Math 2008 Exam 1 Samples

what does it mean to do math?

Doing mathematics means demonstrating mathematical processes. That is, doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense.

what are the doing math verbs?

Analyze, Design, Justify, Apply, Develop, Model, Compare, Explain, Predict, Connect, Explore, Represent, Construct, Formulate, Solve, Critique, Generalize, Use, Describe, Investigate, Verify, and Conjecture

what is something that teachers must have students do to help learn?

productive struggle

what is productive struggle?

a learning process that involves students working through challenging problems that are slightly beyond their current skill level. It's a state of engagement where students persist through obstacles and make mistakes to find solutions

what is understanding?

the measure of quality and quantity of connections between new ideas and existing ideas

what is knowing?

to have or reflect knowledge [Merriam-Webster], which can be facts, information, and skills acquired by a person through experience or education [New Oxford American]

does knowing equal understanding?

NO! (students may know something about fractions, for example, but not understand them)

knowing must involve...

understanding, so our definition of knowing math is enhanced!

instrumental understanding leads to

relational understanding

what are the ideas of mathematical proficiency?

conceptual understanding and procedural fluency

what are the 5 strands of mathematical proficiency?

- conceptual understanding

-procedural fluency

-strategic competence

-adaptive reasoning

-productive disposition

what is conceptual understanding?

Comprehension of mathematical concepts, operations, and relations.EX: Students with conceptual understanding will connect what they know about dividing numbers to make sense of scaling, unit prices and so on.

what is procedural fluency?

skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

what is strategic competence?

ability to formulate, represent, and solve mathematical problems

what is adaptive reasoning?

capacity for logical thought, reflection, explanation, and justification

what is productive disposition?

habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy

what are the implications for teaching math?

1. build new knowledge from prior knowledge

2. provide opportunities to communicate about math

3. create opportunities for reflective thought

4. encourage multiple approaches

5. engage students in productive struggle

6. treat errors as opportunities for learning

7. scaffold new content

8. honor diversity

what is the constructivism theory by jean Piaget?

Learners are not blank slates but creators of their own knowledge

what is the product of constructing knowledge?

networks and cognitive schema

what is reflective thought?

how people modify schemas to incorporate new ideas

what is assimilation?

new idea fits with prior knowledge-blue dots connect to red

What is accommodation?

new idea does not fit with existing knowledge

what is the sociocultural theory by lev vygotsky?

-Mental processes exist between and among people in social interactions

-Learner (working in his or her ZPD)

what is semiotic meditation?

how beliefs, attitudes, and goals are affected by sociocultural practices and institutions

what are learning trajectories?

The selection of content for specific grades reflects not only rigorous mathematics, but also what is known from research and practice about learning progressions

what are the standards for mathematical practice?

-Make sense of problems and persevere in solving them

-construct viable arguments and critique the reasoning of others

-Reason abstractly and quantitatively

- Model with mathematics

- Attend to precision

- Use appropriate tools strategically

- Look for and make use of structure

- Look for and express regularity in repeated reasoning

What do you believe that first graders should know and understand about the number 8?

1. More and less by 1 & 8 is one more than 7 and one less than 9, and two less than 10 and two more than 6.

2. Spatial patterns for 8, i.e., visual arrangement of 8 objects.

3. Anchors to 5 & 10, 3 more than 5 and 2 less than 10.

4. Part-part whole: 8 is 5 & 3 or 2 & 6, or 4 & 4, etc.

5. Double 4 is 8 (special part-whole relationship).

6. Connections to the real world, e.g., Next year I’ll be 8 years old!

7. Measurement—recognize 8 inches? 8 feet?

8. Ordinality (Stable Order)—Countto 8 (know # words in order)

9. Cardinality—Count set of 8 objects & know that the last number word tells how many are in the set

10. Nominal—Write, recognize & read the numeral 8

what is a recounter/number word sayer?

no verbal counting ability

what is a reciter?

Verbally counts using number words, but not always in the right order

what is stable order?

When the number word sequence is a stable list, i.e., repeatable for a child, we will use the term "Stable Order" from Gelman's Counting Principles.

what is a corresponder?

Can demonstrate a one-to-one correspondence between number words and objects, stating one number word per object

what is a counter?

Can accurately count objects and answer the question "How many?" by giving the last number counted, which is called cardinality.

what is cardinality?

answer the question "How many?"

what is a producer?

Can count out and show you objects to a certain number

what is subtilizing?

This ability to recognize numbers by using visual patterns

why do you think subitizing is important for kids?

It can be an important way for students to begin to learn to "count on" because they can name a collection of objects without counting them as individual objects.

Just because a child can subitize does NOT mean...

that they can "count on".

what does counting on involve?

-Being able to start counting at any number, other than 1

-being able to monitor your counts so that you know when to stop.(Subitizing is helpful for this.)

what is a counter and producer?

Combines the previous two levels including counting groupings, separate counted objects from uncounted objects, and count random arrangements. Place-value emerges here, e.g., 13 is 1 ten and 3 ones.

what is a Counter Backwards?

Begins to count backwards verbally or physically by removing objects

-Note that we will explore backwards counting more thoroughly as there are some subtle differences among types of counting backwards

how do you develop number sense by building number relationships?

-subititzing spacial relationships

-benchmark numbers 5&10

-one and 2 more and less

-part-part-whole

what are basic addition strategies?

-Direct Modeling/Count all

-Count on from the first addend

-Count on from the larger addend

- Strategic AdditiveReasoning/Derived Number Fact/Number Conservation

what are basic subtraction strategies?

-Direct Modeling/Count all

-Count back from

-Count off

-Count back to

-Count up to

-Strategic Additive Reasoning/Derived Number Facts/Number Conservation

what are the 5 levels of counting?

1. perceptual counters

2. figurative counters

-counters of motor items

-counters of verbal unit items

-pre-numerical

3. initial number sequence (INS)

4. tacitly nested number sequence

5. explicitly nested number sequence

what are perceptual counters?

Perceptual is derived from senses(usually visual, tactile, auditory)

-Main characteristics of perceptual counters:

-Can count if actual objects are in their perceptual field

-Can usually not count hidden objects

what are figurative counters?

Figurative item is the object re-presented in visualized imagination

- Main characteristics:

-Can count hidden objects by counting figurative items

-Often uses fingers for substitute

- May not yet count own physical actions

what is a counter of motor items?

Motor item involves an action (e.g.,pointing, grasping) that is taken as an item that can be counted

-Main characteristics:

-Can count hidden objects by counting figurative items but can substitute own physical action as items to be counted

-Require some kind of motion, usually with hands; note that fingers can be reused because the movement is what is counted, not the finger

what is a counter of verbal unit items?

-Specialized counters of motor items (more difficult to conceive of an utterance as a discrete item)

-Not just reciting a number word sequence

-Main characteristics:

-No particular motor act is required (can choose from many motor acts)

-Still pre-numerical

what is a pre numerical counter?

-Children at Levels 1 and 2 (perceptual and figurative counters) are still considered to be pre-numerical.

-Pre-numerical means that the child still does not grasp the concept of number as a quantity....numbers are just things that we count.

-When asked to solve an additive story problem, the child would most likely have to use the count all what strategy.

what is initial number sequence (INS)?

First numerical level!

-Child has the ability to unitize

-Child has established Numerical Composites. For example, the number word "four" represents the counting sequence, "1, 2, 3, 4"

-Main characteristics: -Five means the result of counting 1 to 5 and also means 5 items

-So, can count on, which requires a sort of monitoring of counting--counting one's counting acts

What is INS+?

A slightly more advanced stage of INS.

-Main characteristics:

-Even if the larger number appears second in the story, the child may start with it because it gives a faster path to the solution.

- Can enact the commutative property.

- Can count on from the larger number.

what is a tacitly nested number sequence?

Putting up fingers has changed from the INS where putting up fingers were the countable items to now putting up fingers serving as a record of a counting act as well as a countable item.

-Main characteristics:

- Can use fingers to find an unknown number of counts.

- Uses counting on to and counting back to.

what is an explicitly nested number sequence?

Child can disembed. For example, 5 can be seen as 2 and 3, or 1 and 4.

-Child recognizes that a number can be constructed by iterable units of "1."For example, a child sees the number 5 as "1" five times, as well as the counting act of "1, 2, 3, 4, 5."

-Main characteristics:

Can use Strategic Additive Reasoning (also known as "using derived facts")

-A child can solve 8+5 by finding that 8+2 = 10 and 10+3=13.

- Common strategies include "making a 10" or "using doubles".

addition: direct modeling/ counting all

students necessarily count out all qualities starting at 1 to determine the sum. involves a perceptual or a figurative counter.

addition: counting on from first addend

students can count by starting with the addend given first and then add the second quantity while monitoring the count

addition: counting on from larger addend

students can count by starting with the addend given first and then add the second quantity while monitoring the count but they can recognize the larger addend even if its the second one

addition: strategic additive reasoning/derived number facts

students can count by considering relationships between numbers (number conservation -GSE). They can break down the numbers in a strategic manner to determine the sum.

subtraction: direct modeling/count all

Students necessarily count out all quantities in the problem starting at one to determine the difference. Involves a perceptual or figurative counter.

subtraction: counting back from

The student will communicate the starting quantity (minuend) and proceed tocount backwards one whole number at a time to account for the amount to be subtracted(subtrahend). They will end on the actual difference as they acknowledged the initial quantity firstbefore they proceeded to count backwards.

subtraction: counting off

Students are able determine the difference by counting backwards as they remove anumber from consideration after it's been counted. They will end on the last number counted "off" andmust acknowledge that the difference is the preceding smallest whole number.

subtraction: counting back to

Students can determine the difference by counting back to the amount to besubtracted from the initial amount while monitoring the difference at the same time. As an example, astudent asked to determine 8 - 3 can say "7" (extends 1 digit), "6" (extends 1 digit), "5" (extends 1digit), "4" (extends 1 digit), and "3" (extends 1 digit). When they see 5 digits extended, they declare thedifference is 5.

subtraction: counting up to

This is similar to Counting Back To except they start with the amount to be subtracted instead of the initial amount.

subtraction: strategic additive reasoning/derived number facts

Students can subtract by consideringrelationships between numbers. They can break down the numbers in a strategic manner to determinethe difference. For example, subtracting 27 - 9. A student can recognize that 27 = 20 + 7, 9 = 2 + 7, and20 - 2 = 18, so 27 - 9 = 20 + 7 - 2 - 7 = 20 - 2 = 18

Mathematical content standards

Contain the specific skill or concept students should learn at each grade

Mathematical practices

How the students should be able to think and work with math