Learning by understanding

I just read responses on a Linked-In forum about how to teach factoring. The answers were full of steps and technical details. Not one linked factoring to something the students knew, or gave them a reason to learn factoring.This was my response:

Before you even start factoring, make sure students have a reason to use it, that they understand WHY they're factoring. Have them graph a simple polynomial equation, like the square of (x+3), using a T-chart for values, and find the zeroes.

I think it is extremely important for the students to understand factoring in polynomials is the same in factoring, say, 96.

They need to know that polynomials are the result of multiplication, so a good way to start is to have them multiply simple things, like the results of the graph they did and other squares, and then, for example, the sum and difference of 2 terms, to see if they discover the pattern, then give them the same problem, plus some similar ones, to factor. Then move on to things like (x+1)(x+3), saving ones with a coefficient other than one for later.

Using Algebra tiles is another way to visualize what's happening, and using the "box" method, which I like for multiplication of polynomials, because it helps keep them straight, is also a good help to reverse the multiplication, which is similar to the Algebra tiles.

But if they have no clue why they are factoring, it just adds to "when will we ever use this in real life?" which is a very legitimate question. They need to know what those zeroes can be used for, too. I'm not sure all that many Algebra I teachers can carry the discussion that far.

When students understand why they're factoring, what it's used for, how the polynomials are graphed and how they came about, I think they will be much more open to the fun puzzle of untangling them as factors.

Where I need to be

The image on the right was one I found on FaceBook. I think it tells a good bit about my life. Problem is, each time I find "where I needed to be" something happens and I have to move on.

To catch up a little since my last blog post back in October, I was actually glad that I didn't have a job all fall, because there were a lot of family things going on (my husband was very sick, so we moved to a new house without stairs in a neighboring town, and there was a birth and a death that moved us all.)

But in January, I was asked to return to the tiny charter school in Hesperia, where I completed my credential. It was like coming home. I knew all the colleagues except the new Dean of Students, who has been invaluable, and I knew about half my students and they knew me, so we didn't have to start at square one.

I am also teaching the same subjects, Biology and Integrated Science, although different parts of them, since the teacher they had in the fall had taken a different part of the curriculum than last year's teachers. But the most important aspect was that I have learned a lot about Guided Inquiry and Reasoning and Sense Making since then, which turned out to be the right way to address the needs of pretty much all of my students.

Most of our students have come to us because they just couldn't make it in the regular public high school. Some had tried a variety of other charters, home schooling, etc. Many have a great difficulty concentrating, and get easily distracted. If I had been trying to do whole-class teaching, I think I would have lost most of them. But I put them in 6 groups of about 3 students, and provided lots of hand-on labs to introduce topics. I also made many worksheets, often finding illustrations and text on line, and then guiding them with questions to the illustrations and concepts. It took a while for the kids to understand that they were to work TOGETHER in their groups, and that I wasn't going to be standing up front with a PowerPoint, but coming around to each individual group to ask them questions, and guide them on their way (I like the word, facilitate!)

I am more than half-way through the "University Induction Program" at UCLA Ext, to clear my credential, with interesting courses and "Inquiries" into my teaching about what sort of strategies will help my ESL students, and now my students with IEPs. I've also just completed a fun course at CGU in ways to teach Physics hands-on, which gave me a lot of tools and ideas for the Physics part of Integrated Science, and an online course in working with students with ADHD, which is much needed to learn to reach our many "wanderers" and "blurters." And I've also earned a certificate as "Green School Professional." (I've been taking more classes than my students, to learn to teach them better!)

But the tragedy I alluded to in the beginning is that our little school is too little. We need about 20 more students to release some important funds and make us viable. So the charter has been pulled, prospective students are being turned away, and our students are trying to figure out where to look again to continue their education. Some of the students are looking forward to going to a "real" high school, with all the amenities we can't offer, although we do offer gym, a couple of sports, classes in art, music, sign language and astronomy. But many are going to try the individual learning of home schooling or computer-based learning, away from any social aspects of school. Some of my students are sure to get lost, students I was just getting through to. How sad! My younger colleagues (one just got married) need jobs to support their families, older ones aren't ready to retire yet. Our special ed teacher, who isn't much younger than I am) is working on her EdJoin application for the first time ever. She was the life-blood of the school for most of its existence, but is left in the cold like the rest of us.

So far the only jobs I can see for me are even further away than my trip through the Cajon pass to Hesperia. I can manage without a job, but I hate inactivity, and I have discovered that I have much to give my students. So I'll just have to see where life will take me next, and know that that's where I'm supposed to be for a while again.

On Saturday, I will be walking in the graduation ceremony at Claremont Graduate School, with cap and hood and all. I'll post a picture to prove it after it's happened!

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)

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?

What we learn to do we learn by doing

The title was attributed to Aristotle in today's Daily Ray of Hope from the Sierra Club, which included the photo shown here.I'm not quite sure what the duck is learning, but it's nice to have an illustration.

It's strange that so many math teachers think "doing" means doing drills, rather than doing something to make the math they are learning make sense, as in "doing math."

I've been having a lovely time reading lots of books published by the National Council of Teachers of Mathematics, particulary their series, Focus on Reasoning and Sense-making, the Navigation series, to help incorporate the new Core Standards in lessons, maybe using textbooks that don't really get the concept of reasoning, and the new series, Developing Essential Understanding.

Reading these has led me on to other discoveries of my own, since they often use math that I haven't used very often, like polar coordinates, matrices or graph theory. 

I'm enjoying my little break here learning about learning. But I hope I soon have students with whom I can practice what I've learned (learn teaching by teaching!)