The following statement by the Mathematical Science Education Board as recorded in
“Effective teachers are those who can stimulated students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding. To understand what they learn, they must enact for themselves verbs that permanent the mathematics curriculum: “examine,” “represent,” “transform,” “solve,” “apply,” “prove,” “communicate.” This happens most readily when presentations, and in other ways takes charge of their own learning.
This curriculum may establish such a community of learners in the classroom. The actual mathematics in this curriculum is not as important as the instructional strategies such as communicating through writing or speaking, using manipulative, working in cooperative groups, and alternative forms of assessment.
Maps and scale drawings are used in everyday life and mathematics. This curriculum is for students to explore distance and angle measurement and the concept of reading and making maps. Students will be actively engaged in the process of learning as they work in group and individual settings. Students are asked to apply their learning is situations that will require an understanding of the concept of proportionality as it applies to measurement.
This method of instruction may be quite different than methods previously experienced by some students. The purpose of this curriculum is to introduce or reaffirm the instructional strategy and classroom practices used throughout this lesson. In other words, it sets the tone for this study of mathematics for the entire curriculum.
Please note that many of the activities in these lessons have more than one task. These multiple tasks provide flexibility for the teacher.
Although this is an introductory curriculum, you do not need to confine the tasks to the first lesson. You may reaffirm a certain instructional strategy or classroom practice at anytime during the activity. For example, you may wish to complete a cooperative learning activity after a vacation to reestablish the proper atmosphere for learning groups.
In this curriculum, students experience the concepts of scale, similarity, proportional reasoning, and basic geometric constructions. They read and construct different kinds of maps and scale drawings, which call for multiple representations of geometric and numerical data. They become familiar with similar figures by observing patterns and making generalizations. They explore the idea of a path, both in the field and on paper, and estimate both linear and angular measurements in the process of creating paths. They bisect angles, copy angles, and construct triangles using a compass. The coherent mathematical idea underlying this curriculum is the study and application of proportional relationships.
The tasks required in this curriculum are accessible for all students. In order to have success in this unit, students should have had an introduction to distance and angle measure. The unit will flow more smoothly if the students have worked in cooperative groups, used a compass and protractor, communicated mathematical ideas through writing, speaking, and modeling and used technological tools such as calculators and computers. Some students may have had prior experiences in mathematics that they will find valuable in completing the lessons, such as making conjectures, designing maps, and preparing reports. If a significant number of students have not had these experiences, it may be necessary to take additional time to provide them.
Working in groups, students use pattern blocks to create a map of Faulkner’s Island and mark a Wildlife refuge. They indicate the direction and number of units for each move. They look for the best location to locate the Reseate Tern nesting site.
Cooperative groups play an important role in this curriculum. Students interact and work in small groups throughout each of the lessons. Team building and working together are important skills students will need and use in life beyond school. For some students, working with others will be a new experience. Some care will need to be taken to help students develop the skill of collaboration and respect of the ideas of others. Working in pairs is often a good introduction to working with others. Later, pairs can join with other pairs to make groups of four.
When students are working in groups, sharing and listening to others becomes the key to successful mathematical decision making. Your role should be that of a facilitator. Your job includes careful listening and effective questioning to help students stay on or get themselves back on track. Each student should also have a role in the group such as recorder or materials manager so that he or she becomes responsible for his or her own learning. Student roles should change so that each group member has opportunity to fill each role.
Finally, it is important to discuss, either verbally or in writing, how the group has functioned. Questions such as “In what ways did your group work well together?”, “Was everyone in the group given the effectiveness of the group and where improvements need to be made.”
Assessment is a part of each lesson. It is important that students know up front the criteria on which they will be assessed, as well as how much time they will have to complete the task. Students should be graded on their products. Due dates should be set and expected to be met.
When student work is turned in, it should be assessed on its quality. Students may revise work not meeting acceptable standards. Generally, four categories of evaluation should be used Well Done, Acceptable, Revisions Needed, and Re-Start. You may want to allow students to create their own class assessment. These assessment can serve as an excellent self-assessment tool.
There is no set grading system you should use for this curriculum. The philosophy of this curriculum is to have students show their knowledge learned in the lessons by using different types of formal assessment. The following are same examples of grading.
1. Students work in groups, using the mathematics they have learned to make a group presentation.
2. Each student produces a written product to show his or her knowledge of the material. This is the most important part of the assessment of the curriculum.
Homework is an important part or this curriculum. The homework assignments are not routine exercises imitating work done in class. Rather, they are activities that may take a number of days to complete. These assignments may be research oriented, project based, reflective and analytical in nature. Homework is design to extend the class work with meaningful mathematics. New and original ideas may be a product of these assignments. Homework is introduced in class, but the investigations require work outside of the classroom.
A journal is a written account that a student keeps to record what he or she had learned. Journal entries are conducive to thinking about why something has been done. They can be used to record and summarize key topics studied, the student’s feelings toward mathematics, accomplishments or frustrations in solving a particular problem or studying a particular topic, or any other notes or comments the student wishes to make. Keeping a mathematical journal can be helpful in students’ development of a reflective and introspective point of view. It also encourages students to have a more thoughtful attitude toward written work and should be instrumental in helping students learn more mathematics. Journals are also an excellent way for students to practice and improve their writing skills.
A portfolio is a representative sample of a student’s work that is collected over a period of time. The selection of work samples for a portfolio should be done with an eye toward presenting a balanced portrait of a student’s achievements. The pieces of work placed in a portfolio should have more significance than other work a student has done. They are chosen as illustrations of a student’s best work at a particular point in time. Thus, the range of items selected shows a student’s intellectual growth in mathematics over time.
You may wish to have all students include the products of group presentation, and written product in their portfolios. Students should also select the products of at least two additional lessons for inclusion. Bear in mind that the actual selection of the items by the students will tell you what pieces of work the students think are significant. In addition, students should reflect upon their selections by explaining why each particular work was chosen.
The following examples illustrate topics that would be appropriate for inclusion in a portfolio.
a solution to a difficult or non routine problem that shows originality of thought
a written report of an individual project or investigation
examples of problems or conjectures formulated by the student
mathematical art work, charts, or graphs
a student’s contribution to a group report
a photo or sketch of physical models or manipulative
statements on mathematical disposition, such as motivation, curiosity, and self-confidence
a first and final draft of a piece of work that shows student growth