4 – Exemplary Teaching
Silicon Valley Mathematics Initiative
The Mathematics Teaching Rubric
Worthwhile Tasks
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher engages the students in worthwhile tasks.|Often the teacher engages the students in worthwhile |Fairly often the teacher engages the students in |Occasionally the teacher engages the students in |
| |tasks. |worthwhile tasks. |worthwhile tasks. |
|The tasks are all part of a coherent curriculum that |Most of the tasks are part of a coherent curriculum |The tasks are usually a balance between adopted |The tasks are usually a part of an adopted |
|develops student understanding and facility with |that develops student understanding and facility with|textbooks and investigations or non-routine problems.|textbook curriculum. |
|mathematics. |mathematics. | |The mathematics of the tasks focuses on skill |
|The mathematics of the tasks is clear and aligned |The mathematics of the tasks is often clear and |The mathematics of the tasks is usually clear and |acquisition and developing procedural knowledge. |
|with core standards and developed to make sense of |aligned with core standards and developed to make |aligned with core standards and developed to make |A few of the tasks engage student thinking, |
|the world. |sense of the world. |sense of the world. |conceptual development, problem solving as well |
|The tasks engage student thinking, conceptual |Many of the tasks engage student thinking, conceptual|Some of the tasks engage student thinking, conceptual|as skills. |
|development, problem solving as well as skills. |development, problem solving as well as skills. |development, problem solving as well as skills. |A few tasks involve a range of mediums. |
|The tasks involve a wide range of mediums (graphs, |The tasks involve a range of mediums. |Some tasks involve a range of mediums. |Occasionally the teacher selects materials and |
|problems, symbols, equations, manipulatives, |The teacher often selects materials and tasks that |The teacher sometimes selects materials and tasks |tasks that build on and extend from their |
|technology). |build on and extend from their students’ |that build on and extend from their students’ |students’ understanding. |
|The teacher selects materials and tasks that build on|understanding. |understanding. |A few of the tasks foster students’ ability to |
|and extend from their students’ understanding. |Many of the tasks foster students’ ability to solve |Some of the tasks foster students’ ability to solve |solve problems, reason, and communicate. |
|Most of the tasks foster students’ ability to solve |problems, reason and communicate. |problems, reason and communicate. |Most of the tasks involve direct instruction and |
|problems, reason and communicate. |The tasks range across many learning styles and |Some of the tasks range across learning styles and |modalities. |
|The tasks range across most learning styles and |modalities. The tasks may grow out of students’ |modalities. A few of the tasks may grow out of |A few of the tasks may be approached in more than|
|modalities. The tasks may grow out of students’ |conjecture of questions. |students’ conjecture of questions. |one interesting and legitimate way. |
|conjecture of questions. |Many of the tasks may be approached in more than one |Some of the tasks may be approached in more than one |A few of the tasks require more time to develop |
|Often the tasks may be approached in more than one |interesting and legitimate way. |interesting and legitimate way. |solutions. |
|interesting and legitimate way. |Many of the tasks vary in length of time required to |Some of the tasks require more time to develop | |
|Most of the tasks vary in length of time required to |develop solutions. |solutions. | |
|develop solutions. | | | |
The Learning Environment
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher creates a classroom environment where students|The teacher often allows students to actively construct |The teacher regularly encourages students to |Some assigned problems require students to |
|actively construct their own understanding of mathematics |their own understanding of mathematics and learn to use |learn to use mathematics to make sense of the |learn to use mathematics to make sense of the |
|and learn to use mathematics to make sense of the world. |mathematics to make sense of the world. |world. |world. |
|The teacher creates a culture that fosters the development|The culture often fosters the development of each |The culture fosters the development of |The class is usually taught in a direct |
|of each student’s mathematical power and supports the |student’s mathematical power and supports the class as a |student’s mathematical skills and some problem|instruction format. |
|class as a community of learners. |community of learners. |solving. |Usually the focus of the lesson is on skill |
|The teacher provides and structures the time necessary to |The teacher provides and structures a lot of time to |Occasionally the teacher supports the class to|attainment or learning mathematical procedures |
|explore sound mathematics and grapple with significant |explore sound mathematics and grapple with significant |function as a community of learners. |and definitions. |
|ideas and problems. |ideas and problems. |The teacher provides and structures some time |Some times the teacher assigns non-routine or |
|The physical space and materials are purposely arranged |The physical space and materials are arranged and |to explore non-routine problems and grapple |word problems. |
|and allocated in ways that facilitate students’ learning. |allocated in ways that can facilitate students’ learning. |with conceptual ideas. |The physical space and materials are arranged |
| | |The physical space and materials are arranged |and allocated in ways that structure students’ |
|The teacher provides context that encourages the |The teacher provides context that often encourages the |and allocated in ways that attempts to |work. |
|development of mathematical skill and proficiency. |development of mathematical skill and proficiency. |facilitate students’ learning. |The teacher’s goal is to develop the students’ |
|Respect and value of students’ ideas, ways of thinking and|Respect and value of students’ ideas, ways of thinking and|The teacher strives to develop the student’s |numerical skills and proficiency. |
|positive disposition to mathematics are essential |positive disposition to mathematics are often part of the |numerical skill and proficiency. |Students are required to practice during class |
|attributes of the learning community. |learning community. |Students’ ideas and ways of thinking are |and complete homework. |
|The teacher consistently expects and encourages students |The teacher often expects and encourages students to work |sometimes part of the learning activity. |The teacher often expects students to work |
|to work independently or collaboratively to make sense of |independently or collaboratively to make sense of |The teacher often expects students to work |independently and may use some collaborative |
|mathematics. |mathematics. |independently and may use some collaborative |groups. |
|Students take intellectual risks by raising questions, |Students occasionally take intellectual risks by raising |groups. |Often the teacher asks the students to show |
|sharing results and formulating conjectures. |questions, sharing results and formulating conjectures. |Often the teacher involves students by having |work on individual problems. |
|The students display a sense of mathematical competence by|The students may display a sense of mathematical |them raise questions and share results. |Occasionally students are asked to share |
|validating and supporting ideas with mathematical |competence by validating and supporting ideas with | |answers with the class. |
|arguments. |mathematical arguments. | |Students are encouraged to ask questions if |
| | | |they lack understanding. |
Teacher’s Role in Discourse
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher orchestrates discourse in the class. |The teacher often orchestrates discourse in the class. |The teacher directs the class and attempts to |The teacher directs the class. |
|The teacher poses questions and tasks that elicit, engage,| |foster discourse. |The teacher poses questions and tasks that |
|and challenge each student’s thinking. |The teacher often poses questions and tasks that |Sometimes the teacher poses questions and tasks |focus students’ work. |
|The teacher listens carefully to the students’ ideas and |elicit, engage, and challenge each student’s thinking. |that elicit, engage, and challenge students’ |The teacher may listen to some of the |
|discerns mathematical meaning and relevancy from student | |thinking. |students’ responses. The teacher may have |
|responses. |Often the teacher listens to the students’ ideas and |The teacher listens to some of the students’ |difficulty following students’ math thinking |
|Students are asked to clarify and justify their ideas |makes sense of their responses. |ideas. The teacher may have difficulty following |and usually ignores those responses. |
|orally and in writing. |Often students are asked to clarify and justify their |students’ math thinking. |Rarely are students asked to clarify and |
|The teacher decides what to pursue in depth from among the|ideas orally and in writing. |Sometimes students are asked to clarify and |justify their ideas orally and in writing. |
|ideas that students bring up during a discussion. |The teacher may decide what to pursue in depth from |justify their ideas orally and in writing. |The teacher rarely deviates from the textbook |
|Care is given by the teacher to develop concepts |ideas that students bring up during a discussion. |The teacher sometimes deviates from lesson plan to|to pursue ideas that students bring up during |
|thoroughly and insure students’ ownership and |Some care is given by the teacher to develop concepts |pursue ideas that students bring up during a |a discussion. |
|understanding. The teacher decides when and how to attach|and to encourage students’ ownership and understanding.|discussion. |The teacher usually follows the textbook in |
|mathematical notation and language to students’ ideas. |The teacher attempts to attach mathematical notation |Some care is given by the teacher to develop |deciding when and how to attach mathematical |
|The teacher has full understanding of the mathematical |and language to students’ ideas. |concepts and to encourage students’ ownership. |notation and language to students’ ideas. |
|goals of the lesson and decides when to provide |The teacher has good understanding of the mathematical |The teacher follows the textbook in deciding when |The teacher follows the curricula math goals |
|information, when to clarify an issue, when to model, when|goals of the lesson and is often successful in |and how to introduce mathematical notation and |of the lesson. |
|to lead, and when to let a student struggle with |determining when to provide information, when to |language. |The instructional decisions may be aligned |
|difficulty. These decisions are consistent with the goal |clarify an issue, when to model, when to lead, and when|The teacher follows the curricula math goals of |with state standards and with the goal of |
|and pace of the lesson. |to let a student struggle with difficulty. These |the lesson. |covering the topics on state test. |
|The teacher monitors students’ participation in |decisions are often consistent with the goal and pace |The teacher is improving his/her questioning |The teacher modifies expectation from student |
|discussions and decides when and how to encourage each |of the lesson. |strategies. The teacher is developing the pacing |to student. |
|student to participate. |The teacher attempts to monitor students’ participation|and generating good questioning. The | |
|Decisions are predicated on insuring all students will |in discussions and decides when and how to encourage |instructional decisions may be aligned with lesson| |
|learn and be successful in mathematics. |students to participate. |goals. | |
| |Attempts are made to reach all students. |The teacher attempts include all students. | |
Students’ Role in Discourse
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher of mathematics promotes classroom |The teacher of mathematics often promotes classroom |The teacher of mathematics tries to promote classroom|The teacher of mathematics structures the |
|discourse in which students have ownership and |discourse in which students have ownership and |discourse in which students have some ownership and |discourse of the students. |
|responsibility. |responsibility. |responsibility. |Often students are asked questions by the |
|Students listen to, respond to and question the |Students often listen to, respond to and question the|Occasionally students listen to, respond to and |teacher and are expected to respond with |
|teacher and one another. |teacher and one another. |question the teacher and one another. |answers. |
|The students use a variety of tools to reason, make |The students use some tools to reason, make |The students use a few tools to reason, make |The students use a few tools to solve problems |
|connections, solve problems and communicate. |connections, solve problems and communicate. |connections, solve problems and communicate. |and do math tasks. |
|Students often initiate problems and questions for |Students sometimes initiate problems and questions |Occasionally students initiate problems and questions|For specific assignments students create |
|the class to ponder and study. Students regularly |for the class to ponder and study. Students often |for the class to ponder and study. |problems and questions for the class to do. |
|make conjectures and present solutions. Students |make conjectures and present solutions. Students may|Students occasionally make conjectures and present |Students are asked to present answers or |
|explore examples and counterexamples to investigate |explore examples and counterexamples to investigate |solutions. Students occasionally explore examples |solutions. Students learn and practice |
|conjectures. |conjectures. |and counterexamples to investigate conjectures. In a |mathematics procedures, vocabulary and facts. |
|Students try to convince themselves and one another |Students occasionally try to convince themselves and |few situations students rely on mathematical evidence|Occasionally students work in small groups on |
|of the validity of particular representations, |one another of the validity of particular |and argument to determine validity. |problems, discussing answers. In large groups, |
|solutions, conjectures and answers. Students rely on|representations, solutions, conjectures and answers. |In small or large groups, students are occasionally |students go over problems and solutions usually |
|mathematical evidence and argument to determine |Sometimes students rely on mathematical evidence and |an audience for one another’s comments, explanations |facilitated by the teacher. |
|validity. |argument to determine validity. |or questions. |The discourse involves learning procedures and |
|In small or large groups, students are an audience |In small or large groups, students are often an |The discourse sometimes focuses on making sense of |methods to solve problems and activities to |
|for one another’s comments, explanations or |audience for one another’s comments, explanations or |mathematical ideas and/or on using mathematical ideas|memorize facts and vocabulary. |
|questions. |questions. |sensibly in setting up and solving problems. | |
|The discourse is focused on making sense of |The discourse often is focused on making sense of | | |
|mathematical ideas and/or on using mathematical ideas|mathematical ideas and/or on using mathematical ideas| | |
|sensibly in setting up and solving problems. |sensibly in setting up and solving problems. | | |
Tools for Enhancing Discourse
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher supports discourse and a positive |The teacher usually supports discourse by encouraging|The teacher structures the tools that will adapt to |The teacher directs the class as to when they |
|classroom culture by encouraging and supporting |and supporting students to use tools to do and learn |the particular mathematics lessons. |may use a certain mathematics tool. |
|students to use tools to do and learn mathematics. |mathematics. |The teacher finds times to use computers and other |The teacher may do an activity with the |
|The teacher supports the doing of mathematics in the |Often the teacher values and encourages the use of a |technology. |computer or use it for those that either need |
|manner that mathematics is done outside of school. |variety of tools such as computers, calculators and |Teacher attempts to balance the time when students |practice or have finished their assignments. ]|
|The teacher values and encourages the use of a |other technology in addition to traditional paper and|use calculators, paper and pencil, and mental math. |The teacher will allow students to use a |
|variety of tools such as computers, calculators and |pencil and mental math. |The teacher will, at certain times, encourage the |calculator after they have shown they can |
|other technology in addition to traditional paper and|Often the teacher has the students communicate orally|students to communicate using pictures, diagrams, |calculate accurately using paper and pencil. |
|pencil mental math. |and in writing using pictures, diagrams, tables, |tables, graphs, notation, symbols, narratives, |The teacher directs the type of outputs |
|Students communicate orally and in writing using |graphs, notation, symbols, narratives, metaphors, |metaphors, justifications and proofs. |expected such as charts, graphs, equations or |
|pictures, diagrams, tables, graphs, notation, |justifications and proofs. |Students use manipulatives on appropriate activities.|proofs. |
|symbols, narratives, metaphors, justifications and |Often the teacher has students use models and | |When called for in a lesson, the class may use |
|proofs. |concrete materials to make sense of mathematics and |The teacher introduces conventional notation often |manipulatives. |
|Students use models and concrete materials to make |understand concepts. |after students had some kind of concrete experience |The teacher regularly introduces the |
|sense of mathematics and understand concepts. |Occasionally students are responsible for selecting |or when the textbook suggests the teacher do so. |conventional notation to the class. |
|Students are responsible for selecting and using |and using appropriate tools to solve and investigate |Occasionally students communicate their solutions and|There may be special projects or assignments |
|appropriate tools to solve and investigate problems. |problems. |explanations using a variety of tools. |where students communicate using alternative |
| |The teacher attempts to introduce conventional | |materials or tools. |
|The teacher introduces conventional notation at |notation at points when doing so can further the work| | |
|points when doing so can further the work and the |and the discourse at hand. | | |
|discourse at hand. |Occasionally students are expected to communicate | | |
|Students are expected to communicate their |their conjectures, explanations and arguments in a | | |
|conjectures, explanations and arguments in a complete|complete manner using appropriate tools. | | |
|manner using appropriate tools. | | | |
Teaching and Learning Analysis
|4 – Exemplary Teaching |3– Successful Teaching |2– Improving Teaching |1– Limited Teaching |
|The teacher understands that assessment of students, |Often the teacher links assessment with the |The teacher attempts to link assessment with the |The teacher assesses the students on mathematical |
|analysis of instruction and the learning experience are|curriculum. |curriculum. |ideas and problems after the students have been |
|fundamentally interconnected. |The teacher routinely engages in analysis of teaching |The teacher assesses mostly by examining written |taught. |
|The teacher engages in ongoing analysis of teaching and|and learning by observing, listening to, and gathering|products but also by observing, listening to, and |The teacher bases his/her evaluation mostly by |
|learning by observing, listening to, and gathering |information about the students to assess what they |gathering information about the students. |examining written products, which are usually tests|
|information about the students to assess what they know|know and are able to do. |Periodically the teacher will reflect on the lessons and|and quizzes. The teacher may also grade students on|
|and are able to do. |The teacher often examines the effects of the tasks, |learning experiences and compare their effectiveness |homework and class participation. Classroom |
|The teacher examines the effects of the tasks, |discourse and learning environment on students’ |with student progress. |behavior is often a factor in the students’ grades.|
|discourse and learning environment on students’ |mathematical knowledge, skills and dispositions. |The teacher paces the class based on what most of the | |
|mathematical knowledge, skills and dispositions. |The teacher routinely assesses what many of students |students are accomplishing. |The teacher is diligent about following the |
|The teacher assesses what every student is learning, |are learning and the concepts they understand. |The teacher will attempt to differentiate instruction |timeline of instruction based on a set of |
|the concepts they understand and that they are doing |Usually the teacher changes and adapts instruction |and activities for the most successful and most |standards, a continuum and/or the textbook. |
|significant mathematics. |based on the ongoing students’ understandings. |challenged students. |The teacher may provide extra credit to those |
|The teacher changes and adapts instruction based on the|The teacher may revise short and long range plans. The|The teacher may revisit lessons where students have been|students who are the most successful and remedial |
|ongoing assessment of the students. |teacher often challenges and extends students’ ideas. |unsuccessful. |assignments to the most challenged students. |
|The teacher regularly revises short and long range | |The teacher attempts to use multiple representations to |Depending on time available the teacher may |
|plans. The teacher challenges and extends students’ |The teacher attends to an array of dimensions that |address learning styles. |re-teach lessons where students have been |
|ideas. |contribute to students’ mathematical competence. |The teacher follows conference and report card |unsuccessful. |
|The teacher attends to the broad array of dimensions |The teacher routinely describes and comments on each |procedures to keep the parent informed of students’ |The teacher uses mostly direct instruction for |
|that contribute to students’ mathematical competence. |student’s learning to the parent, other involved |progress. |efficiency. |
|The teacher regularly describes and comments on each |educators and the students themselves. |The teacher uses different evaluation measures such as |The teacher follows the district reporting policy |
|student’s learning to the parent, other involved |The teacher uses a variety of assessment measures and |test, quizzes, homework, classwork, reports, |for report cards. |
|educators and the students themselves. |mediums to develop a fair evaluation of students. |participation, etc. to base the students’ grade. |The teacher may use different evaluation measures |
|The teacher uses a large variety of assessment measures| | |but tests have the most weight. |
|and mediums to develop a robust evaluation of students.| | | |
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