Math 413 Mathematics Tasks for Cognitive Instruction

 October 2008 1

NCTM Standards Compared to Connecticut Scope and Sequence

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Connecticut Scope and Sequence

Number Sense Operations Estimation Ratio, Proportion and Percent Measurement Spatial Relations and Geometry Probability and Statistics Patterns Algebra and Functions Discrete Mathematics      

NCTM Content Standards

Numbers and Operations Algebra Data Analysis and Probability Geometry Measurement       NCTM Process Standards Problem Solving Reasoning and Proof Connections Communication Representation 2

NCTM and CT Scope and Sequence

 http://www.nctm.org

 http://www.sde.ct.gov/ sde/cwp/view.asp?a=2 618&q=320872 3

The Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materials TASKS as set up by teacher TASKS as implemented by students Student Learning A representation of how mathematical tasks unfold in the classroom during classroom instruction (Stein & Smith, 1998) 4

What is a Math Lesson?

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 It is a complex function of the work or task that you select  How you set it up?

 How your students will understand what the work you selected demands?

 What they do?

 What you make of what they do?

 What do you do next?

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Defining Levels of Cognitive Demand of Mathematical Tasks

 Lower Level Demands – Memorization – Procedures without connections  Higher Level Demands – Procedures with Connections – Doing Mathematics 6

Levels of Cognitive Demand as Compared to Bloom’s Taxonomy

Highest Levels Doing Math Procedures with Connections Procedures without Connections Memorization Lowest Levels 7

Verb Examples Associated with Each Activity Lower Level of Cognitive Demands



Knowledge

: arrange, define, duplicate, label, list, memorize, name, order, recognize, relate, recall, repeat, reproduce state. 

Comprehension

: classify, describe, discuss, explain, express, identify, indicate, locate, recognize, report, restate, review, select, translate. 8

Defining Levels of Cognitive Demands of Mathematical Tasks

Lower Level Demands  Memorization:  What are the decimal and percent equivalents for the fractions ½ and ¼ ? 9

Defining Levels of Cognitive Demands of Mathematical Tasks

Lower Level Demands  Memorization:  What are the decimal and percent equivalents for the fractions ½ and ¼ ?  Expected Student Response :  ½=.5=50%  ¼=.25=25% 10

Defining Levels of Cognitive Demands of Mathematical Tasks

Lower Level Demands  Procedures without connections :  Convert the fraction 3/8 to a decimal and a percent.

 Expected Student Response :  Fraction 3/8  Divide 3 by 8 and get a decimal equivalent of .375

 Move the decimal point two places to the right and get 37.5 % 11

Verb Examples Associated with Each Activity Higher levels of cognitive demand

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Application

: apply, choose, demonstrate, dramatize, employ, illustrate, interpret, operate, practice, schedule, sketch, solve, use, write. 

Analysis

: analyze, appraise, calculate, categorize, compare, contrast, criticize, differentiate, discriminate, distinguish, examine, experiment, question, test. 12

Defining Levels of Cognitive Demands of Mathematical Tasks

 Higher Level Demands Procedure with connections:  Using a 10 by 10 grid, illustrate the decimal and percent equivalents of 3/5.

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Verb Examples Associated with Each Activity Highest levels of cognitive demands

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Synthesis

: arrange, assemble, collect, compose, construct, create, design, develop, formulate, manage, organize, plan, prepare, propose, set up, write. 

Evaluation

: appraise, argue, assess, attach, choose, compare, defend estimate, judge, predict, rate, core, select, support, value, evaluate 14

Defining Levels of Cognitive Demands of Mathematical Tasks

 Higher Level Demands Doing Mathematics:  Shade 6 small squares in a 4 X 10 rectangle. Using the rectangle, explain how to determine each of the following:  A) the percent of area that is shaded  B) the decimal part of the area that is shaded  C) the fractional part of the area that is shaded 15

Lesson Planning

 An effective lesson plan begins with a relevant clearly written objective.

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Lesson Objective Definition and Purpose

    An objective is a description of a learning outcome. Objectives describe where we want students to go – not how they will get there. Well written objectives clarify what teachers want their students to learn, help provide lesson focus and direction, and help guide the selection of appropriate practice.

In addition, teachers can assess their students learning and their own teaching to determine if the lesson objective has been met 17

Example of a State Standard

Content Standard#1 Grade 4

 Use models, benchmarks and equivalent forms to judge the size of fractions (in relation to ½,1/4, ¾ and the whole and decimals in situations relevant to students’) 18

Number and Operations Standard for Grades 3-5 Expectations Example of the NCTM Standard

 In grades 3-5 all students should Use models, benchmarks, and equivalent forms to judge the size of fractions  Recognize and generate equivalent forms of commonly used fractions, decimals and percents  http://www.nctm.org

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From General to Specific: Going from State Standards to

 

Objectives

While state and national standards provide general content ideas, teachers are responsible for writing their own objectives for their lessons, activities and units.

A teacher’s job is to

translate

the standards into useful objectives that are used to guide instruction.  The

learning outcomes

included in the objectives will then be linked to the state standards.

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How standards, goals, and objectives differ…

   

Specific

–Objectives include specific learning outcomes where standards include general outcome statements.

Goals may be general, for example, understand the concept of fractions.

Long-Term or Short Term

–Objectives are considered short term, they describe the learning outcome typically in days, or weeks.

Goals and standards describe learning outcomes that may be in weeks, months or years.

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How standards, goals, and objectives differ…



Uses

– Objectives are used in lesson and activity plans and IEPs.

 Measurable annual goals are included in IEPs.  Goals may also be found in units of instruction. For example, a goal may be to understand how to add fractions.

 A specific objective may be to be able to add fractions will common denominators.

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Examples of Goals and Objectives Related to State

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Standards

Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.

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The Four Components of an Objective

 Content  Behavior  Condition  Criterion  Concepts taken from

Daily Planning for Today’s Classroom

by Kay M. Price and Karna L. Nelson 24

Content

 Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.



Content-

In the example given the content is

subtraction problems (two-digit numbers from three-digit numbers with re-grouping)

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Behavior

 Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re grouping) on a worksheet, with two errors.

 

Behavior-

the behavior tells what the students will do to show that they have learned .

It is a verb that describes an observable action. In this example the behavior is

“write”

. The student will demonstrate knowledge of subtraction by writing the answers to the 20 problems. (See Bloom) 26

Condition



Condition-

It is important to describe the conditions or circumstances under which the student will perform the behavior.

 In the example objective, the condition is “on a worksheet” not in a real world context.

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Criterion

 Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.

 

Criterion-

The criterion is the level of acceptance performance, the standard of mastery of proficiency level expected. In the objective above, the criterion is with two errors

.

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Examples and Nonexamples of

Content

 Add unlike fractions with common factors between denominators  ________________________________  Add fractions on page 42, 1 to 7 29

Examples and Nonexamples of

Behavior

 Diagram, operate, order, compare/contrast  _________________________________  Know, understand, memorize, learn 30

Examples and Nonexamples of

Conditions

 Given ten problems and a calculator  __________________________________  Given a blank piece of paper, when asked by the teacher (obvious) 31

Examples and Nonexamples of

Criterion

 With no errors  With 80 percent accuracy  Within 10 minutes  To the nearest tenth  __________________________  As judged by the teacher  To the teacher’s satisfaction 32

A Final Thought

    It is very important to begin your lesson or activity with a

clear idea of what you want your students to learn.

Writing a

specific objective

with the four components will cause you to think this through. When teachers experience frustration with a particular lesson, they often have not stated a measurable objective. If you clearly state the objective

,

you will know if your activity or lesson and your intended learning outcome match.

You will be able to tell if your teaching was effective and whether your students learned

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