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 multi-digit 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 Provide more practice, motivation.

Component Reteach specific skill, provide additional

practice. Strategy

Reteach strategy. Incorrect operation

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 CorrectionHow 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 ReviewHow 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 GroupsWhy?What error might students make?How do you correct?

Counting from Different NumbersWhy?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 SeriesWhy? 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 SeriesWhy 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 SeriesThe 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. (This is __. What is it?)

Discrimination practice. (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 Correspondence1. 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.

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 Numerals20-99 What are the preskills? Format 5.7

Writing Numerals20-99 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 Numerals20-99 What pattern of errors might students

make (diagnosis)?

For a Diagnosis and Remedy1. State the diagnosis.2. State the formats that you would

reteach.

3. State the examples that you would emphasize.

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 Numerals100-999 Reading hundreds—What are the

preskills? See format 5.9. Sequencing: What is avoided initially?

Then, what examples are used?

Reading and Writing Numerals100-999 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 Numerals100-999

Writing hundreds numerals—Format 5.10

Reading and Writing Numerals1,000-999,999 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