Transcript Document

t
NUMBERS Mathematics Professional
Development
© NUMBERS MPD
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 higherorder cognitive processing
A good teacher
makes you
think even
when you don’t
want 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?
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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.
2.
3.
4.
5.
6.
Name the polygon.
Describe the polygon using the following terms:
congruent, parallel, perpendicular, angle, measure,
base, height, side(s)
Label the vertices using the letters a-f
Describe the relationship between AB and CD
Identify the congruent sides using the appropriate
mathematical notations for congruency on the figure.
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 QuestionTask 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 partner1. Identify and evaluate each question you
have written according to Blooms level.
Bloom’s Levels





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Level
Level
Level
Level
Level
Level
1:
2:
3:
4:
5:
6:
Knowledge
Comprehension
Application
Analysis
Synthesis
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. They
lead 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 own
ideas, 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 are
those 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
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Inference
Interpretation
Transfer
Predictive
Inference
Interpretation
The 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.
b.
c.
d.
Mean
Median
Mode
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.
Mathematics
Which line segment is 2 times the length of the radius?
L
K
O
N
M
J
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
Pictorial/Graphical
Representation
Process/Explanation
in Words
Expression/equation
Exact Answer
‘06
 Select 10 questions from your group to
 Evaluate
 Revise
or
 Write 10 questions to
 Evaluate
Post 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 inquires
for a day; teach a man to question and
he inquires for life. Adapted from an old Chinese proverb
Remember…
There’s More to Questioning than simply Asking!
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