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.

Rigor mortis or rigor percipiare

OK, the Latin in the title is my own. The second half is supposed to mean "tenacity to learn" in my version of Latin. But the title was inspired by a very thoughtful article today by Linda M. Gojak,  President of the National Council of Mathematics Teachers, called "What is all this talk about Rigor?".

Evidently people have been writing that the Common Core requirements for mathematics include the word "rigor," although she says it is not there. She and a group of math coaches investigated the meaning of the word (as in rigor mortis, but more appropriately “thoroughness”and “tenacity”) to see how it can be applied to the teaching of mathematics. They came up with the following table, which I have borrowed intact from her article.

Learning experiences that involve rigor …	Experiences that do not involve rigor …
challenge students	are more “difficult,” with no purpose (for example, adding 7ths and 15ths without a real context)
require effort and tenacity by students	require minimal effort
focus on quality (rich tasks)	focus on quantity (more pages to do)
include entry points and extensions for all students	are offered only to gifted students
are not always tidy, and can have multiple paths to possible solutions	are scripted, with a neat path to a solution
provide connections among mathematical ideas	do not connect to other mathematical ideas
contain rich mathematics that is relevant to students	contain routine procedures with little relevance
develop strategic and flexible thinking	follow a rote procedure
encourage reasoning and sense making	require memorization of rules and procedures without understanding
expect students to be actively involved in their own learning	often involve teachers doing the work while students watch

This is what teaching should be about, although I wish they'd come up with a better word, since rigor also means "rigidity" and "suffering," according to their research! That sounds more like the drill & kill methods I experienced as a student teacher, and which they define as not having rigor!

The left column should apply to all learning experiences, not just in mathematics. Children are born with curiosity, a need to be challenged and a lot of tenacity. This I experienced this past summer as my year old granddaughter tried again and again to crawl across a very difficult door opening (threshold!) until she figured it out. She was enormously proud of herself as well. I was amazed when my teacher sister-in-law got impatient with my granddaughter's efforts and just lifted her over the threshold. But the child went right back to working it out after that.

We must provide thresholds for students to cross, where they can see intriguing unknowns that awaken their curiosity. Children who are helped to everything must lose their love of a challenge and their curiosity early on. As a high school teacher I find that I have to help students regain their curiosity and encourage them through a challenge until they proudly can see they have overcome it. That is how we all learn!

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.

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!

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!