Wednesday, December 16, 2009
Sir Cumference
Use: This book reinforces literacy while explaining Pi and circumference concepts
http://www.goodreads.com/book/show/253043.Sir_Cumference_and_the_Dragon_of_Pi_A_Math_Adventure
Addition of Fractions
Multiplication of Fractions
Measurement/Partitive Division
Adding & Subtracting with Base 10 Blocks
Tuesday, December 15, 2009
Converting Files to PDF and Posting Them Here
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More Math by the Month
WEEKLY ACTIVITIES
STARS AND STRIPES: K-2
Flag shapes. Using a U.S. flag as a model, generate a list of the different shapes you see on the flag. In small groups, discuss the similarities and differences between the star and the rectangle. Use a Venn diagram to organize your thoughts.
Making stars. Predict how many stars you can draw in one minute. Practice, and then time yourself for one minute. Count the stars by circling groups of ten. How many stars did you draw? How long would it take you to draw 50 stars? Explain.
How many stars on a flag? Look at a U.S. flag for 10 seconds and then put it away. Estimate how many stars you think were on the flag. How did you arrive at your estimate? Use a flag to check your estimate. Share the strategies you used to count the stars accurately.
Around the flag. Use a flag and Unifix cubes. Estimate how many cubes are needed to go around the outside of the flag. After checking your estimate, make another prediction, but using a different-sized manipulative. Try smaller and larger manipulatives. Does the size of the manipulative make a difference? Explain your answer.
STARS AND STRIPES: 3-4
Point to the stars! Ever wonder why the U.S. flag has 5-pointed stars? Do you know about George Washington's idea of a 6-pointed star? Why did Betsy Ross's idea of a 5-pointed star win over Washington's 6-pointed star? There are many ways to compare the two kinds of stars, such as the following:
Which star has more lines of symmetry?
Which star takes less material to make? Compare the areas of the 5- and 6-pointed stars by using graph paper.
Which star is easier to make? Explore making your own stars using different numbers of points. Supposedly, Betsy Ross's idea of a 5-pointed star won over George Washington's 6-pointed star because she wowed audiences with her ability to create a 5-pointed star by making only one cut into a folded piece of paper. You can make Betsy Ross's 5-pointed star using the steps shown on the Web site www.ushistory.org/betsy/flagstar.html. Do you think a 5-pointed or a 6-pointed star should have been chosen for our flag? Explain.
More than material. The U.S. flag is not just material sewn together but also a symbol for the United States and what it stands for. Many rules and regulations exist regarding the flag, and the simple act of folding the flag is no exception. Brainstorm with your classmates to find different ways to fold a paper model of the flag and create a folded flag 1/4 its original size using the least amount of folds. Try 1/8 the original size. Compare the number of folds made for each flag. Now try folding a U.S. flag following the rules and regulations found on www.usflag.org/fold.flag.html. Compare the shapes made after each fold. Hint: If the end result is not a neat triangle shaped like the recorder hats of our forefathers, it is wrong! Remember that one of the rules for a U.S. flag is that it should never touch anything beneath it, such as the ground, the floor, or water.
Time flies. The current U.S. flag has 50 stars and 13 stripes, but the flag had only 13 stars and 13 stripes in 1777. If a star and stripe were added to the 1777 flag for every tenth year up until the present, how many stars and stripes would there be altogether? Compare your results with the number of stars on the current U.S. flag.
Symmetry. Try to make a symmetrical flag by using cutouts of only one star and one stripe. Is it possible? Continue by adding stars and stripes until 50 stars and 13 stripes appear, while continuing to check for symmetry. Describe any trends you notice.
STARS AND STRIPES 5-6
Patterns. The U.S. flag has 50 stars, to represent each of the 50 states. The rows of stars are two different lengths; each row has one more star or one less star than the row above or below it. Without looking at the flag, draw how the stars are arranged. How could the 50 stars be arranged in 5 rows so that every row has one more star than the row above it?
"Super Flag:' The late Thomas "Ski" Desk of Long Beach, California, owned the world's largest flag, "Super Flag." It was hung on cables across Hoover Dam in 1996 to mark the Olympic Torch Relay. Super Flag measures 505 feet by 225 feet, and each star measures 18 feet across. The flag is made of cloth, with steel grommets, or eyelets, placed every 30 inches along the left edge. Determine the area and perimeter of Super Flag. Determine how many grommets were used to provide support for the flag. Go outside and mark off an area the size of Super Flag.
Historical flags. The Flag Resolution of June 14, 1777, stated, "Resolved: that the flag of the United States be made of thirteen stripes, alternate red and white; that the union be thirteen stars, white in a field of blue, representing a new Constellation. " Since 1795, thirty-seven states have been added, and the design of the flag has changed twenty-six times. Use the data on the chart shown on the Web site www.si.edu/resource/faq/nmah/flag.htm to construct a bar graph showing the changes in the number of stars. Analyze and interpret the data on your graph.
Designing. Robert G. Heft of Napoleon, Ohio, the designer of our country's 50-star flag, hopes that his design of a 51-star flag will be accepted if a new state is ever added to the union. Design a flag with 51 stars that you would submit for the new flag.
Betsy Shreero, Cindy Sullivan, & Alicia Urbano. (2002). Star and stripes. Teaching Children Mathematics, 9(1), 22-23. Retrieved December 15, 2009, from Research Library. (Document ID: 161921151).
Another Month
GROWTH AND THEN SUM: K-2
Dominoes-more or less. Get 20 square tiles (1-inch-- square pieces of paper). Pick up 1 tile, which we will call a monomino. You will combine tiles using this rule: Each tile must share at least one complete side with the side of another tile. Place 2 tiles together to make a domino. Now connect 3 tiles to make a triomino. Make a different-- shaped triomino. Create a tetromino with 4 tiles. How many different tetrominoes can you make?
8
Eggs-actly! A farmer has 2 hens. Every day, each hen lays 1 egg. How many eggs would his hens lay in 3 days? How many hens would he need to double the amount of eggs for 1 day? If the farmer needs 28 eggs each week, could he get that many with only 2 hens? If not, how many hens would he need? Draw a picture that shows your answer or find a way to solve this problem without drawing.
15
Boxes and more boxes. Fold a paper in half; open it, and you have 2 rectangles. Refold the paper and fold it in half again. Open the paper and count the rectangles you have now. Refold and fold the paper in half once more but this time, do not open the paper. How many rectangles do you think your paper will have? Open your paper to check your answer. If you folded the paper one more time, how many rectangles would you have? Tell a friend what is happening to the number of rectangles each time that you fold the paper.
22/29
Flower garden. Hold your hand in front of you and spread your fingers apart. Imagine that your hand is a flower and your fingers are petals. How many petals would each flower have? On a large piece of paper, draw a tall line to be a stem. Place your hand at the top of the stem, spread your fingers apart, and trace around your hand. You now have 1 flower on a stem in your garden. If you drew 3 flowers, how many petals would you have? Make a mural by asking class members to make flowers for the garden. Share different ways to find the total number of petals in the garden without counting each petal. How would your answer change if each flower had only 4 petals?
GROWTH AND THEN SUM: 3-4
1
Spring has sprung. New plants are "springing" up everywhere. Just how fast does a plant grow? Plant a seed that will grow fairly quickly, such as a sunflower, pumpkin, or bean seed, and keep it watered. The first day that you see a plant above the top of the soil, take a ruler and measure the height. You will need to choose the unit you measure with carefully because the measurement will probably be very small. Measure your plant every day and record your findings. Describe any patterns you see with the growth. Use centimeter graphing paper and draw a picture each day to show your plant's growth. What do you notice about the pictures when the rate of growth is highest or lowest? Create a table like the one below that shows how much the height changed from one day to the next.
8
Growing patterns. Draw around one square on a sheet of graph paper to create your first shape, Draw around four squares, creating a 2 x 2 array, for your second shape. Color two of the squares to create a chessboard effect. Make a 3 x 3 array for your third shape. Color the squares again to look like the chessboard.
How large will your next array be? Draw a T table like the one below to record your data. What kinds of patterns do you see? Can you find more than one pattern? How does the number of black squares grow as the shape becomes larger? How does that number compare with the number of white squares? Draw and record the data for your shapes as you work toward the size of a chessboard, which is an 8 x 8 array. Use what you have learned to predict the numbers of squares that will appear each time.
15
A grain to grow on. In The King's Chessboard by David Birch (1988), a wise man performed a service for his king and was asked to choose a reward. When the king's chessboard caught the man's eye, it prompted him to ask for 1 grain of rice for the first square on the first day. He would receive 2 new grains of rice for the second square on the second day, making a total of 3 grains. The next day, he would receive 4 new grains, making a total of 7 grains. For each square, the wise man would receive twice as many grains as he received on the square before. The king agreed to the request, thinking that the wise man was not so wise in asking for such a small reward. Use the chessboard from last week's activity How many new grains would the wise man receive on the eighth day? What would be the total number of grains of rice he would have on the eighth day? How many grains of rice would he receive on the sixteenth day? Record your data to show the number of grains received each day and the total number of grains given to the wise man overall. Who was wiser, the king or the wise man? Why?
22/29
Hold your breath. One species of whale can hold its breath for 25 minutes while another species can hold its breath up to 1 1/2 hours, which is 90 minutes. A sea otter can hold its breath for 4 to 5 minutes. You might think that the larger the animal, the longer it can hold its breath. Try this experiment with a few friends and an adult. Find someone who can time you. Hold your breath three different times. Record how long you can hold your breath each time. Compare your longest time with your friends' longest times. Can an adult hold his or her breath longer? Create a graph that compares how long you can hold your breath with the times for sea otters and whales. Explain how you would use that graph to agree or disagree with the idea that the larger the animal, the longer it can hold its breath. (This activity adapted from Corwin and Russell [1989].)
GROWTH AND THEN SUM: 5-6
1
Growth spurts. The chart below shows the average height for boys and girls ages 6 to 20. Use one color of pencil to graph the line for the boys' height and a different color for the girls' height. Find the growth rate by creating a table that shows how much boys' and girls' heights changed from one year to the next. What do you notice about the shape of the graph when the growth rate is highest or lowest? During which years are boys and girls close to the same height? Do both groups experience rapid growth at certain ages? Does one group grow faster than the other during certain years? At what age does growth stop for each group?
8
Growing wealthy seed by seed. In the book Anno's Magic Seeds by Mitsumasa Anno (1995), a man named Jack gets two golden seeds from a wizard. He plants one seed and eats one. Miraculously, Jack is not hungry for a whole year. The planted seed grows and yields two seeds the next fall. For several years, Jack eats one seed and plants the other. He could live forever this way, but then, he gets smart! One fall, he plants both seeds and eats other kinds of food that winter. The next fall, he harvests four seeds, as shown on the chart below in the column labeled "Year 1." Now he eats one seed and plants the other three. How many seeds will Jack have after five years if he continues to eat one seed and plant the others each year? Without filling in every box, predict the number of seeds that Jack will get from his plants at the end of the tenth year. How did you arrive at your prediction? Be sure to read the rest of the book with its many interesting twists.
15
Don't be square! Use a piece of graph paper to draw some squares that each have an area of 1 square unit. Draw a 1 x 1 square with 1 square unit of area. Label the perimeter and area of your square. Draw 10 squares, making each square larger than the one before. You will have a 1 x 1 square, a 2 x 2, a 3 x 3, and so on. Make a table that shows the perimeter and area of each square. Graph the data representing the area and perimeter. Compare the perimeter graph with the area graph. Write a paragraph describing the shape of each graph.
22/29
Tiny panda cubs. The average weight for a human infant is 7 pounds, compared with about 180 pounds for a full-grown man. The fraction 7/180 can be used to describe this relationship. We might say that an infant boy weighs 7/180ths of his normal adult weight. What decimal amount is this? Panda cubs weigh just 5 ounces, compared with the adult male panda that may weigh 230 pounds (3680 ounces!) or more. What part of its adult weight is the panda cub? Try showing this amount as a fraction and as a decimal. Compare the relative sizes of human infants and panda cubs. As a challenge, try to create a graph that communicates the relative sizes of adults and their young (possibly using the computer). Which form (fractions, decimal, graph) makes your comparison easier or clearer?
Gayle Cloke, Nola Ewing, & Dory Stevens. (2002). Growing and then sum. Teaching Children Mathematics, 8(8), 464-466. Retrieved December 15, 2009, from Research Library. (Document ID: 114173591).
Math by the Month
The "Math by the Month" activities are designed to engage students to think like mathematicians. The activities allow for students to work individually or in small groups, or they may be used as problems of the week. No solutions are suggested so that students will look to themselves for mathematical justification and authority, thereby developing confidence to validate their work.
This month's activities are focused on investigating and exploring questions and activities related to using a calendar. Students will explore number sense and operations, logical reasoning, data analysis and probability, and algebra. These activities incorporate not only various Content Standards but Process Standards as well.
WEEKLY ACTIVITIES
CALENDAR MATH: K-2
7
Who's here today? Draw a picture of yourself on an index card. As you and your classmates arrive each morning, place your pictures on a large graph. Are there more girls or boys today? How many more girls or boys are there? How many people are absent? What else can you find out from your graph? As a challenge, record the number of people present each day and make a graph at the end of the week. Is there a day of the week when attendance is lower than the rest of the week? What might explain this? Do you think that this will happen next week? Find out.
14
Guess the day. Write a short mathematics story about a day of the week. Give hints in the story about which day you chose, but don't tell the answer until the end. You could say, "It's two days before the day we have music," or "It's the middle of the week," or "It's the day after the fourth day of the week." Come together as a class, read your stories out loud, and try to guess the days.
21
Today's date. Hang up a piece of chart paper with the date on it. Keep a list of how many times you encounter that number during the day. For example, on October 15 you might have music at 10:15, 15 people might order school lunches, or you might read page 15 in a book. How many times can you find the number in one day?
28
How many days until...? In the book Only Six More Days by Marisabina Russo, Ben is excited about his approaching birthday. "Only six more days!" he tells his sister. After listening to the story, use the calendar to pose a problem to a friend. For example, say, "Only __more days until the end of the month"Your partner can complete the sentence by looking at the calendar. Take turns posing problems. Make sure you explain out loud how you found the answer. For example, you might explain, "I knew it was eight days because it was a week plus one more day." Try one problem that begins like this: "Eight days ago, we
7
Did you know? With a partner, choose a day of the month and find as many mathematical facts as possible relating to that date. For example, for the number five you could list things such as: "Penta means five; therefore, a pentathlon is an athletic contest with five events. A pentagon is a figure with five sides and five angles. There are five Olympic rings!
14
Calendars galore! Approximately forty different calendars are used in the world today. The Gregorian calendar is the one most commonly used. Research different calendars and how they compare in terms of the number of days in the year, what they are based on (for example, planning crops, migration cycles, or annual events), whether the total number of days in the year is an odd or even number, and whether the days of the week fall on the same date each year.
21
The "write" date. Countries around the world have different customs when it comes to writing the date. In the United States, a month-day-year format is common: 12/25/1998 or 12-25-1998. Many other countries use a day-month-year format: 25/12/1998 or 25.12.1998. Furthermore, there is a format developed by the International Organization for Standardization in which the year is listed first, followed by the month and day. Create your own way of writing today's date by using different representations of the numbers or by using pictures. Explain the logic of your method of writing the date and why it might be better than one of the other accepted methods.
28
Calendar patterns. Use this years calendar to count the number of Sundays in each month. Can you find a pattern? Do this with other days of the week. Make a list of the dates. Do you see any patterns? Which months start on the same day of the week? Why does this happen? What effect would a leap year have on this pattern?
CALENDAR MATH: 5-6
7
Today's date. Using any date on the calendar, work with a partner or team to find different ways to express the number. For example, if today's date is October 24, how many equations can you write that equal 24? Use more than one operation.
14
Probability. Make a set of number cards from 1-31 (one for each day of October) and place them in a paper bag. Predict the probability that you will pull out an even number, a multiple of five, and a prime number. What is the probability that you will choose a number in which the ones digit is greater than the tens digit? Set up a table to record your results. Conduct three sets of ten trials to test each prediction. What type of number is the rarest or the most common on the calendar? For example, are there more double-digit numbers, single-digit numbers, or square numbers?
21
Mind-reading mathematics. While looking at October's calendar, tell a friend to choose four days that form a square in which all four numbers are touching. Ask your friend the sum of the four days and then surprise your friend with your mind-reading mathematics skills by guessing the four days. Hint: The first number is n, the second number is n + 1, the third number is n + 7, and the fourth number is n + 8. Therefore, 4n + 16 equals the sum of the four numbers. Using your friend's number, solve for n. For example, if 20 is the given number, your equation would be 4n + 16 = 20. Once you solve for n, you can find the other three days. Ask your partner how you were able to "read" his or her mind! Give only one or two hints before explaining.
28
Calendar computation. Use the calendar for any month. Choose any three consecutive dates and find the sum. Compare it to 3 times the median in the series. Try comparing the sum of five consecutive numbers to 5 times the median. Try seven consecutive numbers, comparing the sum to 7 times the median. Draw a rectangle on the calendar around a 3-by-3 "square" of nine numbers. Draw several more 3-by-3 squares. Compare the sum of the numbers around the outside of each square with the number in the center. Find the mean of the nine numbers of each 3-by 3-square. What do you notice? Why do you think this happens?
Betsy Shreero, Cindy Sullivan, & Alicia Urbano. (2002). Calendar math. Teaching Children Mathematics, 9(2), 96-97. Retrieved December 15, 2009, from Research Library. (Document ID: 209591161).
Friday, December 4, 2009
Tips on Using Real Data and Current Events
Use real events. Students who study news and current events in school do better on standardized tests, develop and improve reading, vocabulary, math, and social studies skills, and continue to follow the news as adults. Use the newspaper, the internet, even the news on television to work current events into your standard curriculum. As the presidential election is fast approaching, look for a lesson plan that will enhance content but also make students more comfortable with the election process. See the election page for a few ideas how.
Use authentic data sets or data from students. Instead of making up a set of numbers to teach a new topic such as box and whisker plots or linear regression, find an authentic data set. This is easier now with the internet. Another option is to ask your students for information about themselves. For example, ask students to report both the length of their feet and their heights, make a scatter plot, and then sketch in the 'best fit' line. Keeping them involved makes the lesson interactive and more interesting. Students are always interested in finding themselves on a graph.
Encourage multiple representations of the same data. Set a good example by using multiple representations. Different representations give us different information. Try to give them a mixture of tables, graphs, words, symbols, and pictures. Some visuals represent the data much better than others. Instruct students to create several, weigh out the advantages and disadvantages of each, and make an argument of which works best in certain circumstances.
Develop good consumers of math. Sometimes advertisements misconstrue data to make it more appealing. Have students examine ads and compare the graphs to the raw data sets. They should be asking questions such as: Are the scales appropriate on the axes? Were some of the data points left? What criterion was used to characterize outliers? Is this appropriate? Did the data come from a survey? What was the sample used? Were the questions biased?
Design activities to predict into the future. The value of mathematical modeling is more than just organizing the data; it is in making predictions about the future that is particularly useful. Examples include 1) looking at population growth in the your city, state, country or the world, 2) looking at world records or Olympic records, and 3) following weather patterns over time. Check out this applet that allows the user to predict the spread of a virus over a population.
Don’t be biased when teaching controversial topics. The goal of education is to prepare students to make their own informed decisions. As teachers, we are obligated to give students the tools to become knowledgeable about particular issues, but should always allow them to form their own opinions. It is best if students don’t even know their teacher’s inclination.
Help students learn to find their own data for research projects. It is a valuable skill to be able to find credible and reliable data, especially on the internet. Direct them to ask themselves questions such as the following: What are the author’s credentials? What is the date of the posting? Who is the site’s sponsor? Why was the information posted?
Use maps to enhance understanding. Reading data from a map can be a useful visual tool for learning and applying mathematics. Maps can be used to integrate current events, again including the upcoming election. But, students need to learn to interpret how the data is represented. For example, when studying the 2000 election, if all of the states won by Bush are colored red and all the states won by Gore are colored blue, the relative areas of the states would suggest that Bush was winning by a hefty margin. But when considering population densities and how the electoral votes are distributed, we can see how this map coloring can be misleading. Investigate this more with your class by using the lesson from NCTM's Illuminations titled "Getting Into the Electoral College."
Source: NCTM (http://www.nctm.org/resources/content.aspx?id=16263)
Tips on Supporting All Students: Equity and Diversity
"Equity" and "Diversity" are very deep topics, and as such, there are dangers in boiling them down to a list of tips. The following is not a list of activities one does to be equitable or to celebrate diversity, and should not be looked at as such. Rather, the goal is to provide a starting point for considering equity and supporting diversity within our classrooms. The following headings are very broad reminders of how we can continue our efforts to achieve the goal of a mathematics education experience that is equitable and celebrates diversity.
- Equity does not mean equal. When considering how equitable one's teaching and expectations are, we must consider the diverse needs and strengths of individual students, as well as the needs and strengths of the whole class. One student may need only a few minutes of extra instruction to master a concept, while the next student may need additional time to work and struggle with a set of manipulatives to develop an understanding which will allow him to own the content. It is not about how much time each student gets, but rather, how to create the appropriate opportunities for each student to learn mathematics.
- Focus on the individual. Learning our students' names is only the first step in developing a relationship with those individuals. The more we understand and respect the individual's background and strengths, the more we understand their particular needs. How do language, culture, gender, and socio-economics shape our students' world? More importantly, how can we, as teachers, understand, celebrate, and utilize the strengths and differences that make our classes unique? A first day handout or classroom exercise might include a survey that asks students to list strengths and rate past successes with mathematics. A simple exercise such as this can give us a good idea of students' feelings about mathematics and about themselves as math students, providing the contextual starting point for classroom interactions.
- Create an environment for success. Do your students know how important their success is to you? It never hurts to remind them! The expectations that we hold for our students send clear messages of how we feel about their education. Holding high expectations for all students shows our confidence in their ability and translates into success for more students. An environment that fosters success can be one in which all ideas and strategies are valued, where students share their thinking, listen with interest, and engage all students in consideration of the ideas presented.
- Identify your biases, and then get over them! Regardless of individual background or upbringing, we all carry own biases and stereotypes. As teachers, we are responsible for helping ALL students succeed, not just the ones that fit into our "box" of people who should do well. Set aside these biases and stereotypes and harness students' strengths to further every student towards the brimming mathematician and problem solver that they can be.
- Create an equitable curriculum that supports diverse needs and celebrates diverse strengths. Not all students learn the same way, so we must vary our approaches to lessons and provide students with manipulatives, visuals, projects, technology and group work to reach as many minds as possible. Give every student the opportunity to shine every day. Most of us have to follow a state or district curriculum, but with some creativity and work, we can meet the state and district requirements while making math interesting, engaging, and attainable for our students.
- Be aware of your questioning and listening techniques. How we ask questions, who we direct them to, and our interest in student responses can have lasting impacts on our students' achievement. We must believe that we can learn from all of our students' responses. We can learn about the students' thinking and often we can learn alternative ways of thinking about the mathematics itself. Are all students asked to engage in rigorous mathematical thought during the course of a lesson? Are all students given the time to think? All students should have the opportunity to tackle rigorous math every day, and carefully examining and altering our questioning and listening techniques can better assure that this happens.
- Walk the tightrope. We need to meet the needs of all our students, but it often feels as though we walk a tightrope to do this. While I am praising and encouraging the student who sits in the front row and knows the answers to even my toughest questions, am I simultaneously discouraging and ignoring that struggling student who sits in the back, never offers answers, and avoids eye-contact when I ask a question? These two students have very different needs, and the one who shouts louder is often more likely to get my attention. After all, the squeaky wheel gets the grease, right? Engaging and supporting all students is not easy, but it is our duty as classroom teachers. I challenge you to touch base with EVERY student, EVERY day. Work to give them opportunities to shine, to show their strengths, every day. You never know when you will turn that corner with a student and have a young scholar on your hands.
Starting the Year Off Right
Tips for beginning teachers:
- Start a lesson plan notebook or journal. Some days your lessons will surpass your expectations; other days, you‘ll wonder what went wrong. Create a notebook with copies of your lesson plans and two sets of worksheets. Keep notes on what works and what doesn’t, writing directly on the lesson plan or one copy of the worksheet. When you go back to revise or recreate lessons for the next school year, you will have a good record of what worked—and what you can build on.
- Don’t sell your class short. Avoid telling your class “This is easy,” “This will be fun,” “This shouldn’t be too hard,” or “This is going to be tough.” If students succeed at a task you’ve labeled “easy,” the accomplishment seems less significant. If they do not, they may feel worse than they otherwise would. What is easy for one student is difficult for the next, so keep all your students on their toes, and celebrate their accomplishments.
- Decorate appropriately. Take a good look at your classroom décor. What messages does it send to students? Does it reflect what you are trying to accomplish? Are your classroom rules prominently displayed? Do your students know where to look for examples of good work? Your classroom décor can say a lot about your personality as a teacher and what you are there to accomplish; don’t ignore it!
- Be consistent. Students tend to remember your rules better when they stay the same and are enforced equally and consistently.
- Create problem solvers. Start each class with a set of questions and riddles that promote logical thinking. Allow students to work in small groups, and emphasize that they should discuss solution strategies and how they got their answers. This activity shows students that your classroom is a place where communication and collaboration is encouraged.
- Who’s doing the math? Be mindful of who is actually doing the mathematics in the classroom. The students should be doing their share of the thinking, explaining, and reasoning. Give students a chance to struggle and wrestle with some math every day! Suggesting a solution strategy too quickly doesn’t give students a chance to solve problems.
- Talk with colleagues. Try to meet weekly with a group of fellow teachers to discuss teaching strategies, share classroom-management techniques, and brainstorm ways to offer more opportunities for students. Consider preparing a monthly math department newsletter for parents.
- Don't jump to conclusions. Regardless of past experiences, try to give each student a clean slate to work from. If you are particularly worried about a certain student, try giving him or her responsibilities from the start. Have the student hand out papers, erase the chalkboard, or collect papers from classmates.
- Use questions. Make your classroom a safe place to ask and answer questions. Try using students’ questions to drive your lesson, with students working to answer each others’ questions.
For more great ideas, check out the Empowering the Beginning Teacher of Mathematics series.
Source: NCTM (http://www.nctm.org/resources/content.aspx?id=6344)
Tips for Keepin' up on Math Skills over Winter Break
Here's a fun list of ideas to encourage math during the Christmas Break:
- Challenge others or challenge yourself. Online math strategy games at Calculation Nation provide a safe environment for elementary and middle school students to challenge themselves and challenge others. Games involve fractions, factoring, symmetry and comparing perimeter and area!
- Play strategy games with friends and family. A great way to spend quality time. Games such as Contig, and other free board games. Play as teams while learning so you can talk about strategy and then move playing individually.
- Talk to your children’s teachers before the break. Ask questions that show you are concerned about their development and maintenance of mathematics skills and fluency. For example, ask, “What do you see as my child’s strengths and weaknesses in math? What could we do while at home to develop or improve his/her weakest areas?”
- Read books that contain mathematics content with your children. There are books at every grade level that can engage students in thinking about math! Some suggestions include Ten Apples Up On Top! (elementary), The Great Number Rumble: A Story of Math in Surprising Places (middle grades), and The Numbers Behind NUMB3RS (high school). Want classroom activities to support math and literature? Check out Exploring Mathematics through Literature: Articles and Lessons for Prekindergarten through Grade 8 .
- Create a number book with your child. Use this template with your preschooler or kindergartner and have them decorate each page with pictures, stickers or stamps (or even glue beads or macaroni) that show the number on the page. For more advanced students, ask them to write expressions that equal the target number. For example, for the number 6, they could write 3X2, 10-4 and 2+2+1+1.
- Do projects with your child. Bake cookies or work on a home improvement project. Real-world applications of mathematical ideas, especially measurement, are everywhere! If you are stringing up lights, work with them to estimate how many sets you will need and calculate the total number of lights used. If you are baking cookies, have them figure out what is needed to make a double batch.
- Exercise your body; mathercise your mind! Take in a sporting event, even if it’s only on TV. Keep track of yards gained and lost from running versus passing plays of their favorite football team or the shooting percentage of their favorite basketball player. Work with them to make comparisons between two of their favorite players and display it graphically. Check out the lesson connecting rate of movement to football on Illuminations, appropriate for middle and high school students.
Cut out snowflakes as decorations. Invite your child to describe the shapes they see in their snowflakes and encourage them to tell you what they know about symmetry. Consider delivering holiday cheer by delivering them to a nearby nursing home! Play with the fractal tool on Illuminations. For a high-school level lesson on creating a Koch snowflake using fractals, see the activity sheets from Navigating through Geometry: 9-12 or from this article from the journal Mathematics Teacher.- Have a problem of the day. Work through one new problem before or after dinner each night. Figure This! has an awesome assortment of interesting problems with hints and solutions, so you don’t have to be a math wizard to facilitate!
- As a family, track your calorie intake or your finances. Are you consuming more food during the holiday season than you would otherwise? Are you spending money on gifts? Becoming aware is important in establishing control. You may also consider how much time each day you spend on each activity such as watching television, eating, sleeping. Make a graph. Then, brainstorm how you can manage to fit in alternative activities to maintain a healthy lifestyle.
- Seek out a volunteer opportunity that appeals to both you and your child. From cooking for a shelter, to collecting food for a food drive, to collecting coats for the needy, there are lots of opportunities to estimate and use math to project how much your efforts mean to others.
- Did your family receive gift cards as holiday gifts? How will you decide to use them during the holiday sales? Are the same discounts available through online ordering as in the store? Which is more – the cost of shipping or the cost of driving to the store?
Toward understanding of "computational fluency"
There is probably no topic that spurs discussion, incites action, and stirs controversy among educational stakeholders more than computation in mathematics. This attention is understandable because computation is a core element of mathematics, school curricula (and our collective memories of learning mathematics), and the knowledge that is important for day-to-day living in the real world.
In its decades of work to establish comprehensive and coherent standards for mathematics education, the National Council of Teachers of Mathematics (NCTM) has worked to articulate, and rearticulate, its vision of teaching and learning about computation. The most recent version of NCTM's Standards (NCTM 2000) uses the phrase "computational fluency" to embody the council's ideas. As has occurred with NCTM's previous ideas about computation, numerous questions and concerns have been raised about the meaning and importance of computational fluency. Examining the mathematical, instructional, and learning elements of computational fluency is the goal of this focus issue. We hope that this focus issue will facilitate movement toward understanding computational fluency. Ultimately, giving meaning to this phrase will require the work of countless people, including students, teachers, curriculum developers, mathematicians, teacher educators, and researchers.
The Standards describe computational fluency as a "connection between conceptual understanding and computational proficiency" (NCTM 2000, p. 35). Conceptual understanding of computation is grounded in mathematical foundations such as place value, operational properties, and number relationships (p. 32). Computational proficiency is characterized by accurate, efficient, and flexible use of computation for multiple purposes (p. 152). These characteristics make fluency-a word that most people associate more strongly with language development-useful for conveying the aim of computational performance. Just as the fits and starts of word-by-word reading detract from comprehension of a written passage, so will inaccurate, cumbersome, and poorly learned computational strategies ultimately detract from the making of mathematical meaning (National Research Council 2001, chapter 1). The Standards call for regular experience with meaningful procedures so that students develop and draw on mathematical understanding even as they cultivate computational proficiency (NCTM 2000, p. 87). Balance and connection of understanding and proficiency are essential, particularly for computation to be useful in "comprehending" problem-solving situations. The articles selected for this focus issue present a variety of perspectives on computational fluency that converge around three main themes.
Dimensions of Computational Fluency
Two articles act as "bookends" for our thinking about computational fluency. "Meaning and Skill-Maintaining the Balance," an article first published in a 1956 issue of The Arithmetic Teacher, gives us a view of what might have been called computational fluency in that era, with an emphasis on the connections between understanding and proficiency. In "Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician's Perspective," a modern-day mathematician states his view of the links between computational understanding and performance. The common theme in these articles sets the stage for investigating both the content and the pedagogy involved in improving students' computational flexibility, efficiency, and accuracy.
Mathematical Foundations for Developing Computational Fluency
Several articles describe the importance of number relationships, place value, and properties of operations as mathematical foundations for computational fluency. "Toward Computational Fluency in Multidigit Multiplication and Division" highlights the importance of students' understanding of the mathematical components of efficient and accurate algorithms. "Subtraction Strategies from Children's Thinking: Moving toward Fluency with Greater Numbers" describes using mathematical foundations to help students bridge existing strategies with more efficient and accurate ones.
Teaching for Computational Fluency
Several articles address important aspects of effective instruction for computational fluency. "Discussion as a Vehicle for Demonstrating Computational Fluency in Multiplication" explores how teachers can assess students' computational fluency from open discussions of carefully selected sequences of computational problems. "Helping English-Language Learners Develop Computational Fluency" extends the instructional strategy of classroom discussion with suggestions for supporting students who are learning English as a second language. Specific instructional strategies for promoting automaticity and accuracy are described in "When Flash Cards Are Not Enough." "Promoting Meaningful Mastery of Addition and Subtraction" also presents instructional practices that promote the development of computational fluency in the primary grades. Continuing the infusion of effective teaching strategies in the classroom, "Developing Teachers' Computational Fluency: Examples in Subtraction" addresses the preservice preparation of elementary teachers.
In addition to the articles, a pullout poster visually represents the dimensions and mathematical foundations of computational fluency. The back of the poster suggests what to listen and look for to determine students' progress in developing computational fluency.
Acknowledgments
We want to thank everyone who contributed to the creation and completion of this focus issue: people interested in mathematics education who submitted articles, NCTM members who reviewed the articles, TCM Editorial Panel members who helped select the articles, and the staff in Reston who provided the assistance required to produce another focus issue for Teaching Children Mathematics. We hope that reading and discussing the articles in this focus issue will be as professionally rewarding for you as editing this issue was for us.
| [Reference] » View reference page with links |
| References |
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| National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. |
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| National Research Council. "Looking at Mathematics and Learning." In Adding It Up: Helping Children Learn Mathematics, edited by J. Kilpatrick, J. Swafford, and B. Findell, pp. 1-16. Washington, D.C.: National Academy Press, 2001. A |
| [Author Affiliation] |
| Timothy A. Boerst and Jane F. Schielack For the Editorial Panel |