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NUMBERS Mathematics Professional Development

© NUMBERS MPD

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Today’s Focus…Today’s mathematics requires more than getting the answer. We must… expand our understanding of

various questioning processes. develop an understanding of

the nature and function of questions related to increasing mathematical content knowledge

Today’s Focus…

learn to write effective questions self assess questions become aware of research in

thinking, metacognition, and higher-order cognitive processing

A good teacher makes youthink even when you don’twant to.

(Fisher, 1998, Teaching Thinking)

Purposes of questioning strategies for instructors…

They help you to: effectively plan class participation activities, design homework assignments, write exams, and match your goals or objectives for

an assignment with the actual components of the assignment.

Read and reflect activity…

Read your assigned article.

Conduct a group discussion for 4 minutes to identify key points. A reporter will summarize your thoughts.

He who learns but does not think is lost

(Chinese Proverb)

Time to practice…

Consider the problem setting below. Construct 5 or more questions that

could be asked using this problem setting.

Language elements needed for effective communication and comprehension in the

areas of mathematics and science:

Academic Language Symbolic Language Content Language

(Vocabulary)

Academic Language Is used to communicate

what students should know and be able to do in state standards.

Determine, simplify, predict, interpret, etc…

The numbers in Set R share a common characteristic.

Set R: 48, 54, 6, 66, 12, 24

The numbers in Set S do not share this characteristic.

Set S: 9, 20, 39, 15, 63, 27, 44

Which best describes the characteristic that only the numbers in Set R share?

F Numbers less than 70 G Numbers greater than 5 H Numbers that are composite J Numbers divisible by 6

Grade 7

Symbolic Language

Is used to demonstrate understanding and interpretation of science and mathematical notations.

Figures and Organizers

Content Language(Vocabulary)

Is technical language associated with the sometimes abstract concepts and skills of mathematics and science.

Quotient, radian, abscissa, etc…

Write an equation you would use to find the mean temperature of this data.

What is the range of this set of data? Which statistical measure would be most affected

if the temperature on Saturday had been 63° F instead of 54° F? Explain.

What type of graph would best depict this data? Explain your answer.

Track the weekly temperatures for your city for one week and compare them to the average temperatures for your area. How do they compare? What can account for vast differences if there are any.

1. Name the polygon. 2. Describe the polygon using the following terms:

congruent, parallel, perpendicular, angle, measure, base, height, side(s)

3. Label the vertices using the letters a-f4. Describe the relationship between AB and CD5. Identify the congruent sides using the appropriate

mathematical notations for congruency on the figure.6. Look at each of the angles. Provide a reasonable

estimate for the measure of each and justify your answer with mathematical proof.

7. Is this a regular or irregular polygon? Write a descriptive paragraph to support your answer. You must also include pictures/drawings.

8. Explain a method you would use to determine the perimeter of polygon.

9. Using a ruler, determine the perimeter to the nearest cm.

10. Describe a method you can use to determine the area. Label your steps in sequential order so that your explanation is easy to follow. You may draw pictures to illustrate the steps.

11. Formulate an expression that represents the area of the polygon.

12. Apply your method, find the area.

13. If the lengths of the sides were doubled, predict how the area would be affected?

14. If the lengths of the sides were doubled, predict how the perimeter be effected?

15. If the measures of the angles were increased, describe how would the lengths of the sides be affected? Using a ruler and protractor, draw a picture to support your reasoning.

16. Measure each angle and find the sum of the angle measures. Compare the sum of the angle measures to the sums of the angle measures for a 3-sided, 4-sided and 5-sided figure. What pattern do you notice?

17. If this were the base of a 3-dimensional figure explain what type of figure(s) could it be and why?

18. If this is the bottom view of a hexagonal prism, what would the front view look like?

19. How many faces, vertices and edges would this 3-Dimensional figure have?

20. Explain how you could determine the volume of the hexagonal prism. Compare your method to another student’s method. How are they alike? Different?

21. How many different lines of symmetry can you draw?

22. Name a line segment that shows a line of symmetry.

23. Use mathematical notation to identify the sides that are parallel.

24. Draw the polygon in quadrant 1 on a coordinate plane.

25. Identify the coordinates pairs for each of the vertices.

26.If you translated the polygon 2 units left and 3 units down, what would be the new coordinate pairs for each of the vertices?

27.If you rotated the figure 90 degrees, in which quadrant would it be located?

28.Draw a 90 degree rotation.

29.Reflect the original figure over the x-axis. Provide the new coordinate points for the vertices of the prime image.

30. What type of transformation would have occurred if the original polygon now lies in quadrant 3? Draw a picture to support your reasoning.

31. If the original imaged is dilated by a scale factor of ½, what would be the new vertices?

32. Draw a similar figure and write a proportion that proves that the figures are similar.

Your Task:

Select a TAKS release item and write 5 questions based on the problem setting.

You have 10 minutes.

Crafting an Effective Question-Task 1: page 4

Ask yourself.... Is the question directly related to a curriculum

objective? Is the question expressed as clearly, concisely,

and as unambiguously as possible? Does the question actively engage learners'

mental energies? Does the question generate critical thought? …

Task 2: Think-Pair-Share…page 5

With a partner-1. Identify and evaluate each question you

have written according to Blooms level.

Bloom’s Levels

Level 1: Knowledge Level 2: Comprehension Level 3: Application Level 4: Analysis Level 5: Synthesis Level 6: Evaluation

The main value of the Taxonomy is twofold: (1) it can stimulate teachers to help students acquire skills at all of these various levels, laying the proper foundation for higher levels by first assuring mastery of lower-level objectives; and (2) it provides a basis for developing measurement strategies to assess student performance at all these levels of learning.

By writing objectives and planning lessons with questions at the appropriate levels, you should be able to develop both objectives and assessment strategies that cover the full range of expectations within each TEK’s Student Expectation you will teach.

Convergent thinking questions are those which represent the analysis and integration of given or remembered information. Theylead you to an expected end result or answer.

Divergent

Divergent thinking questions are those which represent intellectual operations wherein you are free to generate independently your ownideas, or to take a new direction or perspective on a given topic.

Task 4: Think-Pair-Share…

Are some of your questions convergent?

Are others divergent?

Task 5:

Are your questions closed or opened?

Closed or restricted response

What gas is the largest component of air? Response: Nitrogen

What is the sum of 2+2? Response: 4

What is the name of this polygon?

Response: trapezoid

Use: Closed-ended questioning is useful when you want to know what specific knowledge a student has acquired.

Opened-ended or extended response

How would you describe the air? Possible response: Air is a mixture of

gases including nitrogen, oxygen and carbon dioxide as well as dust and pollen.

Write a math story can be written to represent the following inequality: 3n < 98 Possible response: A club with 3

members is going on a trip. The members decided that the club treasury would be used to pay for the tickets which. The total cost of the tickets could not exceed $98.

The most important questions of all arethose asked by students as they try to make sense out of data and information. These are the

questions which enable students to Make Up Their Own Minds.

Write a brief descriptive paragraph using appropriate math terminology that describes the polygon below.

Possible response: It is an isosceles trapezoid having exactly one pair of parallel sides. The sum of the angle measures is 360º.

Task 6: Categorize as Other Types of Questions

Inference Interpretation Transfer Predictive

Inference

InterpretationThe table shows the number of pages read each month

by Chole. If she read only 125 pages in May,which measure of data changed the most?a. Meanb. Medianc. Moded. All of the above

Transfer

Predictive

What type of Question?

Identify each of the following sample TAKS items as either

•inference, •transfer, •Predictive, or •interpretation.

MathematicsWhich line segment is 2 times the length of the radius?

O

JM

N

LK

Mathematics

Mathematics

Social Studies

Science

Reading

Mathematics

A. bc-ef B. af + ad – de

C. de + af + ad D. af+ cd

Social Studies

Mathematics

TAKS Connection…

In your group, review the release TAKS items that have been placed at your table.

Tally the type of response required by of in the chart provided.

You have 20 minutes

Type of Response ‘06

Pictorial/Graphical Representation

Process/Explanation in Words

Expression/equation

Exact Answer

Select 10 questions from your group to

Evaluate Revise

or Write 10 questions to

EvaluatePost the questions on chart paper when

you have completed the activity.

2nd Round Review

Exchange questions with another group.

Groups will review questions according to the criteria provided in the handout.

You have 20 minutes.

Your turn…

As a group, select 1 release TAKS item.

Write 5-6 different questions that can be answered using the problem setting.

Be ready to share your questions with the group.

Social values are revealed through questioning

who can learn and who can teach

learning flows only from a teacher

or whether it can come from other students.

Create the climate for inquiry… How do teachers respond to the answers

their questions provoke? "uh-huh“, “that’s right”, “good” responses can

stop inquiry dead in its tracks. In place of such dead-end situations, you may

pursue an investigation in which simple factual inquiries give way to increasingly interpretive questions until new insights emerge.

Keep questions alive through long stretches of time, coming back to them days, even weeks, after they have first been asked.

Closing thoughts…

What new insights have you gained? What will you implement

immediately? What has been validated? What will you differently as a result of this training?

Ask a man a question and he inquiresfor a day; teach a man to question andhe inquires for life. Adapted from an old Chinese proverb

Remember… There’s More to Questioning than simply Asking!

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