Download Example - Kansas City Public Schools

Kansas City
2/10/2012
Cathy Battles
Kansas City Regional Professional Development Center
[email protected]
The Show-Me Standards – PERFORMANCE (to do)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
6.
7.
GOAL 1
gather, analyze and
apply information and
ideas
1.6, 1.10
GOAL 2
communicate
effectively
within and beyond
the classroom
2.2
GOAL 3
1.
2.
3.
4.
5.
6.
7.
8.
recognize and
solve problems
3.2, 3.5
GOAL 4
1.
2.
3.
4.
5.
6.
7.
8.
make decisions and
act as responsible
members of society
Big
Idea
GLEs/CLEs
Strand
Number and Operations
GL
E
Concep
t
N3b
Content/
Performance
Standards
DOK
DEPTH OF KNOWLEDGE
 Level 1 Recall
Recall of a fact, information, or procedure.
 Level 2 Skill/Concept
Use information or conceptual knowledge, two or more steps, etc.; you do
something
 Level 3 Strategic Thinking
Requires reasoning, developing plan or a sequence of steps, some complexity, more
than one possible answer; generates discussion
 Level 4 Extended Thinking
Requires an investigation, time to think and process multiple conditions of the
problem
Complexity vs. Difficulty
An item may be difficult but
have no relationship to
higher levels of DOK.
DOK is not about
difficulty
 Difficulty is a reference to how many students answer a
question correctly.
 How many of you know the definition of exaggerate?
DOK 1 – recall
 If all of your students know the definition, this question is an
easy question.
 How many of you know the definition of prescient?
DOK 1 – recall
 If most of your students do not know the definition, this question
is difficult.
6
DOK is about what follows the
verb
What comes after the verb is more
important than the verb itself.
 “Analyze this sentence to decide if the commas have been
used correctly” does not meet the criteria for high cognitive
processing.
 The student who has been taught the rule for using commas is
merely using the rule.
7
DOK and
the GLEs & the CLEs
 The assigned DOK to the GLEs & CLEs is the ceiling for
the MAP test only.
 Our classroom instruction will most likely go above and
beyond what is coded to each GLE or CLE
Grade 4
The class went on a field trip. The students
left school at 9:00 a.m. They returned to class
at 1:30 p.m. How long were they gone?
A 8 hr 30 min
B 8 hr
C 4 hr 30 min
D 4 hr
The choices offered indicate that this item is intended to
identify students who would simply subtract 9 minus 1 to
get an 8. More than one step is required here. The
students must first recognize the difference between a.m.
and p.m. and make some decisions about how to make
this into a subtraction problem, then do the subtraction.
Think carefully about the following question. Write a
complete answer. You may use drawings, words, and
numbers to explain your answer. Be sure to show all of
your work.
Laura wanted to enter the number 8375 into her
calculator. By mistake, she entered the number 8275.
Without clearing the calculator, how could she correct
her mistake? Explain your reasoning.
An activity that has more than one possible answer
and requires students to justify the response they
give would most likely be a Level 3. Since there are
multiple possible approaches to this problem, the
student must make strategic decisions about how
to proceed, which is more cognitively complex than
simply applying a set procedure or skill.
Mathematics
The school newspaper
conducted a survey about
which ingredient was most
preferred as a pizza
topping. This graph
appeared in the newspaper
article.
Favorite Pizza Toppings
Pepperoni
Sausage
Cheese
Mushrooms
What information would best help you determine the number of
people surveyed who preferred sausage?
A
number of people surveyed and type of survey used
B
type of survey used and ages of people surveyed
C
percent values shown on chart and number of people surveyed
D ages of people surveyed and percent values shown on chart
Math Content Blueprints
Grade 3
Grade 4
Grade 5
Grade 6
Grade 7
Grade 8
Number &
Operations
30-36%
35-40%
25-30%
26-32%
20-25%
17-24%
Geometric
Relationships
17-21%
14-17%
15-18%
12-15%
16-20%
18-31%
Measurement
15-20%
12-23%
15-21%
12-18%
12-15%
9-13%
Data &
Probability
8-10%
9-12%
15-18%
22-27%
15-18%
10-19%
Algebra
Relationships
18-24%
15-24%
19-25%
17-20%
27-33%
28-34%
EQUIVALENCY
TRUE OR NOT TRUE?
1 2
1 =
3 6
EQUIVALENCY
TRUE OR NOT TRUE?
1 3
3 =
2 6
EQUIVALENCY
TRUE OR NOT TRUE?
1 2
1 =
3 6
1 3
+3 =
2 6
5
4
6
EQUIVALENCY
TRUE OR NOT TRUE?
1
= 0.333
3
1. Incorrect process
24  4  96  7  103  2  206  5  201
2. Correct Process
24  4  96 Change Direction
for Each Operation
7
103  2  206
-5
201
3. Another Correct Process
24  4  96
96  7  103
103  2  206
206  5  201
The first example is called stringing/run-on which will not
be accepted as a correct process.
The second example is an acceptable process. Because
direction changes 24 X 4 is not interpreted as being equal
to 201.
4. Incorrect Process
24
5. Correct Process
4
Change Direction
24
96
for Each Operation
4
7
96  7  103
2
103
206 - 5  201
2
206
6. Another Correct Process
5
24
96
103 206
201
4
7
2
-5
96
103 206
201
The first example is called stringing/run-on which
will not be accepted as a correct process. It would
be interpreted that 24 X 4 = 201 which is incorrect.
The second example is an acceptable process. Because
direction changes 24 X 4 is not interpreted as being
equal to 201.
The third example is an acceptable process.
When researchers asked first- through sixthgrade students what number should be
placed on the line to make the number
sentence
8+4=
+ 5 true,
they found that fewer than 10 percent in any
grade gave the correct answer—that
performance did not improve with age.
How the Brain Learns Mathematics
David Sousa 2008
Number Sentence
 mathematical statement(equation) in which equal
values appear to the right and left of an equal sign or
comparisons written horizontally.
 Examples: 3 + 4 = 7, 8 – 2 = 6,
3 + 4 = 2 + 5, 7 > 6.
Symbolic Representations
Expressions…
Equations…
can be written using
numbers, operation
symbols and variables.
Example: 4a
Example: 3 + 6x
can be written using an
equal sign, numbers,
operation symbols, and
variables.
Example: 6x - 5 = 2x – 1
Example: x = 23 + 7
Equations
 If the problem asks for an equation, but the student
gives an expression, the answer is considered to be
incorrect.
 If the problem asks for an expression, but the student
gives an equation, the answer is considered to be
incorrect.
Equations cont.
 Write an equation for profit of x items if it costs $2.75 to
manufacture each item and the item sells $3.20
 A correct equation: P =$3.20x-$2.75x
 Incorrect equation: Profit=$3.20x-$2.75x
Patterns
You must have at least 3 numbers to determine a pattern.
1, 4, . . .
is not enough to determine a pattern.
There could be many possible answers. (1, 4, 16, 64, . . .
or 1, 4, 7, 10, . . .)
Rules for Patterns
 When students are asked to find a rule
(for a pattern), they should provide a general statement,
written in numbers and variables or words, that describes
how to determine any term in the pattern.
Example:
5, 8, 11, 14, . . .
The first term is 5. Add 3 to each term to get the next term.
 Rules (or generalizations) for patterns can be written in
either recursive or explicit notation.
Describing or Explaining a pattern…
should include the beginning term and the procedure for
finding any subsequent term.
Describing or explaining how to find the
next term in a pattern…
Example: add 5
Example: multiply by 7
Example: multiply 6 times 3 and add 1
Explicit Notation
In the explicit form of pattern generalization, the formula or
rule is related to the order of the terms in the sequence
and focuses on the relationship between the
independent variable (x) or the number representing the
term number (n) in the sequence and the dependent
variable (y) or the term (t) in the sequence.
Example: 5n
Example: 3n – 1
Example: 4x + 7
independent variable (x)
or term number (n)
1 23
Dependent variable (y)
or term (t)
0 2 4
n
Recursive Notation
In the recursive form of pattern
generalization, the rule focuses on the change
from one element to the next.
Middle School
Example: 7, 10, 13…
First Now = 7, Next = Now + 3
High School
OR
an= nth term
Example: 5, 9, 13…
a1 = first term
a1 = 5 ,
an – 1 = previous term
an= an-1 + 4
ARRAY
 A set of objects in equal rows and equal
columns. When describing, the number of
rows should come first followed by the
number of columns. Arrays are used in
describing a multiplication problem. A
pictorial representation of 3 X 2 means there
are 3 rows with 2 objects in each row. If a
student were to draw 2 rows with 3 objects in
each row, it would not be correct.
Discrete vs. Continuous Data
Discrete data is data that can be counted. (You can’t
have a half a person).
Continuous data can be assigned an infinite number
of values between whole numbers. (Time, length,
etc.)
Terminology/Vocabulary
 Use appropriate mathematical terminology
 rhombus not diamond
 Watch for multiple meaning words
 table, plane, even, odd, degree, mean, median, prime
 Homophones
 sum and some
 two and too
Use Sentence Frames for Students with
Language Difficulties or Language Impairments
Function
Beginning
Describing
Location
The
the
is next to
Examples
The square is next
to the triangle.
Intermediate
The
the
the
is next to
and below
.
Advanced
The
is between
the
, beneath
the
, and to
the right of
.
The square is next to The square is
the triangle and below between the triangle
the hexagon.
and the rectangle,
beneath the
hexagon, and to the
right of the circle.
Graphs
If no scales are included on a graph:
a.
Students can assign any scale they wish
b.
It is assumed the scale is 1
A broken axis, with other intervals consistent, means the intervals
between zero and
a.
the first increment are compressed
b.
one are compressed
Meta-analysis research
Best practice families of strategies
1. Finding similarities & differences 45%
2. Summarizing & note taking 34%
3. Reinforcing effort & providing recognition 29%
4. Homework & practice 28%
5. Non-linguistic representations 27%
6. Cooperative learning 27%
7. Setting objectives and providing feedback 23%
8. Generating & testing hypotheses 23%
9. Cues, Questions & advance organisers 22%
Classroom Instruction That Works: Based on meta-analysis by Marzano, Pickering & Pollock
Conceptually Engaging Tasks =
Cognitively Demanding Tasks
High cognitive demand lessons provide opportunities for
students:
 To explain, describe, justify, compare, or assess;
 To make decisions and choices
 To plan and formulate questions
 To exhibit creativity; and
 To work with more than one representation in a meaningful
way.
Silver, E. (2010). Examining what teacher do when they display best practice:
Teaching mathematics for understanding. Journal of Mathematics Education at
Teachers’ College. 1(1), 1-6.
What Makes a Difference
1. The quality of teachers and teaching.
2. Access to challenging curriculum, which ultimately
determines a greater quotient of students’ achievement
than their initial ability levels; and
3. Schools and classes organized so that students are
well known and well supported.
Darling-Hammond, L. (2006) 2006 DeWitt Wallace-Reader’s Digest
Distinguished Lecture – Securing the right to learn. Policy and practice for
powerful teaching and learning. Educational Researcher, 35(7), 13 – 24.
Effective Instruction
Research on effective teaching has not suggested a
direct association between a single method of
teaching and a resulting goal…Research points
to…certain features of instruction that result in
improved student learning.
Hiebert, J., & Grouws, D. A. (2006). Research analysis: Which instructional methods are most
effective? Reston, VA: National Council of Teachers of Mathematics.
Some Features of Mathematical
Practice of Effective Instruction –
T2
TASKS
Conceptual Engagement &
Productive Struggle
TALK
Mathematical Discourse
Supporting Mathematics
Learning
 Research indicates that if effective Tier 1
instruction is in place, approximately 80% of
students’ with mathematical learning difficulties
can be prevented. (Gersten et al. 2009a; Wixon
2011)
Administrator’s Guide: Interpreting the Common Core State Standards to Improve Mathematics Education (NCTM, 2010)
Grade Level Resource Page
 http://dese.mo.gov/divimprove/assess/grade_level_res
ources.html
 http://www.dese.mo.gov/divimprove/assess/Released_I
tems/riarchiveindex.html
Math Glossaries
 http://dese.mo.gov/divimprove/curriculum/documents/M
AgleglossaryK-6.pdf
 http://dese.mo.gov/divimprove/curriculum/documents/M
Agleglossary7-12.pdf
Math Examples
 http://dese.mo.gov/divimprove/curriculum/GLE/example
s/