Math Exam 1 Certified Questions with Accurate Answers (100% Correct).

Math Exam 1 Certified Questions with Accurate Answers (100% Correct). Sociocultural Theory by Lev Vygotsky Mental processes exist between and among people in social interactions Learner (working in his or her Z P D) Semiotic mediation is how beliefs, attitudes and goals are affected by sociocultural practices and institutions Constructivism Theory by Jean Piaget Learners are not blank slates but creators of their own knowledge Networks and cognitive schema are the product of constructing knowledge Reflective thought is how people modify schemas to incorporate new ideas assimilation and Accommodation Implications of Constructivist Theory and Sociocultural Theory There is no need to choose between theories These theories are not teaching strategies but should instead guide teaching assimilation new idea fits with prior knowledge- blue dots connect to red Accommodation new idea does not fit with existing knowledge conceptual understanding comprehension of mathematical concepts, operations, and relations Procedural Fluency skill in carrying out procedures flexibly, accurately, efficiently, and appropriately 1) Which of the following is an example of a statement spoken in the language of doing mathematics? a) "Memorize these steps." b) "Compute this answer." c) "Explain how you solved the problem." d) "Copy down these steps into your notebooks." c) "Explain how you solved the problem." 2) There are many ways to model and solve problems and explore how others develop understanding. Which strategy would foster students examining multiple solutions to try other methods? a) Generalizing relationships. b) Experimenting and explaining. c) Search for a pattern. d) Analyzing a situation. b) Experimenting and explaining. 3) Manipulative materials have the potential to provide opportunities for connection and communication. What statement would be a non-example of how to utilize the materials? a) Distributing materials with guidance on how to use them to construct models. b) Demonstrating at least one connection between the model and the mathematical concept. c) Encouraging students to converse about the model without knowledge of what the mathematical goal they are working on. d) Maintaining a balance between the appropriate amounts of guidance and student exploration. c) Encouraging students to converse about the model without knowledge of what the mathematical goal they are working on. 4) Name at least two examples of classroom culture that could be an environment for students to do mathematics and gain relational understanding of a concept. Persistence, effort, and concentration are valued -Sharing of ideas among students -Students listen to each other -Students look for and discuss connections The teacher asks the students to record as many different patterns they can think of for her number rule. Identify what type of understanding she is asking of her students. Relational understanding Characteristics of teaching through problem solving focus students' attention on ideas and sense making, develop mathematical practices and processes, develop student confidence and identities, builds on students' strengths, allows for extensions and elaborations, engages students so there are fewer discipline problems, provides formative assessment data, invites creativity Teaching through problem solving teaching content through real context, problems, situations, models and exploration What makes a worthwhile mathematical task or problem high cognitive demand, mult. entry and exit points, relevant contexts, have Ss demonstrate/explain understanding, elicit more than one problem solving strategy, task pontential Teaching for problem solving Teaching skills, then providing problems to practice those skills (explain-practice- apply) Teaching about problem solving Polya's four-step problem solving process (understand, devise a plan, carry out the plan, look back) low cognitive demand tasks Procedures without Connections Tasks Memorization Tasks routine problems high cognitive demand tasks Procedures with Connections Tasks Doing Mathematics Tasks Involves understanding, analyzing, applying info and analyzing strategies How to orchestrate classroom discourse facilitate meaningful mathematical discourse; pose purposeful questions; and elicit and use evidence of student thinking Five teacher actions for orchestrating productive mathematics discussion Anticipating- before lesson starts Monitoring- observing strategies students are using Selecting- from observations selecting strategies for public discussion Sequencing- strategically sequences the presentations to emphasize mathematical ideas Connecting-designs questions that connect strategies and mathematical concepts I-THINK Framework Supports Metacognitive Skill Development Individually think about the task. Think about the problem. How can it be solved? Identify a strategy to solve the the problem Notice how your strategy helped you solve the problem. Keep thinking about the problem. Does it make sense? 5) Selecting problem solving tasks that require higher levels of cognitive demand should include which one of the following? a) Use of a specific procedure. b) Are routine and students can use previously learned procedure. c) Use of complex and non-algorithmic thinking. d) Are straightforward with little ambiguity of what needs to be done to solve. c) Use of complex and non-algorithmic thinking. 6) Which statement below best represents worthwhile features of tasks or problems for learning mathematics? a) Problematic, memorization, no connection b) Problematic, concepts and or misconceptions, relevant c) Problematic, require correct answer, rules and formulas d) Problematic, algorithmic, routine. b) Problematic, concepts and or misconceptions, relevant 7) How much to tell or not tell is a constant dilemma for teachers. What statement would be the type of information that teachers do need to tell? a) The level of cognitive demand that the students need to demonstrate. b) Specific problem solving strategies needed to solve the problem. c) Information needed to keep students from struggling to solve the problem. d) Help students clarify their ideas and point out related ideas. d) Help students clarify their ideas and point out related ideas 8) This chapter pointed out the difference between drill and practice. What do these differences look like in a classroom? -Drill is used to increase facility of an already learned procedure -Drill is used to help the retention of facts and procedures -Practice is used to develop alternative and flexible strategies -Practice can deepen conceptual knowledge through more useful connections. Three phase lesson plan structure and characteristics The before, during, and after The before the lesson phase activate prior knowledge, be sure the task is understood, establish clear expectations The during the lesson phase let go!, notice students mathematical thinking, provide appropriate support, provide worthwhile extensions The after the lesson phase promote a mathematical community of learners, listen actively without evaluation, summarize main ideas and identify future tasks 8 step planning process 1. determine learning goals 2. consider your students needs 3. select, design or adapt a worthwhile task 4. design lesson assessments 5. plan the before phase 6. plan the during phase 7. plan the after phase 8. reflect and refine Step 1. Know your focused mathematical learning goals- "What is it that my students should be able to do when this lesson is over? Step 2. Consider your students' needs What do your students already know or understand that can serve as a launching point? What context might be engaging to this range of learners? What misconceptions might need to be addressed? What visuals or models might support the learning? What vocabulary support might be needed? Step 3. Select a task and use the four step guide for activity evaluation and selection (Chapter 3) Does the task address the content goals and needs of my students? Does the task have potential to engage students in the Mathematical Practices? Will the task require students to apply problem solving strategies? Step 5. Before: Getting Ready Elicit prior knowledge. Provide context Establish clear expectations. Step 4. Design lesson assessments Plan assessments that will allow students to demonstrate their understanding of the content goals Step 6. During: Students Work Let go! Avoid stepping in front of the struggle. Notice student mathematical thinking Provide appropriate support and challenges Provide accommodations and extensions. Step 8. Reflect and Refine Make sure all three parts (before, during and after) are aligned and balanced with your learning goals. Review to see if opportunities for students to engage in Mathematical Practices and or problem solving strategies are evident. High level questions are based on the learning goals and posed during and after phases of the lesson. Be prepared with key essential questions that students will extend, analyze, compare, generalize and synthesize. Step 7. After: Class Discussion Promote a community of learners. Listen actively without evaluation. Summarize main ideas and make connections. Methods for differentiating instruction Open Questions- broad-based questions that invite meaningful responses from all students from all developmental levels. Example- One number is about one-half of another. Their product is close to 100. What might the numbers be? Tiered Lessons - a set of similar problems focused on the same mathematical goals. With adaptations to meet the range of learners i.e. Degree of assistance, structure of the task, complexity of the task and complexity of the process 9) A three-phase lesson format provides a structure for students to have inquiry on a topic, engage in the content through action and discussion and time to reflect and make connections. What statement below demonstrates the Before related agendas? a) Be sure the task is understood. b) Let go. c) Provide extensions. d) Identify future problems. a) Be sure the task is understood. 10) What statement represents how a teacher listens without evaluation? a) That strategy worked for you, but will it work for others? b) You only used one representation and one equation is that right? c) Do others agree with Caroline's strategy? d) Why did you solve the problem without any models? c) Do others agree with Caroline's strategy? 11) Flexible grouping means that the size and makeup of the small group vary in a purposeful and strategic manner. Below are examples of considerations teachers must make in putting together mixed-ability groups. What statement reflects the shared responsibility? a) Whether groups stay the same for a full unit. b) Whether groups must understand that each individual must be able to explain the content, process and product. c) Whether group members participate in team building activities to set the standard for collaboration. d) Whether groups present oral reports to the whole class. c) Whether group members participate in team building activities to set the standard for collaboration. Summative assessments Cumulative evaluations Takes place After instruction is completed Generate a single score Measures overall progress Does not shape day to day teaching decisions Formative Assessments Check status of students' development Takes place during instruction or as pre assessment Provides targeted feedback Results and evidence used to inform decisions about next steps Types of assessment strategies and their characteristics Observations, questions, interviews, tasks, rubrics Observations Anecdotal notes and checklists are a systematic plan for gathering evidence during and after portions of the lesson. Questions Probing student thinking through questioning can provide useful data and insights that inform your instruction. Types: info gathering, student thinking, mathematical structures, connections, and relationships, reflection and justification Interviews Diagnostic interview is a one-on-one investigation of a student's thinking about a concept, procedure, process or mathematical practice. Hearing students descriptions of strategies Probe their understanding Identify strengths and gaps Tasks Any product, including problem-based tasks, writing and student self-assessment Rubric Analyzing problem-based learning thru formative and summative assessment provide large amounts of information. A rubric is a scale based on predetermined criteria serves two important functions Permits students to see what is central to excellent performance Provides scoring guidelines for teacher that supports equitable analysis of students' work 12) Three of the statements below are characteristic of formative assessment. What statement would be true of summative assessment? a) Identify where learners are. b) Identify the goal for learners. c) Identify a path for reaching the goal. d) Identify the mastery of standards. d) Identify the mastery of standards. 13) Utilizing problem-based tasks permit every student in a class to demonstrate their knowledge, skill or understanding. What tasks below encourages student reasoning and thinking? a) Leila has six gumdrops and Darlene has 2 more for her. How many altogether? b) Leila has six gumdrops and she shares half of them with Darlene. How many does she have left? c) Leila has six gumdrops and she wants to share them equally with her friend Darlene. How will she do that? d) Leila has six gumdrops and she gives some to Darlene and now she has 4 left. How many did she give away? c) Leila has six gumdrops and she wants to share them equally with her friend Darlene. How will she do that? 14) What description below denotes grading? a) Compare students' performance with established criteria. b) Summarizing student performance with scores and data. c) A scale based on predetermined criteria. d) Systematic plan for gathering student learning. b) Summarizing student performance with scores and data. 3 tiers to RTI Core instruction (tier 1, 80%, large groups), supplemental intervention (tier 2, 15%, small groups), intensive intervention (tier 3, 5%, individual) Modifications changes the task to make it more accessible to the student i.e. Break task into smaller components Scaffold to the original task Accommodations response to the needs of the environment or learner i.e. Write directions in larger font Saying and printing the directions Strategies for teaching students with learning disabilities Structure the environment, Identify and Remove Potential Barriers, provide clarity, Consider alternative assessments, Emphasize practice and summary Emphasize practice and summary Consolidate ideas- study guides for review Provide extra practice- carefully selected problems to use with manipulatives Consider alternative assessments Propose alternative products- verbal response scribed by another someone else or electronically Encourage self-monitoring and self-assessment Consider feedback charts- monitoring their growth Structure the environment Centralize attention- face student and remove competing stimuli Avoid confusion- word directions carefully and specifically Create smooth transitions- limit off task time Identify and Remove Potential Barriers Help students remember- memory aids Provide vocabulary and concept support Vary task size Provide clarity Repeat the timeframe- repeat reminders of time left Ask students to share their thinking- think-alouds or think-pair-share Emphasize connections- provide visual representations Adapt delivery mode- materials, images and examples Support organization of written work-tools and templates Provide examples and nonexamples 15) The goal of equity is to offer all students access to important mathematics during the regular mathematics instruction in the classroom. Which group listed below would need a modification versus an accommodation? a) Students who identified from different backgrounds. b) Students who are mathematically gifted. c) Students who are unmotivated. d) Students who are English language learners. d) Students who are English language learners. 16) Common features across RTI tiers include all of the following. Identify the statement that is related to instructions. a) Prompts, cues and environmental arrangements. b) Research-based practices. c) Strategies and measures that are specific to the context of the school, classroom, and student needs. d) Decisions that are based on data. a) Prompts, cues and environmental arrangements. 17) Describe four specific ways you might address the needs of a student with low motivation. give students choices that capitalize on their unique strengths, nurture traits of resilience (social competence, problem solving skills, autonomy, and sense of purpose/future, make math irresistible, give students leadership in their own learning 4 types of addition problem structures, and how to write an example of each and tools to use in each Join, separate, part-part whole, and compare Join (add to) - change being "added to" the initial ex: megan had 2 dogs. she got a cat for her birthday. how many pets does she have now Separate (take from) - change is being removed from initial ex: john had 7 cookies. he gave 4 away. how many does he have now? Part-part-whole - either missing the whole or one of the parts must be found ex: Jane had 30 minutes for recess. She got extra time added. Now her recess is 45 minutes. How much extra time did she get? Compare - There are three ways to present compare problems, corresponding to which quantity is unknown (smaller, larger, or difference). ex: Kim has 3 fewer books than Kate. Kate has 7 books. how many does Kim have? 3 types of multiplication and division structures equal group problems, comparison, array and area problems 18) Which problem represents the separate, start unknown structure? a) Maryann had 3 library books before she checked out 2 more. How many did she have all together? b) Maryann had 5 library books before she returned 2 of them. How many does she have now? c) Maryann had some nonfiction books and 2 fiction. She now has 8 books. How many did she begin with? d) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim? c) Maryann had some nonfiction books and 2 fiction. She now has 8 books. How many did she begin with? 19) Problems with the join and separate structures, with the start or initial amount unknown, tend to be the hardest for students to understand and accurately solve. Identify the reason for they are more challenging for children to use. a) Children can model the physical action. b) Children can act out the situation. c) Children cannot use counters for the initial amount. d) Children cannot grasp a quantity represents two things at once. d) Children cannot grasp a quantity represents two things at once. 20) Complete this statement, "Constructing models of arrays draws attention to..". a. Factors connection with rows and columns. b. Factors and product. c. Number of rows and columns. d. Connection with measurement of area. a. Factors connection with rows and columns. 21) Present and discuss two reasons for using contextual problems to teach addition, subtraction, multiplication and division. Contextual examples help children use reason with 0 and 1. Where would 0 hops of 5 put you on a number line? Contextual problems are more easily solved with multiple representations- words, pictures and numbers 3 phases for learning facts, and characteristics of each phase 1: counting strategies (7+3 is 7+1+1+1, 7, 8, 9, 10) phase 2: reasoning strategies (4+7, 3+7=10 so 4+7 is one more than 10) phase 3: mastery "i just know it" (4+7=11) 3 approaches to teaching basic facts Memorization Explicit strategy instruction Guided intervention Memorization of facts- 4 operations total over 300 facts- strong evidence that this does not work- inefficient, inflexibility and inappropriate applications Explicit strategy instruction- support student thinking by giving them a choice of strategies to use in recall of facts Guided invention- students select a strategy based on their knowledge of number relationships What not to do when teaching basic facts Don't Use lone timed tests (e.g. 30 items) Use public comparisons of mastery of facts. Proceed through facts in order from 0 to 9. Expect automaticity too soon Use facts as a barrier to good mathematics. Use fact mastery as a prerequisite for calculator use 22) Three developmental phases for learning the basic facts are identified below. Which one is the initial phase? a) Drawing strategies. b) Counting strategies. c) Mastery. d) Reasoning strategies. b) Counting strategies. 23) Research shows that guiding students to use effective reasoning strategies can help them achieve mastery. Identify the explicit teaching strategy below. a) Memorization of facts. b) Guided invention. c) Use of known facts and relationships. d) Object or verbal counting. c) Use of known facts and relationships. 24) All of the following are recommendations that can support students' ability to quickly recall basic facts EXCEPT: a) Drilling activities during mathematics instruction. b) Involving families. c) Using technology. d) Encouraging students to self-monitor. a) Drilling activities during mathematics instruction 25) Identify and discuss three teaching recommendations that support students' development to mastery of basic facts. 1. Works on facts over time- working over months and months on reasoning strategies and sets of facts 2. Involve families- share the big plan on how to learn facts over the year 3. Make fact practice enjoyable- many games designed to reinforce facts that are not competitive or anxiety reducing 4. Use technology- students get immediate feedback and reinforcement