Download Week Two

TEACHING VOCABULARY
AND LANGUAGE SKILLS
Two Areas:


Language of instruction
Mathematics-related vocabulary and language
skills
Language of Instruction



Terms commonly used in directions given by teachers
(directions, actions, names of objects, names of
colors).
Students should be screened to ensure they possess
the language concepts and if not they should
receive remediation.
Remediation:
 Place
in math program with carefully controlled teacher
wording and provide supplementary language
instruction
Math-related Vocabulary and
Language Skills

Terms used to describe characteristics of objects
 e.g.,

square, circle, dime,
Terms used to describe relationships between
objects
 e.g.,
parallel, similar, near, far
Math-related Vocabulary and
Language Skills

Terms used to describe numbers in an operation and
the operations themselves
 e.g.,

sum, addend, difference, add, subtract
Classification terms
 e.g.,
6 boys, 7 girls, 3 cats
Guidelines


Need to integrate brief vocabulary-oriented
instructional activities into math curriculum
Sequence of instruction depends on necessity of
term. Some terms must be taught as preskills, others
can wait until strategy is taught.
 Preskill
-- end with, side, equal, same, other
 Later -- denominator, numerator, subtrahend
Vocabulary Teaching Procedures



Modeling positive and negative examples
Using synonyms
Giving definitions
Modeling Positive and Negative
Examples



Model positive and negative examples of the new
word
Test the students on their mastery of the examples
Present examples of the new word along with
examples of other previously taught words
Presentations:



Quickly paced
Stress important words (this is not)
Present until all students are able to respond
correctly to a group of three positive and three
negative examples
Teaching Vocabulary with Synonyms

Teacher links new word with previously learned
words rather than modeling examples
 Must
carefully select word used as a synonym -- be sure
word is familiar


Tests with positive and negative examples
Provide practice in applying several recently taught
synonyms
Format

Model and immediate acquisition


“Here is a new word. Subtract. Subtract means minus. What
does subtract mean?”
Positive and negative examples
Write 4 + 2 on the board. “Do we subtract in this problem?”
 Write 6-3 on the board. “Do we subtract in this problem?”


Review in context of other words.
What does ADD tell us to do? (plus)
 What does SUBTRACT tell us to do? (minus)

Teaching Vocabulary with Definitions

Teach definition
 Must
carefully select words used in definition -- be sure
word is familiar (i.e., a preskill).


Show positive and negative examples
Contrast it with previously learned definitions
Format

Model and immediate acquisition
A

sum is the answer when you add. What is the sum?
Positive and negative examples
 Write
4-1=3. Ask, “Is 3 a sum? How do you know?”
 Write 4+2=6. Ask “Is 6 a sum? How do you know?”

Review in context of other words
 What
is the DIFFERENCE of 5 and 2?
 What is the SUM of 5 and 2?
Critical Preskills


Equality
More-Less
Equality

Teach first in a context other than addition
 Teach
functional definition
 Present series of positive and negative examples
More-Less



Important in story problems
Introduce as synonym (bigger, not bigger)
Present series of positive and negative examples
COUNTING
Instructional Analysis
Questions to ask yourself for each type of counting:
 What are the preskills?
 What is this a preskill for?
 What sequencing guidelines apply?
 What are potential errors?
 How do I correct them (remediation)?
Preskills


What are preskills?
Give an example of a skill that is a preskill for a
more advanced skill.
Sequence & Integration
General Guidelines
 Preskills are taught before they are needed in
strategies.
 Easy skills are taught before more difficult ones.
 Strategies and information that is likely to be
confused are spaced or separated.
Types of math knowledge errors

Fact

Component

Strategy

Incorrect operation

Random errors
Fact Error

Student incorrectly responds to a memory task in
which s/he is asked to tell the answer to one of the
100 addition, multiplication, subtraction facts or the
90 division facts.
 For
example,
2+2=5
 7 x 3 = 14
5-2=2
4/2=4

Component Error

Student makes error on previously taught skill that
has been integrated as a step in a problem solving
strategy.
 For
example
 counts
incorrectly or forgets the name of a numeral while
completing an addition problem in lower grades.
 forgets to rewrite fractions as equivalent fractions in an
addition problem or forgets to put a zero in the ones column
when completing a multi-digit multiplication problem in
upper grades.
Strategy Error

Student demonstrates that s/he does not know steps
in strategy.
 For
example,
 Student
doesn’t attempt to rename in a multiplication or
subtraction problem.
 Student multiplies top number by bottom number in a multidigit multiplication problem rather than both top numbers
by each of the bottom numbers separately.
Incorrect Operation

Student uses wrong operation -- fails to discriminate
between operations.
 For
example,
 25
- 12 = 37
 13 x 3 = 16
Random Error

Student makes random, inconsistent errors across
different problem types.
 May
be related to motivation.
 Becomes a concern when accuracy drops below 85 to
90%.
General Diagnosis and Remediation

Four step procedure
 Teacher
analyzes worksheet errors and hypothesizes
what the cause might be.
 Teacher interviews student to determine cause of the
error if its not obvious.
 Teacher provides reteaching through board or
worksheet presentations.
 Teacher tests student on a set of problems similar to the
problematic ones.
Specific Remediation

Fact


Component


Reteach strategy.
Incorrect operation


Reteach specific skill, provide additional practice.
Strategy


Provide more practice, motivation.
Precorrect, prompt.
Random errors

If accuracy below 85%, observe closely and work to
increase motivation.
Counting



Why is counting important?
What is rote counting?
How is it different from rational counting?
(What is the preskill for rational counting? Which
sequencing guideline?)
(Rational counting of 2 groups is a preskill for what?
Which sequencing guideline?)
Counting
What is counting from a number?
(What is counting from a number a preskill for?
Which sequencing guideline is this?)

Counting




What is skip counting?
Why should skip counting by 10 be taught early?
What other skill does skip counting facilitate?
Which of the sequencing guidelines do these
exemplify?
Rote Counting
How do you determine where to start rote counting
with young children?
 How do you teach rote counting?
(See Summary Box 4.1 and Format 4.1)

Rote Counting: Error Correction
How do you correct students who leave out a number
when rote counting?
Correction Procedures




“Stop”
Model, lead, test the “hard part” (2 numbers prior
to the error)
Test the whole sequence
Delayed test
Rote Counting: Practice and Review
How can a teacher provide enough practice in order
for lower performing students to master rote
count?
Rational Counting
Again, what is it?
Why start with pictures rather than manipulatives?
Format 4.2—How is rational counting taught?
Rational Counting: Error Correction
What 2 types of errors can students make?
Rational Counting: Error Correction
How do you correct coordination errors?
How do you correct rote counting errors?
Rational Counting: Error Correction
How do you correct coordination errors?
1.
Tell the students to count only when they touch (you
can model too).
2.
“Test”—repeat the exercise.
3.
Continue until students can count correctly several
(3) times.
4.
Delayed “test”—repeat the exercise later.
(Provide lots of practice and review.)
Rational Counting: Error Correction
How do you correct rote counting errors?
1.
Model the hard part.
2.
Lead students on the hard part.
3.
“Test”—repeat the exercise (from 1).
4.
Continue until students can count correctly several
(3) times.
5.
Delay “test”—repeat the exercise later.
(Provide lots of practice and review.)
Rational Counting: Two Groups
Why?
What error might students make?
How do you correct?
Counting from Different Numbers
Why?
How?
What error might the students make?
How do you correct this error?
Counting Backwards
Why?
How?
Rote Counting by 1s from 30 to 100



Preskills: Rote counting from a number other than 1;
skip counting by 10s
Important skill to practice is counting across
"decades."
Demonstrate the relationship between tens
groupings (i.e., sequence of numerals 1, 2, 3. . .21,
22, 23).
Instructional Sequence




Count numbers higher than 100, stay within
centuries and decades,
Count numbers higher than 100, stay within
centuries, but count across decades,
Count across centuries beginning and ending at
number ending with 5
After mastery, change examples to promote
generalization.
Skip Counting: Count-by Series
Why?

Why should counting by 10 be taught early?

What other skill do count by series facilitate?

Which of the sequencing guidelines do these
exemplify?
Skip Counting: Count-by Series
Why is it suggested by we put count-by series in the
following order (sequencing guideline):
10, 2, 5, 9, 4, 25, 3, 8, 7, 6
Skip Counting: Count-by Series
The format (4.5) has 2 parts. What are they for?
How do you teach a new series?
When can the next series be started?
SYMBOL IDENTIFICATION
AND PLACE VALUE
Symbol Identification and Place Value

Three major areas:
 reading
and writing numerals
 column alignment
 expanded notation
Terms

What do the following terms mean:
 Number
 Numeral
 Place
value
 Expanded notation
 Column alignment
Introducing the Concept

Concepts for kindergarten through early 1st grade
 Numeral
identification (0-10),
 Numeral writing (0-10),
 Symbol identification (+, -, =, ),
 Equation reading and writing,
 Numeral and line matching.
Introducing Numeral Identification


When do you start?
What sequencing guideline is critical in determining
the order in which numerals are introduced?
Introducing Numeral Identification



Order of introduction: what numerals would you
separate?
Rate of instruction: how fast can we introduce new
numerals?
How do you introduce new numerals?
Introducing Numeral Identification


Write review numerals (how many times?) and new
numeral (how many times?) on board.
Introduce new numeral.


Discrimination practice.


(This is __. What is it?)
(What order?)
Individual turns.
Introducing Numeral Identification



Why do you need to signal?
How do you signal when students are looking at the
numerals on the board?
How long should you spend on this task?
Introducing Numeral Writing





When can you introduce numeral writing?
Rate of introduction?
What are the stages of introduction (scaffolding)?
What is numeral dictation? What order do you
dictate numerals?
How do you correct student errors?
Introducing Symbol Identification and
Writing
+ - =

How do you introduce symbols?
Introducing Equation Reading and
Writing



What is equation reading a preskill for?
When is equation reading introduced?
How do you teach equation reading?
Introducing Equation Writing



When is equation writing introduced?
How do you teach equation writing?
How do you correct if students write numerals out of
order?
Numeral/Object Correspondence
Students identify the symbol (numeral) and write
that number of lines.
2.
Students count the objects and write the numeral.
Preskills for addition and subtraction using equality
strategy.
1.
Numeral/Object Correspondence

When can you introduce these numeration skills?
Numeral/Object Correspondence



Before teaching students to identify the symbol
and write the lines, what preskill must students
have?
See format 5.3.
What errors might students make in 5.3?
Numeral/Object Correspondence
Why is writing numerals to represent a set of
objects important?
4 + 2 =
llll
ll

Numeral/Object Correspondence



Format 5.4 teaches students to count the objects
and write the numeral.
What are the preskills?
What errors might students make?
Numeral/Object Correspondence



What should you do if students make a counting
error?
What should you do if the students make an error in
numeral identification or writing?
When do you introduce manipulatives?
Place Value



Reading and Writing Numerals
Column Alignment
Expanded Notation
Reading and Writing Teen Numerals





When is reading teens numerals introduced?
What is the order of introduction?
See format 5.5.
When are “irregular” teens introduced?
What is the rate of introduction for irregular teens?
Reading and Writing Teens Numerals



When is writing teens numerals introduced?
See format 5.6.
When might manipulatives be used?
Reading Numerals
20-99


What are the preskills?
Format 5.7
Writing Numerals
20-99



When is this introduced (that is—what are the
preskills)?
See format 5.8.
When dictating numerals in step E, what is the
example selection guideline?
Writing Numerals
20-99

What pattern of errors might students make
(diagnosis)?
For a Diagnosis and Remedy
2.
State the diagnosis.
State the formats that you would reteach.
3.
State the examples that you would emphasize.
1.
Remediation for Written Reversals
(such as 71 for 17)


Reteach writing teens format. At the same time,
reteach writing tens numbers format (without 1s in
the ones place—like 31).
Then teaching writing format with minimal
discriminations—21 & 12, 41 & 14, etc.
Reading and Writing Numerals
100-999



Reading hundreds—What are the preskills?
See format 5.9.
Sequencing: What is avoided initially?
Then, what examples are used?
Reading and Writing Numerals
100-999
Sequencing: What is avoided initially?
(0 in the tens place)
 What examples are used when 0 in the tens place
is included?

Reading and Writing Numerals
100-999
Writing hundreds numerals—Format 5.10
Reading and Writing Numerals
1,000-999,999


What is the sequence for introducing these
numerals?
What are the example selection guidelines when
zeros are introduced?
Column Alignment


Why is this an important skill?
See format 5.13
Expanded Notation


What is expanded notation?
See Format 5.14.
CURRICULUM EVALUATION