Term
| What does mean to DO mathematics? |
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Definition
-generating strategies for solving problems
-applying approaches
-seeing if they lead to solutions
-checking to see whether your answers make sense |
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Term
| What are the 6 principles? |
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Definition
Equity
curriculum
teaching
learning
assessment
technology |
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Term
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Definition
high expectations & strong support for all students
have the same expectation beyond gender & all differences |
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Term
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Definition
| must b coherent focused on important mathematics & well articulated across the grades |
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Term
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Definition
effective math teaching requires understanding what students know & need to learn & then challenging & supporting them to learn it well;
all about creating problem solving models for students to solve using prior knowledge to build intrigue & feel relevance |
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Term
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Definition
| students must learn math with understanding, actively building new knowledge from experience & prior knowledge |
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Term
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Definition
| assessment should support the learning of important math & furnish useful information to both teachers & students |
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Term
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Definition
multiple choice
performance event/task
constructed response |
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Term
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Definition
-only assessment that shows if students truly understand it or not
-create word problem based on what they're working on. if there's only 1 answer to the problem it's not performance event |
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Term
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Definition
| essential in teaching and learning math; it influences math that's taught & enhances students' learning |
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Term
| What does it mean to learn mathematics? |
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Definition
| how does doing the mathematics relate to student learning? look to current theory & research |
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Term
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Definition
-creation of knowledge
-assimilation & accommodation
-reflective thought |
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Term
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Definition
learning depends on:
-learner (working in his/er ZPD)
-Social interactions
-culture |
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Term
| What does a good classroom environment for math look like? |
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Definition
-persistence, effort & concentration
-students share ideas
-students listen to each other
-errors/strategies didn't work are opportunities for learning
-students look for & discuss connections |
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Term
| When is it appropriate to use calculators? |
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Definition
-checking work
-"basics" are understood |
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Term
| Why should students consider each others' solutions? |
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Definition
| there may be alternative answers or ways to getting to an answer |
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Term
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Definition
Jean Piaget:
learners aren't blank slates but creators of their own lerning
-integrated networks/cognitive schema both product of constructing knowledge & tools any new knowledge can come from |
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Term
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Definition
| mental processes between people in social learning settings & from social settings learners move ideas into his psychological realm |
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Term
| How does Sociocultural Theory work? |
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Definition
-info internalized if in learner's ZPD if above range will need guidance
- |
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Term
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Definition
| mechanism individual beliefs, attitudes & goals simultaneously affected & affect sociocultural practices |
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Term
| what does it mean to UNDERSTAND mathematics? |
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Definition
understanding is a measure of the quality & quantity of connections -that a new idea has with existing ideas
-greater the number of connection to a network of ideas--> better the understanding
-knowing doesn't equal understanding
-Richard Skemp named ends of continuum of understanding |
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Term
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Definition
| more ways kids are given to think about a test & emerging idea, the better chance they have correctly forming & integrating it into a rich web of concepts & develop a relational understanding |
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Term
| benefits of relational understanding |
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Definition
effective new concepts & procedures
less to remember
increased retention & recall
enhanced prob-solving ability
improved attitude |
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Term
| how do students demonstrate understanding application through testing, activity, constructing word problems,etc. |
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Definition
| physical materials/manipulative in math abound from common objectives |
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Term
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Definition
1. conceptual understanding
2. procedural fluency
3. strategic competence
4. adaptive reasoning
5. productive disposition |
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Term
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Definition
-knowledge about relationships/foundational ideas of topic
-student makes sense of problems & persevere in solving them
-student looks for & makes use of structure |
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Term
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Definition
knowledge & use of rules & procedures used incarrying out math processes & also symbolism used to represent math
-student is FLEXIBLE in ways to compute answers |
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Term
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Definition
Absence of conceptual understanding results in lack of retention and increased errors.
Weak procedural skills may launch into standard algorithm unfit for certain problem solving process needed |
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Term
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Definition
| Task outside of student’s ZPD can become accessible if it is carefully structured |
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Term
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Definition
| part to whole, basic skills, fixed curriculum, text reliant, consumers, didactic, answers, summative, competition |
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Term
| Constructivist Environment |
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Definition
| whole to part, big ideas, responsive curriculum, experiential base, producers, interactive, evidence, formative, cooperation |
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Term
| What is teaching for problem solving? |
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Definition
| Teaching skills then providing problems to practice those skills |
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Term
| What is teaching about problem solving? |
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Definition
Teaching strategies
An important part of teaching through problem solving |
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Term
| What is teaching through problem solving? |
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Definition
| Teaching content through problems |
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Term
| How to find Quality tasks & problem-based lessons? |
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Definition
A task selection guide:
1. how is the activity done?
2. whats the purpose of the activity?
3. can activity accomplish your learning goals?
4. what must I do? |
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Term
| teaching for problem solving-6 bullet points |
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Definition
Requires all students have necessary prior knowledge to understand teacher’s explanations
Usually involves teacher presenting 1 way to do problem/procedure, which likely will not make sense to many learners, disadvantaging students who could solve it differently
Can communicate that there's only one way solve problem, message misrepresents field of math & disempowers students who naturally want to try on own
Positions students as a passive learners, dependent on teacher to present ideas
Separates learning skills & concepts from problem solving, not improving student learning
Decrease likelihood student will attempt new problem without explicit instruction on how to solve it |
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Term
| What is the 4-step problem solving process? |
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Definition
1. Understanding the problem
2. Developing a plan
3. Carrying out the plan
4. Looking back |
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Term
| Teaching IN a problem-based classroom |
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Definition
Let students do talking
How did you solve the problem?
Why did you solve it that way?
Why do you think your solution is correct and makes sense?
How much to tell & not to tell
Importance of student writing
Metacognition
Disposition
Additional goals |
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Term
| What are some problem solving strategies? |
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Definition
Draw a picture, actity it out, use a model
Look for a pattern
Guess & check
Try a simpler problem
Write an equation
Make an organized list
Make a table/chart |
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Term
| What does it mean to provide relevant context? |
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Definition
-context can be used to learn about cultures
-make info relevant & exciting |
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Term
| how much do you tell and not tell students? |
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Definition
HOW MUCH TO TELL:
-mathematical conventions
-carefully suggest alternative methods
-clarification of students’ methods
AND NOT TO TELL:
-establish teacher-preferred methods
-teacher thinking |
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Term
| first 3 steps to planning & developing a problem based lesson |
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Definition
Determine content & learning goals
Consider Students’ Needs
Select, design, or adapt a task |
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Term
| what are the middle 3 steps to planning & developing a problem based lesson |
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Definition
Design lesson assessments
Plan the before Phase of the lesson
Plan the during phase of the lesson |
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Term
| what are the last 4 steps to planning & developing a problem based lesson? |
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Definition
Plan the after phase of the lesson
Check for alignment within the lesson
Anticipate student approaches
Identify essential questions |
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Term
| what are the main components of a lesson plan |
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Definition
State/local math standards
Lesson goals & learning objectives
Assessments
Accommodations/modifications
Materials needed
Before phase
During phase
After phase |
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Term
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Definition
| different environment or circumstance |
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Term
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Definition
| change in problem or task |
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Term
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Definition
establish goals for all students, but difficulty is adapted to meet range of needs
-degree which teacher provides assistance
-how structured lesson is
-complexity of task/s given
-complexity of process |
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Term
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Definition
The size and makeup of small groups vary in a purposeful and strategic manner
Pairs
Groups of four
Groups based on
—academic abilities—usually heterogeneous groups
—language needs
—social dynamics
—behavior |
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Term
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Definition
Repetitive non-problem-based exercises designed to improve ONLY skills or procedures already acquired
-review of facts/pro. so not forgotten
-focus on singular method excludes flexible alternatives
-false appearance of understanding
-rule-oriented view of math |
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Term
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Definition
| Different problem-based tasks and experiences, spread over numerous class periods, each addressing the same basic concept |
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Term
| when are drill & practice appropriate |
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Definition
DRILL: efficient strategy if skill drilled on is already in place
-automaticity w/ skill/strategy is desired outcome
students who don't get drill-try conceptual approach-drill not answer
PRACTICE APPROPRIATE:
-increase opportunity develop conceptual ideas & elaborate & useful connections
-opportunity develop alt & flexible strategies
-greater chance all students to understand
-clear message math is figuring things out & making sense |
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Term
| The Nature of Children's Ideas (O&F) |
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Definition
Children have ideas about science prior to formal instruction.
Children’s prior ideas are often different from those of scientists.
Children’s prior ideas are strongly held and resistant to change.
Children’s prior ideas are sensible, coherent, and are often influenced in unanticipated ways. |
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Term
| Outcome of Instruction (O&F) |
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Definition
1. Undisturbed/Unchanged outcome
2. Dual/Multiple Perspectives
3. Reinforced preconceptions
4. Confused conceptions
5. Scientifically compatible view |
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Term
| Elements of a scoring guide/rubric |
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Definition
1. One or more traits or dimensions that serve as the basis for judging the student response
2. Definitions and examples to clarify the meaning of each trait or dimension
3. A scale of values on which to rate each dimension and standards of excellence for specified performance levels accompanied by models or examples of each level |
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Term
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Definition
| process of gathering evidence on student's knowledge of, ability to use, & disposition toward math & of making inferences from evidence for variety of purposes |
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Term
| Integrating assessment into instruction |
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Definition
Assessment should enhance student learning
Assessment is a valuable tool for making instructional decisions |
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Term
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Definition
-monitor student progress
-make instructional decisions
-evaluate student achievement
-evaluate programs |
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Term
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Definition
-concepts & procedures
how students understand them
-mathematical processes
evaluation based on NCTM standards
-productive dispositions
confidence, beliefs, dis/likes math |
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Term
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Definition
| Comparing students’ work to criteria or to rubrics that describe what we expect the work to be |
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Term
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Definition
| The result of accumulating scores and other information about students’ work for the pur-pose of summarizing and communicating to others |
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Term
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Definition
| A framework that can be designed or adapted by the teacher for a particular group of students or particular math task, using a three- to six-point scale to rate performance |
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Term
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Definition
4= excellent; full accomplishment
3= proficient; substantial accom.
2= marginal; partially accomp.
1= unsatisfactory; little accomp |
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Term
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Definition
Anecdotal notes
Observation rubric
Checklists for individuals
Checklists for full class |
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Term
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Definition
-always be part of assessment
-not have to test low-level sills
-designed to assess understanding of concepts
-go beyond knowing how to perform algorithm
-allow & require student to demonstrate a conceptual basis for process |
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Term
| Maximizing information gathered from a test |
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Definition
1. allow calculators
2. use manipulatives & drawings
3. include opportunities for explanations
4. avoid using "pre-answered" tests |
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Term
| improving high-stakes tests performances |
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Definition
-teach fundamental concepts & processes
-familiarize w/ different Q formats
-teach test-taking skills |
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Term
| Generative Learning Model |
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Definition
GLM
1. preliminary phase
2. focus phase
3. challenge phase
4. application phase
5. repeat |
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Term
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Definition
curiosity invoked by inquiry activities frequently focusing on tangible objects of immediate interest
-support student independent & cooperative inquiry activities within our lessons |
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Term
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Definition
Purpose: This is where you determine what should be taught.
Examples: Curriculum, student interviews, personal experience…. |
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Term
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Definition
Purpose: This is where the students clarify their conceptions. Cognitive dissonance occurs here first.
Examples: Kwl charts, any discussion that helps the students clarify their views and understanding of a subject. We help “focus” their thinking. |
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Term
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Definition
Purpose: To promote an exchange of student ideas. After “challenging” themselves with new information, most likely from an experiment, cognitive dissonance occurs again. The students are then “challenged” to have a dialogue about their findings.
Examples: This is where the student is “challenged” with possibly new and conflicting information. We want them to resolve this challenge independently. |
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Term
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Definition
Purpose: Students apply what they have learned to a familiar context. This activity could be the primary means of assessment.
Examples: Float your boats at home and explain/record the results. |
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Term
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Definition
NOT ENOUGH to give equal opportunity for all math learners
-IS being sensitive to individual differences
-IS treating students fairly & impartially
-IS examining my beliefs on students' abilities to learn |
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Term
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Definition
-responsible for lower expectations for students in slow track
-frequently denies students access to challenging materials
-for slower tracks is often remedial drill
-exaggerates differences instead of bridging them
-makes moving to higher tracks near impossibles
-doesnt benefit higher-achieving students |
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Term
| Response To Intervention common features across all tiers |
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Definition
Research-based practices
Data-driven
Instructional
Context-specific |
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Term
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Definition
May have difficulty with:
memory
general strategy use
attention
ability to speak/express ideas
perception of auditory, visual/written info
Integration of abstract ideas |
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Term
Tier 3
Research-Based Strategies |
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Definition
Explicit strategy instruction
Peer-assisted learning
Student think-alouds |
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Term
Tier 3
MODIFICATIONS/ACCOMODATIONS |
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Definition
Before
Structure the environment
Identify potential barriers
During
Provide clarity
After
Consider alternative assessments
Emphasize practice and summary |
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Term
| Strategies for Teaching mathh to ELL's |
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Definition
Write and state the content and language objectives
Build background
Encourage use of native language
Comprehensible input
Explicitly teach vocabulary
Plan cooperative/interdependent groups to support language
Create partnerships with families |
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Term
| Reducing Resistance & Building Resilience |
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Definition
Give children choices and capitalize on their unique strengths
Nurture traits of resilience
Demonstrate an ethic of caring
Make mathematics irresistible
Give students some leadership in their own learning |
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Term
| Providing for students who are mathematically gifted |
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Definition
AVOID:
-more of the same
-allowing free time when work completed
-routinely assigning them to teach peers
STRATEGIES TO USE:
-acceleration
-enrichment
-sophistication
-novelty |
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Term
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Definition
| digital tools, computers, calculators, other handheld devices, collaborative authoring tools, software, online digital games, podcasts |
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Term
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Definition
-exploring patterns, conducting investigations, testing conjectures, solving problems
-not when practicing computation skills
-computation beyond students' ability but problem content is appropriate
-when accommodations needed |
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Term
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Definition
tools for developing:
numeration
geometry
probability & data analysis
algebraic thinking |
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Term
| Guidelines for using software |
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Definition
-contributes to objectives
-used for individualized/small group instruction
-combined wiht off-computer activities
-used with management plan |
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Term
| Selecting a software to use |
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Definition
#1=be informed
-whats it do better than can be done without the calculator?
-how are students likely to engage with content (not the frills)?
-How easy is it to use?
-how does it develop conceptual knowledge that supports understanding of concepts?
-what controls & assessments are provided? |
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