The resources below were developed to introduce you to this program and provide you with an individualized approach to using Exemplars problem-solving tasks. Our supplemental math material may be used enrich instruction, assessment and professional development. We encourage you to download and share these resources with your colleagues.

# Tools for Success

## 1. Planning

The four Depth of Knowledge (DOK) levels are defined.

**Level 1 (Recall)** includes the recall of information such as a fact, definition, term, or
a simple procedure, as well as performance of a simple algorithm or application of
a formula. That is, in mathematics a one-step, well-defined, and straight algorithmic procedure should be included at this lowest level. Other key words that signify a Level 1 include “identify,” “recall,” “recognize,” “use,” and “measure.” Verbs such as “describe” and “explain” could be classified at different levels, depending on what is to be described and explained.

**Level 2 (Skills/Concepts)** includes the engagement of some mental processing beyond a habitual or rote response. A Level 2 assessment item requires students to make some decisions as to how to approach the problem or activity, whereas Level 1 requires students to demonstrate a rote response, perform a well-known algorithm, follow a
set procedure (like a recipe), or perform a clearly defined series of steps. Keywords that generally distinguish a level 2 item include “classify,” “organize,” “estimate,” “make observations,” “collect and display data,” and “compare data.” These actions imply making choices among mathematical constructs that come from knowing mathematical concepts and when to apply mathematical procedures. For example, to compare data requires first identifying characteristics of the objects or phenomenon and then grouping or ordering the objects. Level 2 requires multiple actions including processing of information and mathematical ideas. Some action verbs, such as “explain,” “describe,” or “interpret” could be classified at different levels depending on the object of the action. For example, interpreting information from a simple graph, requiring the reading of information from the graph, also are at Level 2. Interpreting information from a complex graph that requires some decisions on what features of the graph need to be considered and how information from the graph can be aggregated is a Level 3. Level 2 activities are not limited only to number skills, but may involve visualization skills and probability skills. Other Level 2 activities include noticing
or describing non-trivial patterns, explaining the purpose and use of experimental procedures; carrying out experimental procedures; making observations and collecting data; classifying, organizing, and comparing data; and organizing and displaying data in tables, graphs, and charts.

**Level 3 (Strategic Thinking)** requires reasoning, planning, using evidence, and a higher level of thinking than Level 1 and Level 2. In most instances, requiring students to explain their thinking is a Level 3. Activities that require students to make conjectures are also at this level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple answers, a possibility for both Levels land 2, but because the task requires more demanding reasoning. An activity, however, that has more than one possible answer and requires students to justify the response they give would most likely be a Level 3. Other Level 3 activities include drawing conclusions from observations; citing evidence and developing a logical argument for concepts; explaining phenomena in terms of concepts; and deciding which concepts to apply in order to solve a complex problem.

**Level 4 (Extended Thinking)** requires complex reasoning, planning, developing,
and thinking most likely over an extended period of time. The extended time period
is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking.
For example, if a student has to take the water temperature from a river each day for
a month and then construct a graph, this would be classified as a Level 2. However, if the student is required to conduct a river study that requires taking into consideration a number of variables, developing questions to answer, collecting data, and presenting findings, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and work should be very complex. Students should be required to make several connections-relate ideas *within* the content area or *among* content areas-and select one approach among many alternatives on how the situation should be resolved, in order to be at this highest level. Level 4 activities include designing *and* conducting experiments and projects; developing and proving conjectures, making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.

*Depth-of-knowledge (DOK) definition per Norman L. Webb & Others, Web Alignment Tool (WAT) training manual, University of Wisconsin-Madison and the Wisconsin Center for Education Research, 2005, page 45. Accessed June 2012. http://wat.wceruw.org/index.aspx*

Each content standard has been classified in one of three ways throughout *Problem Solving for the Common Core*.

In *Problem Solving for the Common Core*, each Common Core content standard has been classified in one of three ways: Aligned, Embedded or Not Applicable. Descriptions for each are found below.

__Aligned__

This classification refers to problem-solving tasks (instructional/formative and summative) that are directly “aligned” to a specific content standard. These tasks can be used for practice and/or assessment. Summative assessment tasks include anchor papers and scoring rationales.__Embedded__

This classification refers to instances where the underlying math concept in the content standard is “embedded” within a task, but the standard is not directly aligned to that task. A student*may*use the underlying math concept in the standard to solve the problem but cannot be*required*to use that math concept, due to the open-ended nature of problem solving. These tasks should not be given as an assessment but rather used with students to practice a particular math concept or skill.__Not Applicable__

Content standards that have been classified as “not applicable” cannot be assessed through problem solving. For this reason, tasks have not been included for these particular standards. For example, the Kindergarten Counting and Cardinality Standard, K.CC.B.4a states, “When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.” This standard specifically describes a principle of counting (one-to-one correspondence) that does not elicit DOK3 tasks.

A Preliminary Planning Sheet is included with each task. This guide provides an essential tool in lesson preparation.

The Preliminary Planning Sheet (PPS) serves as the teacher’s “blueprint” for each performance task and is a useful tool in lesson preparation. This resource enables teachers to foresee what instruction should be done before the task is used for assessment. It may also be used to anticipate which math concepts and skills students might be required to use.

Each PPS includes the following information:

- the
*Underlying Mathematical Concepts*related to the task - some
*Possible Problem-Solving Strategies*that students might use - some
*Possible Mathematical Vocabulary/Symbolic Representation*that students might use - the
*Possible Solutions*that students might find - some
*Possible Connections*that students could make

PPSs are provided with every task. In the summative assessment setting, PPSs are meant to support teachers in assessing student work with the Exemplars rubric. A student may use mathematical vocabulary/strategies/connections/representations that are not evident in any of the anchor papers but are noted on the PPS for the teacher to reference. (Students may also use additional mathematical vocabulary/strategies/connections/representations that are not noted on the PPS or anchor papers, but are mathematically relevant.)

**Accessing Preliminary Planning Sheets**

The PPS for any problem may be accessed and printed from the “Plan” section of a task. The information contained in the PPS is also visible in the task overview. Blank PPSs may be found under the “Downloads” section.

The problem-solving tasks in this program have been classified as either an instructional task/formative assessment or as a summative assessment.

The tasks found in *Problem Solving for the Common Core* have been classified as either an instructional task/formative assessment or a summative assessment.

**Instructional Tasks/Formative Assessments**

Throughout this program, there are four (or more) instructional/formative assessment problem-solving tasks for every applicable Common Core content standard. These are viewed as opportunities for students to learn new mathematical strategies, vocabulary and notation and representations. Students can also explore mathematical connections and self-assess their solutions. These tasks may be done alone, in pairs, groups or as a whole class. Direct instruction may also be used to question and support classroom discussion around the underlying mathematical concepts in a task.Teachers should use these problem-solving tasks to observe and support student understanding. As part of this process, conferencing and editing can occur and students can revisit their work as often as necessary. Teachers can use similar tasks throughout a unit of study to give a student multiple opportunities to use new learning in her or his solution and to gain independence in arriving at a correct answer.

**Summative Assessment Tasks**

Throughout this program, there are summative assessment tasks for every applicable Common Core content standard. These problem-solving tasks are given at the end of a unit of study to assess students’ understanding. A set of anchor papers and scoring rationales are provided with these tasks.In order to achieve a true assessment of what the student understands and is able to do, in words of the Common Core, there should be a wait time of at least one day between the last instructional task/formative assessment and the summative assessment. A similar assessment task may also be given to students much later in the year if a teacher wants to spiral back to determine how much learning is retained.

Summative assessment tasks can be read to the students, and any non-mathematical terms may be defined. Tasks can be reread during the student’s work time, and scribing may be provided for any non-writing or primary students. No coaching or directions can be given for how a task should be completed. A summative assessment

represent a student’s totally independent solution.__must__

**Note: Embedded Standards**

There are instances throughout this program where the underlying math concept in a Common Core content standard is "embedded" within a task, but the standard is not directly aligned to the task. A student *may* use the underlying math concept in an embedded standard to solve the problem but cannot be *required* to use that math concept, due to the open-ended nature of problem solving. These tasks should not be given as an assessment but rather used with students to practice a particular math concept or skill.

*Problem Solving for the Common Core* is a supplemental math program.

*Problem Solving for the Common Core* is not a “test prep” program, but rather a supplement to existing curricula. It is based on research that shows that students who engage in challenging and interesting work will perform at higher levels than those who do not.^{1} (31)

The performance tasks in this program were written according to Universal Design guidelines and developed to support teachers in implementing the Common Core State Standards for Mathematical Content and Standards for Mathematical Practice. This resource is intended to help teachers embed mathematical problem solving into classroom instruction and assessment. Both instructional tasks/formative assessments and summative assessment tasks are provided for every applicable Common Core content standard. Alignments to the Standards for Mathematical Practice are also included.

By publishing authentic problem-solving tasks, Exemplars material engages students and promotes mathematical reasoning, making mathematical connections and communication skills. Our Preliminary Planning Sheets are designed to support teachers as they reflect on the tasks they intend to use. Rubrics and student anchor papers (hallmarks of Exemplars) assist teachers in assessing student performance. Students can also use these to become thoughtful self- and peer-assessors.

^{1. Bryk, Anthony S., Jenny K. Nagoaka, and Fred M. Newmann, Authentic Intellectual Work and Standardized Tests: Conflict of Coexistence? (Chicago: Consortium on Chicago School Research, 2001).↩}

## 2. Instruction

The instructional/formative assessment tasks in *Problem Solving for Common Core* have been differentiated to include a “more accessible” and a “more challenging” version of the original problem.

The instructional tasks/formative assessments in *Problem Solving for the Common Core* have been differentiated to include a “more accessible” and a “more challenging” version of the original problem. This feature allows teachers to meet the needs of students at various levels as they explore and practice new math concepts. The summative assessment tasks in this resource are not differentiated. In order to meet the standard, students need to successfully complete a summative assessment without differentiation.

Individual PDFs of the task overheads may be printed for students at each of the three levels. Once printed, teachers may refer to the symbols in the header to identify the various levels.

**Symbol Key:**

○ - Represents the “original” version of the task.

Δ - Represents the “more accessible” version of the task.

☐ - Represents the “more challenging” version of the task.

Student work and anchor papers are provided only for the original version of the task.

Teachers can make additional alterations as well. For example, under the Common Core Domain Number and Operations, a task could be altered to meet the developmental needs of an individual student. If a kindergarten student only has number sense to 10, a blue block/red block patterning task asking the student to note the color of the 15^{th} block could be edited to the 10^{th} block. Teachers, however, should be careful not to alter the underlying concept(s) of the problem-solving tasks.

What is a mathematical connection? Why are they important? Why are they considered part of the Exemplars rubric criteria?

Exemplars refers to connections as “mathematically relevant observations that students make about their problem-solving solutions.” Connections require students to look at their solutions and reflect. What a student notices in her or his *solution* links to current or prior learning, helps that student discover new learning and relates the solution mathematically to one’s own world.

The National Council of Teachers of Mathematics defines mathematical connections in *Principals and Standards for School Mathematics* as the ability to “recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.” (64)

While the Common Core State Standards do not explicitly refer to the term “connections,” the idea is inherent in a number of the Standards for Mathematical Practice, including:

**MP.3 Construct viable arguments and critique the reasoning of others.**

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counter-examples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and — if there is a flaw in an argument — explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

**MP.4 Model with mathematics.**

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

**MP.6 Attend to precision.**

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

**MP.7 Look for and make use of structure.**

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x[sup]2[/sup] + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)[sup]2[/sup] as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

**MP.8 Look for and express regularity in repeated reasoning.**

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x[sup]2[/sup] + x + 1), and (x – 1)(x[sup]3[/sup] + x[sup]2[/sup] + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

**Connections and the Exemplars Assessment Rubric**

The Practitioner (meets the standard) level of the Exemplars Assessment Rubric defines connections as “Proper contexts are identified that link both the mathematics and the situation in the task.” Examples a student might make at this level include:

- Completing a collecting stamps pattern for seven days and stating that seven days is a week, or that the underlying pattern of collecting stamps is five per day.
- Recognizing that ten pennies is one dime.
- Noticing that an AB pattern from zero needs an even number to complete a repeat.
- Stating that a hexagonal table would allow more places to sit at a party than a square table.
- Recreating a problem by substituting different numbers for the ones in the problem.
- Continuing tables, graphs, patterns, etc.
- Identifying prime numbers, exponents, etc.

The Expert level of the Exemplars rubric calls on students to extend their resolutions by connecting to different mathematical strands. Examples a student might make at the Expert level include:

- A third grader (only having had instruction in perimeter) discovers that as the possible perimeters of a puppy pen approach a square, the larger the area becomes.
- Generalizing and proving a rule for any day. This might include making a table to show the number of birds seen each day for a week. The student states the day pattern is counting by one and the bird pattern counting by two. To achieve Expert, the student uses the found pattern to generalize the rule Day + Day = Birds and uses the function to find the total birds on a number of different days.
- Verifying a solution by solving the problem with a different strategy to document that an answer is correct. For example, a first grader uses a diagram to find how many ears are on six kittens and counts on the number of ears and concludes that six kittens have 12 ears. The student then creates a table to show six kittens and uses a counting-by-two pattern to arrive at the same answer and states, “I know I am correct because I found 12 ears two different ways.” (This same student could also make the Practitioner observations that 12 ears is a dozen, that the counting by two pattern starting at zero always results in an even number, that two ears are called a pair, or that adding a new kitten to the diagram results in 14 ears.)
- Generalizing and proving a rule for any number of plants and insects. The Practitioner might make a table to show the number of plants and insects studied for one week and then find the pattern for plants and for insects. To achieve Expert, the student would generalize the rules using algebraic equations: 2D + 3 = I and 2D = P; defining the variables: D is Day, I is Insect and P is Plant. The student would then use the function to find the number of plants and insects studied on any given day.
- Checking the reasonableness of a solution. A student may solve a task one way and then check the reasonableness of the solution by solving the task another way and verifying it. For instance, to find how many marbles were in a bag before the bag is passed around a table, a student might start solving with the sixth pass and move backwards through the steps until the first pass is reached, finding the number of marbles removed and the number of marbles left in the bag. After a solution is found, the Expert student might evaluate the reasonableness of the solution by either generating a second solution to the task or verifying the first solution. This time the student may start with the first pass, the number of marbles removed and the number of marbles left in the bag and move through the series of steps in the task until the sixth pass is achieved.
- Extending a task. For example, when finding the largest area of a pen that is built with 12 feet of fence a student might find the largest pen that can be made if the pen is rectangular and the pieces are not bent or split. The Expert might extend the problem by considering other options such as building the fence to include a side of a building or buying units of fence that can be bent into a circular shape.

**Guiding Questions**

Many students enjoy making connections once they learn how to reflect and question effectively. Below are a series of questions that students might consider as they are trying to identify connections:

- What could happen next if I add another …?
- Are there other mathematical terms I can use?
- Is there another way I can state my thinking? (5 pennies is a nickel, 100 centimeters is one meter, two eyes is a pair, a square is a rectangle, a trapezoid can look different from the red pattern block)
- Is the solution (all the work including the answer) reasonable?
- How is this problem like another problem I did, and what is the mathematical similarity?
- How is this mathematically like something that is in “real life” and how can I explain the mathematics?
- How can I verify that my answer is correct?
- Is there a general rule?
- Is there a mathematical phenomenon in my solution?
- Can I test and accept or reject a hypothesis or conjecture about my solution?

Students quickly gain an understanding of connections when their teachers use good question “leads” with problem-solving tasks and during class discussions. Some of these question starters could include organize, construct, identify patterns, interpret, compare, relate, draw conclusion, cite evidence, investigate, show and explain.

Listed below are some basic ideas for getting your students off on the right foot with Exemplars.

**Create a safe environment**

- Create a safe and inviting environment, where students feel comfortable taking risks to solve problems as well as to share strategies.
- Make problem solving a regular part of the routine by expanding problem-solving to other disciplines, and by encouraging students to pose problems, to ask probing questions, to research, and to seek solutions.

**Introduce your students to Exemplars rubric.**

- Rubrics are an important component of Exemplars. To be successful, many students need to first understand the rubric that is being used to assess their performance before it is implemented.
- Facilitate this new approach by introducing the idea of a rubric to your students by collectively developing some sample rubrics that address common areas of performance or self-evaluation, such as homework, lunchroom behavior, etc. For more suggestions, click here.
- Once they grasp this concept, you can introduce the Exemplars rubrics.

**Use student anchor papers to understand the rubric.**

- Each summative assessment task includes student work samples and scoring rationales.
- To get students accustomed to the Exemplars assessment rubric and the expected performance criteria, give them an Exemplars task to practice, then share the anchor papers on overhead slides.
- Discuss and critique the anchor papers, and score them with the rubric as a class.

**Conduct group brainstorming and/or conferencing sessions.**

- If your students are new to Exemplars, you might try setting aside time for students to share ideas, strategies and ask questions of each other before solving a task.
- The brainstorming might include identifying different approaches students might use, information that needs to be known, and types of mathematical representations they would use to communicate results.

**Use differentiated tasks.**

- One task may not work for all students.
- That is why Exemplars instructional tasks include a more accessible and more challenging version.

**Scheduling**

Before you feel confident integrating problem solving you might select a time to focus on it, such as, devoting one class per week with/without cooperative groups, assigning problem-solving homework that involves families, or creating a time for “noisy solving” (as you might have for “silent reading”).

**Model expectations**

Articulate, post, and model expectations for individual responsibilities, group work, performance standards and criteria, etc.

**Process**

- Work on specific skills, such as, identifying appropriate operations or needed information, creating graphs or tables, choosing problem-solving strategies.
- Revisit the same types of problems to reinforce use of certain strategies.
- Use math journals and letter writing to help students communicate their thinking.
- Conference with students, use interviews between students, require students to keep a problem-solving portfolio, and to self-assess.
- Guide student work with “reminder” questions, such as:
- I can’t follow this part. Can you make this more clear?
- I see an error in reasoning. Can you find and correct it?
- Your reasoning is excellent. Double-check your calculations.
- You omitted a part of the problem. Can you find it?

Listed below are a set of problem-solving steps that we recommend teachers use with their students. These steps can be posted and taught to students when using Exemplars tasks for instructional purposes.

When beginning problem solving with students it is important to model this process and go through it a number of times as a whole group until each student feels comfortable using them independently. Teachers have also printed these steps for students to keep in their problem-solving folders and refer to each time they start a new problem.

It is always a good idea to teach students how to use different problem-solving strategies for solving problems (step #4 below) such as drawing a picture, working backwards or solving a simpler problem. It is also important to keep in mind that students have many of their own creative strategies for problem solving. Encourage them to use these strategies.

**Problem-Solving Steps**

- Read the problem.
- Highlight the important information.
- What do you know and need to find out?
- Plan how to solve the problem: what skills are needed, what strategies can you use, and what ideas will help you?
- Solve the problem.
- Draw and write about your solution and how you solved the problem.
- Check your answer.
- Share a connection or observation about this problem.
- Be sure to provide your students with a copy of the rubric or use one of Exemplars student rubrics. We publish student rubrics that are written in language more easily understood by younger students. Rubrics may be easily accessed in the “Downloads” section below.
- What instructional materials/technology manipulatives will you need?

Listed below are some suggestions and resources for teaching your students how to make math representations.

- Model different representation for students by doing a problem together and attempting to make appropriate graphs, plots, charts, tables, diagrams and models to accompany the response to the task.
- After students complete their solutions, have them go back and zero–in on their representation. Is it labeled? Titled? Accurate? Appropriate? Etc.
- Feature a representation of the week, and ask students to bring in samples of representations to share from magazines, newspapers, etc.
- Have students assess each other’s representations.
- Create human representations using your students (human Venn diagrams, graph, etc.), and/or have students use their own bodies to show what a line graph might look like, a Venn diagram, etc.
- Play “What’s Missing.” Show students different representations with missing or inaccurate parts. Have students guess “what’s missing” and fix it!
- Ask students to examine representations with no labels and interpret what the data might mean.
- Ask students to bring in mathematical representations that their parents use in their work place.
- Have students experiment with one representation. What happens when you change the scale? Use color? Change information to percents vs. actual numbers? What impact do changes have on the message of the representation?
- Allow students to experiment with different representations using computer programs. Use this as a lead in to a discussion on appropriate vs. inappropriate representations.

**You can learn more about different math representations in this document.**

**You may view a list of grade-level appropriate math representations defined by the Common Core for K–5 in this document.**

Listed below are some ideas and activities for promoting the use of math language in the classroom.

- Plan with all teachers of mathematics in your building the mathematical language and notation that will be required of students at each grade level.
- Give tasks that elicit mathematical language beyond computation.
- Model use of mathematical language and notation in other subject areas.
- Use the templates to assist in the planning instruction of mathematical language.
- Give students highlighters to illustrate the use of their mathematical language. You might use one color to highlight language beyond computations of task and another to highlight other mathematical language beyond computation.
- Give students transparencies to place on their work and use Vis–à–vis® markers to highlight work. (Have clipboards available to clip the overhead and work together.)
- After students have had an opportunity to work a problem, have them brainstorm as a class all the mathematical language and notations that might be appropriate to use when communicating their solutions to that problem. Leave the words posted for easy access.
- Create class dictionaries by content standard as a reference. Have students enter the term and then its definition. Students can also show examples of the term’s use, draw a picture of the term, etc. Use a notebook to create this class or personal reference. By using a notebook, you can rearrange content standard pages in alphabetical order as new terms are discovered.
- Add math words to spelling lists.
- Play Concentration or Jeopardy using cards with words and definitions.
- Share the benchmarks with students.
- Have a math–word–of–the–week.
- Wear a math word and be ready to educate others as to its meaning.
- Collect mathematical language found in magazines, newspapers, etc., and record them by content standard in a notebook. You may wish to use the notebook format here as well.
- Interview people representing a variety of occupations and record the mathematics they use. Students will soon discover that most occupations involve math. Begin with people in your school, the cook, janitor, secretary, etc.
- Have a central display in the school to post unusual math terms and ask students to find their meaning.
- See how long a student can “survive” without using mathematical language (i.e. Telling time, what day of the week, or number of recess milks, etc.). Try to record all the events in a “typical” school day that require and understanding of mathematical language.
- Keep classroom charts of various standards and highlight the vocabulary learned.
- Have quick quizzes.
- Have a Mathematical Language Bee similar to a spelling bee.
- Put mathematical language on index cards and their definitions on other index cards. Give each student a card and ask them to find their partners.
- Offer word searches and crossword puzzles.
- Write a mathematical word on graph paper and find its area and perimeter.
- Provide opportunities for students to peer conference and then share their understanding of mathematical language.

**Download and share these ideas with your colleagues.**

## 3. Assessment

Each summative assessment task in this program includes student anchor papers and scoring rationales.

Anchor papers provide examples of student work that meets or does not meet a Common Core standard. Each scoring rationale explains why.

The summative assessment tasks in this program include student anchor papers at four levels of performance: Novice, Apprentice, Practitioner (meets the standard) and Expert. Exemplars anchor papers are accompanied by a set of scoring rationales that describe why each piece of student work is assessed at a specific performance level. Rationales are given for each of the five criteria in Exemplars assessment rubric (Problem Solving, Reasoning and Proof, Communication, Connections, Representations). The anchor paper is then given an “overall” assessment score or achievement level.

Anchor papers and scoring rationales are designed to provide guidelines and support for teachers as they assess their own students’ performance in problem solving. They can also be shared with students as examples of what work meets the standard and why or as a basis for self- and peer-assessment.

In many cases, there is more than one anchor paper associated with a level of performance. These are intended to demonstrate different strategies a student might use or different misconceptions a student might have.

Exemplars offers rubrics that are designed to help both teachers and students understand what’s required to meet the standard.

Exemplars may rubrics may be assessed from your dashboard or the "Classroom Resources" section below.

**Exemplars Assessment Rubric**

An important component of this program is the Exemplars Assessment Rubric. Our scoring rubric allows teachers to examine student work against a set of analytic assessment criteria to determine where the student is performing in relationship to each of these criteria.

This assessment tool is designed to identify what is important, define what meets the standard and distinguish between different levels of student performance. The Exemplars rubric consists of four performance levels — Novice, Apprentice, Practitioner (meets the standard) and Expert— and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation). Our rubric criteria reflect the Common Core Standards for Mathematical Practice and parallel the NCTM Process Standards.

**Exemplars Student Rubrics**

Rubrics can provide students with valuable information about what is expected and what kind of work meets the standard. They can also be used as a basis for self- and peer-assessment. In addition to our assessment rubric, Exemplars has also created one for students called the Jigsaw Rubric.

A excellent description of how to introduce rubrics to your students resides on Exemplars web site: http://www.exemplars.com/resources/rubrics/introducing-rubrics-to-students.

Exemplars assessment rubric supports the Standards for Mathematical Practice and provides teachers with clear guidelines for evaluating student work and providing meaningful feedback.

The student work in *Problem Solving for the Common Core* is assessed analytically. That is, each criterion of the Exemplars Assessment Rubric — Problem Solving, Reasoning and Proof, Communication, Connections and Representations — is taken into consideration individually when assessing the work. For each criterion, the work is assessed as Novice, Apprentice, Practitioner (meets the standard), or Expert.

The work is then given an Achievement Level Score. In coming to the overall assessment (achievement level), a paper cannot receive a score higher than the lowest score on any of the five criteria. Thus, if a student does not have any representation on her or his work, the “Representation” score would be Novice and the achievement level would be assessed at Novice. If a student has an Apprentice score in “Communication” and all other scores are Practitioner, the student’s achievement level would be assessed at Apprentice. In order to meet the standard, a student has to achieve the Practitioner level or above for each of the five criteria. Because the Exemplars rubric is performance based, it is not possible to take a mode or mean “grade” from the assessed criteria.

While many schools and districts require an overall achievement level for a task, others do not. What is important is to know where the student stands on each criterion and what the next steps are for that student.

Below are sample scoring boxes used to assess a student’s work. (Throughout *Problem Solving for the Common Core*, we have included completed assessment boxes at the top of each piece of student work.) Each box addresses the criteria found in the Exemplars rubric and the corresponding scoring rationales. The sample scoring boxes featured below show scores that would merit the following achievement levels (respectively): Novice, Apprentice, Practitioner, Apprentice, Novice, Apprentice and Expert.

**Key:**

Assessment Rubric Criteria | Achievement Level |
||

P/S | Problem Solving | N | Novice |

R/P | Reasoning and Proof | A | Apprentice |

Com | Communication | P | Practitioner |

Con | Connections | E | Expert |

Rep | Representation | ||

ACLV | Achievement Level |

**Sample Scoring Boxes:**

P/S | R/P | Com | Con | Rep | ACLV |

P | P | N | P | A | N |

P/S | R/P | Com | Con | Rep | ACLV |

P | P | A | P | P | A |

P/S | R/P | Com | Con | Rep | ACLV |

P | P | E | P | P | P |

P/S | R/P | Com | Con | Rep | ACLV |

E | E | E | E | A | A |

P/S | R/P | Com | Con | Rep | ACLV |

P | P | A | N | N | N |

P/S | R/P | Com | Con | Rep | ACLV |

A | P | P | P | P | A |

P/S | R/P | Com | Con | Rep | ACLV |

E | E | E | E | E | E |

***Exception to the Rule**

The National Council for the Teachers of Mathematics has suggested that the “Connections” criterion can be demanding for students because it requires more cognitive thinking and reflection. (For more information and tips on this subject refer to the section “Understanding Mathematical Connections.”) Therefore, there is one exception to the Achievement Level Score. If a student has all Apprentice scores or above but a Novice in “Connections,” the student may receive an achievement level score of Apprentice. The student cannot be a Practitioner (or Expert) because not all of the criteria scores meet the standard.

An example of this can be seen below:

P/S | R/P | Com | Con | Rep | ACLV |

P | P | P | N | P | A |

P/S | R/P | Com | Con | Rep | ACLV |

P | P | A | N | P | A |

The rationale behind this decision is that if a student has correct problem solving and reasoning as well as communication and a correct representation but did not make a mathematical connection, it would be very difficult to assign the student an achievement level of Novice, because the thinking and the solution are correct. This “exception” to the rule is well received by many schools that are looking for a way to give an overall assessment score to a student’s problem-solving piece.

Throughout the school year, Exemplars encourages teachers to keep two student portfolios. The first could be either a pocket folder or binder...

Throughout the school year, Exemplars encourages teachers to keep two student portfolios. The first could be either a pocket folder or binder that contains a student’s instructional tasks/formative assessments. These “working portfolios” should be placed in the classroom where students can access them on a regular basis. The second should be a file that the teacher keeps to store each summative assessment problem-solving task that a student completes.

The working portfolio allows teachers to assess what the student knows using four guiding lenses.

- What do I know this student knows?
- What does this student need to practice?
- What does this student need to relearn?
- What is this student ready to learn (do next)?

Instructional tasks/formative assessments are viewed as opportunities for students to learn new mathematical strategies, vocabulary and notation and representations. Students can also explore mathematical connections and self-assess their solutions. These tasks may be done alone, in pairs, in groups or as a whole class. Direct instruction may also be used to question and support classroom discussion around the underlying mathematical concepts in a problem.

Teachers should use formative assessment tasks to observe and support student understanding. As part of this process, conferencing and editing can occur and students can revisit their work as often as necessary. Teachers can use similar tasks throughout a unit of study to give a student multiple opportunities to use new learning in her/his solution and to gain independence in arriving at a correct answer.

In contrast, summative assessment tasks are given at the end of a unit of study. Summative assessment tasks are identified throughout *Problem Solving for the Common*. These tasks include a set of anchor papers and scoring rationales.

In order to achieve a true assessment of what the student understands and is able to do, in words of the Common Core, there should be a wait time of at least one day between the last formative assessment and the summative assessment. A similar assessment task may also be given to students much later in the year if a teacher wants to spiral back to determine how much learning is retained.

Summative assessment tasks can be read to the students, and any non-mathematical terms may be defined. Tasks can be reread during the student’s work time, and scribing may be provided for any non-writing or primary students. (For more information on scribing, refer to the section “Scribing at the Primary Level.”) No coaching or directions can be given for how a task should be completed. A summative assessment __must__ represent a student’s totally independent solution.

**Portfolio Components**

A student’s working portfolio should include:

- Class pieces
- Scaffold pieces
- Homework pieces
- Edited pieces done after class instruction in the mathematics/problem-solving strategy of the task
- Conferenced pieces with directed editing
- Pieces used as a class to learn strategies, vocabulary and representations
- Pieces used to help students learn to organize and write their solutions
- Tasks used as direct instruction to learn the criteria of the scoring guide
- Tasks for independent student practice

A summative assessment portfolio should include:

- a student’s
__independent__problem-solving work that demonstrates what he or she knows and is able to do

Teachers can capture the mathematical reasoning of non-writing students by ‘scribing” their oral explanations.

A student’s formative assessment and summative assessment portfolio should be an accurate reflection of what the student knows and is able to do. Since many primary students are unable to write their own responses to a problem-solving task, scribing is an appropriate accommodation.

Students are encouraged to use diagrams, labels, ask for help with spelling, and so on to show their thinking. Support can be given to students by having someone translate their solution. Sometimes videotaping the student’s presentation of her or his solution is possible as well as taking photographs of the student’s work as she or he solves the problem. Students can then add captions to these presentations. Other students may wish to audiotape their solutions.

It is appropriate to provide encouragement and ask generic questions to assist a student in clarifying her or his ideas. For example, “What is your plan?” and “Tell me about your thinking, answer or solution” are generic. “Make a chart” and “Your solution is incorrect” are not generic.

If a teacher feels that scribing is needed for a complete understanding of the student’s thinking, time should be allocated for this. When a teacher scribes for a student, they should mark the scribing with the initials of their first and last name. You will notice in *Problem Solving for the Common Core* that all scribing has been marked with the initials “AZ.” for the first and last letter of the alphabet, so there are no identifying factors and student and teacher anonymity is maintained. For this same reason, all references to student name, school or classroom have been removed from each piece of student work.

Another aspect of scribing teachers should be familiar with, is knowing when to use quotation marks, parentheses, and brackets. Quotation marks are used to quote what a student says. For example, “I made 1, 2, 3 puppies. I got 1, 2, 3, 4, 5, 6 ears. I got more ears than puppies.” Parentheses are used to scribe what a student does, for example, (Student pointed to each shape correctly). Brackets are used to scribe what a teacher does, for example [re-read the task].

Scribing is also suitable for elementary- and middle-level students with physical or medical issues. More information about the scribing process and suggested questioning prompts can be found at: http://www.exemplars.com/resources/student-communication/helping-primary-students-communicate

Below are some suggestions for assessing students’ work.

- Indicate on the students’ papers the occasions when students work together versus individually solving a problem.
- Even though students work in groups to research, investigate, gather data, etc., they should each document their own solution for the problem.
- Students can work over a period of time (days) to solve problems. During that time they can add to or change their solutions. You should encourage the “step back from the problem and look at it another time” approach.
- Make sure students include all drafts of their work. Give credit in your grading system for the organization and inclusion of all work.
- Positive redrafting comments:

The following are some comments that you can write on a student’s work and then ask them to go back into their work and revise. However, your comments should be included on the finished piece and all drafts as well.

- I cannot follow what you are doing; can you make it clearer?
- I see an error in your reasoning; can you find it?
- Excellent reasoning... check your calculations.
- It looks to me as if you omitted a key piece; can you find it?

This strategy is particularly pertinent to complex problems with multiple parts.

- Go over a problem with students and then give them a problem at a later time that would use similar strategies or address the weaknesses that you saw in their original solution.
- Once there has been class discussion on the problem ... no revising should be done.
- Teaching samples of student work should not be used as assessment samples of student work.
- (During assessment, teacher intervention should be noted by initialing the student’s work)

Below are some examples of effective feedback that you can give your students when working on Exemplars tasks.

__Understanding__

- Wow! It is clear that you have a complete understanding of the problem.
- You show that you understand and addressed the important parts of the problem.
- Read the problem again. You need to use 10, 5 and 2 to show different ways to equal 20.
- You found lots of ways to equal 20 but not using the numbers from the problem.
- Can you find more solutions?
- What is the solution you are recommending?
- You show that you understand some parts of the problem.

__Reasoning__

- How did you count to 10 on the 5th row?
- How did you come up with your answer?
- Your strategy is effective.
- I am confused. I can’t tell how you arrived at your solution.
- The number sentence matches the cubes and tally marks.
- Where are the equations to show your work?
- You extended the problem by explaining what you would do with the extra money.
- You need to justify your solution.
- Can you make connections to other tasks with this concept?

__Accuracy__

- Your solution is correct and matches the pictures.
- You need to give the total number of flowers on each row.
- Be careful with your counting. Your solution is incorrect.
- Organize your work so each number sentence will be clearly shown.
- You need to organize the way you go about finding solutions.
- You found all of the arrays but need to label them correctly and organize your work.

__Communication__

- You explained that you were counting by 2’s!
- Good use of math language in your explanation.
- Use numbers and math words in your explanation.
- You did a great job with your number sentences but they do not match the problem.
- Your explanation needs to tell the different ways to use the numbers from the problem.
- Great use of math language in your explanations.
- Your communication gives your recommendation and explains reasons for your selection.
- Your explanation needs to give the details about the monies spent, justify the costs, and state how much money you have left.
- You need to explain how the concept works in this problem and why your solution makes sense. What would cause your results to be different?
- You need to explain how the concept works in this problem instead of discussing only the procedures you followed to arrive at your equations.
- Use math terminology when you explain why your solution makes sense.

**Download these examples, so that you may easily refer to them. **