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

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!

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.

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

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.