ANPN STANDARDS FOR MATHEMATICAL LITERACY
There are three "process" standards for adult numeracy. The Massachusetts ABE Math Standards posited Problem solving, Communication, Reasoning, and Connections. Through ANPN's further research, these four standards, often referred to as the "process" standards, were consolidated into three adult numeracy themes: Relevance/Connections, Problem solving/ Reasoning/ Decision making, and Communication. Responses from the ANPN study showed that it was difficult for individuals to distinguish between problem solving and reasoning, both key skills in decision making. The data also revealed that the issue of relevance frequently occurred.
The remaining seven Massachusetts ABE Math Standards have been integrated into four adult numeracy content themes. Number and Number Sense includes two previous standards, Estimation and Number, Operations and Computation. Data is similar to the standard of Statistics and Probability. Geometry: Spatial Sense and Measurement incorporates two previous standards. The two standards of Patterns, Relationships, and Functions, and Algebra correspond to the adult numeracy themes of Algebra: Patterns and Functions. This reorganization is a reflection of the words of adult learners, teachers, and stakeholders as they told the ANPN about the math that they need and use.
In addition to the three process and four content themes, a good deal of data collected dealt with mathematical empowerment. These affective issues continually emerged. Learner self-confidence, attitudes about mathematics, beliefs about what one can and cannot accomplish in mathematics, and math anxiety all were repeated. It is believed that when adults talk about the affective aspect of math, they are referring to their self-confidence in doing math and their sense of competency around tasks involving math. Therefore, there is a section on adults' feeling and attitudes about math, titled Competence and Self-Confidence.
These are not the final standards but merely a framework of themes for developing true standards, a true "honest list" of what is need for adult numeracy. Much more work and reflection remains. This is merely the beginning and we invite you to be a part of it.
ANPN NUMERACY THEMES
RELEVANCE / CONNECTIONS
Adults need to see connections in math -- connections within the domain of math itself, connections to other disciplines, and connections to real life and work situations. Math takes on greater meaning and understanding when it is directly applied in the workplace or in real-life situations. Many adult learners feel that their best math situation was when they learned math at work. This suggests that the math they learned on the job was directly applicable for them.
Adults see little relevance or connections between math and their everyday living and working conditions. Adults often ask, "What is it used for?" about math topics that they, thus far, have seen little relevance or connections in their everyday living and working situations. Adults feel they are more successful when they are able to link any new learning to something they already know. Textbook math, and particularly word problems, seem to have little relevance to what adults perceive as math in everyday life. Adults's real math skills often don't show when they do meaningless word problems. Adults often actually use math successfully in their daily lives, yet fail to see any connections to work problems presented in the class.
IMPLICATIONS FOR LEARNING AND TEACHING
Math must be taught in the context of real-life and workplace situations. When math is taught in context, adults understand that there is a practical application for that skill. Teachers may need to become more knowledgeable about the world of work in order to offer relevant math curricula.
Learner-centered approaches to teaching ensure that learners see the relevance of what they are learning. Math learning for adults should be relevant to their own personal goals, whether it be to attain a GED® certificate, a job, or whatever. Adult learners need to have a voice in what is taught in the classroom. The teacher may think something is important for the adults to learn, but unless the learner sees the relevance to his/her own life, he/she finds little value in the topic. Whether teachers "hook" the students or get them to "buy into" the math, adults will find the relevance when the material is relevant to their needs and goals.
Interdisciplinary approaches to teaching are essential. Math should be an integral part of other content areas.
New math learning should be linked to previous learning. Linkages should be made with other math concepts and skills as well as with other prior knowledge. Not only should the new learning be connected to prior learning, but there should also be a connection between knowing how to perform a skill and being interested in performing that skill. Concepts should be taught before rules. This time is well worth the benefits that will result.
We must help adults see the relevance of learning by seeing the "big picture". When adults see how math skills are interconnected with one another, they begin to see relationships and the relevance of what they are learning.
PROBLEM SOLVING / REASONING / DECISION MAKING
Problem solving, reasoning, and decision making are three very interconnected processes adults engage in continuously, whether they are using numbers of words. The SCANS report defined these higher order thinking skills as follows:
Problem solving entails that persons recognize that a problem exists, identify possible reasons for this discrepancy, and devise and implement a plan of action to resolve it. The problem solvers evaluate and monitor progress and revise the plan as indicated by findings.
Decision making entails that persons specify goals and constraints, generate alternatives, consider risks, and evaluate and choose the best alternatives.
Reasoning entails that persons discover a rule or principle underlying the relationship between two or more objects and apply it in solving a problem. For example, the persons must use logic to draw conclusions from available information, extract rules or principles from a set of objects or written text, apply rules and principles to a new situation, or determine which conclusions are correct when given a set of facts and a set of conclusions.
Reasoning is a key step in problem solving and decision making. Adults use reasoning to analyze information in order to solve problems which, in turn, allows them to make reasonable decisions.
Math skills are integrated in the problem solving and decision making processes. Although it is clear that math skills are integrated in the problem solving and decision making process, the skills needed vary from problem to problem. The math skills needed to solve problems and make decisions are integrated throughout the process, with more than one math operation generally being required to come to final decisions.
Problem solving is a process that includes seeking to understand the problem, the figuring out what information and math skills are important to use to solve the problem. It is clear that the process of solving problems requires an understanding of the situation. Problems cannot be solved without the understanding of the situations.
It is important for adults to have a repertoire of strategies and tools to solve problems. These varied strategies include the use of calculators. While many persons are not yet convinced as to the importance of calculator use, it is important to note that the SCANS report lists "technology" as one of the five important competencies necessary for a literate workforce.
Problem solving and decision making often involve teamwork. On the job and in daily situations at home, problems are solved and decisions made with the advice and input of others. While in school situations, teamwork is not as often encouraged, at home, and in the community, individuals must work together to solve problems and more forward.
Parents, workers, and community members use problem solving and reasoning to reach decisions. Being able to problem solve successfully in the workplace gives workers more confidence, which in turn, gives them more of a voice. Adults use problem solving strategies as parents to "survive". They need to maintain budgets and comparison shop. Adults also use problem solving strategies to better understand how their money is manipulated. Adults, in their role as citizens, also have to solve problems and make decisions using numbers.
IMPLICATIONS FOR LEARNING AND TEACHING
Math content skills must be embedded in processes like problem solving, reasoning, and decision making. Processes such as problem solving are viewed as more than just a topic to be covered in an adult education classroom. Mathematics must be real, not just pseudo-real problems in a book.
Reasoning and problem solving should be integrated in all teaching. Even when teaching basic skills, such as reading, writing, and math, higher level thinking skills such as reasoning, problem solving, and decision making should be incorporated into the lessons. Reading and mathematics become less abstract and more concrete when they are embedded in one or more of the higher level thinking skills. One need not choose between basics and higher level thinking; students become more proficient faster if they learn both simultaneously.
Learners must be provided with opportunities to work in groups. Learners do learn from one another. Working together in groups also gives learners opportunities to hone personal qualities such as self-esteem, sociability, self-management, integrity and honesty. Interaction is not only a key foundation skill,
It is one of the five major competencies needed in the workplace according to the SCANS report. Competent employees are skilled team members and teachers of new workers; they serve clients directly and persuade co-workers either individually or in groups; they negotiate with others to solve problems or reach decisions; they work comfortably with colleagues from diverse backgrounds; they responsibly challenge existing procedures and policies.
Skills like these are not developed overnight, nor are they simply "picked up". Learners need to interact with their peers in problem solving teams within the classroom environment. Adult learners need to work in group situations in order to learn to check reasoning and take advice and suggestions from others. Genuine respect and support of each other's ideas is essential for learners to be able to explain and justify their thinking and to be able to understand that how the problem is solved is as important as its answer. In all adult basic education math settings, the development of critical thinking skills is crucial. Statements should be open to question, reaction, and elaboration from others.
Math is language. Mathematical communication is an overarching process which includes understanding, expressing, and conveying ideas mathematically in order to reflect on and clarify one's thinking, to make convincing arguments, and to reach decisions. Effective workers must be able to interpret and communicate information and communicate ideas to justify positions. In the workplace, much of this information and many of these ideas are mathematical.
Mathematical communication can occur in any relationship and context. In the ABLE setting, communication happens among learners and between learners and their teachers; at work among workers and between workers and their supervisors; at home among family members and between children and their parents; and in the community among individuals and between community members and public officials. Good mathematical communication is like all other effective communication requiring listening, speaking, reading and writing skills along with interpersonal skills.
Communication is essential for understanding. Communication provides the foundation for learning in school and in life. Communication includes knowing when and being able to ask for help both in the classroom and in life. Communication, in math as in other aspects of life, is the bridge to finding and exchanging ideas, to identifying problems, and to seeking and finding solutions to these problems. Communication is essential to working collaboratively at home, at work, and in the community. Communication is the link that makes other math skills effective.
Mathematical communication, the representation of a problem in mathematical language, also happens in the "other" direction, especially as individuals interact with technology. As technology becomes more pervasive, it is necessary for one to be able to distill the elements of a real situation into a mathematical expression, the universal language. In order to communicate the problem to any one of our technological aide, it first must be translated to symbols and then the results from the machine must be interpreted in light of the situation.
IMPLICATIONS FOR LEARNING AND TEACHING
The focus on mathematical communication should be increased. Teaching mathematical communication is integral to the success of math reform efforts. Students always working by themselves with no requirements that they communicate their problem solutions to anyone else is not helpful for their progress. Pairings and small group work are viable alternatives to whole-classes instruction, especially in light of rolling enrollment due to open-entry, open exit classes. Teachers need to use a variety of approaches, models, and manipulatives and have the students involved in talking about their work with each other on a frequent and regular basis.
Good mathematical communication for work, home, and community situations through group discussions should be encouraged. Math should be taught using a well-defined vocabulary of math terms so that what the teacher believes is being taught is what is being received by the student. This should involve verbal and written feedback from the students to confirm that they understand and can express to teachers what they know. As a skill necessary to future employees, students should be able to express mathematical ideas and concepts orally and in writing.
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