The Adult Basic Education Teacher's Toolkit

Computing Skills Toolbox

You can use the following strategies in your classroom in ways you find useful to your students. Connecting Mathematics to Experience

The holistic view that learners should start with real goals and accomplish real tasks from the beginning, and that the learning specific skills and facts should take place within the context of these tasks, allows learners to engage mathematics in a way that is meaningful to them. Mathematics instruction can begin with the central functions of mathematics--to identify and represent patterns and relationships, to solve problems, and to communicate precisely what computations reveal.

Mathematics can build on what the learner already knows and does, which is very important for you, as the teacher, to know. In mathematics, the basis for learning is the learner's experience with actions experienced in his or her daily life. These actions might include:

• activities in which the learner used combining, sharing, and comparing
• the learner's concepts of time and space.
• the learner's recognition of visual, musical, and movement patterns.

Mathematics is best learned when students come to understand its connection to the human experience. For students to learn they can be successful at math, the classroom culture must support collaborative work, discussion and idea sharing, mutual respect for each learner's approach, and students' sense of ownership of their work--this environment becomes essential for mathematics learning. The teacher's role within this setting is not as the knowledge-giver of the traditional mathematics classroom, but rather as a facilitator of students' learning and a participant in using and discussing mathematics.

Mathematics provides a language for quantifying, measuring, comparing, identifying patterns, reasoning, and communicating precisely. This language, like any other language, can provide a means for understanding, analyzing, and communicating across the curriculum. It is just another code; like English, programming languages, or any language, it can be learned.

From: Kleiman, G. (Oct, 1991). Mathematics across the curriculum. Educational Leadership. 48-51.

Activities:

• Have students measure things and compare these sizes (for example: the length and width of their desktops, the height of their chairs, the width and length of the blackboard, and so on).
• Collect information about the sizes of a variety of items from reference books (encyclopedias, almanac, dictionary). The class can come up with objects to look up; you can get them started if necessary with things like, for example, the size of a football field, the size of an acorn, the size of the planet Mars.
• Frame student drawings--how much material is needed to frame this picture?
• Develop a budget for publishing student writings.
• Design a plan for furnishing a space to be used for child care—measure furniture, room size, windows, and so on; draw these elements on graph paper, determine how furnishings will be placed, figure out the sizes of fabric needed for blinds or curtains, and so on.

Recognition of a working mathematics vocabulary is very important. Students must be able to translate words into their appropriate mathematical symbols (Rothman and Cohen, 1989). It is important that you specifically teach mathematics word lists and vocabulary. The student who can recognize a mathematical operation when described through multiple words or phrases can translate these words into a meaningful number sentence and perform the requisite computation.

The following vocabulary and phrases are listed under the basic operations they represent. Work with your students to learn these term--what symbols they translate to and what operation they tell you to perform.

You might want to introduce additional and different vocabulary words that also give meaningful clues for solving a specific word problem. For example, look at this problem:

"During an energy crisis there was a gasoline shortage. Because of this shortage, the price of a gallon of gas increased by 10 cents. A short time later, when gasoline was again plentiful, the price decreased by 5 cents, but soon inflation caused the price to rise again by 8 cents. If the original price was $1.37 per gallon, what was the final price?"

To understand this word problem, students must know the following terms as they relate to math: shortage, increased, plentiful, decreased, and inflation. They must then be able to paraphrase the language in order to understand the meaning and translate the information into an equation. Have students start a math vocabulary notebook, or add a new section in their reading vocabulary notebook just for math terms.

A Strategy for Analyzing Word Problems

Use the following form to help students analyze word problems. (A larger form you can more easily remove is located in the back of this manual.)

Main Idea (in your own words):

Question:

Pertinent facts:

Relationship sentence (no numbers)

Equation (number sentence)

Estimation (without computing)

Computation

Answer sentence

Example of Analysis Form

Problem: A class of 25 students is going on a field trip. The bus can hold 20 riders. How many students will need to ride in the car with the teacher?

Main Idea (in your own words): Some students are going in a bus and a car on a field trip.

Question: How many students will need to ride in the car with the teacher?

Pertinent facts: 1. There are 25 students in all. 2. The bus can hold 20 riders. 3. At least a bus and one car is needed.

Relationship sentence (no numbers) From the total number of students, subtract the number of students who will ride in the bus. The rest will have to ride in the car.

Equation (number sentence) 25 - 20 = ?

Estimation (without computing) The answer will be about ______

Computation 25 - 20 = 5

Answer sentence At least 5 students will need to ride in the car with the teacher.

Modeling How to Read Word Problems

Word problems are confusing because there is so much that needs to be sorted out. When you read word problems with the expectation that you are going to be able to understand or solve them immediately, you are likely to be disappointed. Word problems are not meant to be read that way (understood immediately).

How to Read Passively

Prepare yourself to read math problems by doing something to relax, such as getting a big drink of water, taking a few deep breaths, stretching. Then, when you sit back down to read, read as passively as you can. To read passively, try the following.

• Read word by word with an eye toward deciding what is or is not important.
• Avoid anticipating what the problem is or how you are going to solve it.
• Don't think about math or about formulas. Just concentrate on one word or phrase at a time.
• Write down what seems relevant and tentatively discard what seems irrelevant (tentatively because you may decide you need it again after thinking through the problem). The problem and method of solution often emerges without your focusing on it. Somehow your brain has an easier time accessing this information if you are not trying so hard to make it.

Look at this problem:

In London, there were three gangs operating on August 11,1891. Holmes knew from some inside information that his equal in crime, the clever Moriarity, led a gang with five members. At the same time the treacherous Smerzi headed a gang with seven members and Gilda Z., the trickiest of all, a gang with eight members.

Watson: Have you figured out whose gang pulled off the Great Train Robbery of October 3, 1891 ?

Holmes: Yes!

Watson: But how, Holmes?

Holmes: Elementary arithmetic, my dear Watson.

Watson: Let me in on how you did it.

Holmes: From certain information from Scotland Yard, it was known that originally none of the gangs was large enough to pull off the Great Train Robbery. They must have added another organization that was twice the size of the original gang. Altogether, twenty-one members were involved in the robbery.

Watson: This is all too much for me. l hate math. I can't do it. I block and get anxious. Just tell me whose gang did it, l can't figure it out.

Holmes: The treacherous Smerzi's gang.

Model a method that students can use to read this word problem by going through these steps.

1. First you read, "In London . . ." Don't read any further, but stop and say to yourself, "I really don't care where they are talking about." If you disagree with this and feel the name of the city may have some importance, then write it down on a piece of note paper. Circling or underling works too, but jotting the "important" things down on a separate piece of paper works much better. Somehow highlighting them brings the clues you think are important out into sharp focus, because it separates them from the rest of the material.

2. Next ". . . there were three gangs . . ." Write this phrase down as "3 gangs." You may be tempted to try to just remember that there were three gangs, but it is better not to rely on your memory. If you do, you will find that you have to keep looking back over the problem to refresh your memory. Doing that can easily make you nervous because you start worrying about forgetting. There is also no reason to "clutter" your mind with things that can be written down. Leave yourself free to concentrate on the material itself.

At this point, your note paper may look like this:

3. Continue reading, ". . . operating on August 11,1981." You might guess that this detail is also not important and decide to ignore it. Or you can disagree and wonder how on earth anyone can know in advance whether the date is important or not. The fact is that all of these decisions have some uncertainty about them. The best guideline is, "When in doubt, write it down." If some of the things you write down later turn out to be unimportant, it will not matter, you can just ignore them, or draw a line through them. On the other hand, if you have left out something that you need later, you can always go back and write it down.

This type of reading is slow, careful, and disciplined. You will find you experience the impulse to get on with it and rush ahead. But, in the long run, reading in this way will become a time saver. It avoids anticipation of what the problem is going to be and requires that you go one step at a time.

You should avoid relating too strongly to the material in the story. For example, you may have decided in advance that since you can never figure out Sherlock Holmes mysteries, you are not going to be able to do this one either. Or it is possible that you went to London once and didn't like it, and you will begin to think about all the things that happened to you there.

It is best to make an effort to approach the story with complete neutrality. Your only goal is to gather facts that may come in handy later. Don't try to figure out why you will need them or how they will fit together.

4. Now we read. "Holmes knew from some inside information . . ." This statement is a warning that something important is coming. But there is nothing to write down. We are just breaking sentences down into fragments or phrases.

5. Continuing, we read ". . . that his equal in crime . . ." This phrase is just a descriptive statement--not relevant to the math. Some people are bothered by the word "equal." Because this is supposed to be a math problem, some students might anticipate that "equal" is going to have some special meaning. It is also possible that some students might start thinking about how Holmes can be equal to Moriarity since Holmes is not a criminal. This wondering about meaning indicates you are getting too involved in the story and are anticipating what is to come. Try to catch yourself doing this. Then remind yourself that the purpose is to gather information, not to figure anything out.

It is important to keep the story itself at some emotional distance so you can focus attention on just writing down what you think is important. Don't think about the story (the characters, the plot) too much.

6. Again reading "the clever Moriarity, led a gang with five members," you might think to yourself, "It probably doesn't matter how clever Moriarity was, but I do want to keep track of how large his gang was." You can note this information by abbreviating and writing M = 5. When you read about Smerzi and Gilda, write down S = 7 and G = 8 to remind you of how large their gangs were.

Now you have pulled out all the information in the first paragraph. The most you would have written would be something like:

If you felt the city and date were not important, you would have even less written on your note paper. The nice part of doing this on a separate piece of paper is that you can now take in everything that was in the first paragraph at a glance, rather than seeing a jumble of words that are hard to take in and work with. It is likely that you will never have to look at the first paragraph again. You have also given yourself something to be satisfied and comfortable with, because you have begun to understand and decipher the story.

7. Now comes the dialogue between Homes and Watson: "Have you figured out whose gang pulled off the Great Train Robbery of October 3,1891 ?" One way of responding to this is to think, "This is just a lot of talk," and continue reading. But if you had decided earlier that the date was important, you would also want to write down this new date.

What we are saying is to do the following when carefully, passively, reading a word problem.

• Organize the mathematical information--take out the numbers and put them down in an orderly fashion, dates with dates, gang numbers with gang numbers.
• If there are facts of questionable importance, write them down too.

In this problem, you will see that the dates turn out to be of no significance. If you had written them down, you would stop noticing them by the time you got to the end of the problem.

There is more dialogue between Holmes and Watson, but it is just talk with no information worth writing down until Holmes says, "From certain information . . ." This phrasing is a sign or warning that there is something important to come in the narration. Slow down and read very carefully and deliberately: " . . . it was known that originally none of the gangs was large enough to pull of the Great Train Robbery." There is still nothing to write down, but it is important to think for a moment or two about what this statement is saying. It may be worth rereading several times. Repeat to yourself, "originally none of the gangs was large enough."

STOP! It is tempting at this point to try to anticipate what is going to be said or asked. Doing that is not helpful. It is still necessary to keep being the passive reader who just takes down information. There should be no thinking about solving a problem until you have carefully read and recorded everything that is pertinent to the problem.

Sometimes people take the numbers from the first paragraph and start manipulating them, trying to get them to arrange themselves in some way so that they come to 21, a number noticed when reading through the entire problem. It is not wrong to try to manipulate the numbers, it is just too early.

Don't rush through the next sentence in the story but carefully read, "They must have added another organization. . ." Think about the meaning of this sentence. Ask yourself who "they" refers to. It is saying that in order to pull off the robbery, one of the original organizations must have added others to it.

8. Reading on, we find, ". . . Another organization that was twice the size of the original gang." Perhaps you don't know what to do with that and want to reread it a few times. In fact, you don't have to worry about what to do with this information. You just need to write down the relevant numbers. These numbers represent twice the size of the originals. We know this from the phrase "twice the size of the original gang." You can figure out what all the doubles are (doubles because twice the size means we have to multiply by 2), and write them down next to or under the originals. Again, you may not know how you are going to use this information, but you want to record everything. Your scrap paper should now have on it:

Then it says, "Altogether, twenty-one members were involved in the robbery," and sitting right there, you see a 7 and a 14 which together give you 21. So you put a circle around S = 7 & 14 and conclude that it must have been Smerzi's gang.

If you understand the problem now, you may think "Why didn't I see it before? It's so obvious!" But math problems always seem obvious after you have solved them. That does not mean that it was easy or that there is anything wrong with you if you did not do it.

From: Overcoming Math Anxiety by Sheila Tobias

Mathematics instruction for the adult learner must begin with two assumptions:

1. Adult learners experience some form of math anxiety.
2. Adult learners believe many of the common myths about math. (See the next section for a list of these common myths.)

Definition: Math anxiety is a task-specific learning disability unrelated to success in other learning activities not involving math.

The National Council of Teachers of Mathematics has stated that even some math teachers experience math anxiety.

Fact: Only 60% of those people currently teaching secondary mathematics majored in math and are certified as math teachers (U. S. Dept. of Education 1987).

Fact: Only 7% of elementary teachers who have math as their primary teaching assignment have an area specialization in math (U. S. Dept. of Education 1987).

Before beginning any mathematics instruction, you, the teacher, must recognize that the adult learner does experience math anxiety. As a result, the mathematics instructor must realize the following.

1. You must provide a variety of learning experiences for students to find ones they are familiar or feel comfortable with.
2. The adult learner must experience frequent success in math activities.
3. Help students find their own personal motivations to succeed in mathematics.
4. Short lessons are usually best.
5. All students need lots of encouragement.
6. Discussing math anxiety is one way to deal with the problem.
7. It is essential to use real-life math problems when learning math vocabulary, concepts, and operations.

Most adult learners will believe some, perhaps all, of these myths. It is important to dispel as many of these myths as possible before beginning math instruction.

Plan to conduct several conversations with the adult learner concerning these myths before beginning any math instruction.

Point out that these are myths and give examples to demonstrate that they are only myths. Learning Theory and Mathematics

The theory of constructivism is based on the following assumptions.

• As individuals, we approach each new task with prior knowledge, assimilate the new information, and construct our own meaning.
• Meaning is constructed by re-wording the new information so that it makes sense to us in terms of our prior knowledge.
• We build personal hierarchies of understanding.
• We impose our own unique sense of structure to all new information.

The concept of schema is characterized by the following.

• Schema refers to each individual's unique mental scaffolding that we use to store previously learned information.
• As new information is encountered, we each begin making mental connections between the new information and any prior knowledge we have stored in our brains that seems to relate to this information.
• Schema theory suggests that the first "connection" we make is the identification of what specific element of prior knowledge is similar to the new information.
• The new information is then integrated with the similar prior knowledge stored in our brains.
• Research findings indicate that the strength of the connections is directly related to long term retention of the new information. In other words, if the connection we make between new information and information we already possess is a strong one, the more likely it is that we will remember the new information for a much longer time.

The concept of active processing is characterized as follows:

• Active processing is an essential element in learning any new information.
• Long term retention of new information is directly related to active processing.
• Active processing occurs as learners work with the new information in various formats.
• Active processing includes reading, writing, seeing, talking, listening, and touching the new information in as many different formats as possible.

Instruction activities to emphasize in mathematics and in which to incorporate constructivism, schema theory, and active processing include the following.

problem solving

concrete materials

instructional variety

oral communication

written exercises

paragraph answers

continual assessment

Instructional activities to de-emphasize:

teaching by telling

rote memorization

one method, one answer

memorizing rules

template exercises

routine worksheets or workbooks

• enter GED class
• help children with homework
• have better job opportunities

Generally, the reasons adult learners seek to improve their math skills are similar to the reasons an adult learner wants to improve reading skills. However, educated adults in the U.S. have a higher tolerance for math illiteracy than an inability to read and write. Some Guiding Principles for Holistic Mathematics Instruction

To be student-centered, mathematics instruction should be characterized by the following principles.

• be based on meaningful experiences
• be language oriented
• provide opportunities for reflection on processes
• provide opportunities for personal error analysis
• present concepts as whole units, not fragmented into skills
• provide drill and practice only as needed, not as the basis of instruction
• focus on solving real problems
• emphasis concrete experiences
• use manipulatives as appropriate

The following list includes many ideas to try with your students to help them personalize mathematical concepts.

1. Personalized problems

7 + 5 = 12

Write a story to match the math problem. Example: I bought seven loaves of bread last week. This week I bought five loaves. Altogether I bought 12 loaves of bread in 2 weeks.

2. Write personalized definitions of math terms, for example:

Line—a straight path that goes on forever.

Cylinder—a tube with circles the ends

Think of some more terms to define in students’ own words.

3. Error analysis

Ask students to re-examine problems they have missed and write a brief explanation of why they missed that problem.

Error analysis can also be used as a diagnostic tool for the teacher.

Example: I got 42 for 8 x 5 not 40.

I added instead of multiplying. I don’t know why I missed this problem.

4. Write or verbally explain the step-by-step procedure for a specific problem.

5. Prepare a list of suggestions to do to avoid common errors.

6. Use manipulatives for fraction concepts. Then have students cut paper "pies&" in various fractional parts.

7. Use manipulatives and compartmentalized boxes to demonstrate base-ten and the process for borrowing and carrying numbers.

8. Ask students to list the ways math is used in everyday life.

9. Ask students to write problems based on real-life mathematics situations.

For example: Which is the better buy? a 32-ounce box of soap at $3.49,
or a 16-ounce box of soap at $1.98?

If you pay for the soap with a five dollar bill, how much change should you get?

10. Use the newspaper to plan a specific shopping trip to the grocery to buy food for the week. Prepare a complete list of items with estimated prices. Include advertised specials whenever possible. (Pass out old magazines and coupon booklets from the Sunday paper if possible. Have students cut out coupons for items they want to put on their shopping lists. Use the value of the coupon to compute your final estimated cost for the grocery list.)

These benefits can result when students use writing skills to help them learn math.

1. The process of writing usually helps the thinking processes slow down, providing students with an opportunity to arrive at their own solutions. The act and process of writing gives students a way to understand their own thinking processes.

2. Students make notes, not take notes, and produce interpretive comments and personal reminders.

3. When students are "stuck" on a problem and write out the thought processes, they see their errors and often solve the problem on their own.

4. Teachers benefit as they receive feedback on lessons and become aware of student responses to presentation techniques.

5. Writing can be a tool for peer learning and collaboration.

6. Students gain the opportunity to formulate, organize and evaluate concepts.

7. Students generate a record of their own thinking.

8. Students have more opportunities to integrate common mathematical terms in their speech and writing so that these terms become less of a foreign language.

9. Writing about mathematical procedures promotes metacognition.

The following six techniques can be used to as writing activities for mathematics.

Error Analysis

• prepare a written analysis of errors
• prepare suggestions for avoiding common errors
• maintain a chart showing frequency of types of errors

• write stories for specific math problems
• revise existing word problems by adding conditions
• write serial word problems
• write personalized definitions of mathematical concepts

• write an explanation of a specific solution
• write directions for solving a particular type of problem
• prepare a solution manual
• prepare a test ticket

• journals or learning logs
• math autobiographies
• two column reaction math notebook (use one column for working problems, the other column for writing reactions to the problems).
• letters to the math teacher

• biographies of famous mathematicians
• reports on mathematical inventions

• microthemes on math in various occupations
• interview adults to learn how they use math
• collect examples of math illiteracy
• collect results of opinion polls printed in the paper
• interpret opinion polls
• analyze sources of error in opinion polls

Appendix A | Contents