Saturday, March 19, 2011

Where Have I Been?

Hi there. I have not blogged in a while. Life has taken a different turn lately. Although I am still passionate about math, I am exploring through different channels. I am exploring the concept of "Every day is pi(e) day". Discussions about pi and pie. I have started a new blog of that name. I will still post here from time to time, but my main focus will be at See you there

Tuesday, October 5, 2010

Part One: Mathematics in Education

Mathematics has had many forms over the years in schools. Some vividly remember the "new math" in the 1960s, whose focus on abstract theories spurred a back-to-basics movement, emphasizing rote learning and drills. After that came “reform math,” whose focus on problem solving and conceptual understanding has been derided by critics as the “new new math.”

What is happening in schools today? Truly, there is no one formula. Some schools are still teaching the rote learning and drills of the 60's; Some are embracing trends that come from other countries, such as Singapore, and there is everything in between. The question educators are asking is: what works best, faster, and yields higher test scores?

The National Council of Teachers of Mathematics (NCTM) urges a problem solving approach utilizing multiple representations whenever applicable. The thought is that if students see the problem in multiple ways and form, it will hold more meaning than looking at only one representation. This creates the bridge between the concrete and the abstract, which is sometimes a huge leap for students. This approach is similar to the Singapore Math in that the Singapore approach involves students moving through a three-step learning process: concrete, pictorial, abstract. American math programs, have typically skipped the middle step and students get lost when making the jump from concrete to abstract.

Another shift, or focus, seen in the NCTM standards is the focus on Curriculum Focal Points. These are defined as important mathematical topics essential in grades preK-8. These focal points are core structures that lay a conceptual foundation. NCTM has published them to be used as a guide for organizing content and bringing coherence to multiple concepts that are taught across grade levels. There are three tests for each item before it can be called a Focal Point. Those tests are:
  • Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school?
  • Does it “fit” with what is known about learning mathematics?
  • Does it connect logically with the mathematics in earlier and later grade levels?
The decision to organize instruction around focal points assumes that the learning of mathematics is cumulative, with work in the later grades building on and deepening what students have learned in the earlier grades, without repetitious and inefficient reteaching.

This is good for the future of mathematics. Allowing students to build bridges between the concrete and abstract, and providing a curriculum structure that supports mathematical thinking will undoubtedly help students in the future.

For high school students, NCTM is recommending a series on Reasoning and Sense Making across the curriculum. This is very different from simply memorizing formulas and copying what the teacher does at the board. The problems they propose take entire class periods to solve and require analysis of patterns and making predictions. Students are being asked to think rather than mimic.

So why hasn't the shift to thinking happened in all classrooms? The information is readily available. I think this is a good question for us to look at together. Maybe it's high stakes standardized tests, untrained teachers, unmotivated students, absent parents...

Please leave your thoughts.

Why haven't we made the shift from rote learning and drills to thinking and making connections in the math classroom?

"Making Math Lessons as easy as 1, pause, 2, pause..." Winnie Hu, 9/30/10 New York Times
National Council of Teachers of Mathematics Standards and Focal Points,

Friday, October 1, 2010

Thinking And Mathematics

Thinking and Mathematics. These two terms are not mutually exclusive to mathematicians. However, are they mutually exclusive to young people working on mathematics in a classroom? Is mathematics rote memorization and mimicking another person's processes and thinking? "Mathematical Thinking" is something often said with regard to a person's education, but how often is the student actually encouraged to apply said "thinking"?

I am starting a 6 part series in this blog on the topic of Thinking and Mathematics. Through discussions, I would like to work on understanding how thinking and mathematics are linked in school. Also, it will be interesting to try to figure out how the system of teaching and learning mathematics is evolving or stagnating.

I see the 6 parts as this: (suggestions are welcome)
Part One: Mathematics in Education
Part Two: Why Do So Many Students Dislike Math?
Part Three: Is Math Dead?
Part Four: The Path of Least Resistance
Part Five: Is Change Necessary?
Part Six: Thinking Groups

I would like to start the discussion by asking this question:

What kind of thinking is required of mathematics students?

Wednesday, July 7, 2010

The Bases are Loaded

The base of any number system is the number of different symbols used to compose the numbers. The system we use, base 10, is such because there are ten symbols used to form the numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is assumed that we use this system since we have 10 digits on our hands. People found it easier to count using base 10.

The cool thing about numbers is that we can play in other bases. The binary system, base two, has only two digits: 0, 1. Computers perform their calculations in binary codes.

Here is your challenge for today: Using the examples below of Base Ten, Base Two, and Base Five, write the numbers from one to twenty in base 8. Ready, go!

BTW: The answer to the last challenge is in the comment section of that post.

Base Ten

Base Two

Base Five

Base Eight






























































Saturday, June 12, 2010

Types of Prime Numbers

Time for a little math. I have been writing about lots of things and realized that math was deficient. How about a little fun with prime numbers?

Quick review: Prime numbers are natural numbers that have only two factors: 1 and the number.

Some special types of prime numbers (you didn't think that mathematicians would stop at one definition, did you?):
  • Twin Primes - a set of two consecutive odd primes, which differ by 2. Examples: 3 and 5, 5 and 7, 11 and 13.
  • Symmetric Primes, also called Euler Primes - a pair of prime numbers that are the same distance from a given number on the number line. Examples: Given 6, 5 and 7 are symmetric primes. Given 16, 3 and 29 are symmetric primes.
  • Emirp - a prime number that remains prime when its digits are reversed. (Emirp, of course, if prime spelled backward!) Examples: 13 (31), 347 (743).
  • Relatively Prime Numbers - numbers whose greatest common factor is prime. These numbers are not necessarily prime. This definition is referring to the relationship between numbers rather than the numbers themselves. Examples: 4 and 9, 10 and 27, 8 and 9.

So now, the questions to ponder/work on:
  1. Can you find all of the symmetric primes for 24? (hint: there are more than 3 pairs - and no, I am not telling how many there are. That would spoil the fun!)
  2. How many emirps exist between 1 and 200? (The number when it is listed its initial way - so 13 and 31 would each count as unique emirps.)

Ready..... Go!

Wednesday, June 2, 2010

Dan Meyer - Math Class Needs A Makeover

I couldn't agree more!

Do As I Say, Not As I Do.

Are we past this, or is it still alive and well?

I am going to write about a sensitive subject today. Sensitive, because it may challenge some people's view of the world and the natural order of human relationships. I could also be wrong in the amount of sensitivity; maybe people have moved on from this kind of thinking. I don't really think so.
As the title suggests, I am going to be thinking about the "Do as I say, not as I do" proverb spoken by many parents, teachers, and others in authority. What is this all about, and why do we allow this to go on with a faint smile and nod? It may be because it is how some of us where raised. I was taught that it was wrong to swear by a man who used a curse in every sentence. Even our laws perpetuate this to some degree. Persons under certain ages are not allowed to drink or smoke legally. I am not going to say that I think that persons of all ages should be allowed to do anything and everything. I would like to explore some of the possible consequences of such a system.
What worries me the most is that this way of thinking may be teaching our young people that once a person is of a certain age or in a particular position, that person can inflict "do as I say, not as I do" on someone younger or lower in the hierarchy. Are kids learning that once they grow up, they can transition from being the oppressed to becoming the oppressor?

A Mathematical Way to Think About This

This is something I have thought about for a long time, but I have never really analyzed it. I think the time has come. I am going to look at this problem using an indirect proof. For those of you who are rusty at this, I will lay out the basics.
First, the reason why I have chosen to look at this indirectly is because there are many different thoughts here and a direct path would seem a little bossy. Also, I really enjoy the process of the indirect proof. It makes me explore both sides of the argument. Here are the basics of the indirect proof method.
  1. Assume the opposite of the statement is true. For example, in Geometry, if you were supposed to prove 2 lines are not parallel, you would assume that they are parallel.
  2. List all of the information that supports your assumption. Using the same example, there would be some conclusions that could be drawn because of the parallel lines.
  3. Keep going until you find an indisputable contradiction. Once the contradiction is found, the proof is finished. In the case of the lines, a point of intersection would be concrete proof that the lines are not parallel.
So, here it goes. I would like to prove that the philosophy of "do as I say, not as I do" is not only out of date, but damaging to future generations. **I have to put a disclaimer here for all of my mathematician friends. This is truly done in a sense of fun rather than mathematical correctness, so please view this in that frame of mind!

Counter Assumption: The "do as I say, not as I do" philosophy is good for our young people and teaches them correct behaviors.

Statements that support the assumption:
Note: I am working hard not to be sarcastic here. If it sounds that way, I apologize and would welcome help in rewording anything.
  • I was raised this way and I turned out fine.
  • People aren't really bothered by the inconsistent messages given them by their parents, teachers, employers, etc.
  • People with power and money have earned the right to do what they want, and to tell other people what to do. They have the power, so what they think must be correct.
  • Young people understand that the instructions are for their own good and that being an adult has its privileges.
Now that the statements are thrown out there, I would like to address each one if I can.

I was raised this way, and I turned out fine.
This one is hard to contradict. After all, I do think I turned out fine, but I think the contradiction here is that I do not choose to raise my daughter in this philosophy, nor did I hold my students to this creed. The sense of fairness in me has kept me from it. I hated the injustice growing up and I decided early on that I would not do it.

People aren't really bothered by the inconsistent messages given them by their parents, teachers, employers, etc.
This one is contradicted everywhere. Turn on any major news network and at least 50% of the message will be about the contradictions in the words and actions of public figures. I think another common proverb was developed because of this phenomenon. How many times have people in power been told that "Your actions speak louder than words." (Is it okay to disprove one proverb with another? They are definitely in contradiction of one another.)

People with power and money have earned the right to do what they want, and to tell other people what to do. They have the power, so what they think must be correct.
Consider these as suitable contradictions:
Employer to Employee Sexual Harassment
Jim Crow Laws
Child Labor
(I could continue, but I am starting to feel horrible.)

Young people understand that the instructions are for their own good and that being an adult has its privileges.
I offer this section of an article that I found written by Dan Roloff: "Do as I Say, Not as I Do" or Raising Kids with Moral Intelligence. If you would like to read the entire article, click the link.

In his book, The Moral Lives of Children, Dr. Robert Coles defines our conscience as our "moral intelligence." More simply, he defines moral intelligence as how we behave—"moral behavior tested by life, lived out in the course of our everyday existence."

PBS NewsHour's David Gergen interviewed Coles several years ago and asked him how to encourage morality in our students, our children, and ourselves. Coles told Gergen that adults can only teach values by living them. "I'm trying to insist upon for myself as a parent and a teacher and for all of us, that any lesson offered a child in an abstract manner that isn't backed up by deed is not going to work very well," Coles said. "We live out what we presumably want taught to our children. And our children are taking constant notice, and they're measuring us not by what we say but what we do."

So, is the proof done?

Mathematically, maybe not, but I think I gave it a good try. I have tried to work some things out and maybe some people have something to chew on tonight. I think the good news is that "do as I say, not as I do" doesn't seem as prevalent as in my youth. I felt compelled to write anyway. Just because something is happening less doesn't mean that it has stopped completely or even enough.

Don't get me wrong, I don't think that adults have to stop doing things like having a cocktail at happy hour because their kids can't, but we can do so in a responsible manner. The societal norms that make it okay to have a drink at happy hour, but not in the classroom are well established and I think it gets the job done for most people. It's the other things that I think should be questioned. If your students are required to wear uniforms, should you wear flip-flops? Should school principals turn a blind eye when they see misbehavior in a classroom that a teacher is struggling with, and then reprimand the teacher later? Should the boss be allowed to drink or smoke on company time, a la "Mad Men"? Should a parent curse a blue streak and then punish their child for saying "damn"?

Anyone reading this who thinks "do as I say, not as I do" is fair - I challenge you to ask the people around you, particularly those who look up to you, and give a true ear to what they say. It would be a good thing to do.