Investigate Your World

What I read
http://www.nctm.org/standards/content.aspx?id=264 +Q&A and other focal point topics

If you are not yet familiar with The National Council of Teachers of Mathematics’ (NCTM) Focal Points, now is the time. To summarize, Focal Points are a smaller number of standards emphasized at each grade level of pre-K-8th grade. It is important to note that these Focal Points do not aim to replace any current standards. They do aim to build on NCTM’s current standards. The Focal Points information on the web site show that NCTM believes that learning math is cumulative and the foundations in one grade are building blocks for the foundations in the following grade. The small key emphasis areas allow students time to develop deeper understandings of concepts, fluency in procedure and the ability to generalize (which shows the student can see the big picture). Fewer content areas also allow students time to practice problem-solving, reasoning and critical thinking.

In selecting which key areas to emphasize, the Focal Points had to “be mathematically important,” “fit” with what is known about math, and “connect logically” with previous and future grades. Focal Points are intended to be used however teachers, school districts, states, textbook companies, etc., decide to use them.

I see the Focal Points really emphasizing the nature of mathematics teaching in the following quote:
“Focal Points should be addressed by students in the context of the mathematical processes of problem solving, reasoning and proof, communication, connections, and representations.” (NCTM Web site)

Despite my distrust of standards and the completely unpalatable sensation I get in the pit of my stomach, I truly appreciate these Focal Points, all the work that was put into creating them and the fact that mathematical content should be addressed through solving problems in the classroom.

Lastly, I want to share a short timeline of Standards that I gleaned from NCTM’s web site alone.

1980′s- An Agenda for Action was published, starting the Standards Era.
1989- Curriculum and Evaluation Standards for School Mathematics
2000- The same publication was updated and re-named Principles and Standards for School Mathematics
2005- Standards and Curriculum: A View From the Nation, a document declaring the state of mathematics in the U.S.
2006- Curriculum Focal Points for Pre-Kindergarten through Grade 8 Mathematics: A Quest for Coherence
2006- Navigations, a series of teaching books seeking to expand and illustrate the vision of math instruction using Focal Points
2009- Focus in High School Mathematics: Reasoning and Sense Making

You can purchase the Focal Points (or any of these publications) through NCTM’s web site as a paper publication or as pdf files. You can purchase the whole publication or just your grade level. I am ordering the full paper publication so after I receive and read the entire publication, I may post an update. I would highly recommend purchasing your grade level as a pdf document. It is very affordable, only $2 (non-member) or $1 (member) for one grade level.

Teacher Resource
Click on the link below for free math lessons in PDF for pre-K-5
http://www.nctm.org/resources/content.aspx?id=8768

NEXT: The Common Core State Standards Initiative

When I began this site not so very long ago, my idea was to offer online and printable resources for learners. That is, both teachers and students who wish to take charge of their own learning. The main reason for offering resources to various types of learners is to aid in the pursuit of knowledge and truth which has always been to me, the true and ethical reason for schooling.

Every school or district, whether public or private, offers a different slant on education, different emphases, and different forms of assessment and accountability. This lack of unity often results in some students being better prepared for college, some ready to change the world through compassionate actions and some so weary of school culture that they drop out. Simultaneously, there are “standards” that hope to gain equality in knowledge, process and skills for all children equally but quickly tire out a quality teacher who loses her or his creative abilities as well as losing the time and freedom to teach ones own passions. It seems that teachers, policy makers, testing organizations, tutors and educational organizations all have differing opinions on how children should be taught.

I feel passionately that students should become acquainted with their own history and learn new material in a way that makes sense to them based on what they know about the world. When I say student, I also mean teachers because so many of us go into teaching because we want to keep learning. In order for students to learn new material, their bodies and brains must be developmentally capable of doing so. In a constructivist learning environment, teachers prime their students for the new material, helping them be aware of their current knowledge that most relates to the new material. With brains primed, students are mentally prepared for the new learning. During this learning time, teachers must observe only and not teach! Observe how each student is thinking and how they attack a problem. Make a note of what might come next. Giving hints is okay if students are stuck. You can read more about this method which I attribute to John A. Van De Walle in Elementary and Middle School Mathematics: Teaching Developmentally. After the main lesson is time for thoughtful reflection and discussion with the classroom community.

So, why mathematics? I remember faintly, enjoying math as a young child. I could count and add, subtract, estimate, round, measure. There were easy rules to follow. From about 5th grade on up through high school and even college, I despaired. I lost confidence. Math got complicated because the rules were complicated. The procedures were difficult to follow. For the life of me, I could not “get” percentages or fractions. Memorizing the quadratic equation became the bane of my existence. I remember asking my math teachers in high school and in college specific questions about why and how the math works like it does. I was a question asker. However, when you are told to just memorize equations or the rest of the class sighs whenever you have a question, you “learn” to stop asking them. So, eventually, that’s what I did. I strove for B’s and C’s in my math classes. I gave up on math. I complained about it every chance I got, reinforcing my belief that math was too hard for me.

When I became a teacher in Juneau, Alaska, I was suddenly in charge of a multi-age class of 22 children (ages 7-9) and all their learning. Well, when a teacher loves a subject, kids know it. When a teacher doesn’t like a subject, kids know it, so I wanted to do my best to learn it and be excited by it. I soon discovered that the majority of children in my classroom were already math haters by the time they entered my class. Given the amount of questions, concerns and confessions that came from the children’s families in regards to math, it became clear to me that many parents pass on this distaste, distrust or anxiety of math to their children without even realizing it! I have spoken to teachers in both Alaska and Oregon who have less confidence in teaching mathematics than in any other subject. As a result, mathematics programs are mandated by most schools and the result is that teachers learn how to teach that particular mathematics program (and some are very good programs!) and supplement the mathematics curriculum with things they learned in school that are antiquated to put it mildly.

Now, there’s a difference between ignorance, which can be defined as lack of education, and apathy which can be described as an uncaring attitude. What I see out there, is a great deal of both ignorance and apathy of mathematics, including its history, computation, problem-solving skills and conceptual understanding. If, together, we could change that ignorance to learned, perhaps we could also change that apathy to love and if not love, then at least caring.

I am on a journey to do three things:
1. Discover how to teach constructivist mathematics
2. Analyze educational policy in mathematics today; and
3. Share the information I discover with you.

My goal, beginning today, January 26, 2010, is to post 2 blogs every week day for at least 6 weeks. One blog will be about the books I read or ideas I have about teaching constructivist mathematics. The other blog will generally be a response or review of an article or web content that discusses mathematics in America today. I will try to post pictures and links when possible. Please feel free to comment on any of these blog postings. Also, if you are a mathematics educator and would like to submit a blog entry that relates to your mathematical journey or mathematics in the U.S. education system, please find my email under “Contact.” I would love to include blog posts from guest bloggers to include many different viewpoints. Please also include a short bio. Thanks!

-Tia-

A balanced mathematics program should include the following three parts:

1. Conceptual understanding
2. Computational fluency
3. Problem-solving skills.

The combination of understanding the math taking place, being able to use math rules accurately and efficiently and the use of ingenuity, creativity and flexible thinking to solve mathematical problems at hand, are “mutually reinforcing.” According to Van De Walle, author of Elementary and Middle School Mathematics, introducing mathematically intriguing problems for students to solve as the main mathematical task of your lesson will inherently address conceptual understanding and computational fluency in the process. The carefully chosen problem you introduce to your class should build on previous concepts. Maybe you are teaching a procedure  or maybe you are trying to give students subtraction practice or maybe you are trying to stimulate their spatial reasoning. Whatever skill you are targeting can be addressed by the mathematical task you choose.

The class should be divided into 3 equal slots of time, say 20 minutes each for the before, during and after portions. There should be one problem for the whole class that focuses on a mathematical dilemma, with multiple entry points to the problem. This will help to differentiate for various learning types. Another way to differentiate is to have a challenging problem that builds on the first one for those students who finish early. They should also have an ongoing, long-term math project to work on.

Before the lesson, you must remember to:

1. Activate any prior knowledge the students may have. This will engage those parts of the child’s brain so they are prepared to make new connections
2. You must make sure there are a variety of tools and manipulatives available to all students
3. Make sure the students understand what is expected of them during work and at the end (in this case, a written explanation of work) and finally,
4. Make sure they understand what the problem is asking.

Once they get to work independently or with partners, begin the “during” part of the lesson. The teacher’s job here is not to help or teach, but to listen, engage and assess. Let the student struggle, make mistakes and explain their thinking without your interference. The best discussions come from disagreement, as does the best learning in my opinion and the best satisfaction comes from figuring out a problem on your own without help. During the lesson, you may give hints such as suggesting a strategy to try  but it is most important to first understand what the student has tried already and what their thinking is. Students should be expected to write their 3-part explanation of:
A. What they did to solve the problem
B. Why they did it that way and
C. How they know they have the right answer. This is to prepare for the class discussion in the after phase.

Even if students haven’t completed the problem, when the time is up, don’t cinch on the after discussion. Allow it the full 20 minutes. Most importantly, encourage student to student discussion, have one student answer each others questions or respond with questions of their own to a fellow student. Use equality. Ask every student how they got their answer, why they did it that way and how they know they have the correct answer, whether the student obtained the correct results or not. Asking how a student got their answer should not be a trigger to students to infer an incorrect response. The more varied the results, the more exciting the discussion and often, disagreement. Additionally, you are helping build a mathematical community of learners who take risks and respect each other’s ideas. Encourage  and model respectful disagreement. To be sure every child gets a chance to share, have them partner up and share their results with a classmate as in a think-pair-share model. They are prepared for this because they have written down their strategies and they knew the expectations from the beginning. Take mental or written notes during this discussion time to figure out what tomorrow’s problem will be, based on what the students grasp and what still eludes them.

Some of the positive aspects of this way of teaching math are that it differentiates for learners, it gives the teacher time to assess because you aren’t occupied during the entire math class with a needy child, it gives students who finish quickly multiple options for meaningful work to do that isn’t a punishment or a reward, the multiple entry points of the well-thought-out problem allow all students to solve the problem based on their current understanding of math, the writing encourages the reflective process which reinforces learning, the discussion can be had with confidence because everyone has a frame of reference in which to share; their journal entry. This gives students more self-confidence, students learn to respond respectfully to each other and really listen to their ideas and be able to agree or disagree with another student. The discussion additionally allows for social interaction which improves a student’s own ideas and understandings and it builds a strong math community. When the student tells their family what they did at school that day, you can be sure that it’s easier to remember the depth of that one problem tackled than a page or two of raw math equations.

I hope you can read Van De Walle’s book and adopt these practices into your math curriculum whether you are a classroom teacher, a student, or a parent trying to improve your understanding of how kids think and what they need to be great mathematicians.

Nothing in this world is stagnant. Everything changes all the time. Some things like the weather or a scoop of ice cream left out in the hot sun can change very quickly. Other things such as an orange peel decomposing or a tree growing change very slowly over time. Other things may not have a constant rate of change. Instead, such as a baby growing, the baby will grow very fast in the first years, then it slows down the rate of growth. In the adolescent years, the rate of growth speeds up again and then slows into adulthood.

To help yourself learn about the mathematics of change, one of the best projects is to grow your own bean plant (or a plant of any kind that grows up). Keep a chart and record its daily growth. Create a graph showing the slope, or rate of change.  Learn about functions using a made-up “function machine”, compare rates of growth, examine patterns of change, describe how one variable changes in direct relation to another (such as the area of a circle and the length of its radius), be able to explain to someone else the difference between how big something is and the rate at which it grows, and construct bigger and bigger squares out of tile and describe how the area of the square changes in relation to the increasing length of its sides.

Learning about geometry and measurement helps develop your spatial thinking. It also gives you the tools and confidence to be able to build things, cook and bake, set up chemistry experiments, explore historic and modern architecture, paint with structure in mind, calculate the orbits of planets, measure vast distances in outer space using exponents, create fractals on the computer, perform magic tricks, and even play a great game of pool!

Do you want to make a treasure map? How about build a fort? Do you want to know how to give accurate directions? How will you know what size rug will fit in your living room? What size window box will you have to order or make for your plants? Will the leftovers fit in the container you have? Use geoboards (or a square of wood with nails pounded in at even intervals) for learning about fractions and geometric shapes, use household items such as egg cartons and juice packs as arrays to learn about multiplication, learn how to draw 2D and 3D shapes and other objects like buildings, in a creative “cityscape” project, build a model to scale, start a square foot garden, acquire and learn how to use the three most important tools of a mathematician: a pencil, a straight edge (ruler), and a compass. Also useful is a protractor.

Statistics and probability let you evaluate risks in your own life (such as the effects of exercise, playing the lottery, or smoking), data analysis helps you make sense of information you find in the media in order to make informed decisions, and most of all, a solid understanding of statistics helps you know how to ask the right questions to avoid being misled by advertisers, taken advantage of or straight up duped by companies who are trying hard to sell you their product.

How will you decide which political candidate you will vote for? How will you interpret stories you have heard or read in the news? How will you make informed decisions that affect you, your family and the rest of society? Do you have a good chance of winning the lottery? To function in our world and to be an informed and contributing citizen, you need to understand the basics of analyzing (or making sense of) data and the basics of statistics (it’s not that scary of a word!). Start cutting out all the charts, graphs, and maps you can find in your local paper or a magazine. Familiarize yourself with how to read them. Research the relative costs of health care and who pays what, learn the differences between mean, median and mode and how to solve for them, understand “cost-benefit” ratios, and know how an “average” describes your data.

Number sense means understanding how numbers are built and how they relate to each other. Ideas to understand and make games, puzzles and curriculum out of may include: Bigger than, Less than, Equal to (including decimals, real numbers, fractions, irregular numbers, and percentages), Finding numbers on the number line, combining numbers to see what happens, finding patterns, testing numbers by seeing how they react when we do things to them, and how to categorize numbers (such as even, odd, primes, factors, multiples, square numbers, cubed numbers).

Eventually I will post various activities, games and curriculum plans for you to glean from in order to build good number sense in yourself or a child in your life. In the meantime, you may be able to move forward with just this as your jumping off point. Good luck! Problem to try: Put the following numbers in order from least to greatest: 2/3, 60%, and .5

Mathematics is not arithmetic! Forget for a moment, everything you connect to math: fear, numbers, equations, memorization and right answers. Math is so much more than that. Consider that math’s history and development has been different all around the globe and your understanding of it is likely an ethnocentric one.

Math is the language spoken by the universe. To learn and understand that language is to learn the true nature of the universe. You will discover the intricate connections between math and plants, animals, art, music, architecture, agriculture, planets, stars, statistics, politics, religion, technology and so much more. And what’s more, I’m not going to tell you what you need to know! That’s right, I’m not in charge, you are. I will help facilitate your learning by giving you the tools you need, some questions to ponder and some suggested experiences you may want to try for yourself. You must provide the curiosity, the motivation, the follow-through and the reflection so that you can assess what knowledge you have learned. Good luck!