What are the 5 mathematical processes?
- Problem-solving
- Reasoning and proof
- Communication
- Connection
- Representation
What are the 8 mathematical practices from the common core state standards?
- (PERSEVERE) Make sense of problems and persevere in solving them
- (REASON) Reason abstractly and quantitatively
- (CONSTRUCT) Construct viable arguments and critique the reasoning of others
- (MODEL) Model with mathematics
- (TOOLS) Use the appropriate tools strategically
- (STRUCTURE) Look for and make use of structure
- (REPEATED REASONING) Look for and express regularity in repeated reasoning
What are open-ended problems? Define and give 2 examples.
A problem that can have more than one correct answer
Lex has $10 and she wants to buy a coffee and treat. Lattes are $3 croissants are $3.50 and cookies are $2.50 and chocolate bars are $1.50 and a cheese danish is $2.50. What are some combinations of treats can buy with at least $1 to spare for the bus ticket back?
Paul was making a bracelet and had 4 black beads, 5 blue beads, 4 white beads, and 7 green beads. List 3 different combinations of patterns he can make using these beads for his bracelet.
How can a teacher teach through problem-solving?
-Allow mathematics to be problematic for students
-Focus on the methods used to solve problems
-Tell the right things at the right time
Problems to use are
-investigative
- estimation
- conceptualize large or small numbers
- logic, reasoning, strategize, and test theories
- perform multiple steps or use multiple strategies
Describe Polya’s 4 stage model or problem solving
- Understand the problem
- Devise a plan for solving it
- Carry out the plan
- Look back to examine your solution
Discuss at least 3 problem solving strategies. What are the components of each one?
- Make a drawing or diagram
+Draw what is essential
+ Show the picture first then have students recommend a solution - Look for a pattern
+ Recognizing, describing, extending, and generalizing patterns are important - Solve a similar but simpler problem
+ Gain sight and understanding form solving an easier problem where there are relationships
+ helps them solve challenging problems by applying the strategies they now know
Describe why “looking back” is an important step to include in pre-instructional planning
- Helps students recall the steps and process it took for them to get the answer
- Reflection and discussion helps them explain their own thinking and thought process
- encourages students to check their work
Describe at least 4 strategies for managing your classroom that will allow for support in problem-solving learning.
-Small-group instruction
- Working in pairs
- Stations
- various workspaces
Identify how you would develop the following number ideas with children
- Classification
- Set inclusion
- Pattern recognition
- Seriation
- Group recognition
- Number conservation
Classification: using manipulatives so students can compare and contrast the object and then put them in groups
Set inclusion: class discussion on what sets and. Have students either draw or accurately put objects into sets and explain why they places the objects in each side
Pattern recognition: create a station where students create patterns to make a project such as necklaces, bracelets. I would then have students identify the type of pattern it is or have them follow the directions to make specific types of patterns
Seriation: students would sort objects from biggest to smallest
Group recognition: Students would be given objects that are already sorted into groups which they would then have label and categorize
Number conservation: Having students count objects that are grouped in the same amount but arranged differently
Describe the 4 principles upon which the counting process is based
- Each object to be counted must be assigned one and only one number name
- The number name list must be used in a fixed order
- The order in which the objects are counted doesn’t matter
- The last number name used gives the number of object (Cardinality rule)
Distinguish between the following counting strategies
- Rote counting
- Point counting
- Rational counting
- Counting back
- Skip counting
- Counting on
Rote counting:: knowing the number names but not necessarily in the correct sequence
Point counting: pointing at the numbers and counting as you point
Rational counting: gives the correct numbers their names and is able to answer questions about the number of objects being counted
Counting back: giving the correct number names as they count backward
Skip counting: giving the correct number by counting by 2’s, 5’s, 10’s or other values
Counting on: giving the correct number names as counting proceeds and can start at any number and begin counting
Distinguish between the following set ideas
- Equal sets
- Empty or null set
- Subset
- Union
- Equivalent
- Disjoint set
- Intersection
- One-to-one correspondence
Equal sets: Different names for the same set
Empty or null set: a set that contains no elements
Subset: If every element of set A is also an element of set B then A is a subset of B
Union: the set of all elements that are either in A or in B or in both A and B
Equivalent sets: two sets are equivalent if their elements can be placed in one-to-one correspondence
Disjoint set: sets that have no elements in common
Intersection: all elements are in A and B
One-to-one correspondence: they match up one-to-one with each other
Describe symptoms of a lack of conservation of number? How can teachers help children develop this skill early on?
- A student is unable to identify the same amount of objects in different groups
-Stretching out the row of blocks when counting
- Point counting
Describe how each of the following pre-number concepts contributes to the development of meaningful counting and number sense
- Classification
- Comparisons
- Pattern group recognition
classification: Helps make sense of things around them and helps them become flexible thinkers. Helps them develop a sense for the purpose of early counting skills
Comparisons: Leads to one-to-one correspondence. Students are able to discriminate between important and irrelevant attributes
Pattern group recognition: Helps develop thinking strategies for basic facts
Describe the characteristics of students with good number sense.
- Understand numbers, ways of representing numbers, relationships among numbers, and number systems
- Understand meanings of operations and how they relate to one another
-compute fluently and make reasonable estimates
Identify the 4 properties that make our number system efficient
- Place value
- Base of ten
- Use of zero
- Additive property
Describe the 2 ideas that are foundational to understanding place value
- Explicit grouping or trading rules are defined and consistently followed
-The position of a digit determines the number being represented
Read and write number bases other than 10
to the 0 power= 1
# to the 1 power= that number
# to the 2nd power= number squared
# to the 3rd power= number cubed
Describe appropriate methods and materials for teaching place value and numeration
Ungrounded modeling: beans, cubes, or straws that children put into groups
-Regrouped modeling: materials formed into groups before a child uses them
Proportional modeling: base-ten blocks, beans glued in groups of ten on a stick
Non-proportional modeling: don’t maintain any size relationships like money
Grouping or trading: counting piles of objects; trading or grouped tens, hundreds, and thousands, and talking about the results
Recognize the 5 content standards
- Number and Operations
- Algebra
- Geometry
- Measurement
- Data analysis and probability
Be able to recognize the 8 mathematical practices
- (ESTABLISH) Establish mathematical goals to focus learning
- (IMPLEMENT) Implement tasks that promote reasoning and problem solving
- (CONNECT) Use and connect mathematical representations
- (FACILITATE) Facilitate meaningful mathematical discourse
- (QUESTION) Pose purposeful questions
- (PROCEDURAL FLUENCY) Build procedural fluency from conceptual understanding
- (STRUGGLE) Support productive struggle in learning mathematics
- (EVIDENCE) Elicit and use evidence of student thinking
For one of the mathematical teaching practices be able to describe how you would address it in your classroom
-Pose purposeful questions
-Asking students to look back and recall the steps and procedures they went through to problem solve
