards, I figured the best to do would be to get them up-to-date before grades were to be submitted 3 weeks later, and then use the project I'd planned, and even presented to the students orally, after the winter break.

For the Toolkit course, we read lots of articles and watched videos on sites like Edutopia and BIE, which are great sources on how to organize a project, with links to ideas for projects. Since most projects seem to use math to do its calculations, often in statistics, rather than supporting math standards,  I was looking for ideas that were specifically for math. Here are a few of the links I discovered with good ideas for projects that really support math standards.

• PBL Examples of Math Project Ideas
• Industrial Mathematics Project for High School Students - Projects
• 12 Steps to Great STEM Lessons
• Teach21 Project Based Learning

 As I read on, I decided that I'd like my final project to have something to do with the music of math, which interests me as I am also a musician, and also a physics teacher, where we touch on the production of musical tones while studying waves. I thought it would be a great way to combine various math standards in Algebra II and PreCalc with standards for waves in Physics and performance, composition and historical and ethnic instruments in Music. And there could also be some music-based readings, and the writing of song texts in ELA, and why not something about music in History as well?

Project: Building and Using Musical Instruments
Driving Question: How are musical instruments made so they can be played together harmoniously?
Concept: Students

• Use engineering skills to create musical instruments that can be played together harmoniously
• Use acquired knowledge of the math and physics of music.
• Play the instruments together in a simple composition composed by class members studying music.
• In ELA: read texts and poetry where music plays an important role, including Shakespeare, as well as song texts. Write poems that could be set to music (consider rhythm.)
• In History: discover how music influences history or history influences music
• Brainstorm what they know about music, math and science to find what they need to know.
• Are grouped according to interests, particularly which other participating subjects they are studying (math, physics, music, ELA)
• In groups will learn and use engineering principles to create a musical instrument of different types - string, wind, tuned percussion, etc. based on the knowledge of the physics and math they learn
• Learn the necessary math and physics concepts, with activities and mini-lessons using problems specific for music.
• Teach each other - through presentations, jigsawing or other means - the math, science and music they are not actually studying

Here are 2 major sources I found for this project:

• Mathematics and Music
• The Physics of Music and Musical Instruments

Interestingly, I discovered this short article about creating instruments in the magazine, The Week, a few days after I submitted my proposal. It could be and interesting addition to this project for the students to find out more about the Paraguayan project in Spanish class.

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)

Credentialed Teacher!

Hurrah! I received this email today:

Congratulations, the Commission on Teacher Credentialing issued you the following document on 7/20/2011

Preliminary Single Subject Teaching Credential ...
Issuance Date: 07/18/2011 Expiration Date: 08/01/2016
Authorized Subjects:
Mathematics (Examination), Science: Biological Sciences (Examination), Science: Chemistry (Examination), Science: Physics (Examination)

This has taken me since December 2008, when I took the first test, and has been a struggle to get the required field work, since there were so few open jobs. All of this has been documented here in my blog.

But I still am looking for the job where I will be facilitating students' learning and understanding. Same job market.

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?

Two weeks down and running strong (and tired!)

I feel very comfortable about my new job. My colleagues are very supportive, and happy to have a teacher who can teach biology, chemistry and math. (The French didn't really materialize, thank goodness!)

I love having very small classes so I can work with each student individually when they need it. The 2 remedial math classes have about 6 students each, which is what the students need to be succesful. I am letting 3 students work ahead on their own in Algebra I, so they may be able to catch up with the rest of the class. Others are just filling in the holes in their math knowledge from too many years thinking they couldn't do math. I love the challenge of helping them understand what math is all about. We've been working on the substitution method for solving 2 equations. Today the light went on for one of the students and his face just shone! It has been a very difficult concept for them!

I'm constantly amazed at how much I know about the science subjects. I've not only studied for tests, but lived a whole long life being interested in science and soaking up so much about it that I can use to entice and motivate the students. But they are still new subjects for me, so I have to study the topics thoroughly to present them well. I finally got the electronic gradebook up-to-date with both attendance and quizzes. There's so much to do in the beginning! And I know the names of more than half my students (since there aren't that many) but that is a bit problem for me. Some of the students need so much help and attention, others act bored. Others escape from a difficult home-life. And still others just couldn't make it in big high school classrooms.

All that takes time, of course, on top of nearly 2 hours daily travel time partly through the lovely Cajon Pass. (The time and direction I drive is with very acceptable traffic. I had a couple of foggy days in December, and gusty winds that blew over some tall, lightly loaded trucks. But the wind doesn't bother my little Insight and I keep as far from the trucks as possible.)

This shows the southern end of the Cajon Pass.
and no, I did not take the picture! I'm down on the ground!

New Teacher!

I have just completed my first week of teaching in my new job.
The school is a public charter school, located in temporary classrooms while we wait for the powers to be to give the final construction permit to build the permanent school.
But my classroom is large enough for 6 tables big enough for 6 students each, although usually there are no more than 3-4 per table.
Since they had had a substitute for 2 weeks, it took some time to get them to realize that now we are seriously going to be learning math.
I also don't do much direct instruction, but get them going and then learn by trial and error, from each other, and from me as I go around.
In Algebra II they had been working on factoring with the sub, so we worked on that some more and then took on long division of polynomials, which the sub had skipped. There were many errors, because there are so many steps to do to complete a problem, one of which is subtraction, which involves distributing a negative. A couple of students had learned somewhere to place the change sign in a little circle, which I then taught, and the students took to it immediately. (That I said that I had learned it from other students was probably a plus.) But I told them we'd be learning an easier method - synthetic division - the next day, and that was a fantastic success. There were some holdbacks, but most could see that they could work them much faster that way.
But the big success came when I caught a student writing graffiti on the board, and it turned out to be "Math is great!" And yesterday I met the parents of one of my students, who had come home and reported that they had a new math teacher and she was great!
But I'm exhausted after the first week. Probably hardest is standing up all day long!
But I'm beginning to know where the students are in the material, and have vague ideas about who gets things quickly, who is willing to help others, and most importantly, who do I have to get at constantly to get them to do their work.
Classroom management is going relatively well, (except the day my advisor came for a visit in the period after lunch!) They are good kids, many with large and small learning and living problems, but I think they all want to learn. The school did miserably on last year's state math tests (but really well on English and other subjects) so we have double class periods, so kids can do their "homework" in class while they can ask me for help. (Which is good, except that math teachers only get a single prep hour a week!)

First day of school

Today was my first day of teaching at my new school.
I am dead tired of course, and definitely not used to standing on my feel all day! That I have to get up at 5:30, when I used to be a night-owl, is not fun, but it is lovely driving through the mountains over the Cajon Pass that early in the morning, waiting for the spot where the sun first requires sunglasses!
What was difficult was breaking the habits of students taught by a substitute the past couple of weeks. Getting them do get out books, paper and pencil to do some work.
But I think they're beginning to get that I really expect them to pass, and they have to work to do so!
That's all I can write now. Have to prepare for tomorrow!

Bilingual children learn even better

My children are bilingual English and Danish, since their father is Danish. They also speak French, Spanish and German. Their early years were spent in Denmark, where there are many foreign language programs in TV, and usually only children's cartoons were dubbed. So they got used to hearing not only English and Danish, but also Norwegian and Swedish and many other languages. My son used to pretend talk in various languages, with a perfect accent (but few words and no grammar.) Both have grown up to be very smart and culturally very adaptable.
An article in Education Week show that science behind my experience bears out. Science Grows on Acquiring New Language.

“We have this national psyche that we’re not good at languages,” said Marty Abbott, the director of education for the American Council on the Teaching of Foreign Languages in Alexandria, Va. “It’s still perceived as something only smart people can do, and it’s not true; we all learned our first language and we can learn a second one.”

I started wondering if all these monolingual teachers are afraid to let their bilingual students be "better" than they are, since they can do something the teacher can't.
I am always perplexed to meet students who speak another language at home, but have not been encouraged to keep building it to a level of Academic Competence. While I was studying secondary math education at Claremont Graduate University I found that many of my foreign-born fellow students did not know how to talk about Mathematics in their native language. They miss out on the opportunity to approach problem solving with the logic of two different languages. What a waste!

Other studies also suggest that learning multiple languages from early childhood on may provide broader academic benefits, too.
For example, at the science-oriented Ultimate Block Party held in New York City this month, children of different backgrounds played games in which they were required to sort toys either by shape or color, based on a rule indicated by changing flashcards. A child sorting blue and yellow ducks and trucks by shape, say, might suddenly have to switch to sorting them by color. The field games exemplified research findings that bilingual children have greater cognitive flexibility than monolingual children. That is, they can adapt better than monolingual children to changes in rules—What criteria do I use to sort?—and close out mental distractions—It doesn’t matter that some blue items are ducks and some are trucks.

When I get my own math classrooms, I plan to encourage bilingual students to talk math in both English and their native language. Bilingual immersion programs to this very well, but I think that regular classrooms can also encourage this (as long as there are at least 2 students who speak a particular language.) Math concepts are the same, and a lot of the vocabulary is cognate in some languages.
When I lived in Denmark, whenever we teachers had to do book inventory at the end of the year, I found myself starting to count in Danish, "en, to, tre, fire, fem seks, syv, otte, ni, ti....eleven, twelve, thirteen...." I've heard other foreign born adults do the same. But the counting is the same!
Our language is a very important part of our identity. As part of my CGU coursework, I wrote a blog called Negotiated Identity, where I reflect on a book Negotiating Identities: Education for Empowerment in a Diverse Society we read for class, and brought in a lot of other thoughts and material that I discovered as I read.
Students who are born with two languages are extremely lucky! We have to encourage them to become truly bilingual, also in their academic language.

Still looking but teaching a little, too

Wonder of wonders! I finally started doing a little teaching, or rather tutoring. I have one student in Physics and one in Geometry. Both of them tell stories about teachers who lecture up front and then give homework. The Physics student even had a lab about the recent unit AFTER the unit test. The teacher evidently considers labs to be "activities" to make things a little interesting, not real learning experiences.

The Geometry student flunked Algebra I with one teacher and got an A with the next. Geometry was going the same way before she signed up for a tutor.

As students of education we're taught to differentiate teaching. In order to do that, you have to know who's getting it and who isn't. You can't know that standing in the front of the class and collecting homework once a week, graded by teacher's assistant.

On the other hand, it's a lot to ask of a teacher to know each of 40 students well enough to differentiate. So it's a teacher problem as well as a problem for teachers...and their students.

I've been enjoying Coach G's Teaching Tips on the Teacher Magazine website. On October 5, he told us his secrets for Differentiated Instruction: A Practical Approach, where he writes:

You do, of course, need to provide some whole-group instruction, and you should certainly make it as engaging as possible. But you should also make it as brief as possible. Forget the ideal of every student grasping every lesson. What's more important is that you present key information in a clear, organized way so that students have notes to refer to when the real learning begins--during practice. In fact, in my classroom, where I assigned students to heterogeneous groups for independent (and interdependent) practice, as soon as I was sure at least one student per group grasped a concept, I was ready to move on, since I now had a full complement of assistant coaches.

or how about this great tip from September 27:

Don't Tell Students to Show Their Work--Make Them!
Are you constantly on students to show their work in math (or other) classes, but to no avail? If so, try giving them the answers up front--for class work, homework, even a test or two. Really, what better way to stress the problem-solving process than to limit an activity to that process?! Do this, and you'll really be messing with kids at first--especially if, like many of my students, they care more about getting work done than getting it done right. What are these students to do when the directions for an assignment are, "Show why the given answer is correct," and they can't get it done without getting it done right?

I am just looking for an opportunity to put these great tips to work!

Doing the impossible is harder than I imagined!

I stopped writing here because this thing about getting a credential has become much more difficult than it was when I got the idea to do so. Neither the school I have studied at Claremont Graduate University, or EncorpsTeachers, who have also been supporting me through all of this past year with workshops, study guides, and good advice, had imagined what school administrators already suspected, that they would be hiring very few teachers. In many districts, classes are being filled up to 40 students, even in Middle School, eliminating the need to hire a new teacher, and making life difficult for both teachers and students at the same time. That means that secondary teachers have to get to know 200 students (and their families) and that classrooms built for 25 have desks squeezed in, with no space for separate activity areas, or a way to even access the walls of the classroom.

In the meantime, however, I have completed all the coursework expected of me, except for one course I'll take this Fall (in Statistics) and a concluding course next summer -- if I manage to find a job to complete the Internship training this year. I am hoping that the new government money will open up a job here or there, which may provide me a job (as well as this year's interns) and make classes a little smaller, so that it will be easier to use more creative methods for students to learn as well.

I also took a series of courses at UC Riverside Extension this summer on Science Education to supplement the Teaching Skills tests in Science (CSET) I've been taking this year to expand what I can teach.

So far I've applied for over 30 jobs this spring. I'm hoping that all the credentialed candidates have landed a position by now, as school is starting, and that schools will be more open to taking an Intern as they discover a need for just one more teacher.

Please wish me luck!

Teaching at last!

I finally decided that I had to get into a classroom or go crazy, so I accepted the offer from Claremont Graduate School to become an unpaid student teacher instead. They were absolutely amazed that there are still about 6 of us with math credentials who did not land a job.

I am teaching at a good school with a great Master Teacher. I am observing and teaching one class of seniors who need to pass the California HS Exit Exam (CAHSEE) to be able to graduate, plus 3 honors Algebra II classes, which is sort of the opposite end of the spectrum. The teacher also has a class in AP Calculus BC, which I am technically qualified to teach, but I'm learning more just watching him teach!

Obviously the part you can't learn from a book is classroom management. This is much harder here than when I taught high school English and German in Denmark 15-20 years ago.) I don't know if Danish students have become like American students, or if the American students are just so much more immature than Danish students. I had reasonable success back then treating them as young adults; I just happened to know more than they did about certain things. We weren't even allowed to contact parents outside of parent consultation nights. Here it's expected. Admittedly, our students were on average 1-2 years older than in American high schools. Does that make that much difference?

One older teacher who is retiring this year told me that he thought the HS students now act like Middle School kids a generation ago (despite early onset of puberty.)  Is it because of parents who coddle kids? (I intimidated something like that in a teacher course and one parent immediately defended her parenting, saying that times were very different now.) I think the difference now is that we hear about every single incident that happens anywhere, so the world seems more dangerous. But I think it is more dangerous if the kids don't learn to stand on their own 2 feet, to take responsibility for their own actions (and learning!) and learn to think for themselves, rather than just think in opposition to adults.

At any rate, I have to learn to wait for 5 very long seconds to get the class quiet. I observed my Master Teacher doing it (after he had timed my "5-seconds" to 2!) 5 seconds is very long! But the kids get the message that you mean it. So I have to learn! I have also switched from writing on the white board (I keep saying "blackboard!") to using the overhead, because then I am facing the students. The Master Teacher uses "Equity Cards" to call on students, which I try to remember to do. I've got to learn their names - a total of about 115 students, not counting the Calculus class! In Denmark I never had more than 100. There is an electronic board in the classroom, which we can use to display PowerPoints, use the document camera, and a lot of fun things I hope I learn. But for the time being, it seems to be best to be facing the students! Once I learn Management, then I can try the fun stuff!

What to do when there aren't any jobs and you really want to teach...

This has been a really frustrating Fall for me. After making my decision to become a math teacher at a time when everyone said that jobs would fall into my lap, and then discovering that that is definitely not the case, I have found classes at Claremont Graduate University both inspiring and depressing - the last particularly when the speaker refers to "your students," of course. But at this point, there are still about 10 of us in secondary math, who do not yet have an Internship position.

So I have bought most of the new books available from the National Council of Teachers of Mathematics and used copies of a variety of math teaching materials, in particular Core-Plus Mathematics Contemporary Mathematics in Context from Glencoe. I've beeing reading about Sensemaking and learning through Discovery and Problem Solving, which really "make sense" to me as a way to get students interested in what they are learning. Core-Plus, which has units in Algebra, Geometry, Statistics, etc. each year, instead of separate years, looks like a fantastic way to teach math, except that it would be really hard to implement, since any student who switches schools would be lost wherever else they went. That is probably why there are so many used materials on Amazon.

I've also enjoyed the materials I discovered at the website for Geometer's Sketch Pad, which is a fun way to do geometry (and I understand other math subjects.) I found that they have great online resources for their Algebra and Geometry books, including some in Spanish. I also discovered the Prentice Hall Multilingual Handbooks - all available for almost nothing at Amazon. They included glossaries, etc. in a variety of languages, not just Spanish. But evidently Spanish is the only language that districts want to invest in. I just found a letter from a parent complaining about the creative math texts (dating back to 1996.)

I have also been observing classrooms - I'm required to observe 25 hours, including special ed and bilingual classrooms, and I've observed more than that now. What I see is teachers doing direct instruction and students doing work sheets. In some classes, text books are stacked somewhere in the classroom, or the students have a copy at home, but they are not being used. The ones I've looked at with my inexperienced eyes seems really exciting, if you want to teach by discovery and problem solving. But kids are learning how to solve problems on work sheets - and on standardized tests. I understand that there are pacing guides at the schools, that decide, for example, that next week all Algebra I teachers will be teaching solving two equations with graphing or substitution. No need to use a text book for that. No need to discover anything, when you can just do problems. Teachers tell me that the enormous textbooks just have too much material in them - and of course they're too heavy to carry around! So all the thought that went into making them (so they'd fit most any state's standards!) is just gathering dust.

To make myself more "hireable," I just did a quick (one month) review of physics and took the teaching qualification exam, CSET Physics III last Saturday. Now I have to decide whether to take Physics IV, to qualify as a physics teacher, or Science I and II (including biology, chemistry, earth and planetary science as well as physics) to be able to be a General Science teacher. Or maybe I'll just use the physics I've reviewed to provide more "authentic" problenms for my students.

Finally, I've been writing a few lesson plans, which are assignments for this semester at Claremont Graduate University - writing a total of 3 lessons that will benefit English Language Learners (ELLs) or students with other learning issues, like dyslexia or autism. I've written lessons that involve discovery and sense-making, since I'm not in a classroom trying to keep up with the pacing guide to make sure the kids do well on standardized tests. I'm afraid that my idealism will hit the dust when I finally do get into my own classroom.

After Studying Math

Studying Math
Originally uploaded by bonbayel

My recent journey through Math started innocently enough when I took Biology 101 last summer. Then I wanted to learn something about Organic Chemistry, so I bought the For Dummies book and workbook for that and a set of molecular building balls, which was lots of fun. Then my husband thought it would be interesting to look at calculus again, so we bought Calculus for Dummies and 2 workbooks. . . .

Yesterday I took the CSET in Calculus, Trig and History of Math, the last of the subject matter tests I'm planning to take to qualify as a teacher. The picture shows most of the books I've bought and devoured for this project. There aren't many For Dummies math books I haven't used!

The ones on the floor are mostly about pedagogy and classroom management, which is the next step to become a teacher. Some are required for my classes that start June. Some just looked interesting.

Last night I started a novel and signed up and played around with FaceBook. Today I rescued our beautiful Lantana bush in the corner of our patio, which had fallen down, and then downloaded pictures I took a couple of weeks ago at our family "ancestral estate" (it was a farm then,) now called Lotusland, in Montecito. The quiet before the next storm!

Studying for the last CSET Exam

I'm taking the final California Single Subject Examination for Teachers (CSET) - Math III (Trigonometry, Derivative and Integral Calculus, Infinite Series - and the History of Mathematics!) next Saturday. I have been studying for it for about 2 months - since I took the first 2 exams. My last official class in math was in 1963, so there is a lot of knowledge being pulled out of dusty corners of my brain. Because the interesting thing is that I recall most of what I am reviewing. That doesn't mean it's active knowledge, but I at least recognize the concepts.

I love solving Sudoku puzzles, and I play 3 different solitaires at night to relax my brain before sleeping, so I enjoy the puzzle of solving Trig identities and figuring out Integrals, both of which require puzzle solving skills.

What I don't enjoy is formulas. I'd much prefer to be able to figure out the formula myself than memorize it. My physics professor at college showed us how to set up problems using the different units (like gravity is acceleration, measured in feet (or meters) per second per second,) so you know how the problem should be set up from the units. Or if you know the trig function definitions, you don't have to memorize their values. However in a test situation you can't spend all your time deriving things. I am sure that is why I used every minute of the allotted time for the first 2 exams (taken in one sitting.)

I've been using a variety of sources to review the math, since I really do have to learn it from scratch. These have included college math and physics books, nearly the entire series of math for Dummies books, some Cliff's Notes books, a Calculus book for Economics students (which left out the trig functions, but was an excellent start,) some dedicated CSET review books - and a program I found online Ace the CSET, which is not really all you need to "Ace the CSET" - which is only pass or fail anyway, but a good help. Everything has practice exercises and practice tests, usually with great explanations about how to solve them.

However almost all of them have very vital typos. Some times it's a forgotten negative, which sent me to my calculator yesterday to find out that I was right, or another has been typed up from a hand-written script by a person who didn't have a clue what the material was about. This produces such interesting things as "l n(2x) - i.e. one n times 2 x" instead of "ln(2x) - natural log of 2x." At any rate, you can't trust everything you read, and it keeps me on my feet. It is comforting to know that even text-book writers make the same kinds of errors I make, but that doesn't help on a multiple choice test!

Test-taking and teaching

So will my current intense study of math help me as a teacher (besides knowing the materials, of course?) Will I be able to see the pitfalls more easily, or point out good study habits. Of course, my students will not be dedicating 2 months intensively to one subject! But at least I will understand the pressures of taking multiple choice tests!

I have kept my delight in math throughout (which my husband would not entirely agree with, as I've gotten grumpy here toward the end, and when I've hit something that involves what to me seems very tangle logic to understand.) Originally I figured I'd be teaching English to foreign students, which I also did in Denmark, but I wasn't feeling terribly inspired. When I started studying for the CBEST (Basic Educational Skills Test) my mind woke up reviewing for the math section, and I knew that it was math I was intended to teach!

Math exercise

I thought this was a great way to start a math class!

A Mathematician's Lament

I just read this lovely book in one sitting and then wrote a review about it on a website I just discovered, called Goodreads, which follows here slightly edited. (Note that all links go to the Goodreads site.)

A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart

This brand new book, an expanded essay about why math is taught all wrong in schools, is delightfully short, but a great inspiration while I'm studying for my last CSET Math exam in Trig & Calculus. When I started studying for the CSET a college classmate who has has a long career as a chemist was helping me get my mind around some of the new math concepts. He told me that someone had told him that math was all about definitions. Paul Lockhart couldn't disagree more. It's about solving wonderful, fascinating problems, he says.

The author's thesis is that math is an art - the art of solving problems - and we are teaching the grunt work of math, but not the enjoyment of the art.

He starts with a "Lamentation" about how terrible school math is, as if we were teaching kids to read music notes on paper, without ever letting them listen to, play or compose music. Or an art teacher who teaches color theory and paintbrush techniques so that high school art students can do paint-by-numbers pictures.

He understands that is may be necessary for students to understand the things being taught in school, but he'd prefer it if they figured things out by themselves. The role of the teacher would be to give them the space to discover these things.

But then he concludes with "Exultation" explaining (with a few examples) about how delightful math is. It got me all inspired, since I'm in the process of becoming a math teacher!

I find that math is like a game that needs solving, which fits in to his idea of math as art. The way I have taught before (English and German in Danish high schools) is to get the students figure things out as much as possible. I've rarely had much of a lesson plan, other than the requirements of topics that had to be covered (which was quite free in Denmark.) I don't know possible it will be to teach math (rather than train formulas and definitions) when the students have tests to be taken. But I'm inspired now!