A Closer Look

1
A Closer Look at a Standards-Based Lesson
2
Whats the Mathematics?
Mathematics for the 21st century
3
Goals of Grade Level Modeling Sessions

• Compare and contrast the characteristics of
  standards-based vs. traditional classrooms.

• Examine the components of a standards- based
  lesson through participation.

• Plan for implementation of a standards-based
  lesson.

4
Goal 1.
Compare and contrast the characteristics of
standards-based vs. traditional classrooms.
5
Let's review....
What does a standards-based classroom
look like?
6
What do the students do?
TRADITIONAL STANDARDS-BASED
work alone
work in flexible groups
focus on getting the
right answer
use reasoning to justify their answers
memorize facts
understand and apply concepts
practice procedures
solve problems
7
What do the students do?
TRADITIONAL STANDARDS-BASED
use pencil, paper, and chalkboard
use more manipulatives and calculators
show knowledge orally and in writing, often
using pictures, models, and diagrams to clarify
their words
show knowledge by writing down numbers
8
What does the teacher do?
TRADITIONAL STANDARDS-BASED
textbook guides instruction
standards and objectives guide instruction
spends most of the time facilitating
Guide on the Side
spends most of the time telling Sage
on the Stage
seeks the right answer from students
asks more open-ended questions
9
What does the teacher do?
TRADITIONAL STANDARDS-BASED

teaches specific algorithms and procedures
encourages students to use prior knowledge to
solve new problems
discourages student interaction
encourages students questions and explanations
asks mostly knowledge-level questions
asks more high-level questions
10
Goal 2. THE COMPONENTS OF A STANDARDS-BASED
LESSON

• Launch
• Investigation
• Closure

11
Launch!
Role of the teacher

• Introduce new ideas

• Clarify definitions

• Review old concepts

• Connect the problem to past experiences of
  the students

12
QUESTIONS TO KEEP IN MIND AS YOU PLAN THE
LAUNCH.

• What are students expected to do?

• What do the students need to know to
  understand the context of the story and the
  challenge of the problem?

• What difficulties can I foresee for students?

• How can I keep from giving away too much of the
  problem?

13
INVESTIGATION
Role of the students

• Work individually, in pairs, or small groups to
  solve a problem

• Gather data, share ideas, look for patterns,
  make conjectures, and develop problem-solving
  strategies

14
INVESTIGATION
Role of the teacher

• Talk less, listen more.

• Circulate through the classroom.

• Clarify the problem if necessary.

• Ask questions that encourage
  reasoning and making connections.

15
Common characteristics of
all types of investigations
STUDENTS

• work in depth on a small number of
  problems

• actively use mathematical tools

• consult with peers

AS THEY FIND THEIR OWN WAYS TO SOLVE THE
PROBLEMS.
16
QUESTIONS TO KEEP IN MIND AS YOU PLAN THE
INVESTIGATION.

• How will my students be organized as they work?

• What materials will my students need?

• How will my students record and report their
  work?

• What strategies might my students use as
  they solve this problem?

17
Plan Questions to...

• encourage student conversation, thinking,
  and learning.

• help students see connections to other
  situations.

• focus students thinking if they become
  frustrated or off-task.

• challenge students if the initial
  question is answered.

18
CLOSURE

• Begins when most students have gathered
  sufficient data to progress to solving the
  problem.

• The teacher leads discussion to help
  students enhance their understanding of
  mathematics.

19
Things to consider during discussion...

• How can I help the students make sense of
  and appreciate the variety of methods that may
  be used?

• What concepts or strategies need to be
  emphasized?

• What ideas do not need closure at this
  time?

20
Things to consider during discussion...

• How will I assess individual
  students understanding?

• What will I do to follow up, practice, or apply
  the learning from this activity?

21
Feedback from Teachers using Connected
Mathematics
Impressed by students Mathematical thinking
Got to know their students more
Increased class participation
Surprised by some things students didnt know
Particularly liked the depth of mathematics in
the investigations
High level of student involvement
22
Goal 2.
Examine the components of a standards-based
lesson through participation.
23
LESSON 1.
You will play the role of a student, but keep
your teacher eyes and ears open. We will
analyze the lesson later.
24
TANGRAMS
In this lesson, we will explore the
relationships between the seven pieces of the
tangram.
Launch!
25
TANGRAMS
You will need scissors, a ruler, and a sheet of
lightweight card stock. Cut a square that is 8
x 8. We will cut this square into the seven
tangram pieces.
26
TANGRAMS

• Fold the square on the diagonal.
• Cut the square on the diagonal.

• What part of the square is each piece?

27
TANGRAMS
Set one of the large triangles aside. Fold the
other large triangle in half by matching up the
two acute angles. Cut on your fold line. What do
you have now? These are the first two pieces of
your tangram puzzle. Set them aside.
28
TANGRAMS
Set your other large triangle down in front of
you so the longest side is closest to you. Find
the middle of the longest side and crease it
gently to mark it. Fold the right angle down to
touch the midpoint of the base. Your fold line
should be parallel to the base.
29
TANGRAMS
Cut on your fold line. Set the triangle aside. It
is your third tangram piece. What shape do you
have left?
30
TANGRAMS
Fold the trapezoid in half as shown. What shape
do you see now? Cut on the fold line.
31
TANGRAMS
Fold one of your right trapezoids to make a
square and a small right triangle. Cut on the
fold line.

• These are two more of your
• tangram pieces.

32
TANGRAMS
Fold the other right trapezoid to make a
parallelogram and a small right triangle. Cut
on the fold line.

• These are the last two pieces of your tangram.

33
TANGRAM PUZZLERS
INVESTIGATION
You will use your tangram pieces to solve some
puzzles. You will work alone for the first few
minutes. Then you will work with a partner.
34
MORE TANGRAM PUZZLERS
Its your turn to write some tangram
puzzlers. You will trade puzzles with a friend
later. Show the solution to your puzzle on a
separate sheet of paper.
35
TANGRAMS
How can you state the relationship between the
pieces of the tangram?
CLOSURE
Lets make a list of statements that are true no
matter what the value of any given piece might be.
36
(No Transcript)
37
LESSON 2.
As we go through this lesson, look for
the launch, investigation, and closure.
38
PATTERN BLOCKS
Lets review the names of the pattern
blocks. Sort them by the number of sides.
39
PATTERN BLOCKS
Put all the quadrilaterals in one group. How
are they the same? How are they different?
40
PATTERN BLOCKS
Which ones are rhombuses? Which one is a
trapezoid? Which one is a square?
41
PATTERN BLOCKS
A square has four right angles. Right angles
always measure exactly 90 degrees.
42
PATTERN BLOCKS
Your task is to use that knowledge to find the
measures of all the angles of all the other
polygons.
90
90
90
90
43
PATTERN BLOCKS
You may use more blocks from your bag to help
you.
Trace each block on your paper and label the
angles.
44
PATTERN BLOCKS
Be ready to explain your thinking to the class.
45
PATTERN BLOCKS
Have students come to the overhead to show how
they solved a part of this problem.
46
(No Transcript)
47
Goal 3.
Plan for implementation of a standards-based
lesson.
48
NCTM-Principles and Standards for School
Mathematics states, Childrens literature is
helpful in setting a context for both
student-generated and teacher-posed problems.
49
Problem Solving

• What kinds of problems
• can I present to students
• that will give them
• a chance to grapple with
• important ideas and skills?

50
Reasoning and Proof

• What kinds of situations
• can I pose to children
• so that their reasoning is engaged
• and they have experience
• giving convincing arguments?

51
Communication

• How do I involve students in
• talking and writing
• to help them communicate
• what they are
• studying and learning,
• and hear the ideas of others?

52
Representation
How do I help students use the symbolism of
mathematics to describe their thinking?
53
Connections

• How do I help students
• see the connections
• among mathematical ideas
• rather than seeing concepts
• as isolated and separate
• from one another?

54
If You Hopped Like a Frog by David
Schwartz
55
Planning your lesson...
Use the lesson plan outline to jot down your
ideas. Remember this is just a draft. You may
work alone or with a partner.
56
Standards-Based Mathematics

• Engages students.
• Challenges students at all levels.
• Prepares students for further study of
  mathematics.
• Focuses on important big ideas in mathematics.
• Develops a mathematical disposition which
  emphasizes sense-making.