Chapter 7

Question 1

Progression of early number + counting: spatial patterns and recognizing (1)

Answer 1

perceptual subitizing = know the dot pattern is four, but could not prove why

example: five as a learned pattern from playing dice games

Question 2

Progression of early number + counting: more and less (2)

Answer 2

can visually see and compare the difference - do not need to count and compare candies in two hands (ties back to subitizing)

Question 3

Progression of early number + counting: rote counting (3)

Answer 3

sequence - reciting number names (they cannot count “one to six” because they do not know where six is - they will just keep going)
kindergarteners rote count to 100, forwards and backwards

Question 4

Progression of early number + counting: one-to one correspondence (4)

Answer 4

producer - “put eight in the bowl”
counter - counting a finite set

Question 5

Progression of early number + counting: cardinality (5)

Answer 5

the last number in the count sequence describes the quantity of the set - a counting strategy

Question 6

Progression of early number + counting: hierarchical conclusion (6)

Answer 6

numbers are nested in other numbers - if you have eight pennies, you also have seven pennies if you remove one - do they recount seven, or just remove one?

Question 7

Progression of early number + counting: number conservation

Answer 7

child development - remember with the buttons? the same amount, but one stretched?
When students can decompose, students know that four consists of parts of a whole, 1+3, 2+2
* to conceptualize a number as being composed of two or more parts is the most important relationship that can be developed about numbers.

Question 8

T or F - subitizing happens when you decompose a group of dots shown in a pattern of ten by seeing five in one row and mentally doubling it to ten

Answer 8

True!

Question 9

Verbal counting has at least two skills…

Answer 9

1) producing standard string of counting words in order 2) connect this sequence in a one-to-one correspondence

Question 10

Counting on is the ability to…

Answer 10

start counting from a given number other than one

Question 11

The concept of (more/less) proves to be more difficult for children then (more/less)

Answer 11

less, more

Question 12

Estimation is a very difficult task for young children. How can teachers support this skill?

Answer 12

“Will _____ be more or less than ___ (nonstandard unit of measurement)?”
“Will _____ be closer to ___ (nonstandard UOM) or ____ (nonstandard UOM)?”
“About how many _____ (nonstandard UOM) is _____”

Question 13

What is the learning trajectory for counting?

Answer 13

1) precounter
2) reciter
3) corresponder
4) counter
5) producter
6) counter and producer
7) coutner backwards
8) counter from any number
9) skip counter

Question 14

precounter

Answer 14

The child has no verbal counting ability. A young child looking at three balls will answer “ball” when asked how many. The child does not associate a number word with a quantity.

Question 15

reciter

Answer 15

The child verbally counts using number words, but not always in the right order. Sometimes they say more than one objects they have to count, skip objects, or repeat the same number.

Question 16

corresponder

Answer 16

A child at this level can make a one-to-one correspondence with numbers and objects, stating one number per object. If asked “how many” at the end of the count, they may have to recount to answer.

Question 17

counter

Answer 17

This child can accurately count objects in an organized display (n a line, for example) and can answer “how many” accurately by givint the last number counted (cardinality).

Question 18

producer

Answer 18

A child at this level can count out objects to a certain number. If asked to give you five blocks, they can show you that amount.

Question 19

counter and producer

Answer 19

A child who combines the two previous levels can count out objects, tell how many are in a group, remember which objects are counted and which are not, and respond to random arrangements. They begin to separate tens and ones, like 23 is 20 and 3 more.

Question 20

counter backwards

Answer 20

a child at this level can count backward by removing objects one by one or just verbally as in a “countdown”

Question 21

counter from any number

Answer 21

This child can count up starting from numbers other than one. They are also able to immediately state the number before and after a given number.

Question 22

skip counter

Answer 22

Here the child can skip-count with understanding by a group of a given number- tens, fives, twos, and so on

Question 23

counting on

Answer 23

the ability to start counting from a given number other than one - a landmark on a child’s path to number sense

Question 24

number sense

Answer 24

the ability to think flexibly about numbers including various ways to represent and use numbers - includes number relationships, such as one more and one less, relationship to benchmark numbers such as 5 and 10 (e.g. eight is “five and three more,” or “two away from ten” , and part-part-whole relationships

Question 25

What is a possible list of the kinds of things that children should learn about the number eight (or any number to about 20) while in PreK and Kindergarten

Answer 25

• Count to 8 (number words and sequence)
• Count 8 objects and know the last number word tells how many (cardinality)
• Recognize, read, and write the numeral 8 and pair it with an accurate amount of objects or units
• state more and less by 1 and 2
• recognize patterned sets for 8
• relate to the benchmark numbers of 5 and 10
• state part-part-whole relationships: 8 is the same as 5 and 3, 2 and 6. etc (as well as knowing the missing part of 8 when some quantities are hidden)
• identify doubles
• state relationships to the real world (my brother is eight, my book is 8 inches wide)

Question 26

Three level progression of a child’s understanding of ten

Answer 26

1) An initial concept of ten - understands ten as ten ones and does not see the ten as a unit
2) An intermediate concept of ten - understands ten as a unit composed of ten ones bur relies on materials or representations to help complete tasks involving tens
3) child solves tasks involving tens and ones without using materials or representations - can mentally think about two-digit numbers as groups of tens and ones

Question 27

How to help… a child who does not know the count sequence

Answer 27

• students can correct counting errors
• practice counting out loud (forwards and backwards)
• place number cards in order

Question 28

how to help… a child who does not use one-to-one correspondence

Answer 28

• have child place items they count in an egg carton or ten frame
• pointer to touch each item as they count
• make a plan for counting (e.g. arrange objects in a row and count from left to right)

Question 29

how to help… a child who does not count on

Answer 29

• after child counts out one set and states how many, cover the collection with a sheet of paper or put collection in a cut - the idea is to remove the objects from sight, forcing the child to create a mental image of the objects - let the child peek at the hidden collection if needed but, encourage the child to think about how many before peeking

Question 30

how to help… a child who is confused by perceptual cues such as spacing or size of counters

Answer 30

• pose situations that ask the students to use one to one correspondence to “prove” the two amounts are equal
• use matching to compare sets (e.g. stack counters on top of images to match the sets).

Question 31

how to help… a child who does not understand the cardinality principle

Answer 31

• play board games where students need to move along a path. instead of moving one marker along the path, leave one counter in each space and ask the child how many spaces they traveled
• when counting objects together, end with “we have x [object]s”

Question 32

how to help… a child who has difficulty counting teen numbers or decade numbers

Answer 32

• number games with counting, or count children as they line up
• each student gets a card, and they need to match with the first number of the next decade (e.g. 29 and 30)

Question 33

how to help… a child who writes numbers backwards or reverses the digits in the teen numbers

Answer 33

• vertical number line to show pattern in writing teen numbers
• have students circle numerals not written correctly

Question 34

how to help… a child who is not sure of the magnitude of number from 1-20

Answer 34

• use a walk on a number line to have students count the number of units from the start - place card stock pieces to demonstrate the length as a unit