Volume 3, Issue C ::: September 1999
Teaching to the Math Standards with Adult Learners
by Esther D. Leonelli
For the last 10 years, I have been an advocate for standards-based teaching of mathematics and numeracy to adult basic education (ABE), General Educational Development (GED), and adult English for Speakers of Other Languages (ESOL) students. It has been quite a journey, a learning experience, and the most fulfilling part of my adult education career since I returned to teaching adults in 1985. By "standards-based," I mean a set of values and important ideas used to judge methods of instruction and assessment. With respect to math instruction, I mean both content and methodology based upon the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (1989), which was adapted for ABE instruction by Massachusetts teachers.
I was trained as a secondary math education major in the late 1960s. When I first taught adults, from 1971 to 1973, I tried to incorporate the methods I learned in college. These methods were based upon the "new math movement" and Piaget's work with children, used manipulatives, and included a deductive, but very directed, approach. These did not translate well into my adult education work at that time. I found that my students, who were in an individualized math lab in a Boston program that prepared them for medical training and professions, did not want to be led through these "discovery" lessons. They wanted to be shown the "rule" so they could apply it to the problems in the book and to the math tasks needed for the particular job they were planning to pursue.
I could do this very well. And so I taught math skills one on one, in a linear way, using pencil and paper and remedial arithmetic skills textbooks. I taught basic computation by rote, using decontexualized situations. Once the students mastered computation skill using only numbers, then I showed them how the skills were applied to word problems, which were chosen for the particular skill to be practiced and mastered.
I continued to teach this way when I returned to the adult education classroom in 1985, teaching ABE and GED level students. I was reluctant to use the manipulatives the Cuisenaire rods, the base-10 blocks that were a part of my math education training. I gave up trying to have my students discover the math concepts they were trying to master, although I wished that my students could rely on their own reasoning powers to reconstruct the theory or rule if forgotten. I reverted to teaching math the way I was taught.
My methods worked okay. I relied on textbooks with many practice exercises and the answers in the back of the back. I could get my students to pass the competency-based math tests my center used for the alternative adult diploma credential we granted, one test at a time. That made my students feel confident and good. But I felt something was lacking when they couldn't remember how to divide fractions once we moved on to another math topic. I was disconcerted when they had to return to my class to prepare for the Licensed Practical Nurse (LPN) test or the college entry test. Something wasn't sticking. The math was "learned" for the test and then forgotten. Students depended on me and on the textbook for answers and rules. And, my methods did not work well for the students who were non-native English speakers. Needless to say, I felt that the approach I was taking needed changing.
By 1989, the GED test had changed to include more emphasis on problem-solving and higher-order thinking skills. It allowed more use of estimation skills and required that fewer complicated calculations be done without the use of calculator. A team of GED teachers in Massachusetts took a good look at the test and came to the conclusion that how we were teaching math should change. I joined that team. Around the same time, I attended a multisession workshop on teaching basic mathematics at the Adult Literacy Resource Institute in Boston. This was a mathematical re-awakening for me and an invitation to reconsider my own practice. The workshop introduced me to new national developments in the area of curriculum, methodology, evaluation, and teacher training in school mathematics that moved math teaching beyond the "back to basics" movement of the last two decades. These ideas, along with research and practice in classrooms and constructivist theory, were incorporated into NCTM Standards, the seminal document for the math-standards movement. I read the Standards, became a "believer," and in the process, also became a lifelong math learner.
Through attendance at NCTM conferences I got to see first-hand the exciting changes in pedagogy and assessment advocated by proponents of the NCTM standards. I saw less direct instruction and more modeling of mathematical behavior by teachers. I saw less "drill and kill" practice and more interesting problems for investigation by students. I saw less individual seatwork and more cooperative lessons and conversation in the classroom around math, and fewer answers from the teacher and more sharing by and among students of individual strategies for solving problems. The workshops were intended to engage me in learning more math, which they did. And they showed me how the activities could be engaging for my learners.
In one workshop I was introduced to international developments in math education, the "realistic maths" curriculum from the Freudenthal Institute, the Netherlands. Their approach asks students to make mathematical sense from graphical images of the real world. One of the "geometry" problems that I like to pose to my students came from that workshop (see below). It requires not only visualization, but also the physical handling of concrete materials and group discussion to come up with an optimal solution.
Here are two views of a building:
What is the least number of blocks you can use to build the building?
What is the maximum number of blocks you can use to build the building?
This activity let me view students at work alone and together, solving a concrete problem. When they build their structures I see what they saw. What I learned is that many of my students have never had the opportunity to build and play and visualize. I also realized that I was making a lot of assumptions when I "lectured" or "demonstrated." I had assumed that my students could "read" a picture and could learn to interpret word problems by my teaching of "key" words and formulas.
Standards and Frameworks
The NCTM Standards were based upon the assumption that, in the late twentieth century, American society has four new social goals for school education: (1) mathematically literate workers, (2) lifelong learning, (3) opportunity for all, and (4) an informed electorate. To meet these (1989) societal goals, the Standards state further, that: Educational goals for students must reflect the importance of mathematical literacy. Toward this end, the K-12 standards articulate five general goals for all students: (1) that they learn to value mathematics, (2) that they become confident in their ability to do mathematics, (3) that they become mathematical problem solvers, (4) that they learn to communicate mathematically, and (5) that they learn to reason mathematically. Points 2 through 3 appear in the first three "process standards" of the document: Math as communications, math as problem solving, and math as reasoning. The fourth process standard mathematical connections relates to the inter-relatedness of math topics and the connection of math to other disciplines.
Several instructional themes permeate the NCTM Standards: Concrete and problem-centered approaches to teaching math concepts; emphasis on estimation and visualization in realistic contexts; and using cooperative learning techniques. The Massachusetts ABE Math team found that these practices coupled with the four process standards are completely in harmony with notions of good adult education practice and so they included these in Massachusetts ABE Math Standards.
The "how" of teaching math is followed by the "what to teach." The NCTM Standards content strands are described for three groups of K-12 learners: K-5; middle grades 6-8; high school level 9-12. The ABE math team found that the content of much of ABE and GED mathematics fell within the middle-grade math range and so focussed their content standards on those standards. My own view is that today's GED test covers school mathematics content up to 8th grade. The Massachusetts Numeracy Framework roughly parallels these content stands with seven Numeracy strands.
I find that a standards-based approach to teaching adult basic math fits well with good adult education practice. The approach is learner-centered, involves a solid theory of learning for understanding, and addresses the wide diversity of cultural background, learning styles, and abilities of the learners whom I teach. And, it addresses math content and skills that are relevant for the new millenium.
What I take personally from the NCTM Standards is this:
- Learning (and doing) mathematics empowers adult learners;
- Math (and number sense) comes from real life;
- Mathematics is more than arithmetic competency and a set of rules to be memorized;
- Mathematics is investigation, communication, and a way of thinking about the world.
In terms of content, how and what math I teach, the math must be meaningful and connected to adults but also must stretch them beyond where they are. It must be more than teaching computation. And, since algebra is a "gatekeeper" to entry into and success in further education, my commitment to civil rights and equal opportunity compels me to ensure that adult math instruction includes some algebra (Moses, 1997).
In My Classroom
The learning of math as well as the doing of math includes moving along a continuum from and among the concrete to the representational to the abstract. "Digits" were born to represent fingers and toes; to "calculate" originally meant to use stones to count. In actual practice, mathematicians, scientists, technicians, draftsmen, engineers people who use math everyday often use graphic and concrete models to do math work. So I try to incorporate the use of a "hands-on" curriculum. It starts, as in real life, with concrete models, incorporates graphics and representational activities, includes, as well, games, writing, and the use of mathematical language and symbolism, and finally, integrates technology.
For example, I use a range of visual models to help learners conceptualize fractions, decimals, and percents. They construct number lines using folded paper, to demonstrate halves, quarters, eighths. Pie graphs of a day's activities are drawn with colored pencils or developed on computer from spread sheets. I try to teach decimals and percentages at the same time, so that the students can relate these two concepts. Thus, students use the folded paper which represent fractions to analyze a candy sale's bar graphs, which are calibrated in percentages. They describe the pie graphs in fractions as well as in percentages. Building on students' experiences with percents in everyday life, we construct the meaning of percents in more complex situations.
But I try to do more with manipulatives than just use them to develop and demonstrate concepts. The blocks or tiles or other concrete things are often themselves part of the problem. I recently conducted a bean-bag race in the hallway of the learning center where I work. Two students walked along a track, dropping bean-bags every two seconds, while a third student kept time. The rest of the class, who hadn't viewed it directly, had to look at the bean-bag drops and tell which student walked faster and how they knew. One student spent a number of minutes animatedly explaining to me how the walks differed and how he had analyzed the situation. In explaining his reasoning, he pointed out the differences in the "proportional" distances between the two sets of bean-bags and how that translated into different speeds. As he talked, he got very excited with his own understanding and explanation, and exclaimed at the end of his analysis "and that's math!"
From Real Life
Students from other countries use different procedures than found in many adult education texts, particularly for several of the common computation operations such as subtraction and long division (Schmitt, 1991). Despite this, and although the operations did not make sense as taught to many American-born students the first time around, adult education texts teach only the US algorithm (a rule or recipe for a mathematical procedure or operation). Standards-based math teaching respects students' thinking, background knowledge, and development of their own algorithms for computation. I try to teach my students by listening to their explanations of their own thinking and ways of doing math.
One of my GED students, Leo, was a "street smart" learner. He could apply his own experience in playing the numbers to solving the combinatorial problems I posed in class. (For example, "how many different outfits can you make with three shirts and four pairs of pants?") But he couldn't do a two-digit division problem the "long way," and he felt that would hamper his passing the GED. We spend about 15 minutes after class one day, talking through a long-division problem. In drawing out his thinking, I found he understood the concept of division as repeated subtraction and urged him to use that strategy. In the process, he came up with a method of division that made sense to him and which he could articulate and repeat successfully. Although he claims that he now used a method I "showed" him, it was his own algorithm that he was able to apply confidently in his work, not one that you could find in any GED book.
With my more basic students, those still working on addition and subtraction, I use an investigation of the concepts of carrying and borrowing. Several useful card games, such as Close to 100 and Close to 0, build on the learner's sense and experience with numbers using 100 and 1000 as benchmarks (see box).
"Closed to 100"
The games gives learners a chance to use numbers in the context of a real-life social situation: a card game. Results can be discussed, strategies shared, and which simulates mental math activities that adults need for daily life, such as calculating change from a dollar, adding or subtracting percents, making purchases. Besides being fun, it is learning in a social context.
One of my formerly homeless students graciously shared with me many of his strategies for estimation. He often practiced his multiplication skills by estimating the bricks in a building wall, then counting one-by-one or multiplying row by height to check his number reasoning. Today I give "Elliot's Walk" a true story as a problem to my students to assess their proportional reasoning and communication skills in using math.
Here it is: Elliot took a walk from his apartment to Harvard Square one day and counted his paces as he walked. He figures that his pace is approximately 2 feet long. He counted as high as 3,000 paces and then stopped counting just as he got to the Square. Approximately how far did he walk?
I try to teach for understanding, using a problem-posing, questioning approach that connects the areas in which learners have strengths. For example, instead of directly teaching my learners to do these problems:
1. 3x5 + 6
2. 102 + 10x5
3. 25/5 + 35/7
using the PEMDAS or "Please Excuse My Dear Aunt Sally," (parenthesis, exponents, multiplication, division, addition, subtraction ) rule for order of operations, I ask them to generate their own expressions in my Number of the Day activity. This also gives me a daily assessment of the depth and breadth of my students' grasp of computation, order of operations, and use of symbolic notation. I write "Number of the Day" on the board, and a box with a number next to it. I ask the students to write as many numerical expressions as they can that equal the number of the day. Student responses the day the number was 350 included: 300 + 50; 200 + 100 + 25 + 25; 70 x 5; 50 x 6 + 50; 182 + 26; 150x2 + 50.
My adult learners have access in the classroom to appropriate technology: calculators and computers. Rather than being a crutch, calculators are an invaluable aid in teaching basic mathematics because of their speed, accuracy, decimal display, and memory. Many of my learners already rely on calculators in every day situations as workers and consumers. I use them as instructional tools to assist in the development of concepts, to help reinforce skills, to promote higher level thinking, and to enhance problem-solving instruction. By freeing them from the routine, long, or complex calculations, more time can be spent on conjecturing and reasoning.
It is challenging to change one's practice to the values and practices of the NCTM and ABE Math Standards, particularly when the ideal is not readily shared by other teachers and is not in the experience of our learners. I find many learners look to me for answers when I'm trying to have them develop that capacity themselves. One of my students complained to his counselor that "at the end of the day, we're tired from working, and she expects us to think." It is easy to fall back into old methods of direct instruction and worksheets and workbook pages. Also, teaching using the math standards means covering less material while taking time for discovery. That's hard when students need their GEDs by June. So sometimes this way of teaching brings its own discomforts, to me and to my students.
The only evidence I have for the success of my approach that "less is more" is that I rarely cover all the material in the GED textbooks in the 14 to 15 weeks of my course. Yet most of my students seem to have the confidence to take and pass the test. My experience with teaching a range of learners, from literacy students to GED, has convinced me that teaching for learning in the vision of the math standards is good adult education practice.
Curry, D., Schmitt, M.J., & Waldron, S. (1996). A Framework for Adult Numeracy Standards: The Mathematical Skills and Abilities Adults Need to be Equipped for the Future. Boston, MA: World Education.
Leonelli, E.& Schwendeman, R. (eds.) (1994). The ABE Math Standards Project, Vol 1: The Massachusetts Adult Basic Education Math Standards. Holyoke, MA: Holyoke Community College/SABES Regional Center.
Massachusetts Department of Education (1996) Massachusetts Curriculum Frameworks for K-12 and Adults. Malden, MA: MA DOE.
Moses, R. (1997) "Mathematics, Literacy, Citizenship, and Freedom," presentation at NCTM 75th Annual Meeting, April 19, Minneapolis, MN.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston VA: NCTM.
Russell, S.,J., Tierney, C., Mokros, J., Economopoulos, K., et al. (1998). Investigations in Number, Data, and Space (TM). White Plains, NY: Cuisenaire/Dale Seymour Publications.
Schmitt, M.J., (1991). The Answer is Still the Same It Doesn't Matter How You Got It! A Booklet for Math Teachers and Math Students Who Come from Multicultural Backgrounds. Boston, MA: Author.
Esther D. Leonelli is technology coordinator and a math instructor at the Community Learning Center, Cambridge, MA. She is a co-founder and past president of the Adult Numeracy Network, an affiliate of the National Council of Teachers of Mathematics. Esther moderates the numeracy list, an electronic discussion list for adult education practitioners (numeracy @worldstd.com). During 1998-99 she was an NIFL Leadership Fellow.