What I read
From Science Daily:
“Friends’ School Achievement Influences High School Girls’ Interest in Math”
“Few Gender Differences in Math Abilities, Worldwide Study Finds”
“Believing Stereotype Undermines Girls’ Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds”

From the MathForum:
“Girls’ Attitudes, Self-Expectations and Performances in Math”

From the News Bureau in Illinois:
“Girls’ Confidence in Math Dampened by Parents’ Gender Stereotypes”

From Prufrock Press Inc:
“Helping Teachers to Encourage Talented Girls in Mathematics”

There is a widespread belief in our country that only certain people have the talent to pursue mathematics-related careers and to take advanced math classes. This belief tends to assume that males and Asians are more talented in math than are females and non-Asians. While innate talent is a widely disputed subject, the fact is that if there are some people born with an innate genius in math, those are very few. What matters most is effort, persistence and confidence. It’s time to start letting students in on this “secret” of math.

There are many articles and studies trying to understand when the majority of girls get turned off to mathematics (and science, computers and technology) and how to turn that around. The most apparent issue that came out several times when reading the above articles, is that girls tend to have less confidence in their abilities to do math. When I attempt to track this backwards, it seems that the low confidence is directly connected to math anxiety which stems from two barriers: societal influences and the competitiveness of the math environment.

Let’s talk about the first barrier, societal influences. Girls tend to tune in to their environment and the people in their environment to obtain information about how to act and think. If parents and/or teachers think that boys have a higher aptitude for math, that information is communicated through body language, verbal cues, facial expression, assumptions of girls’ abilities and persistence, and unsolicited help. When adults try to “help” female students with their math, they are often really hurting girls’ confidence in their own abilities. So, note to parents and teachers, lay off the helping and encourage all your students, especially girls to persist in finding their own answers without your help. Another major societal influence for girls are their peers. Girls like to do things together in pairs or groups. If all their friends are going to take an art class instead of an advanced algebra class, which class do you suppose she will choose? The adults in girls’ lives must encourage them to pursue mathematics despite what their peers are choosing. Part of this encouragement is believing in their abilities and having high expectations that they can and will succeed.

The second barrier to girls’ positive attitudes towards mathematics is the competitive environment of the math class. Math classes tend to held in a competitive and fast-paced environment. There is little think time and little opportunity for small group or cooperative learning. While some girls thrive in this competitive math environment, the majority of girls get turned off by needing to come up with answers and solutions to problems as quickly as possible and not having the opportunity to check in with their peers before they decide on a solution. Girls “need classrooms in which they will be heard and understood and where they can discuss ideas before coming to conclusions” (Gavin & Reis-Prufrock Press).

Teachers should do their best to create an environment in the math class that

1. Offers choice between competitive and cooperative work
2. Allows think-time (using think-pair-share or journaling) and disallows shout-outs
3. Encourages risk-taking
4. Includes cross-curricular activities such as math and writing, dance or music
5. Uses assessments such as math portfolios and projects in addition or instead of traditional tests
6. Uses open-ended questions on tests instead of multiple choice
7. Doesn’t give unsolicited help, instead encourage students to persist in finding their own solutions
8. Provides some single-sex learning opportunities
9. Uses co-ed small groups but single sex pairing (since girls tend to defer to boys in partner math situations)
10. Holds Family Math Nights planned and carried out by girls (boys should get this opportunity as well)
11. Gives girls the opportunity to tutor younger children and organize math clubs
12. Ensures math topics are of interest to girls as well as boys (not just sports and construction but also paper folding, spread of diseases, endangered species, scale drawings and patterns)
13. Uses a challenging curriculum that introduces different number systems, Fibonacci numbers, non-euclidean geometry, fractals, chaos theory and other such topics in mathematics
14. Introduces knowledge of female mathematicians such as Hypatia, Marie Agnesi, Sophie Germain, Evelyn Boyd Granville, Sonya Kovalevskaya, and Mary Somerville.

Teacher Resource
Math Publication Best Sellers from Prufrock Press

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 2 in developing meanings for the operations focuses on multiplication and division, what strategies children use, what types of modeling a teacher should use and how the teacher’s understanding of approaches can enrich students’ understanding.

Multiplication and division can both be learned as early as 1st grade. Sound ridiculous? Not if you understand the relationships between all the properties. For example, when teaching multiplication to children, one of the most used approaches is to show how multiplication is merely repeated addition. For example: Eduardo grew 8 tomato plants. Each tomato plant produced exactly 5 tomatoes each. How many tomatoes will Eduardo harvest? The problem boils down to 8 x 5. There are 8 equal groups of 5 sets. The first factor (number) counts how many parts or sets. The second factor tells the size of each set or part and is also known as the multiplicand. In order to solve this problem using repeated addition, you create 8 groups of objects with 5 objects in each group. The resulting equation: 5+5+5+5+5+5+5+5 = 40 should be written down by the student as well as 8 x 5 = 40 next to it. Students will begin to learn the relationship. Equal group problems may also be considered as rate problems.

The major conceptual hurdle to developing meaning for multiplication and division is understanding groups of things as single entities and also that each group has individual objects. Once this mental obstacle is conquered, children of just about any age can begin working with these operations.

Van De Walle talks about four classes of “multiplicative structure,” and that there are two used most often in teaching multiplication. The first is equal groups, which I have written about already. The second class is multiplicative comparison. There are some fantastic example problems of both types that Van De Walle includes in his book. Comparison problems usually compare two people who have two different amounts of something. For example, Trudy built 5 models during art class. Her friend Clarissa built 3 times as many models as Trudy. How many models did Clarissa build?

Consider the two multiplication examples I have given here. Figuring out how to solve each one requires a slightly different way of thinking and certainly a different strategy. Compare this type of thinking to solving a page of textbook multiplication problems. You will be developing meanings for multiplication, not just how to get the right answer. Understanding these meanings are a foundation for algebra as well as other types of advanced mathematics. Van De Walle suggests each math lesson should include 2-3 problems that allow students to use any technique and any math tool they like. For each problem they solve, they must explain, preferably in writing and then later in a discussion, how they solved each problem. Van De Walle also includes a great activity for finding factors in this book, assigning numbers to students and having them use counters, arrays, etc. and then write two equations, an addition and a multiplication for each factor.

Remember how multiplication and addition have a special relationship via repeated addition? A similar relationship occurs between division and subtraction. Division is repeated subtraction. For example, the problem 24/6 = 24-6-6-6-6. A hundreds chart is a useful tool for solving these problems and helping your students skip count backwards as well as forwards will help them use this strategy more efficiently. Other helpful models for practice are counters, number lines and arrays.

When teaching division, teach it at the same time as multiplication or very soon after so students may practice both types of problems and begin to see their relationships. Avoid using the term “goes into,” which carries very little meaning, and instead use “divided by.” Additionally, don’t be afraid of large numbers. Since the strategies and understanding of products and quotients are the same for small vs. large quantities, solve  2 x 4, 13 x 11, and 102 x 15, 6/3, 24/8, 200/5 and even much larger numbers should be introduced at once. To help students see the relationship between multiplication and division, think about how you will word each problem. For example, asking “6 times what is 24?” is more effective than asking “24 divided by 6 is what?”

A note on remainders. There are two methods for remainders in division problems. Method 1 is to leave the remainder as a quantity left over (i.e. 16 R1). Method 2 is to partition the remainder into fractions (i.e. 16 1/16). Van De Walle prefers method 2. In real contexts, you sometimes need a whole number answer so you must round up or down. In a problem where a certain number of cars are needed to transport students for example, using a fraction or remainder answer is useless. There are three ways to decide whether to round the number up of down. If you discard the remaining amount (nobody gets the leftover jellybean) you round the number down. A forced amount (you need an entire car for the one extra student) you must round up. A rounded number will go up or down depending on whether the remaining amount is greater or less than five (We will need 9.6, or about 10 hours to build a compost container).

Finally, when you write or re-write problems for your classroom, make up problems with different contexts. Instead of just using objects such as brownies or pencils. Be sure to use problems that use length, time and volume.

Teacher Resource
Book: Knowing and Teaching Elementary Mathematics: Teacher’s Understanding of Fundamental Mathematics in China and the United States
This book by Liping Ma was highly recommended to me. I have not read it but thought I would pass it on to you. Let me know by posting a comment, if you have read this book and what you think of it. Ma defends the belief that children need to know both arithmetic as well as theory for solid mathematical thinking. Read the editorial review on Amazon by clicking on the book link above or click here.

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 1 is a synopsis of Chapter 10 of John A. Van de Walle’s book, how to help children develop an understanding for the meanings of addition & subtraction and how they relate. Part 2 will discuss multiplication & division.

There are two main methods for teaching addition and subtraction. The first method is to use contextual problems. The second method is to use multiple models such as counters, drawings and number lines. Using these two methods will help children construct a rich understanding of the addition and subtraction operations.

The problems presented should be structured like Cognitively Guided Instruction (CGI) if you are familiar with that. In short, instead of “adding” and “subtracting,” you “join,” “separate,” use “part-part-whole,” and “compare.” Any of the quantities you work with (initial/starting amount, change amount, resulting amount) can be the unknown. For example, here is a part-part-whole problem with the initial amount unknown:

Jayden had some jelly beans. She ate 13 of them on Monday morning. One Monday afternoon she ate the remaining 12. How many jelly beans did Jayden begin with?

Contextual problems must be somehow connected to the children’s lives. You could write your own problems easily by reading some samples presented in the book. Problems could be about a recent experience the class had together such as a field trip, a discussion in another content area, or your read-aloud (or another book you’ve all read).

A typical development for students to make in early math is as follows:

1. Uses counters and counts everything seen
2. Counts on from the first set given
3. Counts on from the largest set
4. Uses facts retrieved from memory and relies less on counters

Part of what will help children progress to the next stage of mathematical development is to teach with models, showing what to do every step of the way, being sure to use a variety of kinds [Designs, picture "stories," unifix cubes, 2-colored counters (or other object), or part-part-whole mat] in order to give students different ways to see the problem and to model how to use each kind of model.

When teaching these first two operations, teach addition and subtraction simultaneously. If you ask what 7 – 3 is, you must also ask what 5+3 is  so students will learn the pattern. Subtraction must be taught as “Think-Addition,” instead of “Count What’s Left.” So instead of subtracting 6 counters from a student’s pile of 9 counters and asking them for the answer by counting what remains or asking what is 9 – 6 , help your students ask the question 6 and what make nine? As kids practice with each other in pairs, they say and write the equations that match with their problems. This is actually the beginnings of algebra.

Lastly, something we often don’t think of because it seems so obvious to us as adults. When teaching addition and subtraction, help children understand the “Order Property” (usually known as the commutative property) and the “Zero Property.” Using understanding of the order property, you learn that when adding two numbers together, the order of the addends don’t affect the result. Using the zero property, you learn that any number plus or minus zero is that number. Van de Walle here says that some students really do get confused because they connect the idea of addition with a number getting larger and subtraction with a number getting smaller, so to some, 12 – 0 should be less than 12, so some students may write an answer of 11.

Teacher Resource
http://www.aaamath.com/
This is a nice little site that offers tutorials and definitions of 25 different math topics for K-8 grades. You can sort by grade or subject. After each tutorial there is an interactive practice application that produces problems and automatically checks answers. I advise you to check out the practice prior to assigning it to a student because I tried the fraction subtraction practice (that’s a mouthful!) and it wouldn’t accept reduced answers. For example, it wanted the answer 2/2 and not “1.” This is reasonable, just something the student should know.

NEXT: PART 2: MULTIPLICATION & DIVISION

Teacher Resources

Math Open References
Need some online tools for learning about geometry? Want to learn how to find the area of an ellipses? Learn about famous geometers? How about refresh your memory on how to use a protractor to measure and create angles? Print blank graph paper? There is so much on this site, you need to check it out. There is also an extensive list of free printable compass and straightedge construction worksheets.

SketchUp
Use this free google application for yourself or your students to create, modify and share 3d models. Good for teaching about architecture & design, geometry and construction. Be sure to go to the “SketchUp for Education Program.” and check out the k-12 gallery for examples of student work, as well as the K-12 case studies. Very cool!

TeacherTube
An online community for sharing instructional videos as well as documents and photographs. Browse this site’s variety of topics or videotape and publish your own. Videos are user-reviewed.

100 Mobile Tools for Teachers
This is a blog entry for mobile phone users though many of the applications can be accessed from your computer like Google Earth (In the latest version of Google Earth 5, you can view the ocean floor and historical images from around the globe!). This site is worth checking out at the very least to scan through the tools. I found several that I hadn’t heard of that seem like they would be very useful. For example, there’s a Google Patent search where you can search for over 7 million patents, their inventors and dates, Edmodo which is like a Twitter for teachers, and Stixy, a sticky note application you can share and use for collaboration purposes.

My Teacher Tools
A very well fleshed out teacher resource site built by Rona Martin, a classroom teacher. Here you can find lesson plans, rubrics, certificates, letter templates, interactive, financial and resource tools, grade books, educational catalogs and much much more!

What I read
Arne Duncan’s Bio on ED.gov
Interview with Arne Duncan by National Journal Magazine
Profile on NY Times site
Interview with Arne Duncan
Meet Arne Duncan on YouTube

Thank you readers for having patience waiting for this blog entry. I have had an interesting time learning about Secretary of Education Arne (pronounced Ar-nee) Duncan and a challenging time processing it all.

The Lowdown

1. Duncan grew up in Chicago & played basketball through college
2. His dad was a professor & mom ran a tutoring center
3. Duncan graduated from Harvard in 1987 with a degree in sociology
4. Duncan & Obama have known each other for over 20 years
5. Duncan’s 2 kids attend public school
6. Prior to his current position, Duncan was CEO of Chicago Public Schools
7. Many improvements to schools & teachers, as well as remarkable test score gains were made during his tenure as CEO
8. Duncan’s policies on education are primarily what is shaping the direction of education today
9. Duncan says preparing kids for success is and “economic imperative.”
10. Duncan believes in a punishment & reward system for teachers

As many of you are aware, once a person steps into the political limelight, the general public tears them apart, criticizes their actions and opinions and are quick to judge. The main reason for this of course is that it’s a political game and everybody is scrambling for attention and power.

There is definitely some criticism out there. Some postulate that Duncan’s success improving the statistics of Chicago’s education system is just another Rod Paige scandal (Paige was appointed Secretary of Education by former President G.W. Bush and vacated the office after it was discovered he had taken unethical action to get his results). I don’t buy into unfounded assertions like this.

Some too, criticize Duncan’s exuberant support of Charter Schools. What will happen to our public schools if all the economically advantaged choose private, alternative and charter schools? Duncan says of charter schools that the good ones are part of the solution and the bad ones are part of the problem. He believes that all families should have choices for where there kids are educated. Right now it is limited to those with economic advantage. Duncan maintains that education is a civil rights issue. I would love to see him expand on this statement.

I appreciate that the education goals of the Obama Administration including Secretary Duncan, are focused on bringing low-income kids up out of poverty through education though I’m not convinced it would be a priority if not for our failing economy. It seems like a good plan for the economy, but is it the best way forward for education? I do believe, given Duncan’s upbringing in his mom’s tutoring center in Chicago serving under-privileged kids and his statement on ED.gov that education is also a “moral obligation,” as well as a “civil rights issue,” that Duncan really has genuinely positive intentions, despite his recent remarks about Hurricane Katrina being the best thing that happened to the education system in New Orleans. I know it is an easy thing to jump on and it was a political blunder on Duncan’s part but to me, it is obvious that he is trying to focus on the positive, something we all need to do just to get by sometimes. I highly doubt that Duncan was glad that Hurricane Katrina devastated New Orleans.

I am absolutely against a punishment and reward system for teachers. Duncan says we need to reward good teachers with financial incentives and get rid of teachers who are ineffective. His saving grace for me is that he says we need to get rid of them after we try to help them and if then they still are ineffective. This is another one of those issues that may sound great in theory, but in practice it is going to be a giant, painful belly flop. Being an effective teacher is a skill that is developed over many, many years. We already have a teacher pay scale that increases with time and experience. Given the average teacher salary, we all know that teacher’s aren’t in it for the money. What will be the system that decides which teachers get merit pay and based on what results? Most of us have figured out by now that judging a student based on a single, multiple choice test can have misleading results to say the least. The most horrendous consequence of all is that the punishment/reward system will scare off new teachers and (possibly) attract teachers who want to get in on the merit pay. Speaking from experience, new teachers already have such unrealistic expectations placed on them and for the most part they are going to naturally be the ones who don’t have lots of success with their students in the first couple of years because they have yet to find their path and to gain the experience they need. Fear of inadequacy and the possibility of having to make a career change mid-life if someone deems you unworthy of being a teacher will drive off teacher candidates.

What does need to happen instead is high quality, useful, meaningful professional development for teachers at differing levels of the trade. Teachers, especially early career ones, are not given sufficient time for collaboration, reflection and learning new content and skills. Early career teachers should have more time for professional development, working with mentor teachers and co-teaching experiences. Late career teachers should be sharing their knowledge and skills through mentoring, conducting research and publishing. School Districts should provide high quality substitute teachers who are familiar with the school to give classroom teachers the time they need to learn and improve. The money for the substitute teachers should come from a tiny portion of the $4 billion dollars from the federal government for education without compromising state rights in education.

For me, watching Secretary Duncan is a wait and see game.

Teacher Resource
Article: Hands-on, Minds-on, authentic learning in math

What I read
http://wheresthemathbellingham.blogspot.com/2010/02/penn-state-math-professors-oppose.html#comment-form

http://ies.ed.gov/ncee/pubs/20094052/pdf/20094052.pdf

There are three issues to address: How Investigations is taught, the meaning of constructing ones own learning, and being mindful of how to move forward in a solutions-oriented way. I hope that members of your group can run some focus groups and invite members of the school community to participate.

Let me begin by saying that I am neither for nor against Investigations as the best math curriculum to teach. However, it is important to understand how Investigations works. Its authors intended it to be a resource as well as a textbook for teachers so they could get professional development in along with preparing for class. This is important to note, because many elementary school teachers find math to be their most challenging subject. In theory this is a great idea, but to teach Investigations well, you have to read the entire book unit you will be teaching, figure out the main concepts and then adapt it to your methods. Then, as with any subject textbook, supplemental materials are a must. Who has the time?! The problem with Investigations as I see it, is the practice ends up being very different from the theory.

My second item of contention is the misuse and misunderstanding of what it means to construct ones own knowledge. In the open letter from the PSU professors you reference in your blog, it states: “This program is based on the highly controversial idea that one bypass basic techniques when teaching mathematics, that the students will themselves discover mathematical truth, while the teacher only plays the role of a facilitator.” Well, yes, it is a highly controversial idea because educators understand it in different ways. We as an education community must come up with a “standard” definition or we will continually argue about issues we later discover we actually agree upon. To me, constructing knowledge is based on the idea that when the brain receives new input, it attempts to understand it by processing it against all the other information it already has. We sometimes call these “making connections.” A teacher, when guiding a classroom of learners will activate background knowledge, create an environment where students do activities, journal, discuss, compare, reflect and defend. In such an environment though the teacher leads them all the way up to the new concept, the “ah-ha” moment will be something they own. A true success that builds self-confidence.

Finally, I have read the study you refer to and Investigations falls neither at the top nor at the bottom of the program studies (Please see comment below). This does nothing to convince me that it is a poor or unreasonable math program as you infer. Furthermore, the study was done only with 1st graders. In the Investigations program, a strong conceptual foundation is built first and in later grades more efficient methods are learned. It doesn’t surprise me that, based on one test to assess their learning, that the first graders learning Investigations didn’t score at the top. In my opinion, the open-minded and solutions-oriented way to move forward, is to work on finding studies, facts, etc. on a program that is worthwhile. Instead of urging the school district to abandon what is already in place, why don’t you take advantage of this opportunity to get a professional research organization in your schools and have them conduct studies. Any of you who are teachers have most likely had training like I did on research methods and you could, working together, conduct your own study so long as its results are uncompromised. I think it’s time that we all step up and take more action in regards to our local schools as well as our school districts and school boards. I sure don’t see anyone else out there who is more capable of improving our schools than you, teachers, parents, staff, principals and concerned citizens. So, my advice, find a better alternative and present it to your school board and the mayor and your school district. Do the work for them and compare the curriculum, show the data and they will have no choice.

All the best, -Tia-

Check out the parent blog:
The Math UnderGround: Seattle & Washington State