ELM 310 and ELM 574 basic terms and concepts

Question 1

Subitizing

Answer 1

This means literally to ‘see suddenly’ In mathematics instruction it is useful for students to see that they can access number in clumps and not just by counting and ones. Humans can perceive and process no more than 5 or 6 at one time without clumping groups together. See subitising article Faulkner & Ainslie, 2017

Question 2

Cardinality

Answer 2

the idea that a set contains a quantity. It can be accessed either through counting or through subtilizing.
See Kathy Richardson and this flash card : ).

Question 3

Perceptual subitizing

Answer 3

Simplest form of subitizing. Visually perceiving quantities up to 4, 5, or 6 in one perceptual unit. this ‘fires’ the quantity processor directly without other processes.
See Clements, 1999; Faulkner & Ainslie, 2017

Question 4

Conceptual subitizing

Answer 4

To clump perceptually subitized chunks into larger numbers. For instance being able to process the quantity 8 by seeing a clump of 3 and a clump of 5.
See Clements, 1999; Faulkner & Ainslie, 2017

Question 5

Why is subitizing so important.

What is the progression of subitizing?

Answer 5

Subitizing is important because the human brain processes number through comparative fullness and is quickly overwhelmed. This analog quantity processor can only process numbers up to 5 or 6 without counting or clumping. Through subitizing we can overcome this limitation by learning to clump numbers into larger unit sizes. 10 separate things becomes 1 thing–a unit of ten. This is the fundamental idea underlying the base system.
I Perceptual. 1.’Fires’ the analog quantity processor. 2.Access cardinality without counting and 3.see numbers in clumps and not just as a series of ones
II Conceptual. 1. Understand what is underneath symbolic adding. 2.Understand that numbers are composed of other numbers. 3.Learn facts within ten at conceptual level. 4. Bamt. Use subitizing w tens frames to teach critical bamt strategies
5. Repeated multiple groups. Use subitizing to develop multiplication facts and understand the structure of the operations.
See Faulkner & Ainslie, 2017

Question 6

Conservation of number

Answer 6

Knowing that a quantity remains constant even if its parts are spread out or rearranged.
For instance ⭕️⭕️⭕️ Is the same value as ⭕️ ⭕️ ⭕
See Kathy Richardson and this flash card.

Question 7

Mental number line within 100

Answer 7

A student at this stage understands that two numbers across a decade can be closer than two numbers within a decade. For instance 28 and 31 are closer than 28 and 23.
See Video of some key Learning Trajectories

Question 8

Place value comparer

Answer 8

Student knows that 31 is larger than 29 because she understands that the tens place value is more important (carries more ‘weight’) than the 1s place. A student at this stage does not yet understand that two numbers within a decade can be farther apart than two numbers across a decade.
See Video of some key Learning Trajectories

Question 9

What does equal mean? Explain.

Answer 9

Equals means Same value as. Equality signifies that two things have the attribute if value in common. These two values do not have to be ‘the same as’ each other they simply need to have the same quantitative value. For instance 1 squared = 1 cubed because they both simplify to 1–but they certainly are not the same thing as each other.
See also faulkner why the common core changes math instruction.
See Faulkner, et al 2016

Question 10

A parent asks why you are making his “gifted” daughter subitize within ten when she already knows her facts within ten. He implies that you are ‘not challenging her enough’ and that you are holding her back. How do you reply?

Answer 10

See subtilizing article, Faulkner & Ainslie, 2017 and respond by explaining how subitizing supports advanced skills–be specific.
See also other subitising flash cards here.

Question 11

Whats the difference between being a ‘counter to ten’ and a ‘producer to ten’?

Answer 11

A counter to ten can count a predetermined amount of objects placed before them. A producer can produce a given amount of objects from a larger pile.
See Video on some key Learning Trajectories

Question 12

Why is matching comparer such an important concept and skill?

Answer 12

Connects to magnitude. Bar model. Number line game. Differences.
See Video on some key Learning Trajectories. Also consider Sharon Griffin and Number Line discussions we have had in class.

Question 13

A parent suggests that their child has moved beyond the number line game and you are holding him back. After all he can count 100. How do you respond?

Answer 13

Explain the idea of the internalized sense of number line to ten. Explain that counting by rote does not indicate whether child understands magnitude and the relationships between numbers. Note concepts such as before and after. Also the idea that numbers have a reliable relationship and that students need to see for instance that 7 is 2 more than 5 because they can decompose the 7 into the 5 that draws them equal to their competitor and the 2 extra that determines how much they are winning by.
Consider Sharon Griffin’s work and Number Line Game discussion from class.

Question 14

Why is the circuit number line game so powerful a tool for teaching place value and the base ten (arabic) numeration system?

Answer 14

Most manipulatives used to teach base ten are ‘static’ and they do not connect quantity to the numerals that represent them in a dynamic and recursive way. The circuit game allows the teacher to provide quantity support ( unit cubes and tens rods for instance) at the same time student is forced to dynamically keep track of their circuits. This exactly illustrates the cyclic nature of the base ten system and links it directly to the repetitive nature of cycling through the digits 0 through 9. As students internalize this flow they can be moved towards a digit representation of the cycles and away from the quantity manipulatives. For instance, every time i complete a cycle i get a digit card to represent how many cycles i have completed.
This is based on work of Sharon Griffin and Robbie Case.
Also - Rewatch Faulkner TEDx lecture.

Question 15

Explain the idea of different forms of a number. Why is this so important?

Answer 15

See Faulkner article why the common core changes math instruction.

Question 16

Explain the difference between
Number
Numeral
Digit

Answer 16

See Faulkner article why the common core changes math instruction.

Question 17

Final exam
What are they key elements of bar models or, as julie russo calls them structures of equality. What are the three types of bar model structures?

Answer 17

ELM 574 Spring 2017 - Don’t worry about this one. I moved this material to later in the semester.

All models must have: 
Structure
A demonstration of equality
Values
Labels/units

The three types of models are
Comparison stuctures ,
Parts equal total structures and
Repeated equal parts structures

Question 18

What does the number knowledge test assess?

Answer 18

An internalized sense of the number line through ten and through 100
Whether students are utilizing the base ten system to organize their thinking or are stuck in 'ones world'
See your class notes.

Question 19

What does it mean to iterate a unit and why is this skill so critical?

Answer 19

This means to repeat a like unit over and over. This concept is so critical because students must understand that units that are counted or added must be of the same size. This is a critical and foundational aspect of both number and operations and of measurement as well as adding fractions.
See class notes.

Question 19

Sharon griffin describes how the brain has a ‘spot’ for quantity and a ‘spot’ for the counting structure. The first big developmental leap for number is connecting the two. The second is utilizing numerals and symbols meaningfully to represent these ideas. Explain how the race to ten nmber line game supports students through these developmental leaps and gets these parts of the brain to ‘fire together’ and thus ‘wire together’.

Answer 19

By using race to ten game teachers can scaffold students 1) through the first big leap of connecting quantity and magnitude by using manipulatives lined up so that relative magnitudes can be compared and 2) by supporting students into using numerals as they internalize the number line relationships.
The number line game is the exact embodiment of this deep connection between counting out quantities and structuring their meaning through a number line that has a reliable and unbending repetitive structure.
See Class notes.

Question 21

What is transitivity?

Answer 21

It is the idea that knowing that one thing is equal to another thing allows you to use either thing in comparison to a third thing. In other words if i know my string is as long as my arm width i can use that string to compare my arm width to my height. In algebra we see this as if a=b and b= c then we know that a= c because both a and c are equal to the same thing, b.
You need to know the properties…

Question 22

What is the commutative property and what is the associative property?

Answer 22

The commutative property states that a+b = b+a. These are not the same thing but they are equal in value. This property holds for addition and multiplication but not subtraction and division.
The associative property states that a+(b+c) = (a+b) + c. This property is particularly i mportant for adding using bamt strategies 8 + ( 5) = 8 + (2+3) = (8+2)+3 = 10+ 3.
This property holds for addition and multiplication but not subtraction and division.
Bonus! It is also useful for adding integers: 8 + -5 = (3 + 5) + -5 = 3 + (5+-5) = 3 + (0) = 3

You need to know the properties…

Question 23

What is a difference and how can understanding this concept help me with strategies for subtraction?

Answer 23

A difference is the result of a subtraction.  It is literally the difference between two magnitudes. This can be seen in a comparison model or the number line game wherein the difference is any part beyond the line of equality. It can also be seen in a number line wherein the difference is literally the space between to values.  The difference between 75 and 18 therefore is the space between those two numbers.  I can "shift" this difference up or down to create an easier problem.  For instance, 75 - 18 can be shifted up 2 spaces to become 77-20.   This difference is quickly understood as 57 and is the same difference as 75-18.
See class notes.