D237

Three aspects of understanding mathematics

1. The student's ability to explain why they solved the problem in a certain way 2. The student's ability to make connections 3. Emphasis on relational understanding

A sixth-grade teacher uses a think-aloud to model how to solve the following three equations. a.) 2x + 4 = 12 b.) 10 = 3x - 5 c.) 3(x + 2) = 30 Which equation can be used on an assessment to measure learning outcomes?

2(x - 4) = 20 The objective of the lesson is to solve two-step equations. The assessment must align with the objective, and the equation provided in 2(x-4)=20 matches the equations students have been practicing.

A fifth-grade class has been exploring volume by using centimeter cubes to build rectangular prisms. They have developed the volume formula of V = lwh. Which activity is a recommended follow-up to this lesson?

A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each.

disequilibrium

A productive struggle- the student is trying to figure out how to make sense of the new knowledge they are exposed to. Once this new knowledge is internalized, the student can use it to make sense of future, related information.

Which statement is not proposed by the constructivist theory?

A safe environment is necessary to gain mathematical power.

Which student error represents a misconception about place value?

A student writes six-hundredths as 0.6.

Activity that could be used to assess students' reasoning abilities

After students solve a problem, you ask that they explain how they arrived at their solution.

Which is a true statement concerning all linear equations?

All of the points on the graph lie on a straight line. Linear functions are a subset of growing patterns and functions, which can be linear or nonlinear. But because linearity is a major focus of middle school math, and because growing patterns in elementary school tend to be linear situations, it is important to understand linear functions. Linearity can be established by looking at other representations. If you make a table for the coordinate points and the graph forms a straight line, the graph is a linear function.

A second-grade teacher has 23 students with varying abilities. Which instructional strategies can be used to help provide learning opportunities for all students?

Allow students to progress through the material at their own pace and use flexible grouping options. When planning for instruction, you must consider all students in the classroom. While pairing a low-achieving student with a high-achieving student is an example of peer-assisted learning, this question is about providing access to learning opportunities for all students. Teachers should instead use flexible grouping options and allow students to progress through the material at their own pace.

Activity that could be used to assess students' communication abilities

Asking students to write out their solutions and then to teach it to someone else.

Activity that could be used to assess students' ability to demonstrate representations

Asking your students to show the results of their data by using a bar graph, a pie chart, and some physical objects of their choosing.

How is a conceptual understanding beneficial to students when learning mathematics?

Because conceptual understanding is about understanding relationships and foundational ideas, this allows the student to apply the ideas and concepts in a variety of contexts.

A math teacher wants to work with a social studies teacher on integrating the content. How can the teacher apply a math connection?

By allowing students to create a timeline of the Native American cultures

A math teacher would like to collaborate with an art teacher by demonstrating a math connection. How can a math teacher incorporate a stained glass window into a math lesson?

By teaching concepts of symmetry

Which strategy will allow the students to visualize the difference between 1/4 and 1/8?

CSA Sequence

The objective of a lesson is to have students develop their own addition problems. Students have been asked to write a "join" problem. Which of the following correctly represents this problem type?

Caesar has $34 and earns $15 more dollars mowing a lawn. How much money does he have?

A fifth-grade teacher is planning a discovery-based lesson that will compare the concepts of perimeter and circumference. Which tool or manipulative can the teacher use to help enhance learning outcomes?

Calculator to explore patterns and relationships

A teacher is explaining the process of finding the slope of a line. Which tool or manipulative could help engage the students in this process?

Calculator. A calculator is considered a technology tool that can be used in the math class. While a calculation should never replace basic computational skills, some math skills and objectives are beyond computation and students can use a calculator to solve these problems. A calculator can be used to solve algebraic and other math concepts and students can always check their work using a calculator.

What are two appropriate uses for a calculator?

Calculators can be used to improve student attitude and motivation. Calculators can be used to explore patterns and basic facts.

A teacher is planning a discovery-based lesson on circumference. Which technology tool can the teacher use to help facilitate this lesson?

Calculators to explore patterns and relationships. Calculators can be used to help students discover patterns and when teaching a discovery-based lesson. Three-dimensional shapes, like the other answer options suggest, do not work here because you are concerned with the concepts of circumference and not volume or surface area.

Examples of math tools

Calculators used for exploration and discovery Rulers and protractors Base-ten blocks

If you want to teach the concept of shape transformations (i.e., rotations, reflections, and dilations), which two tools could you use?

Centimeter grid paper Geometric modeling software Grid paper can be used to help students draw shapes, see relationships between the transformation, and teach concepts such as area. Geometry software programs allow students to create and manipulate geometric shapes.

Community Assets

Common backgrounds and experiences that students bring from the community where they live, such as resources, local landmarks, community events and practices that a teacher can draw upon to support learning.

To create an environment for doing mathematics, what is the teacher's role?

Create a spirit of inquiry, trust, and expectations

A ninth-grade teacher is planning a unit of study on equivalent fractions. Select all tools or manipulatives that can enhance learning outcomes.

Cuisenaire rods Fraction circles Cuisenaire rods are manipulatives that provide a hands-on elementary school way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions, and finding common denominators. Fraction circles are a set of nine circles of various colors. Each circle is broken into equal fractional parts and uses the same-sized whole. The circles included are one whole as well as circles divided into halves, thirds, quarters, fifths, sixths, eighths, tenths, and twelfths. Depending on the manufacturer, fraction circles can be transparent for use on an overhead projector for whole-class activities or opaque for use at students' desks or with a document camera. Fraction rings are clear plastic rings that are open in the middle and can hold the fraction circles in the center. Fraction circles can be used to help students see relationships between fractional parts of the same whole. Students can compare and order fractions, see equivalent fractions, explore common denominators, and explore basic operations with fractions. Fraction rings can be used in conjunction with fraction circles to make connections to time, decimals, and percent. They can also be used to make circle graphs.

Cultural Assets

Cultural backgrounds and practices that students bring to the learning environment, such as traditions, languages, world views, literature, and art that a teacher can draw upon to support learning.

How can you demonstrate a mathematical connection when teaching a unit on making circle graphs?

Demonstrate how ratios, parts to whole, fractions, and percent can be used to create a circle graph.

Which activity could be used to make a mathematical connection within the mathematics curriculum when teaching a unit on multiplying fractions by whole numbers?

Demonstrate the relationship between multiplying fractions by whole numbers and multiplying decimals by whole numbers.

Which two instructional strategies can be used to effectively hold small group and classroom discussions?

Encourage students to "think-pair-share" and compare solutions and strategies. Have students share multiple methods about how to solve problems.

Your fifth-grade class was asked to explore the relationship of various volume formulas and to fill a rectangular prism, cube, and triangular prism with water. Which two strategies could help increase the effectiveness of the activity?

Encourage students to collaborate and discuss findings and strategies once the activity is complete. Before the activity, review the properties of 3D shapes. Explanation: Collaborative learning is a great constructivist teaching strategy that helps students to be resilient. It identifies their resources (peers) and tests their theories to see if they are on the right track, all while developing habits of mind that form the foundation of scholarship. In a constructivist classroom, students should be encouraged to explore multiple strategies and ways to solve a problem. Before students attempt to build a conceptual understanding, it is important to activate their prior knowledge. In teaching mathematics, you should help students relate new information to previously learned material.

Students in a third-grade class are asked, "How many small unit squares will fit in a rectangle that is 54 units long and 36 units wide?" The teacher provides students with Unifix cubes to help solve the problem. How should the teacher use a rubric to assess the progress of the students?

Explain how the rubric will be used for evaluation before students begin the problem

Which instructional strategy can the teacher use that will make use of constructivist teaching while also providing sufficient scaffolding and modeling for students by making known which steps they need to take and when?

Explicit Strategy Instruction

When working with a student with a mild learning disability, which instructional approach should be used to accommodate for the student but should not be used for the general population?

Explicit instruction

A class is working through a unit of study about plant growth. They have planted tomato plants and have placed plants under artificial light, sunlight, and in the dark. They are observing and recording growth over two weeks. How can the math and science curriculum be integrated during this unit?

Gather data about plant growth and then convert it into a graph.

Activity that could be used to assess students' problem solving abilities

Giving the students a problem to solve and telling them to find three different ways to solve it.

A seventh-grade teacher is planning a lesson that will require students to find the slope of the line and explore functions. Which tool or manipulative can the teacher use to help enhance learning outcomes?

Graphing calculator

You are teaching the concept of long division. Which two activities would encourage and promote student communications about their math thinking?

Have students create their own word problem that would require long division. Have students draw a picture when solving long division problems.

If you are teaching a unit on multiplication, which activity would help your students make a connection within the mathematics curriculum?

Have students demonstrate how multiplication and division are inverse operations.

Which activity could be used to make a mathematical connection within the literacy curriculum when teaching a unit on multiplying fractions by whole numbers?

Have students research and write about four applications of multiplying fractions by whole numbers.

A survey was taken on how many years an individual spent in school beyond high school. Which type of graph should be used to represent this data?

Histogram

Activity that could be used to assess students' ability to make connections

In a geometry lesson, you ask students to find ways that the material relates to things they see outside the classroom.

Relational or Instrumental: Having your students complete a ten question quiz

Instrumental. Consider, however, turning this into relational understanding by having students explain their answers to each other.

Relational or Instrumental: Having your students memorize the multiplication table to be able to answer questions faster.

Instrumental. This can be a helpful and even necessary activity, but make sure that students are able to use this for other activities that foster relational understanding.

Relational or Instrumental: Having students compete to see who can complete the most questions

Instrumental: Competition and game activities are fun and can help learning, but make sure that it goes beyond demonstrating instrumental understanding.

How is conceptual understanding beneficial to students in learning mathematics?

It allows students to apply mathematical principles in different contexts. Because conceptual understanding is about understanding relationships and foundational ideas, this allows the students to apply the ideas and concepts in a variety of contexts and mathematical procedures.

Why is problem-based learning (PBL) an effective teaching method that meets the needs of the different learners in your classroom?

It allows students to approach the content in their own way.

Traditional problem-solving lessons involve a teacher explaining the math, and then students practicing that math. This is followed by applying the mathematics to solve problems. Why does a lesson set up as described rarely work?

It assumes wonderful explanations produce understanding.

Which statement is not included in relational and conceptual understanding?

It eliminates poor attitudes and beliefs

When working with children who have learning disabilities, which of the following is important to remember?

Learning disabilities should be compensated for by helping students use their strengths. Students with learning disabilities have strengths that the teacher can encourage in order to help develop a conceptual understanding. Remember that the learning objective should never be minimized for students with learning disabilities and the students' strengths should instead be noted.

What Van Hiele level? Select from a variety of shapes that are similar to each other.

Level 0

What Van Hiele level? Find the similarities and differences of a wide variety of shapes.

Level 1

What Van Hiele level? Sort shapes by recognizing common properties

Level 1

What Van Hiele level? Have students create minimal defining lists for certain shapes.

Level 2

Which of the following statements is true of the van Hiele model for geometric reasoning?

Levels are sequential

A student wants to construct a graph to show the population growth in a city over 10 years Which type of graph would best show this data?

Line graph

A teacher wants to show students their test scores without revealing their names. Which two graph types could be used?

Line plot and stem and leaf plot

5 Adaptations in Planning for All Learners

Make accommodations and modifications Differentiating instruction Learning centers Tiered lessons Flexible grouping

A fifth-grade class is exploring the relationship between various volume formulas and are asked to fill a rectangular prism, cube, and triangular prism with water. Select two strategies that could help increase the effectiveness of this activity.

Mathematical conventions Alternative methods

Equity in Teaching

Offers access to important mathematics with equal access to resources Attains equal outcomes by being sensitive to individual differences Maintains high expectations, respect, understanding, and strong support Maximizes learning potential for all students while providing reasonable and appropriate accommodations to promote access

Which strategy makes use of sociocultural learning to help assist students who are struggling up to a more competent level of understanding by providing them with the knowledge they need when they need it?

Peer assisted learning

You have noticed that one of your students is counting objects with an accurate sequence of number words, but is not attaching one number to each object. Therefore, the student's final count is inconsistent and inaccurate. Which instruction would you plan to help this student develop a better grasp of one-to-one correspondence?

Provide the student with a set of blocks or counters organized in a pattern or row to point at as they say each number word.

Self-Assessments

Provides a record of how students perceive their strengths and weaknesses.

Diagnostic Interviews

Provides a way to gather in-depth information about a student's thinking processes and misunderstandings. These will require that as the teacher you exercise good listening skills as you give the student a problem to work through to verbalize their thinking.

Study the following discussion scenario and think about which two "Talk Moves" the teacher uses. In your classroom you have your students explain how they solved the equation 63 - 27 =__. As your students are explaining their responses, you ask two students to come up with different answers. One student, Meg, has the right answer and the other student, Mark, has the wrong answer. Without saying that Mark is wrong and Meg is right, instead you ask both of them to explain their answers. You then ask Mark what he thinks of Meg's answer and vice versa. After this short discussion, Mark realizes that he got the wrong answer and he is even able to see where his mistake was.

Reasoning and elaborating

Which type of understanding: Relational or Instrumental? Giving your students a problem and asking them to explain how they arrived at the solution

Relational

Asks students to demonstrate their reasoning

Relational Understanding

When a student employs multiple representations, which of the following have they demonstrated?

Relational understanding

Relational or Instrumental: Using problem based learning

Relational. This is a great way to engage students in scenarios that encourage deeper thinking and relational understanding.

Relational or Instrumental: Asking students "Was there something in the problem that reminded you of another problem you have done?"

Relational. This question is a great way to help students make connections either to other things they've learned in class or things they've experienced outside of class.

A fifth-grade teacher is teaching a lesson on dividing fractions. The class is solving ½ ÷ ¾ = ____. Sam says the answer is ⅜ and Ryan says the answer is 4/6. The teacher allows the class to share answers and possible solutions and Ryan suggest that Sam's answer is incorrect. How should a teacher respond to promote critical thinking?

Ryan can you explain your thoughts regarding Sam's answer? A teacher's response should encourage critical thinking and open discussion. Here, the teacher avoids indicating whether the student is correct or incorrect and encourages the student to elaborate and justify his thinking. Through this process, students should strategically compensate for any errors they make by internalizing the situation. By allowing students to compare and analyze possible solutions, other students have the opportunity to correct errors and learn from peer assistance.

Which two statements accurately describe challenges facing students from diverse groups?

Special education students may require extended time to complete a task. Students from diverse backgrounds solve problems differently.

Personal Assets

Specific background information that students bring to the learning environment. Students bring interests, knowledge, everyday experiences, and family backgrounds that a teacher can draw upon to support learning.

Which stakeholder is directly in charge of choosing mathematical standards?

State departments of education

An elementary math teacher is teaching a geometry unit on quadrilaterals. Which instructional approach or student statement demonstrates a discovery-based approach to teaching this concept?

Students are given quadrilaterals and explore their interior angles to discover that all interior angles add up to 360 degrees.

Even though constructivism and sociocultural theory are two different learning theories, they can both be used simultaneously in teaching mathematics. Imagine that you are teaching your students about fractions. You instruct them to explain how to divide five brownies between three friends. Which two scenarios illustrate these two theories working together?

Students are working in groups of three and are figuring out how to solve this problem using some of the concepts about fractions that they have already learned. They must then write their answer using fractions and share it with the class. One student is working by himself to figure out how to solve this problem. He is not sure how to proceed and the teacher assists him by asking, "Have you ever divided candy between your friends before?" The student has. He completes the problem and can then show other students how he did it.

What is an environment that is not desirable for mathematical instruction?

Students are working through practice problems to learn the mathematical strategy.

During October you want to incorporate something from Halloween into your mathematics curriculum. Which two strategies could you use?

Students can estimate how many seeds are in a pumpkin and then make conjectures based on the mass of the pumpkin. Students can find the mass of several pumpkins and arrange them in order.

A first-grade teacher is planning a lesson on adding whole numbers. How can the teacher allow the students to make a math to real-world connection when teaching this lesson?

Students can use a recipe and determine the total amount of ingredients.

Which two activities show a connection to a context outside the math classroom?

Students explore how certain architectural designs influence the stability of a bridge or skyscraper. Students compare oil and gas prices from around the world and create diagrams.

two learning strategies that describe effective math learning

Students learn by rearranging previously learned concepts and making connections. Students learn new ideas through collaboration with the teacher and peers.

Select two learning theories that describe effective math learning.

Students learn new ideas through collaboration with their teacher and peers. Students learn by rearranging previously learned concepts and making connections.

The equal sign is one of the most important symbols in elementary arithmetic and in all mathematics. What is a common misconception about the equal sign?

Students understand that the equal sign means "the answer is." Students will have the common misconception that the equal sign means they have to solve something rather than it showing equivalency.

You are teaching a lesson on adding fractions and want to relate this concept outside of the math curriculum. How could this be taught?

Students use a recipe containing fractions to determine the total amount of ingredients.

A teacher wants students to write out how to use positive and negative chips to model operations with integers. How does this lesson address the Communication Standard from the Principles for School Mathematics?

Students will use precise math language to express their reasoning.

After analyzing high temperatures from July over several years, students created line graphs to display the data. How can a teacher incorporate an observational assessment into this lesson?

Take short narrative style notes on a card to record student understanding while circulating through the class and use that data to modify the lesson.

EL Difficulty: The meaning of some math terms are different than everyday use, thus causing confusion for these students.

Teaching Strategy: Build in math vocabulary lessons

EL Difficulty: EL students have difficulty understanding instructions and objectives.

Teaching Strategy: Pair them with another student

EL Difficulty: EL students are unable to fully explain ideas

Teaching Strategy: Use revoicing and press for details

EL Difficulty: EL students feel isolated from other students. other students

Teaching Strategy: Write and state the learning objectives clearly and avoid using unneccessary terms

A teacher would like to collaborate with an art teacher to integrate a math lesson. Which concept can the teacher use to incorporate a pattern on a quilt into the math lesson?

Tessellations of polygons

When asking questions in your classroom, consider the following:

The "level" of the question: Type of knowledge that is targeted: Pattern of questioning: Who is thinking of the answer: How you respond to an answer:

A teacher is planning a lesson on adding and subtracting positive and negative integers. The teacher will use red and black chips as a manipulative and allow students to develop patterns for understanding the addition and subtraction problems. The following NCTM standard is associated with the lesson: "Represent the idea of a variable as an unknown quantity using a letter or a symbol." How do you rate the activity and alignment to the standard?

The activity is effective at showing students how to add and subtract integers but the standard is not aligned.

A teacher is planning a lesson on finding the area of rectangles. The teacher allows students to use a geoboard to create rectangles and assist in finding the area of those rectangles. The following standard is associated with the lesson: Measure and estimate liquid volumes and masses of objects. How should the teacher rate the activity and alignment to the standard?

The activity is effective at showing students how to find the area of rectangles, but the standard is not aligned.

Imagine that you are observing a teacher who has a group of low-level learners and high-ability learners in her classroom. The objective of her current lesson is for students to create a histogram. She plans to deliver explicit instruction to the low-level learners on how to create a histogram. And she plans to allow the high-ability learners the opportunity to create a histogram after developing a statistical question and gathering data. How effective is this teacher's activity at meeting the needs of all learners?

The lesson effectively addresses all learners and offers complexity and depth for higher ability students. The lesson objective is to have students create a histogram. All students must meet this objective and the teacher can never reduce the expectation for lower-level learners. Teachers can differentiate by considering the four elements: content, process, product, and environment. In this situation, the teacher is differentiating the content by providing the higher-level learners an opportunity to create their own statistical question, develop a survey protocol, and then create a histogram. The process by which students are reaching the objective also differs. The lower-level learners are receiving more explicit instruction, which differs from step-by-step instruction. The teacher will use probing questions and model precise language as students create a histogram. The same product is not being created by all students because the lower-level students are creating a histogram while the higher-level learners are creating a more complex product. This type of lesson delivery provides the higher-level learners with more depth and complexity.

A teacher is planning a lesson on mean and poses the following data set to students: 6, 8, 7, 3. A student responds and says the mean is 6. The teacher then changes the data set to: 6, 6, 8, 7, 3. How will the mean change?

The mean will not change.

Which type of information should not be provided by math teachers

The preferred method

A student correctly reads the number 65 as "sixty-five" and has the ability to form a one-to-one correspondence to demonstrate the value of the number, but when asked to write the number form, the student writes 6 + 5. If any, what type of misconception does this represent?

The student has an idea that multi-digit numbers are independent of place value. Students need a deep understanding of the place-value pattern ("10 of these is 1 of those") to support more efficient ways of working with two-digit numbers and beyond. Place-value is difficult to teach and learn as it is often masked by successful performance on superficial tasks such as counting by ones on a 1-100 number chart. In this example, the student believes that the number 65 is made from 6 ones and 5 ones.

A student submits the following answer: 2/5 - 1/3 = 1/2 How can this student demonstrate strategic competency?

The student uses a manipulative and realizes that a common denominator was not used.

Which two strategies could be part of a lesson that uses think-alouds?

The teacher discusses methods and strategies to help solve problems and provides reasoning for each step. The teacher demonstrates to students how to solve a problem.

Imagine that in your classroom you worked through a series of concrete learning activities to help students understand how to find equivalent fractions using manipulatives. You then transitioned students from using manipulatives to finding equivalent fractions using representative problem-solving. After this, you wish to check for understanding of students' ability to calculate equivalent fractions using multiplication. You call two students to the board to solve the following problem: Find two equivalent fractions for 2/7. The first student provides the answer 4/14 and 8/28. The second student provides the answer 4/9 and 5/10. After seeing the students' work, you now want to use an instructional strategy that reveals student thinking. Which strategy should you use?

The teacher should allow both students to retell their strategies and discuss their method of obtaining the answer.

Two students (Ally and John) are working on how to solve the area of a triangle. Ally says the formula is (b x h)/2 while John says the formula for the area is b x h. How can the teacher respond in a way that will promote understanding and communication?

The teacher should ask Ally to explain and elaborate on her thinking. As a teacher you should serve as a facilitator, especially in a constructivist classroom, so that students will be more willing to share their ideas during discussion without the risk of being judged. When you say, "That is correct," there is no reason for students to think about and evaluate the response. Consequently, you will not have the chance to hear and learn from students about their thought process. Teachers should also use praise cautiously. Praise offered to correct solutions or excitement over interesting ideas suggests that the student did something unusual or unexpected.

All rubrics will have performance indicators. Which of the following is not an important element of performance indicators?

They should focus on the number of correct responses from the student. This is not something that performance indicators in rubrics should focus on. When you focus on the number of correct or incorrect responses, students will likely focus only on how many they can get right and wrong in order to pass, instead of on improving their overall performance by targeting their knowledge and performance gaps. As an example, being a student at WGU, the pre-assessment for your courses is designed to help you know what your knowledge gaps are so that you can fill them in.

Which strategy will help students gain a step-by-step, metacognitive understanding of how to form a whole using 1/4 and 1/8?

Think-Alouds

Observations

This is a way to gather less visible information over a longer period of time.

Tests

This should be used as a way for students to demonstrate procedural knowledge and how to apply it outside the classroom.

In your second-grade class, your students are learning about the compare subtraction model. You use the common experience of eating in the cafeteria as a context outside the mathematics curriculum. Which scenario exemplifies this subtraction model?

Three more students are at lunch than are at recess.

A teacher should make good use of calculators by using them for all the following except which situation?

To perform basic computations such as 7 x 3 when computation skills are the objective of the lesson

instructional strategies that can be used to help and encourage mathematical communications.

Use class time effectively by encouraging students to evaluate and compare their answers with other students Provide opportunities that will encourage the metacognitive process and allow students to explain their thinking

A third-grade teacher is working with a mixed-ability classroom. What is one strategy that is beneficial for all learners?

Use effective wait time and multiple methods when approaching math problems.

When you need to plan a lesson for all learners, which two strategies would you use?

Use graphic organizers to help students organize work. Include opportunities for collaboration and classroom discussion. Graphic organizers can help teachers tier lessons and provide students with structure on how to organize their thoughts. For example, lower-level learners can be provided with a more structured graphic organizer while higher-level learners can be provided with a less structured graphic organizer. Regardless, all learners can benefit from using a graphic organizer to help organize their mathematical thinking. You should never create different learning objectives for different students, only different ways for them to meet those objectives. Discussion and collaboration are heavily emphasized in constructivist teaching since it gives students a chance to share ideas and build knowledge. In addition, these teaching strategies also allow you to accommodate students since they can approach a problem in their own way. However, you will want to make sure that you are still maintaining the same learning objectives and standards for all students.

Which misconception do students have about variables?

Variables can only be used for a single unknown value. Variables are used within expressions or equations to help us understand patterns. In addition to being used for single unknown values, variables may also be used for quantities that vary. Early learners can benefit from using variables in simple problems.

When teaching a unit on geometry and symmetry, which tool or manipulative can be used to enhance learning outcomes?

Virtual dynamic software- Dynamic geometry programs allow students to create shapes on the computer screen and then manipulate and measure them by dragging vertices. A well-known program, the Geometer's Sketchpad, is an open source. Dynamic geometry programs allow the creation of geometric objects so that their relationships to another object is established.

Examples of virtual math tools

Virtual number lines Virtual fraction circles Virtual math games and quizzes Student response systems Software packages like spreadsheets

Examples of non-tools

Vocabulary flash cards Individual whiteboards

Students in a sixth-grade class are using models to explore how to find percents. Which question should help foster critical thinking?

What patterns have you noticed when modeling percents?

When you teach math lessons to your students, which type of information should you not provide to your students? Alternative methods

You should not encourage one preferred method. In using a constructivist teaching approach, students should be encouraged to explore multiple strategies and ways to solve a problem. The teacher should avoid using step-by-step instructions and reverting to a single process or method. Although the teacher may have a preferred method, students should be encouraged to explore alternative problem-solving approaches.

Constructivism

a learning theory based on the idea that knowledge is constructed and not merely transferred. Everyone builds knowledge by making connections in their brains to prior knowledge and prior academic learning that they already have. As they do this, they are also creating meaning about what they are learning and experiencing. When you use constructivist techniques in your classroom, your students will become more active, rather than passive, learners.

A student wants to construct a graph to show each varying temperature of ice water after 5, 10, 15, 20, 25, and 30 minutes. Which type of graph would best show this data?

a line graph

Histogram

a type of bar graph where the categories of data must be displayed in chronological order. In a typical bar graph, categories can be moved around without changing the meaning of the graph. For example, if you surveyed favorite ice cream colors, it would not matter in which order you displayed the data. But with a histogram, the order of the categories cannot be switched around. Histograms display data with categories such as days of the week, months of the year, or numerical categories.

Manipulatives

a type of tool- physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons)

GeoGebra

an open-source dynamic geometry program that allows students to create, manipulate, and measure geometry shapes. As students do these actions, they can see how different geometric shapes relate to each other.

Van Hiele Level 1

analysis students can recognize and name properties and classify them

Tools

any object, picture, or drawing that can be used to explore a concept

A tenth-grade math teacher wants to work with a science teacher on integrating the content. How can the teacher apply a math connection?

by expressing the distribution of water on Earth in a circle graph

Waiting

can be used to allow students to pause and think about the problem at hand.

Facts can be learned by rote memorization. This knowledge is still constructed, but it is not ___________ to other knowledge.

connected

students share multiple methods about how to solve problems

encourages metacognition; One of the great benefits of a problem-based approach is that it allows students to come up with multiple ways to solve a particular problem while also expecting all students to meet the same learning objectives as they solve the problem. You should let your students know that a problem has multiple entry points and encourage them to find a variety of ways to solve it. You can then hold a classroom discussion to allow students to share their approaches.

Think pair share

encourages metacognition; has students first think about how they would approach a problem, and then pair up with another student to share their ideas. This gives all students a chance to talk and to see what others think. This is also a great instructional strategy to use with ELs and students with learning disabilities because it gives them a chance to share their thoughts in a nonthreatening situation.

Allowing multiple strategies is important because __________

every student comes with their own unique way of thinking and diverse sets of knowledge and experiences that help them make sense of the math you are teaching.

With relational understanding, students know...

how a concept they have learned is connected to other things within and outside of math.

Van Hiele Level 2

informal deduction students can now logically order shapes according to their properties and come up with definitions for shapes

Revoicing

involves the student or teacher restating the other person's statement as a question. This can help make sure ideas are understood correctly.

Complete the following sentence. An event in a probability experiment

is a subset of the sample space. A subset of an event is considered to have happened. In probability theory, the sample space of an experiment is the set of all possible outcomes or results of that experiment. If, for example, you teach your students about sample space by rolling a single die, then the sample space is the numbers 1 to 6.

Reasoning

is used when the teacher asks the students what they think of each others' explanations. This invites the students to pause and reflect a little further on what is being said rather than just saying that Mark is wrong.

When teaching about growing patterns, students should be encouraged to do all of the following except __________.

make a pattern when given a table or chart of numeric data. Beginning in the primary grades and extending through the middle years, students can explore patterns that involve a progression from step to step. In technical terms, these are called sequences. Analyzing growing patterns should include the developmental progression of reasoning by looking at the visuals, reasoning about the numerical relationship, and then extending to a larger case. Students' experiences with growing patterns should start with fairly straightforward patterns and then move on to patterns that are more complicated.

Learning basic facts can have better results if a teacher promotes ____________ strategies.

multiple

One-to-one correspondence

one of the first counting skills children develop after learning to count orally. This skill is best developed by helping children practice pointing to objects while saying the number words verbally. Objects should be arranged in sets, patterns, or rows for children, so they can easily keep track of which items they have already counted.

Statements that are true about probability

probability of an event is a measure of the likelihood of an event occurring experimental probability gets closer to theoretical probability with a larger data set probability can be determined in two ways: theoretically and experimentally

A key to getting students to be is _____________to engage them in interesting problems in which they use their _____________ knowledge as they search for solutions and create new ideas in the process.

reflective; prior

CSA Sequence

requires that the students start with concrete representations instead of numerical representations. In the example described, the students will start with physical (concrete) representations of 1/4 and 1/8, then they will provide drawings or pictures (semi-concrete) that represent 1/4 and 1/8, and, lastly, they will work solely with numbers as they solve the problem.

Explicit Strategy Instruction

requires that you, the teacher, are carefully guiding students through the problem-solving process, making known thinking strategies that are involved in figuring out a specific problem. Modeling and scaffolding is an important part of this strategy as it will help students to see the approach that needs to be taken. This approach is constructivist in the sense that the teacher should be building on the students' prior knowledge and experience to help make sense and meaning of the problem.

Constructivism

says that people construct their own understanding and knowledge of the world through experiencing things and reflecting on those experiences. When people encounter something new, they have to reconcile it with their previous ideas and experiences, sometimes changing what they believe or discarding the new information as irrelevant. In any case, people are active creators of their own knowledge. To do this they must ask questions, explore, and assess what they know.

When teaching a lesson on probability, which statement demonstrates correct student reasoning about the probability of choosing two cards with replacement when using a standard 52-card deck?

selecting a black queen both times is just as likely as selecting a red king back-to-back.

Related to supporting multiple approaches, it is important to allow students the time to ___________ with the mathematics they are exploring.

struggle

Metacognition

students are thinking about their own thinking. As students develop this ability they can better detect their own thinking and problem-solving errors.

conceptual misunderstanding

students do not understand the relationship between basic elements and how they function together. Conceptual knowledge will therefore help with procedural knowledge. In this case, there is something that students do not conceptually understand about how numbers higher than three are changed when multiplied.

Rephrasing

students restate other students' ideas in their own words

According to the Teaching Principle defined by NCTM, what does effective mathematics teaching require?

teachers must (1) understand deeply the mathematics content they are teaching; (2) understand how students learn mathematics, including a keen awareness of the individual mathematical development of their own students and common misconceptions; and (3) select meaningful instructional tasks and generalizable strategies that will enhance learning.

Cooperative learning

term that describes an instructional arrangement for teaching academic and collaborative skills to groups of students. Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation, and to promote academic achievement and positive interrelationships. According to the National Council of Teachers of Mathematics, learning environments should be created that promote active learning and teaching, classroom discourse, and individual, small-group, and whole-group learning.

Strategic competence

the ability to self-regulate the learning process and make adjustments to mathematical thinking. This does not occur when a teacher or peer tells a student they have done something incorrect, but rather through the metacognitive process where students can detect their own errors.

Line graph

used to represent two related pieces of continuous data, and a line is drawn to connect the points

Elaborating

used when the teacher has the students explain or elaborate on their answers

Think-Alouds

very similar to Explicit Strategy Instruction in that they both involve the teacher explaining and modeling the process. The difference, however, is that the teacher is trying to show the metacognitive process in approaching and solving a problem. In other words, the teacher will work on a different but similar problem with students. The students will be involved in the Think-Aloud process by helping to answer the teacher's questions about why she did what she did. The students will then imitate the thinking and problem-solving process that the teacher is modeling in one problem and apply it to another.

Van Hiele Level 0

visualization students visually recognize similar shapes and should be able to group them together