The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
parent or family member?
worker?
community member?
parent or family member?
worker?
community member?
I remember teachers asking me if a drop of water is falling, at what point will it pick up the highest speed, the beginning or the end? . . . Like who cares?! This is the stuff they put in there just to mess you up.
. . . they just started this when I was in 8th grade, but we had a business class where you'd go in there, and they'd give you a stock book and, you know, they'd give you so much money and you got to invest in these companies and the guy that came in there actually used the real paper and he'd tell you how much money you made, how much money you lost. You were your own broker and that was real neat. You know you invested, you put so much money in like Nike, for instance, and then McDonalds. You know, you go across the board and he tells you what you made and what you want to invest it in and what you lost. Whoever made the most money, they get something like a free pizza, whatever. And it was using real numbers . . .
My dad owned a bakery for twenty years about the late 70s, early 80s. Starting in the mid 70s he started saying that his employees, the young kids that he was hiring as helpers, baker apprentices -- wasn't much of an apprenticeship program. He said, 'They can't think anymore. Nobody knows what to do when something goes wrong. They just do whatever and go, 'I just followed the instructions.' What he was saying was they needed to predict. If it was really humid, they needed to know that the bread needed to spend less time in the steam box and they needed to know they needed to change ingredients by adding things just slightly. My dad did not know how to articulate it, but I was among those who could think. What I learned was that certain adjustments, certain ingredients needed to be added, not all of them. My dad didn't know which ones to tell somebody, but he could tell that someone was not taking all the raw data in and making a judgment on all the raw data: increasing time, predicting what was going to happen because the temperature was 75 degrees and it was 80% humidity and the bread was going to have to spend five extra minutes in the steam box. You have to change ingredients, less salt. He couldn't articulate it, but nobody who was working for him during that time period could interpret those changes critically. You talk about critical thinking. But it is basically day to day understanding of adjustments that is as important as knowing how and when to use it.
Math for me is the same as for W. I grew up with math and numbers. Raised in the South, being poor, we had a love for money, a need for money, so it was kind of natural. When I became older, math became like a second language. I relate to it like a second language. It's automatic. I've never had difficulty in math or anything pertaining to numbers. I've always loved it. I've always found great success and accomplishments dealing with math and numbers.
"Workers can't be afraid of numbers!"
GM Parts Plant Manager
Operation sense or understanding how the four operations (addition, subtraction, multiplication and division) work means "recognizing conditions in real-world situations that indicate that the operation would be useful in those situations." (NCTM, 1989)
"Multiplicative reasoning is basic as it leads to the understanding of multiplication, division, and proportional reasoning. The notion of unitizing or forming units of units where a person begins to group objects together and consider them as sets or wholes, e.g., five candies per bag, six bags, gives 30 candies. The five candies are considered a unit." (teacher)
Proportional reasoning was mentioned as critical to "people's ability to understand and communicate about (rather than compute) what an average or percent is... Anything that is decreasing/increasing or changing magnitude relies heavily on deep understanding of proportions, rates, ratios, relations and relative comparisons." (stakeholder)
An automotive parts plant manager from New England ticked off aspects of number proficiency needed in his workplace. The workers are usually on a forklift making some quick calculations such as knowing how many boxes of filters to load if a dealer orders 100 and there are a dozen per box. They also need to be able to read and retain an identification (SKU) number ... to be able to break up a number, repeat it, recognize it and locate it. Finally, he said how important it is for workers to be able to use logic in solving problems in the workplace.
A banker noted the importance of calculators in banking, but was concerned that employees were now doing things that people used to do in their heads, (like adding and doubling), but not using the calculators for more complicated problems. He would like to see more of that.
A restaurant owner, stressing the need for mental math ability, says she noticed over the past ten years, employees' skills have gotten worse... I've had to change the equipment because people didn't know how to do math. I have to put calculators around the restaurant and change registers. If some item costs $5.25, and you give them a ten and then you say you have a quarter, they're lost and have to start over!"
A learner states, "Everyday at work I use math. I'm a cashier and gas station attendant without a cash register. Therefore I have to figure out change on my own and if people get the wrong change back, they become highly upset and critical. In order to make change at work I usually use (mental) addition, subtraction, and multiplication."
In the workplace the common thing would be -- we measure everything. If someone is measuring something and they are taking samples and they see that is it going out of the acceptable margin. I mean, they need to be able to stop everything and get someone to find out what is happening. Upper and lower limits, check that against a blueprint and see that the upper and lower limits have been put in properly, then compare that with how it was running earlier that day. Someone has got to understand; it could be a supervisory level. Someone's got to be putting in the data properly. It is computerized, of course, but it is only as good as the information that someone is putting in. We try to tell people, numbers have got to make sense. They have got to make sense. You have to be able to see if for twenty-five days it has been operating like this and suddenly it looks like this. What does your gut tell you? There is something wrong here. Something is going out, that kind of thing. Look at the chart. People should have enough understanding and knowledge to read all the results of all our quality measure that we post on a regular basis. And if they see something that they don't understand, they need to question that. Say, 'what is that?' How does that impact that? Where does that come from? Everyone of those things impacts something else. In a workplace setting, there is this domino effect. Everything builds on the next thing. From the time that raw material comes in the door and one process after another through departments takes place. You are adding value, but you are also adding costs. We need to catch those numbers if they start to go out of alignment. Again going back to -- people need to see the big picture first. The numbers stand out, how things operate. So they see the big math picture. You get to the big math picture through all these tiny calculations.>blockquote>
Overview
Adults make decisions based on data in their daily lives and in the workplace. According to Equipped for the Future, "Adults are also interested in learning and strengthening the skills associated with using information to have an impact on the world. They identify the need to develop the problem solving and critical thinking skills that have to do with analyzing and reflecting on information in order to make good decisions. . ." (p. 24) Reading charts and graphs, interpreting the data, and making decisions based on the information are key skills to being a successful worker and an informed citizen. Being an informed citizen includes understanding statistics and probability as well. Adults41
cannot make reasonable decisions unless they understand from where the statistics come.
Charts and graphs are essential in the workplace. According to SCANS documentation, tomorrow's workers must have reading skills that enable employees "to read well enough to understand and interpret diagrams, directories, correspondence, manuals, records, charts, graphs, tables, and specifications. Without the ability to read a diverse set of materials, workers cannot locate the descriptive and quantitative information needed to make decisions or to recommend courses of action." (p. xvi)
Data from charts and graphs are used to make decisions. Graphs are useful tools in that they organize data so that information becomes clearer. This organized information can then be used to draw conclusions, to make decisions, or to influence others. Data is organized in a variety of fashions, from charts and graphs, to computer-generated spreadsheets.
In comparing the comments made by stakeholders --often individuals in managerial positions -- with those made by adult learners, several interesting distinctions were noted. Stakeholders tended to use the data in charts and graphs to make inferences and decisions. Adult learners, on the other hand, were more inclined to use charts and graphs in a more literal way - simply to gather information. Adult learners who created charts and graphs used them to help themselves while stakeholders (management) used charts and graphs to influence others. In addition, adult learners claimed either to not have a use for charts and graphs or felt they used them when they needed information. Stakeholders shared a concern for the lack of "chart literacy".
Key Findings
Data collection, analysis, and graphing are essential in the workplace. SCANS proposes five competencies needed by employees for success in the workplace. The competency Information clearly suggests that data analysis and graphing are necessary skills for tomorrow's employees. An example of the level of proficiency for the competency Information includes the ability to ". . . analyze statistical control charts to monitor error rate. Develop, with other team members, a way to bring performance in production line up to that of best practice in competing plants." (p. xx)
Many industries, manufacturing in particular, now use statistical control processes (SPC) to monitor their processes in order to ensure quality products. Often the front-line employee is required to collect the data used for charting the manufacturing process; therefore, employees at all levels should be knowledgeable about and comfortable with using a variety of charts. As more and more quality teams -- consisting of a variety of employees -- are charged with the task of ensuring quality products, employees will need to have an understanding of probability and sampling. During a focus42
group session, when asked how math is used in the workplace, an adult learner responded, "Sometimes I have SPC graphs. It kind of determines if something is wrong with the machine and pretty simple things, nothing really major. One out of 6 is bad, shut the machine down. A lot of times we don't count the parts. They'll just be in a can and we write down on an overlay. You know, if you red tag some parts that are no good, you tag -- nothing really major."
It is interesting to note that only stakeholders seem to be acutely aware of the need to have the ability to read and interpret statistical process control charts. "There are two other areas and we mentioned this before about statistical process control. Our industry is moving into really using numbers to determine whether the production process is functioning or not. And they are using the concept of time as well as numbers. How fast it takes to do a particular process. Budgetary hourly rates is another phrase that is kicked around. . . " Yet another stakeholder commented, "In the workplace today, employers want everyone to understand quality. Any chart or graph that shows production uses statistics."
Other forms of charting are also used in the workplace to make decisions as well as gauge accuracy. One learner shared how he uses blueprints to determine whether a part is within tolerance: "Basically at work every day, you know, just looking at parts, I use a blueprint. That gives you a tolerance, a couple thousandths here, couple thousandths there, sometimes 5, -2, and if you've got a part that's right on the borderline of tolerance you want to decide if you want to just keep on running it or fix it or basically see what kind of problem it is going to cause." And another adult learner has an idea of what will be required of her in the workplace: For myself, I m hoping to get into social work. So I need to be able to read and understand graphs (for statistics). That way, I ll be able to compare past trends with current trends and hopefully predict results. Also making my own graphs. At least I think that s what statistics will be like.
Statistical knowledge is important in problem-solving and decision-making. Adults, often without even realizing it, make decisions based on statistical information. It may be via the television, radio, or it may be through print materials. The following adult learner made her decision based on what she had seen in a magazine. "That reminds me of a fee that I thought was too high. There was a newsletter I wanted to subscribe to, but it charged $35 a year, and I couldn't understand why a little paper would cost so much. But then on the inside it showed a circle graph, with sections like a pie, and it showed what the money was spent for. Then I could see that it really was a reasonable amount to pay." It is important that adults know that they are using statistical information in their reasoning. A stakeholder weighed statistical information to decide whether it was relevant to him: "I was reading an article about what to look for if a kid is using drugs. And I thought about my 14-year old grandson and how different it is now that he is a teenager. I had to take my random knowledge of a person and decide whether it was statistically relevant to drug use." Another stakeholder agreed that sta43
tistical information does influence decision-making. "You're seeing different information that you need to reason and draw conclusions based on this. Is this a good sale, not a good sale, and so forth depends on the whole reasoning process -- looking at graphs and charts -- looking at your paycheck and whatever -- just being presented with information and attempting to draw conclusions."
Statistical information is used to communicate information and sometimes influence others. Understanding the flood of statistical information allows adults to make more informed decisions. A teacher said it very well, "[When] we understand math, we can use it to take control of our lives. Do our own figures so as not to be the victim of scams." A stakeholder explained how she thought a nice graph could be an influencing agent: ". . . Not only did he do nice histograms and circle graphs but inside he did it by ethnic, by cities that the students are from, and so forth, and then he has a final chart on the money that he's asking on the back. I think it is a nice piece of work. On the back is a financial chart which could hopefully affect the budget."
Graphs, tables, and statistics make data easier to understand. Adults create graphs for clarity and understanding, for themselves as well as for others. Sometimes seeing the data in chart form makes the decision making process easier since the information is clearer. The following adult learner provides an example of how charts helped them see the issues more clearly. "When I bought my car, I put a $1000 down payment on it. I owed them like $8000 on it for five years which I was gonna end up paying a lot of interest . . . so I made a plan to pay as soon as possible so I save the interest. So I figured out how much I could spare each month. I did a budget so I would send them like 2 payments, 3 payments a month. I did a graph to see how far it would go. So like in a year and two months I called the company and I asked them how much money I save from all the interest from me paying early and I end up with almost $2000 and I ask them if I can take this $2000 I saved and add it to what I pay and they said I could."
Even when the charts and graphs are not initiated by adults, they do tend to make the information easier to digest. From an adult learner, "I watch this thing on PBS. It's like this physics show -- and it's actually enjoyable. They give images and graphs; they give you things about different theories with physics. It is hard to understand, but the way they put it on TV, it's very simplified. You can actually see it with your eyes." There are those, however, who don't agree that charts make understanding information easier. A stakeholder commented, "Right now all the changes that are going on. There's ATM. You can't go to a bank anymore necessarily and talk to a person. Statements are getting more, well, they are supposed to be easier to read . . . things are constantly changing on them, you know."
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Charts and graphs are also used for record keeping such as spreadsheets and data bases. According to the SCANS report (p. xviii), employees of the near future will need to be able to use spreadsheet programs for tasks such as monitoring expenditures
There is a concern for the lack of understanding and ability to read and interpret statistical information, including charts and graphs. There is also worry about the use and misuse of statistical information. While adult learners did not have this concern, stakeholders and instructors agree that adults do tend to have difficulty deciphering what the numbers and charts mean. Stakeholders shared their concerns: "I think transferability is really hard for adults. To know a concept is one thing, but to be able to look at a table and say, 'I understand this table, or I can read this table, or I can interpret what this means' is hard to do." "I tried something in the workforce here about a month ago based on quality where we measure quality based on the number of errors per 1000 lines shipped. We benchmark ourselves against Toyota, Ford, Chrysler, and all the other automotive people. I had a lot of blank stares going back from the audience. They could not associate what I was trying to talk about . . . statistics to the competition. They simply had a block saying, 'But we're different.' Math is math and I was having a hard time with that comparison. They were not able to connect. I kind of lost them on that one. So we are going to try again going back to more graphs I guess is probably the best way to try to communicate this. You know, 'A picture is worth a thousand words.' I'll try that out next time."
Yet another stakeholder believed that understanding statistical data involved much more than just looking at numbers in a literal sense. "I'd like to also suggest that, underlying understanding of statistics and some key aspects of both measurement and number sense lies the fuzzy yet critical domain of proportional reasoning. Consider, for example, that people's ability to understand and communicate about (rather than compute) what an average or percents, understand notions of sampling and representation, make sense of and make choices about probability and risk (e.g., likelihood of accidents, errors out of XXX products), and anything that is decreasing/ increasing/changes its magnitude, relies heavily (though of course not exclusively) on deep understanding of proportions, rates, ratios, relations, and relative comparisons, which are all parts of the same conceptual system that mathematicians (and I'm not one of them) call Rational number concepts. "
Adults use charts, graphs, and statistical information in their roles as workers, parents, and citizens. As workers, adults used data to monitor the quality of the products being made. They also make decisions based on the data. A stakeholder uses blueprints and statistical process control charts: "People that I work with have to know at least the basic math skills in order to perform the SPC and we do a lot of blueprint reading -- a lot of math involved in reading a blueprint." A learner explained how he uses a variety of skills to do his job. "My daily job was office manager. My responsibilities were to do a daily cash reconciliation, post accounts receivable, accounts payable,45
keep up with hourly employee time cards and keep track of everyone's vacation and sick time. I always made decisions using amounts, money, graphs, and basic addition and subtraction skills. I used all these skills on a daily basis to reconcile and solve any problems regarding my specific job requirements."
As citizens, adults need to understand the data that they are continually being bombarded with -- through all forms of media. This stakeholder clarified the importance of understanding data as it relates to elections. "I want to switch from workplace to community and society and all that data that we get inundated with -- try to make, you know, what's going on in the world -- what does it mean to win a primary and say that is 14 electoral votes and, all of a sudden, you're supposed to be the front runner and how do you gauge the real significance of that. Then the next week you're blown out of the water supposedly because something else happens. I remember the election last year when the polls and the data, they became the driving force themselves as you watched one go up and one go down. How do people really assess that because it is such a big part -- you're talking about one arena, but locally, we're having a school tax referendum. People are being surrounded by numbers in which they've got to make decisions. This was cut in half. Well, what does it mean that this was cut in half? Half of what? Is it really that big of a difference . . or whatever. So I think something in adult ed. I think we're tending to look at the work stuff a lot and a lot of the sort of consumer needs, but I think some of the data in terms of broader societal issues. We're tying to get people more engaged." And another stakeholder, when asked what were the three most important math concepts that should be taught, included statistics: " . . . As members of a community we use statistics to understand our community better and to help create a better environment for ourselves."
Implications for Teaching and Learning
Introduce more work-related charts and graphs and other statistical information to better prepare adult learners for the world of work. According to the Massachusetts ABE Math Standards [pg. 50], to become successful employees, adult learners need to have the opportunity to "systematically collect, organize and describe data; and construct, read and interpret tables, charts and graphs". Adult learners need much more than simple activities where they are asked to find literal bits of information in charts and graphs. They need opportunities to collect their own data, then create their own charts and graphs. In designing their own charts, adult learners begin to understand how data can be represented. Employees at all levels are being required to read and interpret charts and graphs, so adult learners need to be prepared. As one stakeholder put it, ". . . Being able to be chart literate and being able to read those charts and graphs that we produce and we put up in our plant everywhere; all our quality charts -- the lowest level, entry-level employee should be able to read those."
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Provide hands-on experience collecting, organizing, and interpreting data. It is not enough that adult education classes give learners practice in simply reading and finding literal information based on charts and graphs. Providing adult learners with the actual experience of gathering data, deciding on how to represent the data, and interpreting the results will give them a deeper understanding of statistical information. According to The Massachusetts ABE Math Standards (p. 50), adult learners should be able to "make inferences and convincing arguments that are based on data analysis; and evaluate arguments that are based on data analysis." Adult learners need opportunities to interpret charts and graphs and discuss their findings and implications with others. A stakeholder added, ". . .Need basic level of mathematics to survive, for public discourse -- the use, abuse, and misuse of statistics today -- how and why -- more observation -- reading charts and polls. Why they're done and how they're used."
Connecting to the Four Purposes
Stakeholders interviewed for this project were concerned about many adults' inability to read and interpret statistical information. This suggests that many adults, at least when it comes to statistical information including charts and graphs, need to become more literate for access and information. The National Adult Literacy Survey includes the literacy tasks of reading and interpreting statistical information under the heading of quantitative literacy. In fact, while most adult learners viewed charts and graphs as a medium for accessing information, there were a few exceptions as illustrated by this interaction between three adult learners and the focus group facilitator. BB [facilitator]: "How about if you're reading the paper and you see a graph comparing the number of high school dropouts in 1965 and 1995. Can you read and understand information presented that way?" M [first learner]: "You need to know the number of students in '65 compared to the number of students in '95." C [second learner]: I can't read graphs, no." BB: "Would it be important to you to be able to?" C: "No." M: "Yes, it would be important. S [third learner]: "It is if you're doing a test." BB: "Any other reason?" S: "No, not really." C: "Yes, actually it would be important to know." S: "My brother-in-law uses his computer to graph his income, you know?" M: "The light bill is a graph."
The creation of charts and graphs based on data collection is one method of giving voice to the data. Literacy as voice requires that adults be able to communicate to others; charts, graphs, statistics are each a means of communicating what the data is suggesting.
As seen in earlier examples, adults use statistical information to guide their decision-making. They often create charts and graphs to clarify the problem, then make decisions based on the interpretation they give. The following adult learner provides an example of how adults use charts and graphs to take action and make decisions: "We had to reduce our hours at work. We made a big chart on the chalkboard. We compared four-hour shifts, eight-and ten-hour ..."
47 GEOMETRY: SPATIAL SENSE AND MEASUREMENT
I need math when I redid my house, measuring dry wall. That was a problem but we did it. How much did I need? How much to go buy? ... I cut dry wall to remodel my kitchen. When we put it up it didn't fit. It's uneven but most people can't tell.
Overview
As told by the learner quoted above and noted in The Massachusetts ABE Math Standards, "adult learners who attend basic mathematics classes at any level share a wealth of pragmatic experience surrounding geometric and spatial concepts. They've probably built a bookcase, laid out a garden, applied wallpaper or tiled a floor, all the while discovering informally the rules which formally govern the study of geometry itself. For many adult students, geometry is one math topic that immediately makes sense to them and gives them confidence in their ability to learn." (p. 51) It is also true, however, that many adults associate geometry, like algebra, with failure. "In seventh grade I started to have trouble with geometry. I still have trouble with the GED geometry. I don't know why we have to learn it. It's so confusing." And "the hardest part for me is geometry."
Measurement, a foundation skill for geometry, is also an essential life skill, one that adults use in many different but familiar contexts: "on-the-job, for home improvement projects, in the daily task of food preparation." ( Massachusetts ABE Math Standards, p. 53) Or as one learner states, "Measuring. You can put it under workplace, family. You're always measuring something. You can be at home or stuff where you're measuring out ingredients, whatever, like cooking.
Key Findings
Measurement is not an end in itself. It is a tool used in many contexts: home, work and community. We measure many different attributes of physical objects and time in many different ways in many different situations and contexts. As learners state, Measuring, well, cough medicine, anything like that. Temperature, yeah. You re not using it, but it s on the thermometer, so that s a form of math. 98.6 is normal, right? So that s math. And When I worked in a factory, we made fan belts ... We had to measure them if they re too long or too short. We had to use a cutting machine to adjust, if too long or too short. What I had to do if they were too long, I had to cut em or sew em together ... measured by two sticks to check if they were right. Measured in meters I think it was. And During the windstorm our fence got blown down. We had to go back out, measure everything, and, you know, put it up. How far apart everything would be and then figure out how much fencing we needed.
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Measurement is essential to our sense of ourselves and our orientation to the world. For example, as one teacher states, I work with 80% welfare mothers, 20% low income ... They work with people in nursing homes ... Some don t have that concept, measurement. They have to do measurement. How tall someone is. How much they weigh. 5'10" can be like 4'10" to them. Kind of things we take for granted about how we see our world. We try to bring them to something that s real in their lives so that they will tie into it and try to generate some trust in that.
Because measurement is used so often and in so many contexts, many learners have great confidence in their measurement skills. For example, We sell fry food and chicken and fish. The fish, we sell a lot of fish, fried fish. For example, the fish is $3.99 a pound. And the people say, I want $15. I don t have to go to machine and check how many pounds is $15. We think and we fry a lot of fish and we separate and exactly for the customer $15. I do that every day. And, I am night manager in a restaurant, so I have to order every night. I have to do a balance for everything. Everything by the pounds, like sugar. See how many pounds I have and how many pounds I need to order.
For ESOL learners, teaching measurement is very important as a cross-cultural component of mathematics and second language learning, since many of these learners have used the metric measurement system much more than the U.S. system. For example, as one learner states, In addition, it s very difficult for us to use the American system. I think about mathematic, and American system about labor, pounds, miles, yards, because in European you have metric system. I heard United States try to change to this system, 2000. You change now?
Learners and stakeholders recognize that measurement skills can be critically important. As one learner states, I worked for Nabisco as a mixer. You had to know the correct scale and formulas. I kept messing up. I lost my job. It doesn t look too good on the record. If you don t know math, you can t succeed. Or, I just remember, in my job, I use in my country ... nurse. When the doctor tell me you have to give 25 milligrams, I have to. This is very important with medicine. You have insulin, you have to know. And, carpentry work. Making sure the measurements are correct so that you don t waste too much and so that you don t have to do twice the work, by having to redo the work over. And as one employer states, In the workplace, the common thing would be--we measure everything. If someone is measuring something and they are taking samples and then see that it is going out of the acceptable margin. I mean, they need to be able to stop everything and get someone to find out what is happening.
Time management is another critical measurement skill. As one employer states, Time management, that is math. People have no sense. It is a work ethic, or break down of. We used to have a schedule ... Some people don t get what being to work on time means ... being late and putting stress on all that must be completed by the time49
the door opens ... forgetting how long it would require to do all the food prep. Or as one learner states, In my work I must count the time I work. The time I begin work and the time I stop working. I write it or not. Good to check. That week if the boss count it exactly, I can check it. And as another learner says, You need to figure out how much time to plan so you can get to places on time. You need to figure out when the bus comes, how long it takes to walk to the doctor s appointment, and everything. And from another, Just time basically, time in a day.
Some adult learners identify geometry (along with algebra) with failure. " The hardest part for me is geometry. Learning geometry was really hard."
Other learners recognize their excellent everyday skills in geometry, although they may or may not use the term geometry in relation to these skills."Three or four years ago, I took up quilting and realized how much geometry you need when you go to modify patterns or create patterns. Estimating only goes so far when you re dealing with little, tiny pieces. When they re fairly big, you can estimate and you re okay. I did addition, subtraction, estimating, did a lot of work with angles. Who knows what else. I don t even remember my geometry well enough to remember the terms but some of it took going back to books and some of it was pretty straightforward." And," I use math to figure how to shoot pool shots. I know where you need to hit the ball so it will go in the pocket. "
Some adult learners don't see geometry as useful. For example, as one learner states," Like geometry is so esoteric, with the angles and things. You think to yourself, what am I gonna use it for. You think, what's the point? ... This stuff they put in there just to mess you up."
However, geometry is and can be related to all aspects of life: home, school, work and community. Geometry and spatial sense can be used to describe the physical world. For, as another learner sees, "Geometry is everywhere. Not everything is square." And, "We have four horses. Once we had to figure how much hay we d need for a year. Then we had to figure if the hay would fit in the barn. So one horse eats about a bale a week, so multiply by four and so on. Then we had to know how big the bale was. And how big the loft was." Or as another learner states, "I remember six months ago we had moved from a house in Annandale to Alexandria and the house we had moved in, it was no carpet on the floor. So we had to use our old carpet. It was good, not that bad. We had that carpet. The house we used to be before, it was bigger. This one was smaller. The carpet was bigger than we need ... We had to measure the room and the hallway exactly what it was. We had to cut the pieces of the carpet that we had to fit exactly the room. So we had to make a map of the paper and how we gonna cut it. How many feet. How many centimeters. Exactly how it s gonna be and it was good using the math. We did some mistake. We had some left over. The next time, maybe we do fine."50
Implications for Teaching and Learning
Use exact and estimated measurements to describe and compare phenomena to increase the understanding of the structure, concepts and process of measurement. " Despite the fact that competency in measurement is vital, some adult basic education learners have difficulty selecting and determining appropriate units of measure as we ll as using the appropriate tools of measurement ... Teachers should use concrete activities (with non-standard and standard units) to help ABE learners develop an understanding of the many measurable attributes of physical objects (length, time, temperature, capacity, weight, mass, area, volume, and angle). This is the natural way of building a vocabulary for measurement, and for comprehension of what it means to measure. " (Massachusetts ABE Math Standards, p. 53)
Address the impact of measurement skills on self-efficacy and self-reliance. As one learner points out, "Recently, I transformed my storage room into a walk-in closet. I needed to decide how large I wanted the clothing racks to be which depended on the size of the closet, the amount of space I needed left over for walking space, a bureau, and other items that are stored inside. Basically, math is everywhere, and to be independent and survive on a limited budget you need to be able to do things yourself and find the best values along the way."
Extend measurement skills to concept areas such as volume, proportion, and problem solving. As one learner points out, "I bought a couch last fall and miscalculated the footage on it. Then I had to rearrange my whole living room. It needed to go where the entertainment center was, but it was also big. It really got complicated, and I really was surprised at how challenging it was to get everything to fit." Or, after Oregon flooded, "We had a situation where we had to move some stuff out because of the flooding. We had to rent a truck. We needed to figure out how big a truck to rent. Was it really going to be more cost effective to rent one big truck or two little trucks? We had to figure out how may boxes we would need, and how many boxes would fit in a trailer."
Increase the awareness of acceptable tolerances (margins and upper and lower limits) and the consequences of being within and outside of these tolerances. To return to the workplace, where "we measure everything ... They are taking samples and then see that it is going out of the acceptable margin. I mean, they need to be able to stop everything and get someone to find out what is happening. Upper and lower limits, check that against a blueprint and see that the upper and lower limits have been put in properly, then compare that with how it was running earlier that day. Someone has got to understand ... someone s got to be putting in the data properly. It is computerized, of course, but it is only as good as the information that someone is putting in."
Start from the learner 's strengths and make the instruction practical and useful for learners to overcome their fears regarding geometry. Provide opportunities51
for learners to make connections between instruction and real-life situations common to their lives. As one workplace learner states, "I think like in these examples you have in here where we used real workplace kind of examples rather than some kind of theoretical examples that didn t apply. How many rolls of paper fit in the truck? What s the area of the truck, the volume of the truck. Real examples like that--that means something to your everyday job." Or as another learner says, "Learning volume here is easy because I can see it as something in front of me. It's easier for me to figure out if it's hands-on." Or as another learner states, "I like learning volume and shapes because in landscaping you can visualize in your head the shape to determine how much fill or sod. You need math to compute the job." Or, "the best learning was when I am at work using my tape measure."
Focus on hands-on problem-solving and give special attention to developing spatial sense in order for learners to develop an understanding of geometric principles. As one participant in the Virtual Study Group states, Spatial reasoning which in my mind includes not only geometry, but measurement and the ability to visualize. It is often the visual and concrete models that can help people understand and learn what we want to teach about number and statistics. In addition, being able to realize that this kind of reasoning, this part of mathematics, often helps students who have talents in this direction realize and accept that they do have mathematical potential.
52 ALGEBRA: PATTERNS AND FUNCTIONS
My high school algebra class was really hard. I didn't know what I was doing and I felt like I was the only person without a clue.
I've used every math skill I ve learned with the exception of algebra.
Algebra is the gatekeeper.
Overview
Should algebra be on the "honest list?" The Conference on Adult Mathematical Literacy voted on "informal algebra" as one of four basic topics to include in adult numeracy education. NCTM's Curriculum and Evaluation Standards, The Massachusetts ABE Math Standards and other reform movements include it as a critical skill. But say the word "algebra" to any group of adults (ANPN focus groups, for example) and the reaction is negative, with personal stories of frustration and sheer agony spilling out. Is the general perception that "there's really no use for it" a signal that we drop it from the list of instructional topics or is the answer simply to improve instructional practice? This area of mathematics presents a challenging dilemma, because what appears to be a case of the experts want it but the people say no, may be more a case that the two groups are talking about apples and oranges.
When adults reflect upon what it means to do algebra, they tend to recall formal methods of equation solving, age problems, and a lot of x's and y's. But mathematics educators at all levels have begun a dialogue with a very different emphasis, one that "moves away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as a problem-solving tool." (NCTM, 1989, p. 150) They are not talking about the mechanical high school algebra but algebraic reasoning that allows us to think about and express patterns, relations and functions and which ultimately gives many more people access to technology (e.g., spreadsheets and relational data bases).
Key Findings
Many frustrations are connected to past experiences with algebra. "I remember my father standing over me at the dining room table attempting to drill into my head the algebra x, y, and x + y. I couldn't understand how anyone could understand it and why anyone would want to." (Stakeholder) So many adults cite algebra as a major stumbling block in their earlier mathematics education, the place where they got stuck. "Math is pretty easy (division); but algebra, forget it." "Algebra in ninth grade was hard for me. The teacher would do a problem and I was lost. I really felt out of place." "Algebra is hard." Whatever the reason, many learners report an incredible disconnect at the53
point when algebra is traditionally introduced. "I never had any trouble with math in school until I got to algebra." " Math was pretty decent, and then when you got to algebra it was like they totally switched it all the way around."
There is a widely held notion that algebra is not practical, relevant or useful. "What is it used for? You don't use it unless you're teaching it or you're going into some kind of manufacturing type deal where you actually make diagrams, but otherwise it's no use. I use math everyday, fractions and so on and so forth, but I just don't use algebra or geometry."
Algebra is a bridge between arithmetic and more broadly generalized mathematical situations. Mathematics is the study of patterns. "Learning to recognize and analyze patterns and number relationships connects math to the world." (Massachusetts ABE Math Standards, p. 42) These generalizations can be expressed in the notation of formulas and graphs.
Many life and work experiences can be expressed in algebraic terms. While most adult learners do not see the relevancy of algebra, and many teachers see the academic relevancy, employers and other workers do see application to today's workplace. "Our union has a formula to calculate union dues. Dues are based on a weighted average, because not everyone in the union made journeyman at the same time. Some journeymen make $14 an hour, and some make $22 an hour. So you have to take so many $14 an hour, so many $16, so many $18, so many $20 and take a weighted average. And there's another equation: twice your weighted average plus your life insurance plus your union payments..." (learner) One person mentioned an unusual algebra application. "I read an article about how they determine how long someone will actually stay in prison or jail. There is a formula that they use that factors in things like good behavior and work release. With this formula, the sentence that someone would get from the court can be reduced a certain number of days, so the inmate can get out earlier than originally expected." (learner) The Massachusetts ABE Math Standards hold that "life experience has afforded adult basic education teachers with a broad base of real-world ties which can be readily linked to the concepts of equation, function, variable, and graph." (p. 46)
Algebraic thinking skills are crucial if adults are to compete in the global economy; therefore, all adult learners should have the opportunity to improve in that area. While the SCANS Report doesn't get very specific in math content areas, there is support within the document for going beyond the very basic skills and including thinking skills. And beyond that, the "five competencies" that build upon the foundation skills imply the need for some algebraic competence, especially in the areas of "information", "systems" and "technology." SCANS stresses the need for organizing, interpreting and communicating information and employing computers as a tool for those tasks as well as the ability to "discover a rule or principle underlying the relationship54
between two or more objects and apply it in solving a problem." Identifying and expressing pattern, relation and function are the algebraic skills imbedded within these competencies. When the teachers who wrote The Massachusetts ABE Math Standards say "the opportunity to study algebra should be available to any adult basic education learner who may have missed it due to past educational experiences," (p.46) they are calling for a chance for adult learners to compete in the global economy.
Algebra impacts the competency of workers, parents and citizens. Workers who are involved with technology cited several examples of algebra use. "I did marketing analysis using Lotus 123 to forecast exponential marketing sales vs. quotas to determine sales regions. In the manufacturing area I was 20 years old and three top managers couldn't figure out an algebraic formula and one of them very jokingly said, Here, see if you can solve this, as they all laughed at the thought. However, in one minute, I solved the problem to determine their daily production which they couldn't do. " (learner) " We're teaching students at the Great Lakes (Naval Base) all different facets of shipboard life..., and one of the big things we teach up there is electronics... Ohm's law, it is simple basic algebra."
We heard from many parents enrolled in adult basic education classes who don't like the fact they can't help their children. " They're (her children) always asking me why I don't know about algebra. So I told them about what happened in the past. (My daughter) can see how automation is coming in with computers and all - and she knows if you don't know it, you'll fall through the cracks, be on the street." " Back when I was growing up, we didn't have to (take) things like algebra and geometry, and now you have to know these things. Like my son comes home with papers that I can't even do." "My kid's doing algebra in 6th grade now. I'm trying to help him out."
When asked to talk about math skills needed to be a successful community member, one learner replied, While basic math skills, e.g. adding and subtracting, are obviously important, it is also useful to have a working knowledge of algebraic language. I am surprised by the opportunities I ve had to use algebra mind, which I was unaware of before I acquired it. For instance, I know now that God didn t create complex mathematical problems; the Math Wizard did. Because I have an understanding of how these formulas happen, I am able to apply that knowledge to creating formulas that work for me. Algebra mind has also made a contribution to my daily thought process. I find myself thinking more critically and analytically, which is a nice side effect of all the problem solving ... In my opinion, any person who thinks more systematically and analytically is bound to be a better citizen ... crime rate, parenting, population, welfare and many other social concerns. Better thinkers think ahead. Algebra mind should be taught as part of social reform.
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Implications for Teaching and Learning
Improve algebra instruction by providing effective staff development . Teachers need to relearn algebra through the lens of patterns, relationships and functions, how it's applied now to the real world, and need to know a diversity of approaches. The learners' curriculum should reflect these same elements.
Introduce all learners to algebraic concepts by making links to the learner's experiences. "I learn better if I start off with something I already know. When I worked in marketing...If you go back to the basic formula and link it to an easier way. Because the more I learn the easier it gets. Link it to something you already know and you'll get it, you'll remember it."
Pay attention to instructional pace, vary teaching strategies and strengthen the development of concepts to improve algebra instruction. Learners suggested that the source of the trouble might have to do with the pace of the instruction. "To me, an improvement might be to slow down...and maybe dwell a little more on it." "There are so many concepts to grasp at one time, you need more time." One stakeholder suggested that a better understanding of the large concepts might have helped. "I remember finding it (9th grade algebra) the first time I was frustrated with math. And I would get the right answers but not the way the instructor got them. I have since learned that I am highly intuitive. I probably skipped some steps along the way but it was frustrating and difficult for me to go back and fill in to tell him how I got there. I have since (worked with) some algebra materials which were sort of an aha, that s what I was supposed to be learning. I think it would be more fun and exciting to study those math concepts now because I would have a much clearer idea of why I was learning and what I was doing and there'd be things like missed calculations that computers and calculators could help with so I could really focus more on the concepts."
Connecting to the Four Purposes
Algebra supports the key purposes for literacy. How can algebra be a door-opener rather that the gatekeeper to higher education and well-paying jobs? Skills and knowledge in the area of algebra help adults access information that is presented in written and oral mathematical symbols. Conversely, the ability to represent information and relationships with algebraic symbols, graphs, or everyday language strengthens voice. The ability to reason algebraically (to think logically), to recognize patterns and generalizations provides a scaffold for problem solving and decision-making.
Effectively taught, algebra can be a source of empowerment for adults as parents, workers, and citizens. It should not be the "sieve" and the hurdle that keeps people out.56 COMPETENCE AND SELF-CONFIDENCE
You have to go back many decades ago when I was in grade school. That's where I first think about math -- same thing, the times table. You were under the gun to be able to verbalize. I can recall hitting the wall at 6 times something or other and that was the end of it. I couldn't go farther. Fear was a motivation. In those days, if you didn't produce, you flunked. So I learned methods of getting to the same answers but not the standard subject methods. I never could figure out how, say a, b, or c, had values to it in algebra, and that was sheer agony. I got into geometry. I was in a private school, in a very small class. I knew where I was in the class, maybe that was (my) self-image, but the problem was some of the fear of math, so the first time I flunked it. Then I changed schools and never told anybody I'd taken geometry before and I got an A the second time through it 'cause I had to take it to get the math credit. And I like geometry but that's about where we stop. . . I was supposed to take physics and did not and I avoided math. I've learned my own methods from dealing with where I needed to do math.
Overview
In focus group after focus group, adult learners and stakeholders openly shared their positive and negative experiences with math. They shared their best experiences with math, and from whom they learned math. The also shared their worst math experiences. When given a choice -- a good experience or a bad experience -- the results were striking. Many more learners, and a good number of stakeholders, described their lack of confidence and competence in math. From the discussions, it became clear that many adults fear math and especially lack confidence in their ability to handle the math taught in classroom situations.
Key Findings
The loss of self-confidence in math, the lack of understanding of particular math concepts, and fear of math inhibits power. Adult learners and stakeholders alike often remembered experiences that discouraged them from enjoying math and appreciating its potential. In focus group after focus group, adult learners reflected on particular experiences when they had difficulty learning math: "In second grade I had trouble with multiplication. The teacher just forced it down my throat and expected me to memorize it and I just couldn't do it." "Math was okay until seventh grade when we started fractions. I had no idea what was going on and the teacher would not explain it." "In seventh grade I started to have trouble with geometry. I still have trouble with the GED geometry. I don't know why we have to learn it. It's so confusing."
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Many adults do not feel confident, competent, or comfortable in math. "I don't like math. We don't get along. I just don't like it. Adding and subtracting was okay, but when you get to dividing, fractions, and algebra, it just gets hard." "All math is frustrating." "I've always been really dumb in math."
Many adult learners are frustrated because they do not feel competent in math. Comments like "Doing homework with the kids is most frustrating for me because my kids do math better than me," or "Working with my daughter at home is really frustrating. I won't let her use a calculator because I don't want her to get dumb like me," or "I want my kids to be able to do math like I couldn't" reveal that sometimes this frustration shows up when dealing with their children's math homework.
Adult learners and stakeholders alike, in some cases, fear math. As stated by one stakeholder, I was taught to fear math. Or as stated by learners, I like math but I m scared because I can t divide. Math makes me terrified and tense.
For some learners, frustration with math spread to frustration in other areas of school as well. For example, one learner related, "I am not good in math. When I was in school and we started on decimals and fractions, I could not catch on and my teacher wouldn't help so I got behind in class and could not keep up with everyone else so I just gave up completely on all of school so that no one knew that I couldn't do it and (I) quit school. I don't even like it anymore."
Is this lack of confidence in math because people are limited or lack the ability to learn? Certainly not. The causes are more likely found in poor learning environments and lack of recognition of different learning styles and needs. As one learner stated, "I didn't understand math because the teachers wasn't explaining it right from the base. They was getting to, they were starting the middle part of it and I didn't understand the first part of it and I never could understand math." Or as another learner noted, Mine was high school, too. It was close to what he was saying with the problem with a teacher ... It would be at the point where maybe I wouldn t grasp something and you try to ask questions, and it seemed that, if you were in the minority that had trouble, it was always see me after class. It seemed you were being pushed to the side a little bit. Maybe I didn t get enough where you drill with it and it gets repetitive, but it was like the majority got it, so that was good enough, so we went on to the next thing. So that was when I just fell behind.
Good learning environments -- within the family, at work, or in school -- produce different attitudes toward math and can help to overcome fear and lack of belief in one s ability. Confidence builds competence in math and competence builds confidence. The learner quoted above who could never understand math when talking about his current ABE experience stated, "Now I understand it perfectly. I used to hate math; now I love it." Or as another learner related, "Here, when I ask them ... I feel58
that builds up your confidence level and self-esteem ... they always try to help. It makes me work harder than ever. " Or as another learner states, "I hated math but the football coach made it interesting to me. So I had to learn to deal with it. The coach made learning math fun." Or within the family: "It (math) was my best subject. My grandfather helped me a great deal when I needed help. He was very helpful to me because he was a carpenter. " And "my mother was the one who would help me the most with math. I would work at it by myself until I would get frustrated ... so I would ask her to help me and she would sit down and explain things to me. "
Sometimes confidence in math comes after gaining self-esteem as an adult. One stakeholder explained, "Now that I am older, I feel much more confident in the math I need to do. I wouldn t voluntarily go back and take a higher math course, (but) math doesn t frighten me. I have more confidence in myself ... This was nothing I was taught. It is something I picked up." Math skills are also acquired on the job. For example, when learners were asked to describe a good math learning situation, they responded, "The people that helped me was my boss at work;" "The best learning was when I am at work using my tape measure;" and "I worked in a Chevrolet parts department and learned more math on my job than in school."
Those learners that feel comfortable with math have confidence in their ability and respect for the domain of math. "I bought a house last year. The price of the house sounds pretty inexpensive, but when you add up the interest on it, the points they charge you, the closing fees, the maintenance. It's like on a 30-year loan, you end up paying three times as much as the house is worth. You gotta compute simple interest, compounded interest, all that sort of stuff. First I took what I made a month. I took an average, then I deducted all my expenses, then I had a budget saying what I could afford to pay a month. Simple math. Only you divide that if you have a roommate or whatever. Just basic planning and basic math skills, averages. When they first tell you, 'Just put down 5% or 10%, then pay this much a month, you take it like that and you don't know what it really costs you. You gotta figure everything else. That's what math does, it makes you organize, makes you think in a certain manner." Confidence in math increases power, voice, and the ability to act. For example, "I went into the store. The lady had an item that was supposed to be discounted, I think like 50 or 60% off ... And she brought it up and it didn t sound right to me, and I was in a hurry that day and I really didn t have time to figure it out on my own. So after I got home, I figured it out. I went back to my math book and looked at the amount that was supposed to be discounted and it was wrong. And so I kept my sales receipt and I took it back and I got six dollars back. She was wrong. "
The more adults learn, the more confident they become, and the more enjoyable the experience of learning becomes. "The more you learn, the more fun it is." Stakeholders explained how knowing how to do math improves confidence. "In 1946, in high school, I was taking geometry and the teacher made it so clear that I actually understood-59
what I was doing and thought that I was really good in math!" "It's funny, most of the stuff I thought of (about learning math) didn't have to do with school ... I learned a lot of my math as a kid from my dad and it was one of the few things that he did with me so it was important for that reason. I have clear memories of the two of us looking at a word problem together and my dad always drawing pictures of whatever was going on in the problem. And my growing up with a sense that I could figure anything like that out. All I had to do was draw a picture of it. I have a memory when I was in the sixth grade, when I was getting kind of bored with math, my dad saying, 'Oh, well, do you want to learn algebra?' So I have clear memories of him basically teaching me how to write sentences in math and leaving holes in the middle of them and plugging letters in. So I had this great confidence in my ability to do it, partly because I had a dad who assumed I would be good at it and who kind of instilled me with that."
Implications for Teaching and Learning
Teachers need to become comfortable presenting math concepts using a variety of strategies and approaches. This suggests that teachers need staff development where they can share with each other successful teaching strategies. Teachers need to become comfortable using manipulatives, calculators, computers, whatever it takes for learners to grasp math concepts. A stakeholder suggested, "I have five children in their twenties, but they missed out on math somehow. I don't know where math education has gone to but I think we need to go back and educate the teachers." Another stakeholder shared her own personal experience with becoming comfortable with other approaches: "I thought of last year. I was teaching reform calculus for the very first time. We had completely switched over to reform calculus, Harvard Reform Calculus which involves a lot of manipulatives, a lot of graphing, calculator work, a lot of computer work, emphasis entirely on understanding and not on manipulation. I came up through a very traditional background which most of us all did -- the rote, go home, be able to do 50 problems and all that stuff. All of a sudden, I was faced with teaching these students all these concepts without showing them the manipulations but actually get them to understand the concepts not just how to do it. I sat back with other instructors. We met on a very regular basis to share our woes. And we said, 'Don't know why we ever -- now it makes sense.' We just sort of did it by manipulation before. We now understood. So I learned probably as much calculus as my calculus students did. Now that's a terrible confession. And it was amazing the complex concepts that my students could understand before I had even shown them the algebraic manipulations to do something. We didn't show them the manipulations until afterwards and it was just phenomenal and so I learned a lot of math, they learned a lot of math. I learned a lot about how to teach math."
In recent years, more research has been done in the area of math learning. This research has revealed that individuals learn math differently. Some are comfortable with learning step by step procedures while other learners tend to jump to the big picture60
and work forward and backward to solve problems. Teachers need to let go of the need to make all learners solve problems the same way that they were taught in school. An adult learner shared how this step by step method does not work for him: "It (math) was hard because they wanted you to write everything down, every step you took. I did a lot of it in my head. I couldn't write how I'd done it 'cause I'd done it in my head. I had the right answer but I couldn't write down this is the step, this is the step, because I'd done it in my head. They'd try to show me the steps but it didn't always look like how I had done it."
Success needs to be built into the adult education classroom. Adult learners need to have success early on and often when they begin a math class. This success enables them to develop confidence in their ability to do math, which in turn paves the way for further positive math learning experiences. All individuals benefit from positive feedback, but it is particularly important that adults who have experienced failure in math class previously now find success in the adult education classroom. As one adult learner put it, "It (math class) can be just as interesting as a social studies class, and the positive feedback is the best thing. When my teacher told me when I was doing something, that I really was a good thinker, I felt so good! That sounds really dorky, but it's the truth." When asked what recommendations should be made in math instruction, stakeholders offered these comments: "We need to build confidence in our students;" "People lose confidence in math because they don't realize that they are already doing lots of math and that can be built upon;" and "The other thing is, that I really noticed, more than any other subject area, when students feel competent in math, their self-esteem really grows and their confidence in themselves as learners really grows."
A stakeholder voiced her concern, not about the adult education classroom, but the K - 12 system. Because she felt that her daughter might not get supported to be successful in math, she had to take on this responsibility as a parent. "The other thinking that was important for me as a kid and that I've tried to do as a parent -- not so much math skills -- the belief that my kids will be good at this and can be good at this. That it is logical, simple, always do-able. That's probably the other piece out of this as a parent. Especially for my daughter because I assume a little less that school will give her that message -- but the belief that she's good at it (math)." Another stakeholder reflected on her past experience in math: "I have sadness that no one ever said to me, 'You're really good in math.' I was good in math."
Math content skills need to be presented in the context of real-life situations. When learners can immediately apply what they have learned, the learning crystallizes and the learners gain confidence and competence in their math ability. When asked about his experience learning math, one adult learner responded, "I picked up some stuff from me helping my father. I used to know, like all the money. We owned a restaurant back home and every day he gives me the money and at the end of the month, I used to add it up, let him know what we got. That helped me a lot." A suggestion from61
a stakeholder was to change the perception that math is difficult: "We need to start over and completely redesign the way math is taught. There should be more application. The current application attempts to teach application with too much theory and in too convoluted a way. The perception is that math is too hard and that perception must be changed."
Connecting to real-life situations and understanding the why behind math processes improves math ability. A stakeholder, even though she enjoyed math as a child, shared this revelation, "But I always felt good about math. I always enjoyed math. It was in college then, in teacher preparation work, that I picked up a lot of why I was doing it and the background and it would click and then it made more sense at that point than previously when I was just repeating a process."
Adults use math in their daily lives but often do not connect their real world math to the math in the classroom. When asked whether they use math, adults who are not confident in math will often say that they don't use math yet they earn wages, spend money, buy gas for their cars, and so on. Connecting math to their real-life situations helps adults understand that they do use math. A stakeholder suggested, ". . . Building on the familiarity that they (adults) do have. I think that one thing, we estimate, we talk about adults' wealth of experience, but that experience has given adults math intuition -- the way that they've dealt with it -- so we can find strategies to go with their own intuitive way that they've walked through things. It is very natural for them. I think you'll find this with some of the women students, that they've developed some strategies that they can build off, solve their problems." Other stakeholders added, "People lose confidence in math because they don't realize that they are already doing lots of math and that can be built upon." "Most people don't perceive themselves as being math literate. We get through the classes, we get a 'B' in algebra, maybe we get an 'A' in trig, but when we get out, we perceive people who are math whizzes are engineers, or physicists, people who are in math fields. Most people in this country do not perceive themselves as being math literate and so we shy away from it."
There needs to be a level of trust in the adult education classroom. The adult education teacher needs to build an environment that is comfortable for adults and one in which adults can be open. Adults need to feel comfortable sharing their frustrations and lack of math skills. One adult learner shared how he felt a need to "expose" his lack of times table knowledge to his teacher, "Well, math is hard for me, but I can learn it. The hardest part for me is the geometry and the division. But, if I knew my times tables right off the top of my head, I could get all of it. I don't know all my times tables right off the top of my head. I could work the problem out but my time tables slow me down. When I go into the classroom I'm going to have to explain to the teacher on the outside of the door that I'm not good in math and she'll have to explain it to me over and over again. I will get it eventually, but it will take time. Once I've got it I feel good about myself when I get it because math is the hardest thing for me." A stakeholder explained62
what she values in an instructor: "With my staff, what I think is successful -- what is just as important as the academic background a person brings with them to the classroom, the knowledge that they have -- is the ability to interact with students on a personal basis. Being able to sit down and talk to them as human beings, let alone just as students that sit in front of you and I have to teach them this material. Most of the students that I have had or I have talked to whether in the alternative high school or adult have had negative experiences in educational programs. As a result, that is why they quit. Coming in, they say, 'Well, people care about me here'. That just raises their self-esteem ... gets them on target again ..."
Connecting to the Four Purposes
For adults to be literate enough to accomplish their goals, they need to understand and use math. When adults do not have a handle on math, they have difficulty coping -- whether it be in their role as parent, worker, or community member.
The following adult learner lost his job because he was not able to read scales to access information: "I worked for Nabisco. As a mixer you had to know the correct scale and formulas. I kept messing up. I lost my job. It doesn't look too good on the record. If you don't know math you can't suceed."
This adult learner expressed his feelings about how it is difficult to have a voice because he lacks confidence in his math ability: "I guess I'm not too comfortable because I lack a lot of knowledge (in math)."
These adult learners were not successful in making decisions for work. "Dairy Queen wouldn't hire me because I couldn't make change in my head. I couldn't give the answers in an oral quiz to making change questions." "In the bank when I had to 10-key, do some things with checks. . . couldn't put it in the machine. If I messed up checks, I couldn't work. I quit the job." "One time I was at work and this guy came up to my register, and I rang all this stuff up. And when my register opened, he gave me like $20 or something. And he's like 'Oh, wait a minute, I have the change.' And I'm like looking at him and I'm like, Oh my God, I have to figure the whole thing out and it took me about five minutes. The register doesn't do it for you. Dealing with money and stuff is important. Sometimes when they give me extra change and stuff, I just ignore it and then say, Oh, I'm sorry. "
Math skills, knowledge and abilities are gatekeepers. To bridge to the future: to get a job or a better job, to go on to college or to create a brighter future for their children, adults need to understand math. "My daughter asks me about that (percentages) so I'm teaching her a little about that. They're always asking why I don't know more about the algebra and geometry and stuff. So I told 'em about what happened in the past. She can see how automation is coming with the computers and all -- and she knows if63
you don't know it, you'll fall through the cracks, be on the street." And this adult learner now understands that math is critical to moving ahead. When asked what was a frustrating math experience, he replied, "Math in general. In school we had to know it all at once. We had to go on or else. Some like me never went on."64 CONCLUSION AND RECOMMENDATIONS
The National Institute for Literacy issued a wake-up call to the adult education community with the publication of Equipped for the Future. The voices of the 1500 adult learners motivated those who work in the field of adult education to respond, to critically evaluate, and ultimately rework the curricula, instructional and assessment practices, program structures and supports currently in place. Similarly, the Framework for Adult Numeracy Standards documents the voices of adults who tell us about the significance of mathematical skills and knowledge in their lives. These voices create an expanded definition of numeracy, one that includes much more than computation or passing standardized tests. This emerging definition encourages all of us to give voice to our mathematical understandings, to take joy in accessing information and making meaning via a solid sense of number, data, geometry and spatial reasoning, and algebra. It insists on the importance of making informed decisions and solving problems through the use of quantitative and spatial reasoning, and most importantly, it defines a numeracy that is situated in modern, relevant situations and addresses skills necessary to cope with present and future societal demands.
The adult education community needs to commit itself to this broadened definition of numeracy. Let's step back a little further. The adult literacy/basic education field needs to include numeracy in its agenda period. Not as an afterthought, but in a basic and systemic way. Numeracy must become an integrated component of all ABE, GED, ESOL, family literacy and workplace learning environments. Policy makers, administrators and curriculum developers on the national, state and local program levels need to include mathematical literacy in their mission of providing a second chance to adult learners. Classroom teachers must step forward as leaders in reform and take on the challenge of creating mathematically empowering learning environments. Failing to take action means that we're willing to turn a deaf ear to what parents, workers and community members have told us. Because we cannot consider that an option, we now turn our attention to what it might take to make lasting reform happen.
Recommendations for System Reform
The Framework for Adult Numeracy Standards is a consensus statement about what the content of adult education classes should include. Each of the seven numeracy themes include key findings and implications for learning and teaching that are intended to serve as a map for adult education practitioners and policy makers as they take the important next steps toward reforming instructional programs for adults. The ANPN Working Group members and several study and focus group participants have begun to speculate about how we're going to make those next steps happen. What will it take to create learning environments that support quality numeracy instruction? We have polled the members of the ANPN Planning Project for System Reform and have also65
drawn upon the focused discussions which took place at the Conference on Mathematical Literacy. We are in agreement that several necessary conditions and system components must be in place for our vision to become a reality.
The Necessary Conditions
- Reform has a chance only if it is supported at all levels: national, state, and local. It has to be both bottom-up and top down with ongoing opportunities for people at the different levels to dialogue.
- Learners must be meaningfully involved in the development of each system component.
- Teachers must be meaningfully involved in the development of each system component.
- Realistically adequate resources must support all system components.
- Numeracy must be included in literacy reform.
The System Components
The following is a discussion of each system component needed to support the changes called for in the framework and suggestions for initial strategies for beginning the improvement in these areas.
Communication/Dissemination
A group writes a document . . . a tree falls in the forest. All stakeholders need to hear our "sounds," and be invited to make noise with us. Strategic dissemination is key all along the way. And so is two-way communication that both explains and invites feedback. We knew this as we began to talk about reform two years ago, so we established a communication structure through the Adult Numeracy Practitioners Network (ANPN). ANPN publishes a quarterly newsletter (The Math Practitioner) and has two very active Internet communication mechanisms: the ANPN Homepage and the NUMERACY listserve. We need to disseminate the framework and all subsequent products and processes through the network and through other existing adult education channels.
For the system to be reformed teachers have to know what the work environment requires and what is needed by the community in order to better prepare adult learners. There must be ongoing dialogue with industry if, in fact, the curriculum is to change to66
meet the needs and expectations of the workplace. There must also be communication with community stakeholders -- for their input into math reform and for them to become comfortable with the new approaches to math.
Adult education is able to react to system reform much more quickly than the K-12 system, but it should not have to work alone. Communication must occur across schools, K - 12 as well as community and technical colleges. Other institutions that work with adult learners, specifically community and technical colleges, need to be at the table during the discussion of system reform, and they must be making every effort to move forward along with adult basic education.
While effective communication needs to occur across schools, communities, and industry, it must begin within the adult education system. According to the Conference on Adult Mathematical Literacy (p.7), "although State Literacy Resource Centers and other agencies have been established to act as clearinghouses, materials from the national level filter very slowly to teachers at the local level and teachers remain largely unaware of standards, new teaching materials, technologies, and curricula in use outside their programs. Further, improvements in adult (numeracy) education are hindered when adult educators do not have opportunities to network, exchange ideas, and collaborate."
1996 - 1997
- Expand circulation of ANPN newsletter (from 1,000 to 10,000). Disseminate to local learning programs via state ABE offices and literacy resource centers and to national stakeholders. Focus discussion in the newsletter around the numeracy themes. Publish excerpts of the document in the September, 1996 issue to begin the dialogue around numeracy themes.
- Continue Internet dialogue. Make the Framework for Adult Numeracy Standards available on the ANPN Homepage.
- Rework the July, 1996 draft framework for dissemination to ANPN members, state directors, resource centers and the mathematics education community.
- Present draft framework at at least five state or regional ABE conferences.
Numeracy Content Learning Standards
The Framework for Adult Numeracy Standards should be used to develop a more specific set of learning outcomes that in turn guide the development of curricula, instructional materials and assessment. Questions that need to be considered are: What is67
the "grain size" or specificity level of a learning standard? What generative and specific skills support each standard? How do we address levels of proficiency? Do we use one set of standards with benchmarks or do we create three or four? How do we refine and validate the standards?
1996 - 1997
- Form an ANPN Working Group to co-develop Adult Numeracy Standards. Expand the Working Group to members from ten states who are supported by and report progress to their state ABE offices. Develop working relationships (possibly subcontracts) with NIFL-funded projects so that the work remains in sync with and informs the EFF Project.
- Participate in EFF Working Groups and meetings.
- Begin simultaneous teacher research projects that begin to explore and implement components of the numeracy standards in adult education classrooms across the country. Use this research as a way to begin classroom reform. The experience of NCTM, the MA ABE Math Team and several state curriculum framework development groups is that engagement and experimentation must begin before the final document is complete. Ownership by a critical mass of practitioners will create a real base of support.
Curriculum
The numeracy learning standards will guide adult learning programs as they update their curricula to respond to current and future demands of workplaces and communities. Local programs and states will utilize the standards to evaluate and, in some cases, develop their own curricula. State resource centers or collaboratives might pool resources to develop model curricula. Major funding should be sought to support the development of this component.
1997 - 1998
- Seek a partnership with the National Science Foundation, the National Council of Teachers of Mathematics, the Departments of Education and Labor, private industry and foundations to fund the development of a comprehensive adult numeracy curriculum based upon the learning and performance standards.
- Call together the developers of the major National Science Foundation- funded K-12 mathematics curriculum projects (Interactive Mathematics, Connected Geometry,
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and Investigations in Number, Data and Space, for example) to consider the possibilities of adapting these projects to adult learning environments.
Assessment
Once clear numeracy learning standards are developed as statements of what adults should know and be able to do, we need to think about developing new performance and assessment tools. What kinds of evidence will show that people have met the standards? How will we test what we value? How do we align assessment with content standards and the curricula based on those standards? Too often, the testing instrument, whether it be the TABE or the GED, drives the curriculum. When a learner's only goal is to pass a test, it is difficult to teach math concepts needed for real life. Unfortunately, it often doesn't matter what adult educators consider the most important concepts or topics to teach. Students are very likely not to think of them as important unless they are reflected in the tests and assessment. No matter what assessment is designed, unless it is incorporated into individual instructor's assessments, GED exams, job placement tests, and the like, it will get lost in the need of both students and instructors to address what does appear on those assessment tools.
As more and more industries are beginning to consider national skill standards that are portable, adult educators need to follow closely to see what implications these standards might have for adult learners. How might industry skill standards affect how we should be assessing adult learners? Evaluation and assessment needs to communicate to learners a range of what they may need to learn and to certify that learners have mastered competencies so that employers and colleges will honor learners' educational accomplishment. We should assess and certify adult learner's workplace readiness with respect to math. While we're looking at industry standards, we need also to look to business and industry to find out just what skills are needed to succeed in the work environment.
Assessment tools need to be designed to evaluate the ability to "process". How well do individuals and groups problem solve? How are their reasoning skills? Is it possible to design concrete benchmarks for the process skills -- problem-solving/reasoning, communication, connections/relevance? How do we know when an adult is reasoning mathematically?
1996 - 1997
- Establish a numeracy assessment work group to collect promising assessment practices from across the field and outside of ABE (K - 12, industry, and from other countries including Australia, the Netherlands and Great Britain where interesting work in the area of numeracy assessment is happening).
1997 - 1998
- Establish a work group to develop assessment models that provide good evidence of mastery of the skills and knowledge called for in the learning standards.
Staff Development
Ongoing staff development is critical for math reform. One stakeholder stated the need for staff development this way: "We have the same teachers we had yesterday, so without staff development, we will have the same teaching we had yesterday. We need to help teachers teach in a way they were not taught."
Too often in adult basic education programs, teachers are expected to be proficient in all areas -- math, reading, writing, social studies, science -- yet most adult educators are trained in only one of these domains, and rarely in the math domain. Therefore, many adult education teachers are uncomfortable with math. Teachers need professional development in order to improve their own math skills as well as change their attitude and perspective toward math. We need to help teachers understand math concepts, not just be able to follow the steps. If we continue to teach the way that we were taught, math will continue to be an elitist subject, only accessible to a very limited number of individuals. Until teachers see the value of math to real life situations, they will be unable to help their learners connect to math.
Along these same lines, teachers have to become knowledgeable and comfortable with the workplace. Teachers often teach the same concepts in the same way that they were taught. If teachers have no awareness of what math skills are needed for work, they cannot possibly begin to prepare their learners for work environments. There needs to be communication between business and adult education with opportunities for experiencing hands-on what workers need for today and the future.
Staff development requires more than simply having adult education teachers attend training sessions. Teachers need opportunities to develop ongoing peer relationships where they are supported as they explore new strategies in their classrooms. Any kind of reform or change is scary. Teachers should not be expected to try to reform their practice without support.
1996 - 1997
- Support existing and establish new statewide teacher math teams to plan and conduct staff development to learn new math content and pedagogy. Draw upon the expertise of NCTM and ANPN members and include employers in content discussions. Use the ANPN newsletter, homepage and electronic discussion groups to share learnings between states. Bring representatives of teams together in April 1997, at the ANPN/NCTM annual meeting.
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Instructional Practices
Changes in instructional practice go hand in hand with staff development. Adult education teachers need to learn new methods of presenting math topics in the classroom. For example, many teachers have not been trained to use manipulatives to present math concepts. Typically, they were taught to follow the algorithms which is what they, in turn, teach their students. Yet many adult learners related that they learned best during hands-on activities. A part of staff development should be building teachers' comfort level with using hands-on materials.
Rather than the perception that the teacher is the one with all the knowledge, the new perception for math reform is teacher as "guide on the side" or facilitator. Teachers need to incorporate into the adult education classroom many opportunities for learners to discover math for themselves. Supporting learners to work together and share their experiences will be necessary for system reform in math.
Conference participants suggested that "diversity in the classroom" was a reality which should be kept in mind as we talk about system reform. Diversity includes differences in linguistic and cultural backgrounds as well as diversity of learner goals and learning styles. For teachers to be prepared for this diversity in every adult education class, there must be effective staff development and support systems in place.
1996 - 1998
- Create and provide opportunities to encourage teachers and programs to share promising practices. These opportunities may include teacher inquiry projects, math study circles, and peer coaching projects.
Instructional Materials
The math materials developed for adult learners need to be seriously revised. Most of the materials presently available "often reflect some of the worst traditions of K-12 mathematics education" (Proceedings of the Conference on Adult Mathematical Literacy, p. 6). There seems to be very little awareness of the MA ABE Math Standards and little evidence of any effort to provide learners with practice in process skills, such as reasoning and problem-solving. Very few materials offer open-ended or cooperative problem-solving activities for adult learners.
1996 - 1998
- Meet with and hold focus groups of publishers of curriculum and assessment materials. Strategize about ways to develop materials based upon the new standards.
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Resources/Funding
Adequate funding and resources are needed for change. Obviously, staff development requires funding earmarked for teachers to be able to seriously work on improving their math teaching abilities. For staff development to truly have a positive impact in the classroom, per student funding needs to be increased. Added funding is also necessary to ensure that adult learners have opportunities to use tools such as computers and calculators.
Appropriate salaries and support for more full-time teaching positions, would recognize and support teachers to invest more in their profession and to actively seek to change the system.
At the Conference on Adult Mathematical Literacy, funding was called out as a critical factor in system reform. The lack of adequate funding negatively impacts adult education programs in four ways: 1. Limiting achievement of goals, by causing programs to be understaffed and under-equipped, and by creating time and financial restraints; 2. Limiting preparedness of teachers (and tutors), by restricting availability of and access to pre-service, in-service, and professionalization opportunities; 3. Inhibiting development of instructional materials and not allowing for adequate experimentation with new methods and resources; and 4. Limiting research initiatives (either academia-based or program-based). Research is essential to provide insights into teaching and learning processes of adult students, and to better understand the factors affecting the application of what they have learned in real-life contexts. (Proceedings of the Conference on Adult Mathematical Literacy, p. 7)
1996 - ongoing
- Ongoing work with the National Institute for Literacy and EFF to strategize about obtaining funding to support system reform.
Research
There is little research on how adults do mathematics or how they acquire new mathematical skills. There is also very little on the effectiveness and impact of adult numeracy instruction. This information is essential to guide us to improve instructional programs. If we are to enter upon major reform, we must be supported by a research base.
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1996 and ongoing
- ANPN will seek to partner with the new National Center for Adult Learning and Literacy and other centers of research on adult learning and mathematics learning.
Program Standards, Outcomes and Evaluation
Participants at the Conference on Adult Mathematical Literacy recognized that accountability demands affect math instruction and reform. There is concern that current methods for ensuring accountability -- the use of data such as standardized test results -- do not truly reflect or measure learner's accomplishments. As long as funders expect this kind of accountability, teachers will be inhibited in their ability to truly address the needs of their adult learners.
Some major industries are now focusing on developing industry-specific skill standards. If these standards are implemented and used as a basis for hiring, these standards will have to be a part of the discussion on math reform. Will there be some basic math content and skill that all industries feel is critical to employee success?
System reform must build in an accountability system so that we're not in the same place twenty years from now. We need to develop standards that everyone can understand and accept. Then we need to be held accountable for preparing our adult learners so that they can meet these standards.
1997 - 1998
- Continue dialogue with all members of the Equipped for the Future Project and the US Department of Education to connect the newly developed standards to the Department of Education's Indicators of Program Quality.
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Draft document last updated 8/9/96.
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