Manipulative
Concepts K-3
Attribute blocks
Balance scale Capacity containers
Counters
Cubes
sorting, classification, investigation of size, shape, color, logical reasoning, sequencing, patterns, symmetry, similarity, congruence, thinking skills, geometry, organization of data Weight, mass, equality, inequality, equations, operations on whole numbers, estimation, measurement
Measurement, capacity, volume, estimation
counting, comparing, sorting, classification, number concepts, fact strategies, even & odd numbers, equality, inequality, operations, ratio, proportions, probability, integers number concepts, counting, place value, fact strategies - especially turnaround facts, classification, sorting, colors, patterns, square & cubic numbers, equality, inequalities, averages, ratio, proportion, percent, symmetry, spatial visualization, area, perimeter, volume, surface area, transformational geometry, operations on whole numbers & fractions, even & odd numbers, prime & composite numbers, probability
Cuisenaire rods
classification, sorting, ordering, counting, number concepts,, comparisons, fractions, ratio, proportion, place value, patterns, even & odd numbers, prime & composite numbers, logical reasoning, estimation, operations on whole numbers
Dominoes
counting, number concepts, fact strategies, classification, sorting, patterns, logical reasoning, equality, inequality, mental math, operations on whole numbers
Numeral cards
Pattern blocks
Ten-frames
counting, classification, sorting, comparisons, equality, inequality, order, fact strategies, number concepts, operations on whole numbers, fractions, decimals, logical reasoning, patterns, odd & even numbers, prime & composite numbers
patterns, one-to one correspondence, sorting, classification, size, shape, color, geometric relationships, symmetry, similarity, congruence, area, perimeter, reflections, rotations, translations, problem solving, logical reasoning, fractions, spatial visualization, tessellations, angles, ratio, proportions fact strategies, mental math, number concepts, counting, equality, inequality, place value, patterns, operations on whole numbers
Worthwhile Mathematics Tasks Mathematics teaching should pose tasks that are based on --
sound and significant mathematics; knowledge of students’ understandings, interests, and experiences; knowledge of the range of ways that diverse students learn mathematics
and that
engage students' intellect; develop students' mathematical understanding and skills; stimulate students to make connections and develop a coherent framework for mathematical reasoning; promote communication about mathematics; represent mathematics as an ongoing human activity; display sensitivity to, and draw on, students' diverse background experiences and dispositions; promote the development of all students' dispositions to do mathematics.
Teacher's Role in Discourse Discussion should be orchestrated by
posing questions and tasks that elicit, engage, and challenge each student’s thinking; listening carefully to students’ ideas; asking students to clarify and justify their ideas orally and in writing; deciding what to pursue in depth from among the ideas that students bring up during a discussion; deciding when and how to attach mathematical notation and language to students’ ideas; deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty; monitoring students’ participation in discussions and deciding when and how to encourage each student to participate.
Student's Role in Discourse Classroom discussion should promote children to
listen to, respond to, and question the teacher and one another; use a variety of tools to reason, make connections, solve problems, and communicate; initiate problems and questions; make conjectures and present solutions; explore examples and counter examples to investigate a conjecture; try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers; rely on mathematical evidence and argument to determine validity.
Tools for Enhancing Discourse The teacher, in order to enhance discourse, should encourage and accept the use of --
computers, calculators, and other technology; concrete materials used as models; pictures, diagrams, tables, and graphs; invented and conventional terms and symbols; metaphors, analogies, and stories; written hypotheses, explanations, and arguments; oral presentations and dramatizations.
Learning Environment The teacher should create a learning environment that fosters the development of each student’s mathematical power by --
providing and structuring the time necessary to explore sound mathematics and grapple with significant ideas and problems; using the physical space and materials in ways that facilitate the learning of mathematics; providing a context that encourages the development of mathematical skill and proficiency; respecting and valuing students’ ideas, ways of thinking, and mathematical dispositions.
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Questioning Strategies The art of teaching is based on effective questioning strategies. Asking good questions is an informative process that needs development, refinement, and practice. Teaching through questioning is interactive and engages students by providing them with opportunities to share their thinking. The classroom should be a community of collaborative learners whose voices and ideas are valued. In order to obtain more information from students during classroom discussion, we need to develop an open-ended questioning technique and use a more inquiring form of response, encouraging students to defend or explain both correct and incorrect responses. Here is an example of closed and open questioning for the same situation: Closed - What unit should be used to measure this room? (limiting) Open - How could we measure the length of this room? What choices of units do we have? Why would some units seem more appropriate than others? (probing -encourages students to think about several related ideas) Good questioning involves responding to students in a manner that helps them think and lets you see what they are thinking. Response techniques involve:
Waiting. Time is a critical component. An immediate judgment of a response stops any further pondering or reflection on the part of the students. Requesting a rationale for answers and or solutions. Students will utimately accept this procedure as an expected norm. Eliciting alternative ideas and approaches.
The teacher strives to
pose questions and tasks that elicit, engage, and challenge each student’s thinking; ask students to justify their ideas orally and in writing.
Examples
Category 1 questions focus on helping students work together to make sense of mathematics. o "Do you agree? Disagree?" "Does anyone have the same answer but a different way to explain/tell me about it?" Category 2 contains questions that help students rely more on themselves to determine whether something is mathematically correct. o "Does that make sense?" "What model/ picture/number/pattern shows that?" Category 3 questions seek to help students learn to reason mathematically. o "Does that always work?" "How could we prove that?" Category 4 questions focus on helping students learn to conjecture, invent, and solve problems. o "What would happen if...?" "What would happen if not…?" "What pattern do you see?" Category 5 questions relate to helping students connect mathematics, its ideas, and its applications. o "Have we solved a problem that is similar to this one?" "How does this relate to/seem like ...?"