When teachers elicit students' ideas at the beginning of a unit or an activity, making the transition from these ideas to student-centered investigations is often a challenge. How can students' ideas lead to productive hands-on, minds-on, and meaningful investigations? During this workshop, we'll consider how teachers can move students' thinking from exploring what they already know, to asking a question about what they want to know. We'll also consider how the ideas that emerge during an open-ended exploration or a brainstorming session might be "finessed" toward the learning goals intended by the teacher.
The Great Bean Bag Adventure
What does a seed need to sprout? We begin the Adventure
by focusing on an old favorite--the bean--and we invite you and your colleagues
What is your role in your metaphor? What is the role of your students? How does your metaphor shape your approach to teaching and learning? How does it shape the expectations of your students?
Students often enter into a new lesson with a wide range of knowledge about the particular math or science concept, and their understandings are often revealed by comments they make during open-ended explorations and class discussions. How have you dealt with the wide range of knowledge and understandings in your classroom?
When teachers give students freedom to develop their own questions for investigation, students' ideas do not always coincide with the intended learning goals, methods, and/or materials. What do you think is the appropriate balance between student ideas and teachers' goals? How do you maintain that balance?
What are your colleagues' metaphors for teaching? Conduct your own "teacher-on-the-street" interviews by asking several teachers at your school about their teaching metaphors. Keep track of what they tell you, and bring your results with you to Workshop 2. (If you have access to the Web, you can enter the metaphors on our Web site, and your data will be used in an upcoming article about teaching metaphors!)
Suggested Grade Level: K-3
Students explore part-whole understanding by using pattern blocks to fill in a pre-determined shape.
The green triangles, blue rhombuses, red trapezoids, and yellow hexagons from a Pattern Block set.
For each pair (or small group) of students, create an activity sheet like the following:
The dimensions for the "big shape" are:
width of side 1=7.5 cm
width of side 2=2.5 cm
Try a similar activity, but have students record their answers in an equation rather than in a chart. Students may decide to write out their equations with pictures, words, or colors of the different shapes. For example:
1. You can challenge older students to figure out the maximum number of each type of Pattern Block that will fit into the big shape. Then, using fractions, they can determine the exact number of Pattern Blocks that will fit.
To figure out exactly how many HEXAGONS will fit into the shape, for example, students will give the hexagon a value of 1. If the hexagon is equal to 1, then the trapezoid is equal to 1/2, the rhombus is equal to 1/3, and the triangle is equal to 1/6.
Students can then use these fractional equivalencies to write equations from the data on their activity sheets (above), and their answers--all the same--will represent the number of hexagons that fit into the big shape. For example:
|= 3(1/6) + 1(1/3) + 3(1/2) = 2 1/3 hexagons|
|= 2(1) + 2(1/6) = 2 1/3 hexagons|
|= 1(1) + 1(1/2) + 5(1/6) = 2(1/3) hexagons|
You can repeat this exercise three more times, giving the trapezoid, the rhombus, and the diamond each a value of 1.
2. Another related activity you can do with older students is to have them solve problems like the following:
In grades K-4, the mathematics curriculum should include the study of patterns and relationships so that students can --
"Physical materials and pictorial displays should be used to help children recognize and create patterns and relationships. . . . The use of letters and other symbols in generalizing descriptions of these properties prepares children to use variables in the future. This experience builds readiness for a generalized view of mathematics and the later study of algebra."
National Council of Teachers of Mathematics, (NCTM). 1989. Curriculum and evaluation standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics. (pg. 60)