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Kansas City 2/10/2012. Cathy Battles Kansas City Regional Professional Development Center [email protected]. The Show-Me Standards – PERFORMANCE (to do). GOAL 1. GOAL 3. recognize and solve problems. gather, analyze and apply information and ideas. 1.6 , 1.10. 3.2, 3.5. GOAL 2. - PowerPoint PPT Presentation
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Kansas CityKansas City2/10/20122/10/2012
Cathy BattlesCathy Battles
Kansas City Regional Professional Development CenterKansas City Regional Professional Development Center
[email protected]@umkc.edu
1.2.3.4.5.6.7.8.9.10.
1.2.3.4.5.6.7.8.
1.2.3.4.5.6.7.
1.2.3.4.5.6.7.8.
GOAL 1 GOAL 3
GOAL 2 GOAL 4
1.6, 1.10 3.2, 3.5
gather, analyze and apply information and ideas
recognize and solve problems
communicate effectively within and beyond the classroom
make decisions and act as responsible members of society
The Show-Me Standards – PERFORMANCE (to do)
2.2
Strand
Big Idea
GLE N3b
Concept
Content/Performance Standards
DOK
GLEs/CLEs
Number and Operations
DEPTH OF KNOWLEDGELevel 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.
6
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.
7
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.
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
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 minB 8 hrC 4 hr 30 minD 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.
Grade 4
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 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
Mushrooms
Cheese
Pepperoni
Sausage
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 BlueprintsGrade
3Grade
4Grade
5Grade
6Grade
7Grade
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?
EQUIVALENCY
TRUE OR NOT TRUE?
EQUIVALENCYTRUE OR NOT TRUE?
EQUIVALENCYTRUE OR NOT TRUE?
1. Incorrect process 20152062103796424
2. Correct Process
201
5-
2062103
7
96424
3. Another Correct Process
2015206
2062103
103796
96424
Change Direction for Each Operation
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
201
5
206
2
103
7
96
4
24
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.
5. Correct Process
2015 -206
2
103796
4
24
6. Another Correct Process
201 206 103 96
5- 2 7 4
206 103 96 24
Change Direction for Each Operation
When researchers asked first- through sixth-grade 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 Sentencemathematical 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
EquationsIf 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 PatternsWhen 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 5Example: multiply by 7Example: multiply 6 times 3 and add 1
Explicit NotationIn 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 + 7independent variable (x) or term number (n)
1 2 3 nDependent variable (y) or term (t)
0 2 4
Recursive Notation
Middle School
Example: 7, 10, 13…
First Now = 7, Next = Now + 3
OR
In the recursive form of pattern generalization, the rule focuses on the change from one element to the next.
an= nth term
a1 = first term
an – 1 = previous term
High School
Example: 5, 9, 13…
a1 = 5 ,
an= an-1 + 4
ARRAYA 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 DataDiscrete 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 Intermediate Advanced
DescribingLocation
The is next to the
The is next to the and below the .
The is between the , beneath the , and to the right of
.
Examples The square is next to the triangle.
The square is next to the triangle and below the hexagon.
The square is between the triangle 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 researchBest 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
TasksHigh 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 Difference1. 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 InstructionResearch 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 Pagehttp://dese.mo.gov/divimprove/assess/grade_level_res
ources.html
http://www.dese.mo.gov/divimprove/assess/Released_Items/riarchiveindex.html
Math Glossarieshttp://dese.mo.gov/divimprove/curriculum/documents/M
AgleglossaryK-6.pdf
http://dese.mo.gov/divimprove/curriculum/documents/MAgleglossary7-12.pdf
Math Exampleshttp://dese.mo.gov/divimprove/curriculum/GLE/e
xamples/