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Preceding Pages
- Looking Ahead
- Part A: Beliefs About Best Practices for Teaching and Learning Mathematics and Science
- Part B: Beyond Beliefs: The Guiding Principles
- Structure for the Discussion of Guiding Principles
- Guiding Principles
Current Page
Guiding Principle #3. Students reason effectively in mathematics and science.
Instructional Implications
Content StandardsA. Students understand and demonstrate that ideas are more powerful if they can be justified.
B. Students use different methods of thought to justify ideas.
C. Students recognize instances in which attitudes influence reasoning.
Following Pages
- Guiding Principles
- References
Reasoning skills may be viewed as the power source in the construction of knowledge, both the learner's personal knowledge and the discipline's collective body of knowledge. In personal knowledge construction, students may either accept new information and connect it to their previous learning, or they may modify or reject the new information. This same process can be seen in the construction of the discipline's body of knowledge. When anomalies are encountered or new conjectures are proposed, the community of learners must accommodate this new information and eventually reject, modify or accept the implications.
Both individual and collective disciplinary knowledge construction demand creative and critical thinking. Creative thinking is characterized by fluency, flexibility, originality and elaboration. It allows for the generation of conjectures and hypotheses, leaves room for anomalies, finds new problems, and poses alternative problem-solving strategies. Critical thinking compares and tests conjectures, hypotheses and anomalies.
Instructional ImplicationsThe development of logical reasoning is tied to the intellectual and verbal development of students. Students must have the opportunity to formulate arguments, present them to classmates, respond to criticisms and critique the arguments of others. The statements of both teachers and students must be open to question, reaction and elaboration. Students must recognize that the reasoning and thought processes involved in solving a problem are as important as the solution itself. Teachers need to use questioning techniques which elicit reflective responses and assist students in recognizing that much can be learned by analysis of errors.
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Snapshot First graders are engaged in Mary Baratta-Lorton's lima bean activity (Mathematics Their Way). One side of each white bean has been painted black. The children have counted the ten black-white beans in their cups.
Each child is given a recording sheet and asked to draw and color the beans as they appear when they are dumped out of the cup. Children repeat this eight times, making observations and later conducting a group discussion about what they noticed.
Several observations are shared in the discussion. The number of beans has always stayed the same (10). Sometimes 1 and 9 make 10, and sometimes 5 and 5 make 10. If ten blacks roll out of the cup, then no white ones will be visible. Some children notice that if they can make a 4 and 6, then they can make a 6 and 4. They see that numbers are made up of other numbers.
When someone asks if bigger numbers also break into smaller numbers, the class begins a "40 board" to record their discoveries about the number 40. This activity continues over the next couple of weeks, as students use other materials to show that their number sentences or beliefs are true. Forty is shown as a number made up of many numbers.
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Content Standard A. Students understand and demonstrate that ideas are more powerful if they can be justified.
________________________________________________________Ideas may be justified through deductive or inductive reasoning. Deductive reasoning, the reasoning of Euclid, establishes a set of rules from which conclusions can logically be made and forms the basis of mathematical arguments. An understanding of deductive reasoning is necessary for distinguishing good logic from bad logic in analyzing arguments.
Inductive reasoning is used to make generalizations from observations. This is the reasoning of science and everyday life. Inductive arguments can never be absolutely true; they stand only until new evidence is presented.
Performance Indicators Primary
1. Make observations. (S-K3)
2. Use various forms of logic. (S-K5)
3. Discover relationships and patterns. (S-K6)
Intermediate
1. Use various types of evidence (e.g., from logical processes, from measurement, or from observation and experimentation) to support a claim. (S-K4 and M-J1 combined)
2. Draw conclusions about observations. (S-K3)
3. Demonstrate an understanding that ideas are more believable when supported by good reasons. (S-K5)
Middle
1. Support reasoning by using a variety of evidence such as models, known facts, properties, and relationships. (S-K6 and M-J1 combined)
2. Show that proving a hypothesis false (i.e., that just one exception will do) is much easier than proving a hypothesis true (i.e., true for all possible cases). (S-K7)
3. Construct logical arguments. (S-K8)
4. Demonstrate that multiple paths to a conclusion may exist. (M-J2)
Secondary
1. Distinguish between different forms of logic.
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Snapshot Fourth-graders have been studying graphs: bar graphs, line graphs, circle graphs etc. They have collected several types of information and displayed it in various ways. The students notice that sometimes there is more than one way to represent a set of information, sometimes one way is better than another for a given set, and sometimes the choice of graph type changes the way in which information will be interpreted.
These observations spark a class discussion about how graphing practices can be used to distort information and lead to an incorrect conclusion. At home, students ask their parents to help them search newspapers and magazines for examples of distorted information to share with the class the next day.
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Some types of reasoning involve the application of specific rules,
whereas others do not. Types that do are commonly used to provide
evidence for claims or to prove things; other types (such as
reasoning by similarity, brainstorming) are not used to provide
evidence or to prove things. Formal logic may be of limited help
in finding solutions to problems if we are not sure that general
rules hold or that particular information is correct; most often,
we have to deal with probability rather than with certainty.
B.
Students use different methods of thought to justify ideas.
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Primary
1. Examine and describe strengths or weaknesses of simple arguments (S-K1 and M-J1 combined)
2. Distinguish between true observation and conclusions about observations. (S-K2)
3. Participate in brainstorming activities. (S-K4)
Intermediate
1. Give alternative explanations for observed phenomena. (S-K1)
2. Practice and apply simple logic, intuitive thinking, and brainstorming (S-K6) 3. Relate the unfamiliar to the familiar.
Middle
1. Examine the ways people form generalizations. (S-K1)
2. Identify exceptions to proposed generalizations. (S-K2)
3. Apply analogous reasoning. (S-K9)
Secondary
The application of formal logic to real world situations can be difficult. Claims made in print, radio, and television by reporters, editors, hosts or advertisers should be regularly critiqued for quality and accuracy of argument. Students should analyze the arguments for explicit or implied premises, logic and evidence. Based on the analysis, they should be able to judge the soundness of arguments and evaluate claims.
1. Judge the accuracy of alternative explanations by identifying the evidence necessary to support them. (S-K1)
2. Analyze situations where more than one logical conclusion can be drawn from the data presented. (S-K6 and M-J1 combined) 3. Develop generalizations based on observations. (S-K3)
4. Determine when there is a need to revise studies to improve their validity through better sampling, controls, or data analysis techniques. (S-K4)
5. Produce inductive and deductive arguments to support conjecture. (S-K5)
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Snapshots A sixth-grade teacher wants students to experience the benefits of camouflage. Before class she scatters 80 colored toothpicks (20 each of red, blue, green, and yellow) in a 10m x 10m area in the grass. The students are asked to predict which color will be easiest to find and which will be the most difficult to find. In doing so, they must justify their predictions.
Working in small groups, students have a timed period during which they try to find as many toothpicks as possible. Upon completion of the search, students organize and present their data in a suitable mathematical or graphical way.
Students analyze their data and establish hypotheses to explain their observations. The teacher then invites the class to reflect on ways this activity is and is not analogous to animal adaptation.
~~~~~ High school students, concerned about traffic congestion at a nearby intersection, decide to design a method to study the situation. Their study requires them to consider the definition of traffic, when and how to measure it, what kind of data will be necessary, how to record and interpret the data, and possible methods for easing congestion. Students also learn about the structure and function of town government in order to make contact with the appropriate public officials. Further, they must build a convincing argument to persuade the town to act on the problem. After several weeks of study, students present their results to the town council with the recommendation that a traffic signal be installed.
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Content Standard C. Students recognize instances in which attitudes influence reasoning.
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Primary
1. Distinguish between important and unimportant scientific and mathematical information in simple arguments. (S-K2 and M-J2 combined)
Intermediate
1. Describe how feelings can distort reasoning. (S-K2)
Middle
1. Identify basic informal fallacies including intermingling of fact and opinion, lack of explicit premises, and over-generalizing. (S-K3)
2. Analyze means of slanting information. (S-K4)
Secondary
1. Explain why agreement among people does not make an argument valid. (S-K2)
2. Describe an example that demonstrates what people study and how they study can affect the theories they develop.