Transcript NCTM Scope and Sequence

Math 413
Mathematics Tasks for Cognitive
Instruction

October 2008
1
NCTM Standards Compared to
Connecticut Scope and Sequence











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?
The Lesson Plan

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?
5
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

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.
13
Verb Examples Associated with Each
Activity
Highest levels of cognitive demands

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.
16
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 shouldUse 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
19
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.
20
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.

21
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.

22
Examples of Goals and
Objectives Related to State
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.
23
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)
25
Behavior

Students will write answers to 20 subtraction problems
(two-digit numbers from three-digit numbers with regrouping) 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.
27
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.
28
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.
33