Research and Math Tools

Question 1

Mathematics:

Answer 1

The Science of Pattern and Order

Question 2

Mathematically Proficient Students:

Answer 2

Explaining to themselves the meaning of a problem and looking for entry points to its solution.
Analyze givens, constraints, relationships, and goals. Make conjectures about the form, meaning of the solution and plan a solution pathway.
Check their answers to problems using a different method, and continually ask themselves, “Does this make sense?”
Can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Question 3

Decontextualize:

Answer 3

Ability to abstract a given situation and represent it symbolically and manipulate the representing symbols.

Question 4

Contextualize:

Answer 4

To pause as needed during the manipulation process in order to probe into the referents for the symbols involved.

Question 5

Quantitative Reasoning:

Answer 5

Entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Question 6

Standards for Mathematical Practice:

Answer 6

Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

Question 7

Mathematical Info:

Answer 7

In multiplication, how an odd number times an odd number always generates an odd answer, an odd number times an even number is always an even number, and an even number times an odd number is always an even number.

Question 8

Mathematical Pattern:

Answer 8

What are the relationships between these examples?

Question 9

Mathematics Begin:

Answer 9

With posing worthwhile tasks, creating an environment where students take risks, share and defend mathematical ideas. Students are actively engaged in solving problems, and teachers are posing questions.

Question 10

Language of Doing Mathematics:

Answer 10

Compare 
Conjecture 
Construct 
Describe 
Develop
Explain
Explore 
Formulate 
Investigate 
Justify 
Predict 
Represent 
Solve 
Use 
Verify

Question 11

Expectations in Doing Mathematics:

Answer 11

1. Persistance, effort, and concentration are important in learning mathematics. Engaging in productive struggle is important in learning!
2. Students share their ideas. Everyone’s ideas are important, and hearing different ideas helps students to become strategic in selecting good strategies.
3. Students listen to each other. All students have something to contribute while being evaluated.
4. Errors or strategies that didn’t work are opportunities for learning. Mistakes are opportunities for learning.
5. Students look for and discuss connections. Students should see connections between different strategies to solve a particular problem.

Question 12

Conceptual Understanding:

Answer 12

Making mathematics relationships explicit and engaging students in productive struggle.

Question 13

Standards for Teaching Mathematics:

Answer 13

1. Knowledge of Mathematics and General Pedagogy
2. Knowledge of Student Mathematical Learning
3. Worthwhile Mathematical Tasks
4. Learning Environment
5. Discourse
6. Reflection on Student Learning
7. Reflection on Teaching Practice

Question 14

Mathematics Strategies:

Answer 14

Tree Diagrams

Grids

Question 15

Constructivism:

Answer 15

Jean Piaget’s work; The notion that learners are not blank slates but rather creators (constructors) of their own learning. Integrated networks, or cognitive schemas, are both the product of constructing knowledge and the tools with which additional new knowledge can be constructed.

Question 16

Assimilation:

Answer 16

Occurs when a new concept “fits” with prior knowledge and the new information expands an existing network.

Question 17

Accommodation:

Answer 17

Takes place when the new concept does not “fit” with the existin network (causing what Piaget called disequilibrium), so the brain revamps or replaces the existing schema.

Question 18

ZPD:

Answer 18

Zone of Proximal Development (Refers to a “range” of knowledge that may be out of reach for a person to learn on his or her own, but is accessible if the learner has support from peers or more knowledgeable others).

Question 19

Semiotic Mediation:

Answer 19

The “mechanism by which individual beliefs, attitudes, and goals are simultaneously affected and affect sociocultural practices and institutions”.

Question 20

Metacognition:

Answer 20

Conscious monitoring (being aware of how and why you are doing something) and regulation (choosing to do something or deciding to make changes) of your own thought process.

Question 21

THINK framework:

Answer 21

Talk about the problem.
How can it be solved?
Identify a strategy to solve the problem.
Notice how your strategy helped you solve the problem. Keep thinking about the problem.
Does it make sense?
Is there another way to solve it?

Question 22

Teaching through problem solving:

Answer 22

Focuses students’ attention on ideas and sense making.
Develops mathematical processes.
Provides a context to help students build meaning for the concept.
Allows an entry and exit point for a wide range of students.
Allows for extensions and elaborations.
Engages students so that there are fewer discipline problems.
Provides formative assessment data.
Is a lot of fun!

Question 23

Getting Lesson Ready:

Answer 23

Activate prior knowledge
Be sure problem is understood
Establish clear expectations

Question 24

Students Work:

Answer 24

Let go
Notice students mathematical thinking
Provide appropriate support
Provide worthwhile extensions

Question 25

Class Discussion:

Answer 25

Promote mathematical community of learners
Listen actively without evaluation
Summarize main ideas and identify future problems