Kansas City 2/10/2012 Cathy Battles Kansas City Regional Professional Development Center

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Kansas City 2/10/2012 Cathy Battles Kansas City Regional Professional Development Center battlesc@umkc.edu

DO NOW/WARM-UP Start with the number of feet in a yard Multiply by the number of sides of a quadrilateral Divide by the number of inches in a foot Multiply by the number of cm in a meter Divide by the number of years in a decade Add the number of angles in a pentagon

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 1GOAL 3 GOAL 2 GOAL 4 1.6, 1.103.2, 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 N1b Concept Content/ Performance Standards DOK GLEs/CLEs Number and Operations

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.

8 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.

9 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 min B 8 hr C 4 hr 30 min D 4 hr Level 2. 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. Level 3 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. 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 Level 2

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)

EQUIVALENCY TRUE OR NOT TRUE?

EQUIVALENCY TRUE OR NOT TRUE?

EQUIVALENCY TRUE OR NOT TRUE?

EQUIVALENCY TRUE OR NOT TRUE?

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.

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 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 = 2 3 + 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) 123n Dependent variable (y) or term (t) 024 2n - 2

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. a n = n th term a 1 = first term a n – 1 = previous term High School Example: 5, 9, 13… a 1 = 5, a n = a n-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 FunctionBeginningIntermediateAdvanced Describing Location The is next to the The is next to the and below the. The is between the, beneath the, and to the right of. ExamplesThe 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 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 organizers 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  The quality of teachers and teaching.  Access to challenging curriculum, which ultimately determines a greater quotient of students’ achievement than their initial ability levels; and  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 – T 2 TASKS Conceptual Engagement & Productive Struggle TALK Mathematical Discourse

Primary Example