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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? - correct answer The activity is effective at showing students how to add and subtract integers but the standard is not aligned. 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? - correct answer Students will use precise math language to express their reasoning. Which stakeholder is directly in charge of choosing mathematical standards? - correct answer State departments of education According to the Teaching Principle defined by NCTM, what does effective mathematics teaching require? - correct answer 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. 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? - correct answer The activity is effective at showing students how to find the area of rectangles, but the standard is not aligned. Which type of understanding: Relational or Instrumental? Giving your students a problem and asking them to explain how they arrived at the solution - correct answer Relational Asks students to demonstrate their reasoning - correct answer Relational Understanding Relational or Instrumental: Having your students complete a ten question quiz - correct answer Instrumental. Consider, however, turning this into relational understanding by having students explain their answers to each other. Relational or Instrumental: Having students compete to see who can complete the most questions - correct answer Instrumental: Competition and game activities are fun and can help learning, but make sure that it goes beyond demonstrating instrumental understanding. Relational or Instrumental: Asking students "Was there something in the problem that reminded you of another problem you have done?" - correct answer 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. Relational or Instrumental: Having your students memorize the multiplication table to be able to answer questions faster. - correct answer 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: Using problem based learning - correct answer Relational. This is a great way to engage students in scenarios that encourage deeper thinking and relational understanding. Constructivism - correct answer 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. Three aspects of understanding mathematics - correct answer 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 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? - correct answer 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. How is a conceptual understanding beneficial to students when learning mathematics? - correct answer 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 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. - correct answer reflective; prior Facts can be learned by rote memorization. This knowledge is still constructed, but it is not ___________ to other knowledge. - correct answer connected Learning basic facts can have better results if a teacher promotes ____________ strategies. - correct answer multiple Allowing multiple strategies is important because __________ - correct answer 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. Related to supporting multiple approaches, it is important to allow students the time to ___________ with the mathematics they are exploring. - correct answer struggle disequilibrium - correct answer 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. A student submits the following answer: 2/5 - 1/3 = 1/2 How can this student demonstrate strategic competency? - correct answer The student uses a manipulative and realizes that a common denominator was not used. Strategic competence - correct answer 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. two learning strategies that describe effective math learning - correct answer Students learn by rearranging previously learned concepts and making connections. Students learn new ideas through collaboration with the teacher and peers. Cooperative learning - correct answer 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. Constructivism - correct answer 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. 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? - correct answer 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. Which type of information should not be provided by math teachers - correct answer The preferred method 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. - correct answer Mathematical conventions Alternative methods 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? - correct answer A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each. Select two learning theories that describe effective math learning. - correct answer Students learn new ideas through collaboration with their teacher and peers. Students learn by rearranging previously learned concepts and making connections. 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? - correct answer It assumes wonderful explanations produce understanding. To create an environment for doing mathematics, what is the teacher's role? - correct answer Create a spirit of inquiry, trust, and expectations What is an environment that is not desirable for mathematical instruction? - correct answer Students are working through practice problems to learn the mathematical strategy. Which statement is not proposed by the constructivist theory? - correct answer A safe environment is necessary to gain mathematical power. Which statement is not included in relational and conceptual understanding? - correct answer It eliminates poor attitudes and beliefs Students in a sixth-grade class are using models to explore how to find percents. Which question should help foster critical thinking? - correct answer What patterns have you noticed when modeling percents? 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? - correct answer 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. Revoicing - correct answer involves the student or teacher restating the other person's statement as a question. This can help make sure ideas are understood correctly. Rephrasing - correct answer students restate other students' ideas in their own words