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?

Sources:

"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, www.nctm.org

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?

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
1	1	1	1
2	10	2
3	11	3
4	100	4
5	101	10
6	110	11
7	111	12
8	1000	13
9	1001	14
10	1010	20
11	1011	21
12	1100	22
13	1101	23
14	1110	24
15	1111	30
16	10000	31
17	10001	32
18	10010	33
19	10011	34
20	10100	40

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!

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:

Slavery

Apartheid

Employer to Employee Sexual Harassment

Jim Crow Laws

Genocide

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.

Math and Religion?

I enjoy listening to a radio program called "Speaking of Faith". I like the overall feel of the program. There are discussions about a variety of topics and it is a "judgement free zone". (My words, not theirs) I have learned a little about other faiths and some of the things people are doing in the pursuit of meaning in the modern world. I did not expect something to stimulate my math brain at the same time.

Last Sunday, as I was contemplating getting ready for church, the program came on. I always listen a little to see if the content will be worth the consequences of having to rush to get ready. (It usually is.) I chose to listen to the program in it's entirety. I was intrigued by the content and the discussion.

The show was called, Who Ordered This? New Mysteries of an Expanding Universe.
Astrophysicist Mario Livio works with the Hubble Telescope's findings on phenomena like dark energy and white dwarfs. We explore edges of discovery where scientific advance meets recurrent mystery — questions richer than any of their current answers.

I invite you to visit the Speaking of Faith website and listen in or download the podcast if you like things like fibbonacci numbers, the golden ratio and marveling about the mathematical beauty of our world. The link takes you directly to the story.

http://speakingoffaith.publicradio.org/programs/2010/who-ordered-this/

This is the link for the discussion blog as well. This is what originally caught my eye. I enjoy thinking about mathematics in nature. Interesting ideas!
http://blog.speakingoffaith.org/post/617552387/mathematics-in-sunflowers-shubha-bala-associate

Fun Night with Steven Strogatz

Monday night, I had the opportunity to meet Steven Strogatz. He is a mathematician, author and a professor at Cornell University. I have posted some of his writings here on my blog in the past. Currently, the work that I am a fan of are his articles written for the New York Times website. I will post the link again because if you haven't seen these, I think you will enjoy then when you do.

Imagine how excited I was when I found out that Math for America was hosting an hour-long lecture where Mr. Strogatz would talk about his series of online articles! Of course I was thrilled, so I dressed up and went. The lecture lasted an hour, but it felt like 20 minutes. He started by talking about the meeting with the NY Times editor who asked him to write the series. He discussed the process by which he decided his topics, audience and voice of the articles. The session was interactive, too. We were all given the opportunity to share what we would have done if we were in his place.

Hearing about his creative process and having another chance to see some of the content was entertaining and educational. The playful tone you read in the articles is true to the man when you meet him in person.

If you haven't, read some of the articles. You won't be sorry, and you might learn something or see something in a way that makes you think. I have included the link to the most recent article. It's time to kindle or rekindle your love of math!

Steven Strogatz in the NY Times.

Why do we write students off when they can't behave?

If we believe that all students can learn, why do we write them off when they can't behave?

Collaborative Problem Solving. It sounds like the latest educational or even psychological fad. After years of working in schools, I have realized that I often view new ideas with the mindset that this may be something I have heard before under a different name. Or worse, it will sound great in theory, but the practice will fall flat. This thinking has made me skeptical of many new ideologies in the past. Why don't I feel that with this one?

I know why. After years of working with all kinds of students, I hold the hope that through good teaching, all students can learn. Why would we think this only applies to content? I believe it applies to behaviors as well. CPS is not unlike some other programs designed to help students with challenging behaviors to overcome them. Dr. Marsha M. Linehan, the Director of the Behavioral Research and Therapy Clinics in Seattle Washington, has made huge strides with people with bipolar disorder. Her method includes a large component of teaching skills. The assumption is made that people are doing the best they can right now AND they can do better. This is mirrored in Dr. Greene's idea that student do well if they can. Often, the reason they can't is that they are lacking skills.

There are 3 main steps and one preliminary step to CPS. I'll explore each step in detail in further blogs, but here is an overview.

The preliminary step involves working to identify the skills that the student is lacking before having the conversation. The work for this step is done based on observations of the student's behavior. Dr. Greene uses a form called the Assessment of Lagging Skills and Unsolved Problems (ALSUP). It includes a long list of ideas for problems and also a section for brainstorming what the triggers are for undesired behaviors.

The first with the student is Empathy. This reminds me of Validation. During this step, the goal is to find out as much as you can about what is going on with the student. You are trying to understand the student's concerns and perspective. Getting them to talk may be difficult at first since trust needs to be established and the student will realize over time that you really want to know what is going on. The best way to do this is to continue to ask questions and try not to tell the student what you think the problem is.

The second step is Define the Problem. The problem is defined as "two concerns that have yet to be reconciled." This is both the student's and the adult's concerns combined. Both sides are equally valid. In this step, it is important to define the problem by clarifying concerns rather than moving straight to solutions. This may be hard since as teachers, we are major problem solvers and we want to help.

The third step is the Invitation. I'll bet you thought it was going to be "solutions". This step is were solutions will be discussed, but I like the name of the step because it reminds me to invite the student to be a part of the problem solving process. If they aren't invited to help solve the problem, then we are back to the beginning with adults telling students what to do and we are no longer being collaborative.

Plan B Continued - Collaborative Problem Solving

As I read over my blog from last time, I realized that I neglected to explain why Collaborative Problem Solving (CPS) is called "Plan B". The idea is simple. Plan A is what we are currently doing with challenging students. Since it is not working, we need to move to "Plan B".

In the book, Lost at School, Dr. Greene fully defines Plan A and even a Plan C, but the main focus is CPS, or Plan B.

Plan A - This is a typical school plan. Lists of consequences used when students mess up. The system is designed to be a deterrent for bad behavior. This plan works for the large majority of the students. The students who possess the skills to adapt to changing situations and who are typically successful in school. The more challenging students are not being served by Plan A. They are getting suspended, expelled, etc. The list of consequences that they have experienced is very long and yet they continue to misbehave. As educators, I see a choice. Give up on them or move to Plan B.

Plan B - I am excited about exploring CPS both here and in the classrooms I visit. I am curious and more than a little hopeful.

Plan C - This one is interesting in that at first it seems like it is not a plan at all. This plan involved ignoring behaviors. After reading further, I realized that Plan C can actually be done in conjunction with Plan B. To give an example: While a student is working hard on the skills s/he needs in order to stay in class without violent outbursts, we may ignore the fact that all of the homework is not complete. I kind of think about it like divide and conquer. We may not be able to fix everything at once, but if we chip away at it, one skill at a time, we will see progress.

Plan B

So, one of the books I am reading was a recommendation from one of the math teachers I work with in NYC. It is called Lost at School by Ross W. Greene. I am not halfway yet, but I feel compelled to share a little of what I am learning.

After being a teacher for a number of years and now in my work with new teachers, one main challenge eventually boils down to a few students who are especially challenging. It is easy to start blaming - the student, the parents, the year's previous teacher, etc. for the issues that the student is having. The thing that impresses me about the content of this book is that all of the blaming is set aside. After all, the blaming may make for good talk in the teacher's lounge, but it does nothing to tackle the problem, and the problem is huge. These students and their teachers are at risk.

The students in question are often called "at-risk". I think that at-risk students lead us to at-risk teachers if the problems are not addressed properly. The situation is very frustrating, especially to new teachers. If the situation doesn't improve, our teachers are at-risk of quitting, becoming complacent, and becoming victims of poor attitudes.

Now, to "Plan B". Dr. Greene defines Plan B as Collaborative Problem Solving (CPS). I haven't finished the book, so I am not going to try to fully analyze it yet, but the basic idea is that challenging students are in need of skills. They don't want to misbehave, they lack the skills to behave appropriately. And, these skills can be taught.

I can't wait. I will read on.

Test Taking Strategies

It is now the season for test prep! Teachers and students all over the country are preparing for or taking their state mandated exams. Assuming that the content is under control, I have been thinking about test taking strategies. I have some that I have compiled from some different sources. If any of you have other things that you like to use, please post!

1. Practice. Practice tests help students and teachers identify areas where improvement is needed. Allow time for students to take full-length versions of the released tests. Optimally, this should be done at least a month before the test to allow for time for targeted review.

2. Read the directions carefully. It may seem obvious, but some students completely ignore the instructions, skim them or don't listen as they are read. Help your students break this habit.

3. Write on the test. Students should be active test takers. They should always work on the test paper. This helps cut down on the guessing in a multiple choice formatted test.

4. Look for "turn words". Show students how to pay close attention to words that change the initial meaning of a question. Look out for "except", "not", "at least", "at most", and "all of the following".

Additionally, there is a good "during the test" strategy to help students learn how to tackle the test and how to check it over when they are done. We always tell our students to check over their work, but they don't always understand what we mean. Some of them will simply check to make sure they have bubbled properly. While this is a good thing to do, it is not the only thing that can help.

Taking the test:

1. The first "pass". While students take the test, have them code each problem. A "√" next to a problem means that they are confident they got the right answer. "Circled" items are shaky. They may have been able to eliminate answer choices, but they are not sure they have it right. A "?" next to the problem means that they have no clue how to solve the problem or answer the question.

2. The second "pass". After completing the test, the student goes back to circled items only. The goal is to give this problem the level of the "√" problems. With fresh eyes, the student may be able to eliminate choices or remember a formula they had previously forgotten.

3. The third "pass". The third time through is the time to deal with the "?" items. The goal is to be able to apply some knowledge to the problem. If nothing can be done, students either guess or leave the item blank based on the format of the test. If the format does not penalize wrong answers, the student should guess. Help your students choose a wise letter. If "C" is the most popular answer on most of the tests, the student should choose "C" for all of their guesses.

I hope you find this useful. Please share more strategies that have worked for you and your students.

Treva

Resources: How to Thrive as a Teacher Leader by John G. Gabriel; The 3-pass system, Leander ISD, Leander TX

Fun and Interesting Math Reading

I love it when I find math discussions in The New York Times! This Opinionator series is very interesting. In addition to the article, it is fun to read what people have to say about the author's opinions and his mathematics. Enjoy!

http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/#more-43909

The Importance of Math Teaching

I am inspired to write today. This is somewhat unusual since I consider myself a mathematician and not a writer, but I am inspired none the less. I am starting this blog to create a forum for me to express the feelings and observations I have and experience as a mentor and advisor for math teachers. It is my hope that the discussions here may be of use to others who find mathematics education important.