This document discusses several key concepts in mathematics education:
1. It outlines five process standards and six shifts for teaching mathematics effectively.
2. It discusses constructivist learning theory and how students build new understanding by connecting to prior knowledge through assimilation and accommodation.
3. Effective teaching strategies include encouraging multiple problem solving approaches and building opportunities for reflective thought.
4. Different types of understanding and the benefits of relational understanding are defined.
5. Key aspects of effective mathematics instruction are summarized such as teaching for, about, and through problem solving.
This document discusses several key concepts in mathematics education:
1. It outlines five process standards and six shifts for teaching mathematics effectively.
2. It discusses constructivist learning theory and how students build new understanding by connecting to prior knowledge through assimilation and accommodation.
3. Effective teaching strategies include encouraging multiple problem solving approaches and building opportunities for reflective thought.
4. Different types of understanding and the benefits of relational understanding are defined.
5. Key aspects of effective mathematics instruction are summarized such as teaching for, about, and through problem solving.
This document discusses several key concepts in mathematics education:
1. It outlines five process standards and six shifts for teaching mathematics effectively.
2. It discusses constructivist learning theory and how students build new understanding by connecting to prior knowledge through assimilation and accommodation.
3. Effective teaching strategies include encouraging multiple problem solving approaches and building opportunities for reflective thought.
4. Different types of understanding and the benefits of relational understanding are defined.
5. Key aspects of effective mathematics instruction are summarized such as teaching for, about, and through problem solving.
This document discusses several key concepts in mathematics education:
1. It outlines five process standards and six shifts for teaching mathematics effectively.
2. It discusses constructivist learning theory and how students build new understanding by connecting to prior knowledge through assimilation and accommodation.
3. Effective teaching strategies include encouraging multiple problem solving approaches and building opportunities for reflective thought.
4. Different types of understanding and the benefits of relational understanding are defined.
5. Key aspects of effective mathematics instruction are summarized such as teaching for, about, and through problem solving.
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PST201F
Five process standards Problem solving
o -Reasoning and proof o -Communication o -Connection o Representation Six shifts o Creating an environment equal opportunity to learn o Focusing on balance of conceptual and procedural fluency o Active engagement in 5 NCTM standards o Using technology to enhance understanding o Incorporating multiple assesments aligned with instr. Goals o Helping students recognise power of sound reasoning What does it mean to do mathematics? Means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense. Constructivism-rooted to Jean Piagets work, learners are not blank slates but rather creators of their own learning. Intergrated networks or cognitive schemas are both the product of constructing knowledge and the tools of which additional knowledge can be constructed. As learning occurs these netowrks can be rearranged, added to, or modified. Reflective thought effort to connect existing ideas to new information.People modify their existing schemas to incorporate new information. Assimilation- new idea fits in with prior knowledge, expand current network. Accommodation-new concept doesnt fit in, revamp or replace existing network. Encourage multiple approaches, build opportunities for reflective thought,build new knowledge from prior knowledge,engage students in productive struggle. Understanding- defined as the measure of the quality and quantity of connections that an idea has with existing ideas. Relational understanding-knowing what to do and why. Instrumental understanding-doing something without understanding. Five representations of mathematical ideas o Pictures o Written symbols o Manipulative models o Oral language o Real world situations Conceptual understanding-Knowledge of relationships or foundational ideas of a topic.Comprehension of math concepts operations and relations Procedural fluency-knowledge and use of rules in carrying out mathematical processes Skill in carrying out procedures. Strategic competence- ability to formulate , represent and solve math problems. Adaptive reasoning-capacity for logical thought, reflection. Did I do it right? Productive disposition-Can do attitude,see mathematics as wonderfull, sensibleand belief in ones efficacy. Benefits of relational understanding o Less to remember o Effective learning of new concepts and procedures. o Increased attention and recall. o Enhanced problem solving ability. o Improved attitude and beliefs. Teaching for problem solving-Teaching a skill so a student can later solve. Teaching about problem solving-teaching how to problem solve,teaching the process or strategies.* Teaching through problem solving-Learn mathematics through real contexts, problems situations and models. *Understand the problem Devise a plan. Carry out the plan Look back.
Problem solving strategies Draw a picture, act it out ,use it as a model Look for a pattern Guess and check Make a table or chart Try a simpler form of a problem Make an organised list Write an equation. A problem is defined as any task or activity for which the students have prescribed or memorised rules. Before phase of a lesson 1. Activate prior knowledge 2. Be sure the problem is understood. 3. Establish clear expactations. During phase of a lesson 1. Let go 2. Notice students mathematical thinking 3. Provide appropriate support 4. Provide worthwhile extensions. After phase of a lesson 1. Promote a mathematical community of learners 2. Listen actively without evaluation 3. Summarise main ideas and identify future problems. Why is it better for students to tell or explain than the teacher? Firstly, students are grounded in their own understanding. Second, as students communicate there math ideas there solidify their understanding. Third, there are implications for creating a community of learners. Students will question peers but not teacher. Groupwork-individual accountability,shared responsibility, Practise-different problem based tasks or experiences spread over numerous class periods, each addressing the same basic ideas. Provides an increased opportunity to develop conceptual ideas and usefull connections Alternative and flexible strategies. Greater chance for students too understand. Clear message maths is about figuring and making sense.
Drill-refers to repetive non problem based exercises designed to improve skills already acquired. Provides increased facility with procedure-only a procedure already learned. Review of facts or procedures so they not forgetten Limitations focus on a singular method and an exclusion of flexible alternatives. False appearance of student understanding. Rule orientated or procedural view of mathematics.] Transmission of knowledge or show and tell approach: The definition of transmission is to pass something on from one place or person to another. In the past, a traditional approach ignored the students mental level of interest that is the learners developmental level to understand the learning content. Not all learners are on an equal level of understanding. Furthermore, the traditional approach followers assumed that there is a fixed body of knowledge that the student must come to know; therefore, it is up to the educator to transfer / (transmission) what they know to the learners. Students are expected to blindly accept the information they are given without questioning the instructor. The teachers role is to transfer thoughts and meanings to the passive student leaving little room for student-initiated questions. In the case of the show and tell approach, too often, learners are passive. They are given rules or examples that they are unable to use their mathematical skills to solve real-world problems. They rarely create mathematical ideas for themselves, but rather they depend on the teacher for direction. Over time, they may come to believe that they cannot do mathematics themselves. Assessment-process of gathering evidence about a students knowledge of, ability to use and disposition towards maths and making inferences from that evidence. Why we assess- Monitor student progress Make instructional decisions Evaluate learner achievement Evaluate programs What should we asses- Conceptual understanding and procedural fluency. Strategic competence and adaptive reasoning. Productive disposition. Rubric -framework that can be designed or adapted by the teacher for a particular group. Performance indicators-task specific statements that describe what performance look like at each level. 3 basic ways to use formative assessment to evaluate understanding- observation,interviews and tasks. Assessment tools for gathering evidence on knowledge and ability- tests,diagnostic interview Assessment tool for gathering evidence on learners disposition- student self assessment- observation,interviews and journals
Fractions- fractions are difficuilt because there are many meanings.Written in an unusual way. Instruction does not focus on conceptual understanding of fractions. Over generalise whole number knowledge. Models -length , area , set. Benchmarks- important reference points which are used to compare the relative sizes of fractions through estimating, ordering and placing them on a number line. Benchmarks are reference points which help to gauge information about the fraction, the most important being, 0; ; and 1 Algorithm is a step-by-step solution to a problem. It is like a cooking recipe for mathematics.
Geometry- Van hiel levels
Level 0 visualisation- shapes and what they look like. Level 1 analysis-classes of shapes rather than individual shapes. Level 2 informal deduction properties of shapes. Level 3 deduction-relationships between properties of shapes. Level 4 rigor deductive axiomatic systems for geometry.
Characteristics of levels Products of thought at each level are the same as the objects at the next level Levels are not age dependant Advancement through levels need geometric experiences. When instruction or language is higher level than the student they will not understand the concept . Eulers formula: V + F -E = 2 definition of a prism is :