Modifications in Math

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Math seems like a paradox. We all need it; we cannot be self sufficient in life without the ability to understand and use money, days of the month, time of day, how much water to boil, how long to let something cook in the microwave. At the same time, many of us have ambivalence - almost a love-hate feeling about math.

Many of us had a devil of a time in some of our math classes!

As we look back, there are so many things that go with teaching math that many of us never use -- don't know when to use -- haven't seen anyone else use. And there are a ton of things we wish we could use, but never get a handle on how or when to do the processes. (Don't let this secret get out, but there are also a lot of math things I finally learned and started to use once I was teaching math to others.)

When we think about helping youth learn math, we can ask a number :~} of questions. The inquiry comes down to: understanding, child by child,

Where does math fit in the student's life for this year and as an ultimate destination.

What the youth can do this year	What the youth has inside him or her as potential math ability
Short term goals: days of the week, coin names, count items with one-to-one correspondence	Long term or overarching goal - functional math skills
NOW	SOME DAY

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What is math and how does it help a student? It is not really multiplying, dividing, adding, subtracting and algebraic equations. It is the study of pattern and the use of patterns to solve problems. It is embedded in life - and in life skills. Independent living is dependent upon having or achieving "math sense." It is so much a part of our lives that we don't think about it until we see someone who is missing it -- can't make change for a dollar, doesn't know what size of container to get for the left overs, can't double a recipe -- or for that matter, doesn't know which place on the cup to stop at for 1/3 of a cup of liquid.

What does math ability look like? Math success comes about when youngsters have the tools to:

form and remember associations	understand basic relationships
make simple generalizations	see and use patterns

Even if some or all of these pieces are present, a student may not be able to connect them together in a consistent way, or on consecutive days. The following example describes a youngster named Emily who has Down's, and tests with an IQ in the moderately delayed ranged. It may illustrate these points about math readiness.

Emily

Emily is now 14 years old. She is still working on coin identification. Last year she got a $20.00 bill in her birthday card. "WOW!" she said, "A dollar." Just to find out how much she understood, I offered to switch her bill for two of my one dollar bills -- and she jumped up and down in excitement at the idea.

Last week she wanted to get a coke at the store. She asked for two moneys. I gave her two nickels, and she told me, "No, Mom, the big nickels (meaning quarters)."

When asked to count sequentially, she is able to get through 11, 12, 13, and then somewhere after that it all breaks down. If asked to add up a series of coins, she will begin by using the quarters - 25, 50, 75 . . . but once the quarters are gone, every coin is counted as if it were worth a penny. She touches the dime and says 76, pulls the nickel toward the pile while saying 77, and then adds the last dime with a resounding, "One dollar". She gets the amount correct this time, but will she next time? Unlikely!!

Emily can tell you her birthday, but not her date of birth. She says "I am 14 and my birthday is May 3." When asked if she was born in 1985, she will say, "Yes, I was." The next year is just as acceptable. "Were you born in 1976," I ask? She replies, "Yes, I am."

Where does Emily live? She can tell you that her address is 180 South Kent. She does not know that addresses are consecutive, so she cannot tell you that the person on one side is 170 or that the person on the other side is 190. She does not know that people on the opposite side of the street will have odd numbers in the address and she does not know that across the main highway the people live on North Kent. At the same time, she does know relationships -- that we are the fourth house on the block, that our house is the biggest, that we have the longest driveway, and she knows when the van is heading toward home and familiar territory, for she will say, "See? Going home now.

Emily has a delicious sense of humor. She is constantly making word associations that are deep, meaningful and funny. They seem to come out of nowhere. Over and over again she will sum up a situation with a humorous comment, and we will all laugh at the relationship she has seen and called to our attention.

At least twice a day Emily takes things out of some of her cupboards, refolds them, organizes them and returns them to her cupboard. She goes through her books and orders them from shortest to tallest and then puts them back on the shelves. She loves to keep them ordered, loves to see them marching from one end of the shelf to the other, in exactly the right sequence. She does not use the Dewey decimal system, for she can't read the books and cannot put them in order by alphabet letter or subject, but order them she does!

In one way or another, she sees relationships, associations, generalizations and builds on patterns, but I'm willing to bet that she will still cheerily trade her twenty dollar bill for a couple of ones next year when she turns 15.

What kind of math for life are we talking about? Where does math fit into Emily's life? . . . or the life of a person such as myself, or you? How much math do we teach for the sake of math? These are important questions, and they are addressed by a 1998 official statement and recommendation of the National Council of Supervisors of Mathematics

What do the experts think math is all about?

Current Math Standards

But what does that mean for Emily and the math part of her instructional day? What does it mean for the students in your class?

Math is an important art of living an independent life. It is important for students to feel successful while taking math classes - and it is important for them to have success in preparing for the myriad of ways that numbers, relationships, using patterns and problems solving. These are critical steps for insuring math success.

1. Assess the student's math skills.

Placement scores from the district or State achievement tests are a good start. Once a general level of numeracy is established, it is important to work, one-on-one with the student. Set up tasks that will allow observation of the way the student approaches a problem, and when possible, have the youth talk about what he or she is thinking as a task is performed. At the fundamental levels, look for classification -- Emily classifies coins by size and cannot overcome the notion that dimes really are worth less than pennies. Ordering is difficult for many youngsters. It involves seeing a pattern and using it consistently, then being able to turn around and apply a different rule and rearrange materials to fit the changed rules. Emily can put clothes together by color, but she cannot make the leap to hot or cold climate apparel. She can match socks, two by two, but cannot then put them in the drawer according to dressy or daily. She can make a row of X blocks or a row of O blocks, but putting them X O X O X O is too hard. One-to-one correspondence is an emerging skill. She can count blocks for about five items, and then she begins to group blocks and say one number, skip a block while counting, or say two numbers before changing to a new block. It comes as no surprise when we check on conservation skills, to find that they are not yet present. Emily can be fooled into drinking a smaller amount of soda by offering her a tall thin glass instead of a low fat glass of liquid. Her sisters do it all the time. Since these are emerging skills, and Emily has not yet mastered them, our conference with her will focus on math readiness. We can insist that she learn addition and subtraction facts, use flash cards or jump rope games to get her to learn the drills, but we cannot hope to build a math castle without a foundation. The most basic skills are still our focus, because that is the point where Emily can have success -- and it is the area where she is motivated to push and press and learn.

Of course, most students will have better skills than Emily.

The next areas for testing include the underpinnings for successfully completing and understanding operations - addition, subtraction, multiplication, division -- and basic axioms -- associative, commutative, distributive properties and inverse operations. These tap into the ability to recognize and use patterns and to generalize the patterns and associations from one set of experiences to another. Remember those basics for success in math?

form and remember associations	understand basic relationships
make simple generalizations	see and use patterns

If the student has all of these pieces in place, it would seem that math success ought to be assured. No so!

Computation adds another dimension. This is where rigor, drill, practice, and order fit into math. Many youngsters have the ability to see relationships and make generalizations, but the way they process information makes math success uncertain. When a student repeatedly gets the same kind of problem wrong, it is often the result of not knowing HOW to do the problem -- not understanding. Listen to a Bob's discussion while he solves the problem. In this problem, 32 + 47, Bob says, "Three and two are five and four and seven are eleven, so the answer is 16."Bob does not see 32 as a distinct number and there is much work ahead, including teaching tens and units.

On the other hand, Mark says, "Thirty-two and forty-seven, hmmm. In the first column, the sum is 12 - put down a two and carry the ten over. . ." Ask further and it becomes clear that Mark reversed the two in his mind and saw it as a five. Mark has the necessary mathematical understanding, but it won't show up until he finds the tools to recognize when reversals are occurring and sets up a system to prevent this from occurring.

Both students get the problem wrong, but the reasons behind the errors are the critical findings. Most important, from an assessment perspective, what we do for Bob will be completely different from what we help Mark learn to do for himself.

Steps in math assessment usually include:

General readiness - those basics - relationships, 1:1, conservation, ordering or sequencing

Understanding of operations - conservation, commutative, associative, distributive

Specific skills - knows math facts, recalls and uses steps in computation, understand the processes

Problem solving - sees the relationships and utilizes them in real life, in word problems, in verbal exchanges

Fluency and accuracy- does the process correctly over and over again, can use speed, almost like solving the problems is second nature, consistently gets the right answers.

	Using Timed Tests
	Tests usually give us information about power or speed. Power tests give students plenty of time to show ability and are looking for understanding. Speed tests assume that the student is already able to do the work, and just check for fluency.
"No - No"	Using timed tests to measure depth of student mastery is great. Using timed test to determine power may "shut down" a student's ability to think and work, and contribute to math anxiety, decreasing student motivation to work and succeed in math.

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2. Honor the developmental nature of learning to understand math. Piaget, Vygotsky, and many modern educators believe that being able to understand and utilize math is developmental. At the same time, there are a number of children who seem to have a knack for "knowing" or seeing patterns, relationships, generalizations in an age defying way, and without formal training.

Who didn't feel a sense of astonishment when the male character in "Rainman" (the movie) knew how many toothpick fell on the floor? Who could help but be astounded when the boy in "Little Man Tate" (the movie) instantly knew answers to questions that were stumping other "genius" youth? Some students seem to have a knack for understanding math, just like some children seem to be instantly successful at reading, while others struggle with each step forward.

Developmentally appropriate practice supports understanding that students need to be ready to take full advantage of material. We can promote that growth by getting students ready in every way possible. We cannot take the cognitive leap for the youth, and if we go on as though the leap has occurred, we may ultimately be wasting time, or worse, may make the tasks seem so impossible that the child's mind shuts down to avoid the anxiety of being forced to see what has not been "viewable" in previous attempts.

What we want to do is prepare the student, so when that moment of readiness appears, we can take full advantage of the energy and excitement that comes along with the cognitive leap. We want to utilize all the other ways to support the breakthrough by providing math worthy experiences that include patterns, opportunities for problem solving, counting, game playing, manipulating, until the "Ah-Ha" arrives.

Math for Emily and students who are developmentally delayed as well as youth who are not yet ready for formal "book" math - playing Bingo, Connect Four, tic tac toe, Monopoly like games with money counting, space counting, Yahtzee with dice counting, Nintendo games that require logic and keeping track of time, space, timing. We may find that dance or dance exercise promotes rhythm, awareness of timing and counting. Opportunities to buy things using money, to count and recount money in the process of waiting for "enough" to purchase something can be very stimulating and facilitates memory for coins and numbers. Remember to use the phone, too. Many students love to communicate with the phone, and the play phone has a great number pad. We can use puzzles, dot to dot, music and singing to enhance her movement toward cognitive readiness to use numbers as separate entities.

Remember some of the following as worthy practice:

a) We all grow - and we continue to grow most of our lives, so if the student isn't ready today, practice patience.

b) Growth has it's own individual calendar and one of the best ways to enhance growth is safety. When we feel safe, we reach out for stimulation. When we feel anxious, we tend to retreat.

c) Play is one of the natural tools for growth. It is appealing and motivating, so it supports many hours of on-task practice and experiences.

d) In that same way, role playing is powerful. Asking a child to "be teacher" is a powerful way to involve the creative energy and focus of the child toward "understanding" so they can relay the ideas to others.

e) The child probably does not really "know" what is best for the self, but attending to the nonverbal cues, natural excitement, and things the child is drawn to can help us make the most supportive plan for growth.

f) Good timing, snacks and exercise can support learning. We utilize them in earlier grades, and often forego them as children get older and have better physical discipline, but they are still very powerful.

g) Feeling powerful and "in control" is part of feeling safe. When students are empowered to learn, to set times and tasks rather than being forced, they learn much faster. When children feel trusted and supported, they can give us their best. When they feel important and valued, respond in kind. If a child is not responding positively, it is crucial to model those responses so the youth can learn.

3. Honor the individual. In the second grade, students seem to have a lot of persistence, motivation, belief that they can accomplish, joy in learning, need to know. It is often necessary for a teacher to bandage up blistered hands, for children who are learning to swing on the monkey bars will go back out and defy the painful blisters in order to master the skills. This type of single-minded focus is the hall mark of developmentally appropriate practices. When a student has that kind of intensity and focus, the right things are being taught. If that is not the response we are getting, we should seriously search for that "magic" place.

It is appalling to think about how difficult it is to learn to walk. Children fall down again and again, legs quivering, bumps from coffee tables and bruises from abrupt loss of balance, not withstanding. How does a newborn get from that helpless state to a point of successfully walking and talking inside 18 months -- without a manual, lesson plans, formal training, grades for effort?

When a child says "NO" find out why.

When a student says "I CAN'T", believe him or her and back up to the place where success can occur..

When a student expresses discouragement, go beyond encouraging and listen to what the child wants to explain.

If a skill eludes a student, try the following:

Link math instruction to the student's current conceptual understanding. Give students problems that pertain to their own lives. Teach word problems as games and have students develop their own rather than solving preset ones. Concentrate on the success and what is going well. Allow students to find personal methods for solving math problems and then allow them to teach the tricks to other students. Encourage students to use manipulatives, calculators, computer games to enhance depth and rate of learning. Find ways to generalize math operations to current, every day use of the skills.

How about drill? Rote memorization can really help, but drill can kill -- kill interest in math and in its place create a sense of boredom, carelessness, or worse, -- rebellion,

Adapting for Auditory Learners

i. Be certain to explain the directions orally before having student begin to work

ii. Provide a verbal summary of each step.

iii. During the activity, use key words to focus the student's attention to upcoming tasks, for example, "Now watch."

iv. Allow students to discuss the work as it progresses; to ask peers questions, to share insights, to provide help to one another verbally. [Sometimes you can tell who these students are by watching who "buzzes" as soon as instructions are given -- since these students often turn to others to get a repetition of directions or to say out lout - self talk - what the instructions seem to be].

Adapting for Visual Learners

i. Use concrete manipulatives or demonstrations and modeling to show students what they will need to do.

ii. Have students practice what they observed.

iii. Use mental pictures to punctuate verbal instructions.

iv. Consider using nonverbal signs to move the pace of the lesson along or as an aide to transitions in the lesson.

v. When possible, make picture cue cards or overheads showing the sequence of tasks. Use pictures and graphs to explain processes.

Adapting for Kinesthetic Learners

i. Expect students to gain insight from using manipulatives.

ii. Discuss (two-way talk) what the student is finding and extend learning by modeling some of the outcomes when a student is stumped.

iii. Pair the student with others who learn through touch and let them work in cooperative groupings, sharing insights.

iv. Use a number of different mediums for learning - rods, blocks, puzzles, glitter, food, so the student generalizes the concepts rather than identifying learning with one setting. (Adapted from Mercer and Mercer, 1988)

4. Believe that all can learn math, and that we can develop strategies to help students succeed. But also, remember that math does not come from a book, that it is not always sequential, that we cannot force learning and love of a subject. We love to do things that we find challenging, that we may be able to get good at doing and that feel hopeful for us. We do not love to fail and we do not love to feel threatened. Sometimes it helps to recall what math is really about, so once more, we define math as being able to:

form and remember associations	understand basic relationships
make simple generalizations	see and use patterns

The longer we push scores on tests, the harder we make it for students to become math literate. The achievement tests are summative - they look to see how we are doing as teachers, as a nation. When math becomes fun, possible, exciting, self stimulating, students will cheer rather than groan when the teacher says "time to do math." If no one else understands it, let it be enough that you do.

Empowering the Student - Collaborative Classrooms - Loving to Learn

Students are collaborators and active participators. Students set goals and plan learning tasks; during learning, they work together to accomplish tasks and monitor their progress; and after learning, they assess their performance and plan for future learning. As mediator, the teacher helps students fulfill their new roles.

Students prepare for learning in many ways. Especially important is goal setting, a critical process that helps guide many other before-, during-, and and after-learning activities. Although teachers still set goals for students, they often provide students with choices. When students collaborate, they should talk about their goals to clarify and solve the problems actively. As students become actively involved they can design learning tasks and self monitor time on task and progress in learning and constructing personal learning. While teachers plan general learning tasks, students assume responsibility for planning their own learning activities. Ideally, these plans come from goals students set for themselves. Thoughtful planning by the teacher ensures that students can work together to attain their own goals and capitalize on their own abilities, knowledge, and strategies within the parameters set by the teacher.

Students are more likely to engage in these tasks with more purpose and interest than in traditional classrooms. Self-regulated learning is important, too. Students learn to take responsibility for monitoring, adjusting, self-questioning, and questioning each other. Such self-regulating activities are critical for students to learn today, and they are much better learned within a group that shares responsibility for learning. Monitoring is checking one's progress toward goals. Adjusting refers to changes students make, based on monitoring, in what they are doing to reach their goals.

Self-assessment is intimately related to ongoing monitoring of one's progress toward achievement of learning goals. In a collaborative classroom, assessment means more than just assigning a grade. It means evaluating whether one has learned what one intended to learn, the effectiveness of learning strategies, the quality of products and decisions about which products reflect one's best work, the usefulness of the materials used in a task, and whether future learning is needed and how that learning might be realized. Collaborative classrooms are natural places in which to learn self-assessment. And because decisions about materials and group performance are shared, students feel more free to express doubts, feelings of success, remaining questions, and uncertainties than when they are evaluated only by a teacher. Furthermore, the sense of cooperation (as opposed to competition) that is fostered in collaborative work makes assessment less threatening than in a more traditional assessment situation. Ideally, students learn to evaluate their own learning from their experiences with group evaluation. Adapted from the writing of M.B. Tinzmann, B.F. Jones, T.F. Fennimore, J. Bakker, C. Fine, and J. Pierce NCREL, Oak Brook, 1990 - What is the collaborative classroom?

Conferencing with the Student

1) Begin the conference by establishing rapport and giving the student time to talk about math and attitudes or feelings about personal level of competence. In that period of time it is inappropriate to correct shared feelings -- [think of it like telling a person he or she is wrong about a choice of a favorite color or food].

2) Give the student a forum for sharing current successes - "strut their stuff" time.

3) Ask the student to provide the next challenge - and if the student is uncertain about the next step, offer a problem that dovetails with the current success and a challenging new skill.

4) Allow the student to attempt to solve the new challenge, using personal skills, and make note of the strategies used and the "talking" or verbalization of the issues involved in solving the challenge.

5) Jointly make a goal that moves the student into the new challenge, and offer guidance on strategies to be employed in learning the new skills.

6) Jointly develop an assessment plan.

As possible, allow students to work together in pairs to conference and build new goals, and eventually, provide a weekly time for these strategy sessions to take place. Once a week, briefly review student progress [can be done in about 60 seconds while students are setting goals or working independently.]

Ideas about mathematical and educational restructuring and white papers regarding research efforts in restructuring, as well as a wonderful article about collaborative classrooms student as self responsible learner.

Books with excellent ideas:

Kenschaft, P. C. (1997). Math Power: How to help your child love math, even if you don't. Reading, MA: Addison-Wesley.

Mercer, C. D. and Mercer, A. R. (1998). Teaching students with Learning Problems (5th ed.). Upper Saddle River, NJ: Merrill.

Assignment Options

Remember: You are building this course to suit your needs. This is a cafeteria style presentation of assignments. Please do those that will strengthen your skills and enhance your ability to teach and provide services. Keep track of your points.

1. Write a one minute essay, reviewing the material presented in this on-line reading [25 points]. Feel free to discuss these ideas in WebCT or discuss them with a learning buddy in your local area [25 points].

2. Describe your math experiences and ways you utilize those experiences to teach others. [25 points].

3. Develop a set of interventions for enhancing math instruction. Feel free to use the following chart to organize your response. [50 points for each set developed]

Concept:________________________________________________

4. Conduct an informal math assessment with one student and report findings. [50 points]

5. Conduct a Student Conference and help the student develop an individualized plan for moving forward with math skills. [50 points].

6. Visit web sites on math standards and review guidelines for teaching math to students. In your search, review at least four sources on helping students learn to love math. Provide a summary of your findings. [75 points].

7. Review the math tests used by your district to assess student math success. Try to gain access to the formal assessments used as well as any district criterion referenced tests. If none are available, try accessing those used by your State Department of Education. Identify a personal favorite and also critique any that are inappropriate for use. [100 points]. Sites to begin the search for math assessments: Harvard group , math practice test, web math resource list.

8. Find and adapt or personally develop a list of functional math skills that promote independence. Then using that list, develop a series of five or more strategies for teaching those skills. If finding a set of skills seems difficult, consider looking at the categories on tests like the Vineland or Adaptive Behavior Scale. Broad categories might include Consumer, Homemaking, Recreation, Employment, Travel. [25 points for each strategy]

The following format is only a suggestion, to get you started. Feel free to use it or develop your own.

But remember... There are no ORANGE KANGAROOS in DENMARK.

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E-mail J'Anne Ellsworth at Janne.Ellsworth@nau.edu

Course developed by J'Anne Ellsworth

Copyright 1998 Northern Arizona University
ALL RIGHTS RESERVED