Planning for Mathematics Instruction

1. Planning for Mathematics Instruction Main Reference: Teaching Mathematics in Grades K-8 Research-based Methods, 2nd-ed, Edited by Thomas R. Post. Teaching Elementary School Mathematics: Methods and Content for Grades K-8 by Frederick H. Bell

2. Stages in Teaching • Planning • method • evaluation

3. Teacher as Decision Maker • Content • development approach -- instruction proceed from what students know toward knowledge and skills beyond their present understanding. • Teachers have a great deal of autonomy within the prescribe curriculum. • behavior of the learners: how the students spend their time. • behavior of the teachers: motivation, reinforcement, retention, and transfer

4. Effective Mathematics Instruction Good Teachers • are clear about their goals, and are able to articulate to students, fellow teachers, parents and administrators. • Are knowledgeable about the the content they teach • knowledgeable about a wide range of instructional strategies • communicate to students what is expected and why; help students search for meaning in mathematics

5. Effective Teaching • High expectations, high academic performance • amount of time actively contributed to learning -> achievement • explain what expected to learn and demonstrate the steps • students tutoring other students • achievement rises when questions asked that require apply, analyze, synthesize and evaluate

6. Achievement rises significantly when homework assigned regularly and were conscientiously done. • Frequent and systematic monitoring of students’ progress • emphasis on academic courses, more advanced the subject matter, more rigorous the textbook --. More learned Source: What Works: Research about Teaching and Learning. U.S. Department of Education, 1986.

7. Effective Teaching Practices • Instruction is guided by a preplanned curriculum • high expectations for student learning • students are carefully oriented to lessons: objectives explained; linked to previously studied; key concepts/skills reminded • Instruction is clear and focused • Learning progress is monitored • Reteach what not understood • Classtime is used for learning

8. Smooth and efficient classroom routines • Instructional groups (whole/small) formed to fit instructional needs. • Standards for classroom behavior are explicit; • Personal interactions between teachers and students; • Incentives and rewards for students Source: Onward to Excellence: Making Schools More Effective, Northwest Regional Educational Laboratory, 1984

9. Three Goal Structures • Competitive • Individualistic • Cooperative

10. Example Competitively: 3 to 4 in groups, students in each group compete to see who can count the most triangles Individualistically: students count as many as they can, those who counts 90% succeed. Cooperatively: Students in groups asked to find as many as they can; encourage helping each other.

11. How to Plan a Lesson • Set the stage - motivation • Tell the objective(s): what can they do after the lesson • Give direction: work together? Seatwork? • Provide learning context: connections between lessons • Illustrate the key concept or skill • Help them to carry out the assignment: move around, … • Promote reflective thinking • Clarify any extended expectations: what do they at home?

12. Considerations in Planning Mathematics Lessons (Bell, 1980) • Mathematics Content: select and name the topic; identify the facts, skills, concepts, or principles; be sure each topic is properly sequenced. • Learning Objectives: Identify and choose appropriate cognitive objectives; Select desirable affective objectives; Share the objectives; illustrate the application of each mathematical concept • Learning Readiness: Identify prerequisite and assess students’ mastery

13. Teaching/Learning Resources and Activities: Locate, obtain, and evaluate required materials, then select • Teaching/Learning Strategies: Select and use appropriate strategies; create learning environment; assessing student learning; evaluate and improve teaching effectiveness.

14. Learning Objectives • Motor-skill learning: coordinating one’s sense and skeletal muscles to learn to talk, walk,... • cognitive learning: accumulation of intellectual knowledge • affective learning: developing attitudes, values, likes and dislikes, preferences, and commitments.

15. Cognitive Learning ObjectivesBloom’s Taxonomy • Knowledge: remembering and recalling information in nearly the fame form that it was presented. • Comprehension: students can make some meaningful use of it; correct using it when told to do so. • Application: ability to use it in an appropriate situation without being told to do so.

16. Analysis: ability to subdivide information into its components so the relative hierarchy of ideas is identified and the relationships among the ideas are apparent • Synthesis: ability to combine elements to form an unique system or structure: e.g., finding patterns, discovering principles, .. • Evaluation: making judgments about the usefulness and value of ideas, procedures, creations, inventions and methods.

17. Cognitive Objectives Children will give the definition of even numbers Children will identify the numerators and denominators of proper fractions Evaluation Item What are even numbers In the fraction 2/3, which number is the numerator Preparing Cognitive Objectives Knowledge of Arithmetic Source: Bell, 1980

18. Cognitive Objective Children will identify even and odd numbers Children compute the sum of two proper fractions. Children will draw triangular shapes Evaluation Item Which of these numbers are even numbers: 8, 11, 19, 3, 16, 27, 22, 10? Find the sum: 1/2+1/4 Draw a little triangle and a big triangle Comprehension of Arithmetic

19. Cognitive Objective Children will explain why the sum of two odd numbers is an even number Children will describe the relationship between addition and multiplication of natural numbers. Evaluation Item Why is the sum of two odd numbers always an even number? Give three different examples showing that multiplication of two numbers is the same as repeated additions. Analysis in Arithmetic

20. Cognitive Objective Students will construct addition and multiplication tables for clock arithmetic Students will develop procedures for adding and multiplying numbers in base two Evaluation Item Prepare addition and multiplication tales for clock arithmetic Develop sets of rules for adding and for multiplying numbers that are written in base two Synthesis in Arithmetic

21. Cognitive Objective Students will determine the advantages and the disadvantages of handheld calculators as an aid in compuation Students will explain the value of zero as a number in our system of mathematics Student Activity What are some of the reasons why calculators should be used to do computations? Limitations? Disadvantages? Suppose we did not have a zero in our number system. What limitations would this place upon our ability to use mathematics? Evaluation in Mathematics

22. A Sample Lesson Plan Pp. 133-134 Bell, 1980