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Math Literacy News: Volume 4 Winter 1996

In This Issue...

The Ohio Literacy Resource Center is thrilled to be part of two grants focusing on math education for adults. The first is an Adult Numeracy Practitioners Network (ANPN) grant from the National Institute for Literacy (NIFL). This planning grant will look at math skills needed for a parent, a worker, and a citizen. There are groups in Ohio, Oregon, New England, Virginia, Illinois, and cyberspace researching this topic. We are lucky to have been chosen as one of these groups and also to have one of the five national stakeholders on the project from our state, Denise Schultheis, ABLE supervisor. She will certainly be an asset to this project!

The OLRC has also received a 353 grant from the Ohio Department of Education. This grant will put teams of teachers throughout the state into focus groups to look at the NCTM standards as they apply to adults. Math Kick-Off Days in each region will be held next August to share the results of these focus groups. Team leaders and focus group topics are: Paula Mullet of South Euclid, geometry and spatial sense; Mary Lou Swinerton of Medina, anxiety and number sense; Patty Bilyeu of Bellaire, algebra; Charley Flaig of McDermott, estimation; Susan Cann of Lancaster, problem solving; Delores Jones of Harrison and Lois Borisch of Cincinnati, connections and statistics; Sonja Brown of West Alexandria, measurement; and Diane Ninke of Toledo, connections/life skills. The groups will be meeting throughout February, March and April to develop units for all skill levels. I look forward to working with this terrific group as project coordinator for both these exciting grants!

Nancy L. Markus

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ON-LINE RESOURCES

Listserv
The ANPN sponsors a listserv for adult math teachers. There have been many great ideas as well as interesting discussions on the list recently. Topics range from Paulo Freire and how he relates to adult math education to specific ideas for teaching fractions!
If you wish to subscribe to the numeracy list, send an e-mail message to: majordomo@world.std.com
The message should read: Subscribe NUMERACY (your e-mail address here)
Several of the ideas in this newsletter have been obtained via the numeracy listserv.


World Wide Web
The ALM (Adults Learning Math) newsletters #1 and #2 are at this website. ALM is an international group of researchers in adult numeracy education.

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FRACTIONS
Fractions: An Alternative Approach from Bright Ideas: by and for the Massachusetts Adult Education Community. By Sally Spencer, CARE Center, Holyoke:

"Fractions are a common stumbling block for adult learners. The following problems were designed for the Summer Math Program at Mt. Holyoke College. Summer Math promotes mathematical understanding and confidence building in high school girls by encouraging discussion and visualization. I have found this alternative approach facilitates conceptual development and self-reliance in adult learners as well."

"You will find the following problems are most effective when students are encouraged to both explore these problems with a student-partner and explain their own problem solving processes. "Right" answers, "correct" methods, and "rules" discourage student investigation, while questioning encourages higher level thinking. To spark discussion, I use questions such as:

Can you explain your pictures? Can you repeat that in your own words? Can you say more? Can you solve it another way?

"Remember, questions don't mean that there is an error, only that you're interested in your students' thought processes--and that you really do want to learn from your students."

Problem A
Task #1: Eva spends ½ of her income on food and rent, 1/4 of her income on clothing, 1/12 on entertainment, and saves $1200 per year. What is her yearly income?

Task #2: Maria spends ½ of her income on food and rent, 1/3 of her income on clothing, 1/12 on entertainment, and saves $1200 per year. What is her yearly income?

After you read Task #2 but before you solve it, consider the following questions: What changed from the first task to the second? Do you think Maria earns more or less money than Eva? Why? Now, solve the problem.

Problem B
Task #1: For each of the following pairs of fractions, use diagrams to determine which fraction is larger.
3/8 and 5/8 ; 3/8 and 3/5; 5/6 and 7/8; 4/5 and 1/5; 4/5 and 4/7; 2/3 and 1/5; 7/10 and 3/10; ½ and 1/3; 3/5 and 2/7

Task #2: From these comparisons and any other pairs of fractions you might try, what generalizations can you make about comparing fractions?

Task #3: For each of the following pairs of fractions determine which fraction is larger and explain why. You may use a diagram of explain in a sentence.
3/8 and 5/12; 13/5 and 21/10; 4/7 and 4/9; 4/7 and 2/3; 14/35 and 7/18; 2/3 and 5/7

Task #4: List three ways to determine which fraction is larger.

Task #5: Put the following fractions on a number line:
13/10; 3/4; 3/5; 10/13; 1/9; -7/11

Editor's note: After spending a semester doing research on how students learn fractions, I can't stress enough how important it is for students to be allowed the time to develop some of their own ideas about numbers. When we give a student a rule, we don't allow him/her to own it for him/herself. (Remember Piaget from your undergraduate years?) "Predict, Solve, and Reflect" should be part of each student's classroom time. Your questions can help clarify his/her thoughts. Students need a visualization of what fractions mean; this number sense takes some exploration including manipulatives. To expect a student to compute using fractions without spending time developing concepts such as, what does 1/4 mean?, which fraction is bigger?, what do we mean by a greater fraction?, etc., dooms the student to confusion and the teacher to teaching the concept over and over. Linguistics are a big problem for many students when they begin to work with fractions; why do we say greater when we mean more area? An other problem students have is identifying the unit we want to work with. In some problems a whole is a unit, whereas in other problems it might be a set of objects. Letting students work with chips, M&M's, or other pieces as sets help develop a deeper understanding of fractions from the "fraction circle." More ideas for fractions, decimals, and percents follow.


Graph Paper and Money Trays
M. Adele Megann (mamegann@freenet.calgary.ab.ca) has used the following with her students:

"One way to show the connection between fractions and decimals is to use graph paper. I make a master with several blocks of ten by ten outlined in black. One bar is one/tenth, or 0.1. One block is one/hundredth, or 0.01. This is especially useful with addition. Use different colors for each addend."

"You can use place value blocks for tenths, hundredths, and thousandths. You might find a picture of a thousand-cube to make up worksheets."

"Money trays with everything removed except hundreds, tens, ones, dimes and pennies can be used to do addition, subtraction, multiplication and division of decimals."

"Finally, use graph paper when working with percentages. Discuss the origin of the word and the symbol to give context. After fractions and decimals, students find percents easy!"


Fraction Circle
Eileen Simons (SouthWoods@AOL.com) explains a bit more about the "fraction circle."

"Have you ever seen a percent circle? It is a circle with 100 small lines on its circumference, and then 10 long lines. The short lines are hundredths and the long ones are tenths (dividing the circle into ten equal parts). The advantage of the circle is that it clearly ties fractions to decimals to percents. If you cut the circle into quarters or fourths, it is the quarter circle that students have used in fractions and it has two long lines (2/10) and 5 more short lines (5/100) which is wonderful for showing .25 .2 + .05. I also tie the percent sign to the cent sign and tell my students that 2% = 2 C. They seem to be able to "see" that better and have less trouble turning percents into decimals."


Common Denominators
Mark Schwartz (Mark_Schwartz@ed.gov) has some final thoughts on fractions:

"One basic concept that I learned from my students revolves around the idea of finding a common denominator in adding and subtracting fractions...this (concept) is surprisingly fundamental and robust and hardly ever addressed."

"Consider why a common denominator is needed at all. Really think about it. For example, I like the idea of using objects, manipulatives, and other tactile approaches. I like the idea of using money, and showing 7 pennies and a dollar bill as a percent/fraction relationship. However, embedded in that example is the "hidden" construct of the pennies and the dollar ALREADY using a common and "equal" basis for the fractional relationship. In fact, most manipulatives do this. .... {when the teacher} rephrases things like ½ as a symbolic statement of "take something, break {divide, etc.} it into 2 parts and take {or do something with} 1 of those parts.....the reality is that it is 2 equal parts. When you write 3/4, the understood event is that the denominator represents 4 equal pieces. If you let students play with this idea, or if you lead them into a discussion of it and let them lead themselves out, the discussion usually trails through all sorts of concepts--not just fractions--and it becomes an A HA! experience to realize the basis for needing a lowest common denominator. This can really be demonstrated with objects. For example, if you start with various lengths of strips of paper and you don't use the same length strips of do use the same length strips but cut them into non-equal parts, you can get into some amazing paradoxical situations about fractions, which are remarkable learning experiences in addition to learning fractions."

"You can readily shift into grouping of objects, factors, sets of factors, prime factoring, etc. This sets up the framework not only for a methodology for finding the Least Common Denominator for any set of numbers, but also sets up ideas, without identifying them as such, about algebraic factoring."

Editor's note: We as teachers may assume that fractions are equal parts, but check to see that the concept is there for your students also! The time spent trying to understand the concepts above may seem unrealistic, but consider how much time is spent trying to teach and reteach LCD and addition and subtraction of fractions with minimal success.

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MATH CONNECTIONS/COMMUNICATION
Journals and Student Teachers
From Joyce Claar (Jclaar@aol.com) this suggestion was sent to the numeracy network:

"Periodically let students write in their journals about a particular math process (e.g., how to reduce a fraction or how to solve a proportion). If a student can describe the process clearly and in sequence, then he/she has clearly mastered it."

"Also try letting students be teachers. Assign a specific math process to each student. Have him/her plan and give a 10-minute demonstration lesson in front of the class, including introducing vocabulary, providing 3 computation example and 3 word-problem examples. Once a student teaches something, he/she really knows it forever. Provide a critique sheet for students to use to evaluate the student-teacher, with guidelines such as: clarity of speech, effectiveness of examples, eye contact, use of board, etc. Let only one student give a lesson on any given day so he/she feels special and there is no comparison to another student teacher!"


Writing Directions
Maria Magiassos is a member of the Ohio Literacy Math Planning Committee. She teaches eighth grade math at Warren Western Reserve Junior High School and also does special projects through the Warren City Schools ABLE program. These workplace programs include computer literacy and skill trades math. Her suggestion is "Bringing Writing Skills into the Mathematical Classroom:"

"On a 3 X 5" notecard have students draw (using measurement tools) a figure such as a circle, rectangle, square, etc. Using what they have drawn, have the students write directions on how to draw that figure in the same place on another notecard. Directions must be clear and the measurements accurate. When the student finishes, have them exchange directions. Each student now has a set of directions and must try to draw the figure on a blank notecard. After this attempt, have the students get together to see how close they came and to discuss how the directions could have been more clear."

"This lesson helps students write clearly and also reinforces measurement skills. It uses very few materials and develops communication skills both written and oral."

Editor's note: I like the use of measurement skills in this project. Measurement skills (standard 8) are often ignored because they are so difficult to master.


Writing Word Problems/Group Activity
Dianna B. Kennedy, also of Warren City Schools Special Programs, uses this strategy in her workplace program for autoworkers and their spouses:

"Students work collaboratively to write and solve their own word problems. The lesson provides students with practice solving word problems, gives them the opportunity to write, uses critical thinking skills, and encourages team work!"

"Students are divided into groups and given sale ads from local stores featuring items they might buy. They use the ads to write a specified number of work problems. These problems can be whole number, fraction, or percent problem depending on the group. After the group writes the problems, they must be solved on another sheet of paper (answer key). When all the groups have finished, they trade problems. Each group should get a chance to solve all the problems. Each group then gets the opportunity to explain the solutions for the problems they wrote. Encourage them to put the problems on the board and solve them for the whole class."


Math Humor
Editor's note: Dianna also uses math humor and math cartoons. We can't duplicate syndicated cartoons in this newsletter, but I am sure that many of your friends will look out for appropriate "Far Side," "Snoopy," etc. for you. Listed are a few jokes that may be used in your class:

1.) Introducing the formula for the area of a circle, usually someone will remember it and say "pi R squared." The reply should be, "No, no. Pie are round, cornbread are square."

2.) "Fractions are my friends. Not only are they my friends but they're a cheap date because you can take them out at a fraction of the cost!"

3.) "What's five plus five?" -- "10"
"What's five Q plus five Q" -- "10Q"
"You're welcome!"

4.) "What did decimal say to the other?"
-- "Did you get the point?"

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MATHEMATICAL INVESTIGATIONS
The following problems were used during Massachusetts Math Kick Off Days this fall. These mathematical investigations are ideal for small, cooperative group work. You might want to start or end a class with one of these, or send one home as the "problem of the week."

How Many Twos and Fives?
The number of $2.00 bills I need to pay for my purchase is 9 more than the number of $5.00 bills I need to pay for the same purchase. Use bills (drawings) to figure out the number of each bill type. How much is my purchase?

Rectangles
How many rectangles can you count? Where?

Mental Math
Fold the sheet in half and read the first 8 problems orally to your partner. Have him or her write the answers on a sheet of paper. The have your partner read the second 8 problems to you and write the answers on your paper. A variation might be to have the student write out what he/she hears. It encourages them to pay closer attention to the words.

Problem set #1.

Problem set #2.

Another mental math technique is to give your class "chain problems." These can be tailored to your class's level. To help students who are not auditory learners, I write the problems on the board, but do not allow the students to write anything on their papers. This is a good activity at the start of class, before or after break, at the end, or anytime when there are a few minutes to fill and perhaps not everyone present! They are easy to make up, but a few are listed below:

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BLOCK ACTIVITIES

Problem Solving
Delores Jones, of Southwest ABLE/JOBS in Harrison uses her blocks from the Math Kickoff Day this past summer to work on the following problems. While we can usually solve problems such as these "in our heads," students benefit from actually fitting blocks together.

What is the volume of each figure below? Use your blocks to build and check.

Put two figures together to make figure 5. Which two would fit together? Use your blocks to build and check.

Which two figures fit together to make figure 6?
Which two figures fit together to make figure 7?
Which two figures fit together to make figure 8?
Which two figures fit together to make figure 9?

Extension: How would the volume change in figure 2 if you:
doubled the height?
doubled the base?
doubled the base and height?

Get It Together: Group Activities
Some other block activities come from Get It Together: Math Problems for Groups Grades 4-12, This book, published by Lawrence Hall of Science, is reproducible for home, workshop, and classroom. Copies are available at each ABLE regional resource center. While teachers are encouraged to check this book out and read the reasons and rationales, here are some of the block activities to get you started. Cooperation and communication are the key to these activities.

Every problem in the book has the same struc-ture and the same rules. Students need to learn the rules which may be reviewed frequently and displayed prominently. The clues below can be copied onto cards and the groups are made up of the same number of students as cards. Place the cards into an envelop. You will need one envelope for each group.

RULES: Each group has an envelop to start with. When it's time to begin, open the envelops, find the clue cards, and pass them out to members of the group. When you get your clue, you may only look at your own clue. You may not look at anyone else's. You may share your clue by telling others what's on it, but you may not show it to anyone else. If you have a question, you must check with your group first. If your group agrees that everyone has the same question, you may all raise your hands and t he teacher will come.

These block activities require colored cubes. Every problem can be solved using a subset of eleven cubes: two each of red, blue, yellow, green and orange, and one purple. These activities reinforce the concept areas of geometry and spatial reasoning in three dimensions, logic in a geometric setting, and vocabulary such as cube, face, edge, side, touching, above, below, each and every.

Remember, each number can be copied onto a card. * denotes an optional clue, helpful but not necessary to solve the problem.

Build it #1

Build it #2

Build it #3

Build it #4

Note that each player can touch only one color of block; if you don't use the optional clue, no one can touch red. That's intentional!

Build it Between

There are many other cooperative problems in this book. Other groups of problems use a hundreds chart, number shapes, stick figures, pattern blocks, polygons, city blocks, Venn logic, spinners, mysteries, measurements, constructions, number patterns, Martian for beginners, around the world, and back to nature. It is a wonderful start for meaningful problem solving in cooperative learning situations.

What Shape is My Building?
Another activity that uses exactly twenty four cubes for each building is designing a new apartment building. You want to show your fellow workers your design using wooden cubes. Problem: How many different ways can you design a box-shaped building using exactly 24 cubes?
Hint: You might want to try 16 cubes also.

Make the Rectangle
With the color cubes or tiles, make a rectangle that is 25% yellow and 1/3 green. Things to think about: Are there other rectangles that have the same properties? How many cubes are in these rectangles? Is there a minimum number of cubes you can use? A Maximum?

Perimeter and Area Activity
Make four different shapes that have an area of 4 square units, using either the tops of cubes or tiles. Find the perimeter of each. What is the maximum possible perimeter for a shape with area four? What is the minimum possible perimeter?

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ALGEBRA RESOURCES
During a Ohio Math Literacy Planning Committee Meeting, members brought reviews and recommendations for different math books currently published. The following three suggestions were given for algebra help. I am sure that Patty Bilyeu and her focus group will have many other ideas after exploring algebra for all levels this spring!

Algebra
G&G Publishers
2 American Boulevard
Hopwell Junction, New York 12533
(914) 221-8638 or (914) 298-8879

This recommendation was given by Patty Bilyeu of Midwest Ohio Vocational School District - Adult Center in Belmont, Ohio. This is an inexpensive ($2.95 per copy) book for beginning algebra students. Its approach teaches algebra concepts one at a time. Its drawback is the printing style which could be difficult for some students.

Programmed Math for Adults
by Sullivan Associates
Phoenix Learning Resources / 1988
2349 Chaffee Drive
St. Louis, Missouri 63146-96775
1-800-221-1274

This series is recommended by Sonja Brown of Miami Valley Career Technology Center in Clayton, Ohio. It is especially appropriate for students who are afraid of algebra. She highly recommends Book 12, Understanding Algebra and Book 13, Using Algebra. Sonja follows these books with Steck-Vaughn Working with Numbers-Refresher, pages 178 to the end of the book. For those students in a workplace environment, Sonja also recommends Book 15, Trigonometry. (Don't let the title scare you!) These books have reading levels from grades 2 -4, and they minimize verbal instructions so students with reading and language deficiencies can still learn math skills. The books are around $12.00 each for 1-4 copies and $9.00 each for 5 or more copies.

Breakthrough to Math: Level 3 Algebra
by Ann K.Tussing
New Readers Press
1-800-448-8878

Finally, both Charley Flaig of Northwest ABLE in McDermott and Delores Jones of Southwest ABLE/JOBS in Harrison recommend:

These small books are not intimidating, with good examples and an easy-to-read format. There are only a few problems per page and the reading level is grades 4 - 5. They are recommended for all levels because of the low reading level and will help build confidence with algebra. Other books in this series are Basic Skills With Whole Numbers, Fractions, Decimals, and Percents, and Geometry. Components include student skill books, student workbooks, teacher's guides, mastery checkups, placement inventory, and student profile.

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BOOK REVIEW
Math Curse by Jon Scieszka and Lane Smith (1995 Viking Press). Math phobia is the topic of this "children's" book. The day Ms. Fibonacci, the math teacher, says "You can think of everything as a math problem" is the day that one student becomes obsessed with math. Common activities such as choosing a shirt to wear or eating cornflakes become insurmountable. Creative solutions abound in this tongue-in-cheek, delightful book. This book would be applicable to any adult education class and would certainly start discussions as to what exactly math is and why it bothers us so much. Highly recommended!

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FINAL THOUGHTS
What do we really mean when we talk about standard 1, Problem solving? The following is an excerpt from The Massachusetts Adult Basic Education Math Standards, Vol 1.

Rationale
Problem solving in today's world is not a simple task. We are faced with a fast-paced, complex, multi-dimensional universe where we must continually develop new skills and abilities. The need for new strategies for problem solving is real; and never so real as in the mathematics problem solving adults must be able to accomplish every day. Critical thinking skills must always accompany any form of computation. Real facility with calculators and other technologies is becoming a commonplace expectation. Employers demand that workers be able to quickly apply basic skills knowledge to a variety of ever-changing job situations. Adult basic education learners face these requirements daily. They know through experience that yesterday's mathematical proficiencies do not meet the requirements of today. Problem solving methodologies in the mathematics curriculum must help learners meet the real needs of the real world.

Application
Adult basic education mathematics teaching must involve authentic tasks centered in authentic problems using multiple problem solving strategies. Isolated, non-contextual computation drills are not enough. Teachers must engage learners in true-world problems.

Good problem solving techniques allow the learner to develop essential critical thinking skills, confidence, and important communication abilities.

Mathematics instruction must also provide the adult basic education learner with multiple opportunities for success in problem solving. A lifetime of negative experiences or memories of the educational process has produced in many students a major lack of self-esteem and self-confidence. This lack prevents the individual from attempting important risk-taking tasks involved in thinking and communicating about mathematics.

Instructional strategies must include methodologies which are framed by the broad empirical and cultural knowledge base of each learner. Adult students come to the basic mathematics classroom with a wealth of non-standard problem solving strategies which must be recognized and given credibility. And at all turns, the goal-centered nature of the adult mathematics learner must be acknowledged and accommodated; when this student cannot see his problem solving experience as moving him closer to his objectives, the risk of failure is high.


Summary
In the adult basic education classroom, curriculum design must include approaches which allow the learner to:

  • explore and employ multiple strategies for problem solving;
  • determine, collect, and analyze appropriate data with respect to the original problem or in new problem solving situations;
  • have access to and the ability to use appropriate problem solving tools including the use of calculator, computers, and measurement instruments;
  • generalize problem solving strategies to a wide range of adult-oriented, real-world situations. Note: We would love to include ideas that work in your classroom. Send your ideas to the Ohio Math Literacy Newsletter c/o OLRC (see address below). Don't worry about being the first to submit an idea, or even one that is original. We want ideas that work for you! As an added incentive, we will hold a drawing for a book or math manipulative (of our choice) from all who submit information for the newsletter, whether we use it or not. This month's winner of a classroom set of pattern blocks is: Delores Jones of Harrison. We hope that her students at the Southwest ABLE/ JOBS program appreciate the wonderful teacher they have and that they benefit from the pattern blocks! (Note: There will be more about pattern blocks during our Math Kickoff Days this coming August!)

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    Math Literacy News is a publication of the Ohio Literacy Resource Center and is edited by Nancy Markus.




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