Mathematics Curriculum Framework
Achieving Mathematical Power - January 1996
These are the underlying beliefs and tenets central to the vision of mathematical power and content standards for mathematics education in Massachusetts.
- Students explore mathematical ideas in ways that maintain their enjoyment of and curiosity about mathematics, help them develop depth of understanding, and reflect real-world applications.
- All students have access to high quality mathematics programs.
- Mathematics learning is a lifelong process that begins and continues in the home and extends to school and community settings.
- Mathematics instruction both connects with other disciplines and moves toward integration of mathematical domains.
- Working together in teams and groups enhances mathematical learning, helps students communicate effectively, and develops social and mathematical skills.
- Technology is an essential tool for effective mathematics education.
- Mathematics assessment is a multifaceted tool that monitors student performance, improves instruction, enhances learning, and encourages student self-reflection.
Students explore mathematical ideas in ways that maintain their enjoyment of and curiosity about mathematics, help them develop depth of understanding, and reflect real-world applications.
- Though math learning follows a certain progression, it is not a purely linear process, but is recursive, with children needing to rediscover and refine "old" concepts and skills as they build "new" ones. Also, as with all learning, the development of math skills is unique to each child.
-- Early Childhood Today
Developing mathematical power is a complex process. The mathematics that students learn depends not only on what is taught, but also on how it is taught. Curriculum cannot be separated from the instructional practices used to teach it. Instructional strategies should encourage students to engage intellectually with important mathematical ideas, to embrace the aesthetic value of mathematics, and to use mathematical principles to solve problems in their daily lives.
Teachers must have high expectations for all students and be familiar with each student's knowledge base in order to plan developmentally appropriate work. The intellectual, social, and emotional development of students should guide our choices of mathematics experiences for our students.
- What experiences has this student encountered at home, on the playground, in the cafeteria or classroom, or in earlier years that affect how the student learns?
- What is the learner's background in mathematics education? What mathematics knowledge has the learner acquired? What challenges the student? Has the learner overcome obstacles with aspects of mathematics?
The following vignette depicts how a teacher explores ratio with his students at different developmental stages:
- Mr. Manning plans to explore ratio with his middle grades mathematics class. He knows some of the students are not developmentally ready to grapple with proportionate reasoning in terms of ratio and percent, while others have begun to explore these concepts of number sense and relationships in the context of data analysis. Knowing that all students have developed an understanding of fractional expressions of number, Mr. Manning chooses to approach ratio and percent through a probability investigation based on the hand game of Paper, Scissors, Rock. This provides accessibility for all students, while respecting their individual developmental levels.
Experimenting with ideas, inventing constructs, and exploring our curiosities are at the foundation of all learning, including mathematics. Asking the right questions of students promotes creative thinking, prompting them to look deeper into their imaginations. Students can be encouraged to reflect on their learning and articulate their reasoning through questions such as:
- How did you work through this problem?
- Why did you choose this particular strategy to solve the problem?
- Are there other ways? Can you think of them?
- How can you be sure you have the correct solution?
- Could there be more than one correct solution?
- How can you convince me that your solution makes sense?
A rich matrix of ideas should be explored thoroughly throughout each academic year. Students should have regular opportunities to revisit important mathematical ideas throughout each school year and from one year to the next.
If students are to develop mathematical understanding, then they should engage in tasks of inquiry, reasoning, and problem solving that reflect real-world mathematical practice. In addition, hands-on exploration can deepen understanding of abstract concepts by encouraging the practice of process skills and communication, and allowing for reflective thinking. For example, when students explore prime numbers, they may engage in a manual approach such as the Sieve of Eratosthenes, then extend their understanding by modeling prime and composite numbers when they build rectangular arrays with cubes, as well as communicate and reflect upon their ideas about prime numbers when they create human pyramids. When students later use technology to explore number theory, they have already acquired a knowledge base of the concept.
Students learn best when they can connect their classroom learning to real-life experiences, and when they can experience the same concept or idea in multiple contexts. For example, students often make place value errors when working with a multiplication algorithm. To address this, another approach to learning the algorithm would be exploring multiplication by means of students' using place value blocks to represent area, then using addition to add the units, rods, and flats to measure area, thereby inventing for themselves the multiplication algorithm and discovering the meaning and importance of place value in mathematics.
The vision for mathematics in Massachusetts will become reality only if students deepen their understanding of mathematics by means of activities, investigations, and projects that promote inquiry, discovery, and mastery. For example, students may make pinwheels to explore rotational symmetry. A group might canvas the cafeteria, collecting data from the student body to test a hypothesis. Real-life meaning and mathematical structure will permeate the mathematics curriculum.
Investigations that a teacher introduces to a class should target important mathematical ideas, illuminate the connections among mathematical ideas, and identify relationships between the ideas introduced in the investigation and the concepts with which students are already familiar. It makes sense to embed problems or investigations in and draw resources from the various cultures and backgrounds of students. The questions that follow are suggested as one way to help teachers plan investigations.
- Have I identified and defined the mathematical content of the investigation, activity, or project?
- How does the investigation address the learning styles and diverse backgrounds of all students within the classroom?
- Do I have a plan to initiate thoughtful discussion of or reflection on the concepts explored and of the relationships uncovered in the process to help students clarify their understanding and integrate it fully into their existing network of ideas?
- Have I carefully compared the network of ideas included in the curriculum with the students' knowledge?
- Have I allowed time to note discrepancies, misunderstandings, and gaps in students' knowledge as well as evidence of learning?
- How is the investigation designed to test students' false assumption, confirm accurate findings, and extend the students' knowledge?
All students have access to high quality mathematics programs.
- "My mom and I can't do math."
-- Alma, a second grade student from Dorchester
"Let's get together and talk about math."
-- Alma's teacher to Alma's mother
All students must have access to high quality mathematics programs that support successful learning of mathematics and help them to develop a mathematical sense and intuitive understanding. It is our responsibility to ensure that students are fairly represented in each mathematics program and have equitable access to resources. Everyone learns best in an environment that acknowledges, respects, and accommodates each learner's background, learning style, and gender. For example, a teacher's listening attentively to all students' ideas helps to foster in students a sense of control of their future. Or, by making special efforts to achieve classroom integration when students self-segregate, a teacher enhances students' respect for others' backgrounds and learning styles.
This vignette shows how one Massachusetts teacher seized the moment to explore ratio in the context of gender equity. She relates:
"When students were surprised that gender prejudices existed, I was relieved.
I suggested that we conduct some kind of investigation to see whether their perceptions of my lack of bias were accurate.
We decided that a sample of just that one classroom would not suffice. Rather, the entire grade comprising ninety-two students became involved. For two weeks, students monitored the number of girls versus the number of boys on whom I called. Because there were so many more boys than girls in one class, they were quick to decide that ratios were needed (daily, because of absentees). I capitalized on this for a quick review of ratio fundamentals!
When it seemed that I, indeed, may have been favoring the boys, I wanted my students to know that it was unacceptable to me. Gradually, I noticed that there was a more balanced distribution of raised hands between the boys and girls."
- "The underrepresentation of certain groups in science, technology and mathematics education is well documented. The Statewide Systemic Initiative seeks to effect change in student participation by transforming school curriculum, classroom instruction, teacher education, and by increasing parental support so that all students have opportunities suited to their needs to learn science, technology and mathematics."
--National Science Foundation, Equity Framework in Mathematics, Science, and Technology Education
All students should see themselves as mathematicians, capable of using their evolving mathematical power to solve new problems. Some students may ultimately progress further in their mathematical learning than others, and learning may take different amounts of time for different learners. However, if each student is offered an accessible approach to learning mathematics, consistent with his learning style and experience, then all students can learn mathematics. This means establishing high standards of expectations and helping students when they are struggling with mathematics. It is not enough to enroll students in higher level classes; everything possible must be done to engage their interests. Each member of every class should participate meaningfully.
The diversity in communities and classrooms should be treated as an advantage that can help all learners in Massachusetts schools. The presence of diverse learners in Massachusetts classrooms presents an opportunity for all students and teachers to learn about the rest of the world and appreciate the talents and culture of each individual. Since different cultures sometimes use alternative mathematical strategies or perceive the relationships of objects and events in the world in ways other than the mainstream culture, their strategies and understandings can enrich the understanding of all students. For example, Cambodian children learn a different algorithm for division. If given the opportunity to explain their method to the rest of the class, then everyone broadens their cultural experiences, deepens their understanding of the concept of division, and recognizes the varied approaches to mathematics.
If the women, minorities, and individuals with disabilities who have made important contributions to the field of mathematics and embarked on careers that utilize mathematics are presented as role models, then all students see for themselves its practical applications. Knowing that increased opportunities await students who are mathematically empowered, families, educators, and communities should encourage students to continue their mathematics learning through grade twelve. Guidance counselors, students, and families should be fully aware of the impact that mathematics has upon future access to higher education and employment opportunities.
Mathematics learning is a lifelong process that begins and continues in the home and extends to school and community settings.
The need to make sense of the world begins before kindergarten and continues beyond formal schooling. PreKindergarten students begin to form ideas about mathematics as part of the natural process of exploring their world. Building with blocks gives them an opportunity to begin developing an understanding of shape, size, position, and symmetry. Gathering items such as rocks, shells, toy cars, or erasers for their collections leads to discovery and exploration of patterns and classification. Such informal explorations are important developmental precursors to an understanding of mathematics. They are the beginning of lifelong skills that enable us to learn more abstractly.
Preschool activities give children opportunities to solve problems and discover information for themselves in an environment where they can explore freely and safely. Young children and school-age students need repeated experiences exploring materials, time to talk about their experiences, and freedom to experiment and learn from each other. In addition to the mathematics that students learn in PreKindergarten through grade twelve classrooms, there are other arenas in which the learning of mathematics can be strengthened.
For example, families explore mathematics together, as described in this vignette:
Teddy is working with his grandmother, sorting laundry. Teddy picks up several socks and puts them in a row. His grandmother says,"Teddy, you're sorting socks!" Teddy replies,"These are the Three Bears' socks - Mamma Bear's sock, this big one is Papa Bear's and this one is Baby Bear's!" Grandmother says, "I need to sort these socks to put them away. Help me sort them." His grandmother holds up the 'Mamma Bear's sock' and Teddy finds a matching one and they fold together. Teddy and his grandmother continue until all the socks are sorted into pairs. They count the pairs of socks they have sorted. Later on in the morning, Teddy's grandmother reads aloud Teddy's favorite story, Goldilocks and the Three Bears. Later on in the day, Teddy continues sorting laundry with his grandmother. Teddy picks up several socks of many different sizes. He starts placing them randomly on the floor--large red sock and small blue sock. His grandmother suggests another way to sort the socks. "I wonder if we can find all the socks that are the color red," she says as she holds up a red sock. Teddy helps find five other red socks. He then counts the red socks on the floor: "One, two, three . . ." Later in the afternoon, Teddy's grandmother reads aloud One Fish, Two Fish, Red Fish, Blue Fish by Dr. Seuss.
- Learning about math, science, social studies, health, and other content areas are all integrated through meaningful activities such as those when children build with blocks; measure sand, water, or ingredients for cooking; observe changes in the environment; work with wood and tools; sort objects for a purpose; explore animals, plants, water, wheels and gears; sing and listen to music from various cultures;and draw, paint, and work with clay.
-- National Association for the Education of Young Children
Community programs, museums, and businesses can provide valuable learning experiences that enhance and extend classroom activities. Careful planning in these contexts will provide opportunities for developing students' curiosity, creating skepticism, and promoting self-esteem, as shown in the following vignette:
Mrs. Jones, who teaches both science and mathematics to her sixth grade class, is beginning a unit on models. She wants to work with her students on proportion and scale and decides to take a trip to the Museum of Science to introduce these ideas to her class. Although Mrs. Jones is familiar with the museum, she has never used it to study models, so she visits the museum before the field trip. (Massachusetts teachers have free admission and parking privileges.) She finds the museum full of models--living organisms and mathematical, physical, life-sized, and scaled models--everything from mollusks to machines.
Before the trip, Mrs. Jones introduces the concept of models to the class and asks her students to make a list of familiar models. She asks the students to think of models they have seen in the museum on previous trips, and continues the discussion by posing questions such as: What are the differences between models and real objects? Can models be misleading? When are scale models useful? Is there a change in structure when a model is scaled down or up?
At the museum, students are asked to find examples of a real object, a larger than life model, a life-sized model, a smaller than life model, a physical model, and a mathematical model. Students then create lists of what the models do and do not tell them about the object. They also respond to a series of questions. Does the model suggest something that might not be true? (e.g. a globe of the earth in which the height of the mountains is in a different scale from the planet?) Why is the model portrayed in a particular scale? Can the scale be misleading?
Back in class, students compare their findings and discuss their observations and ideas. At the end of the unit, each student uses clay to create three scale models of an object: life-size, one-half life size, and twice life-size. Mrs. Jones uses this project to assess the students' knowledge and understanding of models and scales.
- The learners' inquiries into the formation of graphs lead to more work . . . they were instructed to bring in graphs of interest to them . . . . There were bar graphs on everything from consumption of candy bars per person per year, to circle and bar graphs on the concentration of wealth in the United States . . . and most important, [on] how the less educated were suffering more. This last one had a great impact upon my learners, for they noted the importance of a high-school diploma and its connection to a better income.
-- Adult Basic Education Math Standards Project
Adult mathematics learners often seek further education to meet a specific goal, perhaps to advance their career, or to help their children, or for self improvement. This framework supports them by offering hands-on problems based on real-life situations, using a range of manipulatives and technologies. Adult learners should be active participants in defining personal learning objectives and deciding measures of success. Instruction should include opportunities to question, discuss, and write about ideas.
Mathematics instruction both connects with other disciplines and moves toward integration of mathematical domains.
Upon this gifted age, in its dark hour,
Falls from the sky a meteor shower
Of facts . . . they lie unquestioned, uncombined.
Wisdom enough to leech us of our ill
Is daily spun; but there exists no loom
To weave it into fabric . . .
-- Edna St. Vincent Millay
To help us understand the world, we draw upon a knowledge base that spans disciplines and experiences, forming networks of thoughts and ideas. Students may relate their knowledge of functions to mechanics, patterns to music, or statistics to the economy in Massachusetts. In all cases, they are connecting their understanding of mathematics with other disciplines and their world.
A move toward an integrated approach within the mathematics curriculum is recommended. Beginning with PreKindergarten, an integrated approach to mathematics might include activities that combine sorting, measurement, estimation, and geometry.
In middle schools and high schools, it will mean helping students make connections between algebra and geometry, but also among ideas from discrete mathematics, statistics, and probability. In Adult Basic Education, it will mean establishing connections between mathematics and daily life at home, at work, and in the community.
An integrated mathematics curriculum gives students a more accurate picture of the nature of mathematics, contextualizing the essential connections among various fields of mathematics. Integrating the domains encourages students to approach problem solving in more than one way, making them more powerful problem solvers. Students will be able to solve problems numerically, algebraically, and graphically. The use of technology and computer software facilitates connections and integration. For example, graphing calculators make it possible to switch from equations to graphs to data analysis. If a problem can be approached either visually or numerically, then it may be more accessible to a visual learner struggling with abstraction. As solutions are shared, these same visual learners will have the opportunity to explore the problem numerically.
Furthermore, an integrated mathematics curriculum reduces review time, since students will continually encounter ideas from all parts of mathematics. More time will be available to teach recommended mathematics topics, such as statistics and probability, which have not been part of the traditional mathematics curriculum for all students.
- Socrates advocated reflection as opposed to observation, an activity dependent upon a principle that is important to any theory of reflective method: what we are trying to do is not to discover something of which until now we have been ignorant, but to know better something which in some sense we knew already; to know it better in the sense of coming to know it in a better and different way.
-- Paul Bitting and Renee CliftImages of Reflection in Teacher Education
Part of the task for schools and districts will be to develop an approach to a more integrated curriculum that is appropriate for the local circumstances. The following vignette shows how it worked in one high school in the Commonwealth.
During the 1991-1992 school year, a team selected new integrated textbooks. Members considered two critical questions during their evaluation. The first consideration was, did the textbook include a variety of examples and applications at many different levels so that students could proceed from simple to more complex problem-solving situations? And second, were algebra and geometry truly integrated rather than presented alternately, like a layer cake, within the text?
Since then, teachers and students use a variety of approaches to explore mathematical ideas. For example, when working with the Pythagorean theorem, students do a hands-on proof, actually building squares on the legs of right triangles. They explore the theorem algebraically and by using a geometric proof. Teachers usually present for about five minutes at the beginning of class. Students work on problems both in cooperative groups and alone. Often they work with a partner, and then share results in a foursome. Students are asked to write and explain their thinking. At times, a group project is required.
Assessment in this high school is often unorthodox. For example, one year students made a design that used several different figures, computed the area of each figure, and recorded how they tackled the problem. Although it took one week for teachers to review the projects, they gained many insights about students' understanding. In the cases where teachers could not understand a student's work, the student was asked to explain it.
Part of what made the integration efforts successful at this school is a history of open communication. Teachers are willing to observe and be observed by one another. They discuss individual students among themselves and with others, such as resource teachers. The results of the teachers' efforts are an improved mathematics program for the students and professional growth for the teachers.
Working together in teams and groups enhances mathematical learning, helps students communicate effectively, and develops social and mathematical skills.
Independent learning helps to develop self confidence. However, students deepen their understanding of mathematics as they interact with the ideas, theories, and opinions of their peers and teachers. Being able to communicate mathematical ideas in a variety of ways helps students to "develop, test, and evaluate possible solutions" as suggested in the Common Core of Learning. Team work encourages members to interact with others, enhances self-assessment, encourages the exploration of multiple strategies, and helps prepare students to be members of the workforce. The variation offered by group work leads to enriched solutions and offers an ideal venue for informal assessment. Here is a vignette that describes one teacher's use of team work in elementary school.
Ms. Gutierrez' third grade class is enthusiastic this morning as students actively learn division facts. One group is working diligently with base ten blocks, discovering patterns in division. In the opposite corner, another group is comfortable on the reading rug, brainstorming ideas to devise a rule for using division by ten and one hundred to help them solve problems mentally. After lunch, students from both groups will pair up and present their findings to each other.
A teacher may group students in mathematics for a number of reasons -- students may be working together on different projects, the teacher may work with a group of students who need special help, or highly motivated students may be working on enrichment activities. Some groupings may be temporary and some may be of longer duration.
One of the most complex and difficult tasks for teachers, schools, and districts will be how they group students in order to promote the achievement of mathematical power for all students. Here are some considerations:
- High expectations and standards should be established for all students, including those with gaps in their knowledge bases.
- Students should be encouraged to achieve their highest potential in mathematics.
- Students learn mathematics at different rates, and different students' interests in mathematics vary.
Support should be available for all students based on individual needs. Appropriate opportunities for enrichment and advancement need to be provided for students at all achievement levels. The mathematics should be challenging and expectations for all students should be high.
Technology is an essential tool for effective mathematics education.
- All students should use computers and other technologies to obtain, organize and communicate information and to solve problems.
-- Massachusetts Common Core of Learning
Students in all courses need access to tools for learning mathematics. These tools include measuring instruments, manipulatives, graphing calculators, and computers. The calculation powers of computers and calculators free students from the rote tasks of computation but do not replace the thinking that underlies mathematical operations. Computer software for modeling and visualization of mathematical ideas such as statistics and probability or fractals and chaos can open a whole new world to students and help them connect these mathematical ideas to their language and symbol systems.
"Students at one Massachusetts high school explore open-ended lab projects, using a geometry construction software program that enables them to collect data, make observations, and develop conjectures. They must support their findings by writing lab reports that summarize the exploratory process and conclusions they have drawn. This makes the transition to deductive proof a much more natural extension of the learning process. Students are encouraged to construct and justify their own understanding of the geometry explored when they participate in a follow-up discussion, facilitated by the teacher." 1
Technology tools, when integrated in a mathematics program, raise the level of mathematics to which students can be exposed, improve their self-confidence, and facilitate increased student-teacher interactions. The availability of calculators, computers, and other technology has changed forever the way that people are able to think about and do mathematics. New technologies have changed our culture into an "information society." These changes have "transformed both the aspects of mathematics that need to be transmitted to students and the concepts and procedures they must master if they are to be self-fulfilled, productive citizens." 2 Some mathematics becomes more important because technology requires it, some becomes less important because technology replaces it, and some becomes possible because technology allows it. This integration of technology in our global society today impacts the lives of our students tomorrow.
- When technology is implemented in our schools, change occurs. Electronic formats enable access to information from almost anywhere in the world, crossing age groups and cultures with ease . . . . Students move into the world possessing the skills to gather, manage, and assimilate the vast resources of information at their fingertips.
-- The Switched-on Classroom, Massachusetts Software Council
For students with special needs, technology can be especially helpful in assisting students in regular and special classrooms, the home, and the community. New software, hardware, and assistive devices can all be used to help students succeed. Technology can enhance a student's access to the curriculum, increase opportunities to interact with peers, and increase avenues of communication.
Mathematics assessment in the classroom is a multifaceted tool that monitors student performance, improves instruction, enhances learning, and encourages student self-reflection.
- The assessment should examine the extent to which students have integrated and made sense of information, whether they can apply it to situations that require reasoning and creative thinking, and whether they can use mathematics to communicate their ideas.
-- National Council of Teachers of Mathematics
The goal of classroom assessment is to provide information about students' evolving mathematical understanding, skills, and knowledge, so that teachers can give feedback to students and make decisions about where to go next with their instruction. Mathematics assessment has been primarily short answers to short questions. While standard assessment is one method of evaluation, a broader interpretation of mathematics suggests that assessment must take on many new dimensions.
Performance-based assessments are especially congruent with the goals of mathematics instruction. The three types of performance-based assessments discussed here are open-ended written assessments, portfolios, and observation.
Open-ended written assessments present questions to students that invite multiple approaches to problems, allow for creative expression of mathematical ideas, and encourage comparative analysis and reflection. By soliciting written responses, students are encouraged to communicate their strategies, develop their hypotheses, and explain their solutions by using prose, graphs, or drawings. An example of an open-ended question can be as simple as asking students to explain their reasoning or to justify their answers, or as multifaceted as asking them to design and conduct their own probability experiments.
Portfolio assessments imply that teachers have worked with students to establish individual criteria for selecting work for placement in a portfolio and judging its merit. Charts, models, constructions, and students' reflections on their work can all be included within mathematics portfolios. The contents of a mathematics portfolio should be indicative of each student's abilities and understanding, representative of his efforts, and indicative of progress over a period of time.
Observation as a means of assessment reflects a student's appreciation of mathematics, or the strategies he commonly employs to solve problems, or perhaps his preferred learning style. Formal observations are pre-planned, target specific mathematical skills, and refer to established criteria. These criteria for performance are usually expressed within a range and made known to the student.