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Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the First Grade Level

Monday, July 21st, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 2 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 2: Observations at the First Grade Level

The second anchor paper and set of assessment rationales we’ll review in this series is taken from a first grade student’s solution for the task, “A Birdbath.” In this piece, you’ll notice that the teacher has “scribed” the student’s oral explanation. This practice was also used with the Kindergarten task that was published in the first blog. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

“A Birdbath” is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 1.OA.6. This task would be used toward the end of the learning time allocated to this standard. “A Birdbath” provides first grade students with an opportunity to apply different strategies to find the sum of addends six and 14 by decomposing six into five and one and decomposing 14 into 10 and four, or by finding the sum of six and four and adding that sum to 10. The student can use counters, ten frames, a Rekenrek, number lines or a tally chart to support her/his numerical thinking.

First Grade Task: A Bird Bath

Leah counts the birds that came to her birdbath. In the morning, Leah counts six birds that came to her birdbath. In the afternoon, Leah counts fourteen birds that came to her birdbath. Leah says nineteen birds came to her birdbath. Is Leah correct? Show all your mathematical thinking.

Common Core Alignments

  • Content Standard 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Kindergarten Level

Thursday, July 10th, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 1 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress towards being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in-hand while reviewing a piece of student work.

Blog 1: Observations at the Kindergarten Level

The first anchor paper and set of assessment rationales we’ll review in this series is taken from a Kindergarten student’s solution for the task, “Boots.” You will notice that the teacher has “scribed” the student’s oral explanation. This method allows teachers to fully capture the mathematical reasoning of early writers.

“Boots” is one of a number of Exemplars tasks aligned to the Counting and Cardinality Standard K.CC.5. This task would be used toward the end of the learning time allocated to this standard. Prior to “Boots” being given, students have already completed a number of tasks with questions that state, “How many ears?”, “How many shoes?”, “How many balloons?”, etc. “Boots” gives students an opportunity to bring a stronger understanding of the concept how many to their solution.

Kindergarten Task: Boots

Five students wear boots to go outside for recess. When the students come in from recess they must put all boots on a rubber mat to dry. The teacher counts seven boots on the mat. The teacher thinks some boots are missing. Is the teacher correct? Show and tell how you know.

Common Core Task Alignments

  • Content Standard K.CC.5: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle or, as many as 10 things in a scattered configurations: given a number from 1-20, count out that many objects.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6

Bridge the Gap Between Common Core, Your Classroom and the Real World

Tuesday, March 25th, 2014

Written By: Elaine Watson, Ed.D., Exemplars Math Consultant

To most nonscientists, mathematics is counting and calculating with numbers. That is not at all what a scientist means by the word. To a scientist, counting and calculating are part of arithmetic and arithmetic is just one very, very small part of mathematics. Mathematics, the scientist says, is about order, about patterns and structure, and about logical relationships.

By, Keith Devlin, Life by the Numbers

The word “scientist” above could be replaced by the word “doctor, lawyer, engineer, accountant, CEO, military officer, government worker, homeowner, citizen …” In other words, anyone who uses numbers to make decisions needs to look beyond the calculations and be able to discern what the numbers are telling them.

Math textbooks have developed “word problems” in response to the question so often asked by students as they learn to follow algorithms and solve equations in order to find the correct answer: “When am I ever going to use this in real life?” The question is often answered by the jaded teacher, who has heard it from each new generation of students in this way: “You’re going to use it on the test!” This answer seals the students’ belief that what they learn in math class is not applicable to the real world, but merely a set of exercises that need to be done in order to pass the course.

Past mathematics standards documents have focused on the hard content, the factual and procedural content students should learn, which is of course important. The focus on the soft content, the habits of mind and thought processes that are practiced by students when solving a problem, has traditionally either been relegated to the end of the standards document as an afterthought or omitted altogether.

The Common Core State Standards in Mathematics (CCSSM) recognize that the soft content, the practices students used to approach and solve a mathematical task, are as important as the hard standards. Soft does not mean unimportant. In the same way that a computer (hardware) cannot function effectively without appropriate software, CCSSM Content Standards cannot be accessed and used without students using the supporting Practice Standards.

The Practice Standards have to be learned, and practiced, alongside the Content Standards, but because of the “soft” nature of Practice Standards, they are harder to pin down. Phil Daro, one of the three authors of the CCSSM, describes the Practice Standards as “the content of a student’s mathematical character.”

It is important to remember that it is the students who practice the Practice Standards. Teachers should model the practices in their instruction, but more importantly, teachers should explicitly plan lessons that include teacher pedagogical moves, student activities and tasks that will elicit the Practice Standards in students.

The tasks created by Exemplars are excellent examples of rich problem-solving that naturally elicit the Practice Standards. Below we will look at the Grade 2 task “Barnyard Buddies” and discuss how it meets each of the eight Mathematical Practice Standards as well as content standard 2.OA.A.1.

Barnyard Buddies

A farmer has 8 cows and 10 chickens. The farmer counts all the cow and chicken legs. How many legs are there altogether? Show all your mathematical thinking.

CCSSMP.1 Make sense of problems and persevere in solving them.

There is no hint in this task as to how to go about solving the task. It is not a generic type of problem with which the student has had previous experience. The student must make sense of the task before being able to develop an approach for solving it. Some approaches may be more efficient than other approaches.

CCSSMP.2 Reason abstractly and quantitatively.

In order to solve the problem, students will need to use an approach in order to organize their thinking and keep track of the quantities involved.  One approach is to draw 4-legged animals and 2-legged animals and count.  Another approach is to create a table. Both of these approaches have created an abstraction (mathematical model) of the situation. The student work below shows how two students modeled the problem.

Student 1 created abstractions of the chickens (square with 2 legs) and cows (circles with 4 legs).

Student 2 simply drew the legs without the bodies, which was a step toward greater abstraction. She or he then went on to use an even more abstract approach by noticing that there was a pattern and deciding to use a table. This student work is also a good illustration of Practice Standard 8: Look for and express regularity in repeated reasoning.

CCSSMP.3 Construct viable arguments and critique the reasoning of others.

This task will elicit a lot of different ideas as to how to approach it. Students will need to persuade others as to why their approach will work the best. In order for students to exhibit this practice standard, a classroom culture needs to be developed where student discussion of their work is the norm. The teacher’s role is to encourage the discussion and question and guide as needed.

CCSSMP.4 Model with mathematics.

In order to solve this task, students will need to go through the steps of the Modeling Cycle. They formulate an approach, compute, and then check their answer to see if they have correctly counted all 8 cow’s legs and all 10 chicken’s legs. If their answer makes sense, they report it out. If it doesn’t make sense, they need to go back through the cycle, determining where they went wrong. Were their pictures correct? Did they have the right number of each type of animal and the correct number of legs on each type of animal? If they used a table, did they skip count correctly by 2 and by 4? Did they add correctly? The cycle continues until they are satisfied that their result is a viable answer for the problem.

CCSSMP.5 Use appropriate tools strategically.

Tools are not necessarily physical, like a ruler or a calculator. On this problem, the student’s drawing or table can be considered a tool, since it helps make sense of and solve the problem.

CCSSMP.6 Attend to precision.

Precision is needed in the drawings or table, in the counting, and in the addition. Students also need to be precise in labeling their answer. If a student answers with only a number without the label “legs,” they are not attending to precision.

CCSSMP.7 Look for and make use of structure.

The student needs to visualize the structure of the situation. In this case the structure involves a given number of animals with 4 legs and a given number of animals with 2 legs. That structure will inform how the student approaches and solves the problem. If the student notices that 4 consists of 2 copies of 2, this will help in counting, since he or she should be proficient at counting by 2s.

CCSSMP.8 Look for and express regularity in repeated reasoning.

The student is repeatedly adding 2 or adding 4 for a given number of times. The student can count by 2s while pointing to each chicken. For the cows, students can either count by 4s, or they can count by 2s when pointing to the cows and touching each of the two pairs of legs on every cow.

Support for Common Core Content Standards

In addition to eliciting the Common Core Practice Standards, Exemplars tasks are also aligned Common Core Standards for Mathematical Content.

To solve “Barnyard Buddies,” students need to model the situation by using some type of drawing to represent the 10 chickens and the 8 cows as well as the number of legs on each animal.  Creating such a representation is an early form of algebraic thinking. After developing the pictorial model, students then need to count the total number of legs. Most students will skip count by either 2 or 4. Some students may organize their counting by making groups of 10 (2 cows and 1 chicken or 5 chickens).  Whichever approach students use for counting, they are recognizing a numerical pattern, which is also an underpinning of algebraic thinking.  This type of thought process is best matched by the Common Core Domain Operations and Algebraic Thinking.  Within this Domain, “Barnyard Buddies” aligns with the cluster, Represent and solve problems involving addition and subtraction. The specific content standard addressed is 2.OA.1.

2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Download a copy of the “Barnyard Buddies” task complete with anchor papers and scoring rationales to try with your students!

 

How Problem Solving Prepares our Kids for Success Beyond the Classroom

Tuesday, February 25th, 2014

By: Ross Brewer, Ph.D., Exemplars President

Students actively engaged in Exemplars.

Once upon a time, Americans might have been content to live in a  community much like Garrison Keillor’s Lake Wobegon, “where all the  children are above average.” That’s because historically American  kids, and our schools, were above average; however, for decades,  America’s education system has been losing ground internationally.  In an era when knowledge-based competition comes from every corner  of the globe, average is no longer good enough for American students  or workers. American jobs are becoming increasingly vulnerable as  technology becomes more sophisticated and overseas workers better  educated. Both of these are happening at an accelerated rate.

Because of the disruptive changes occurring in our knowledge-based economy, the good jobs—jobs that pay high wages—that will survive    are those that require higher cognitive skills.

For years, economists and educational experts have been warning  about the impact that the increasingly rapid development of  technology is likely to have on unskilled workers. MIT professors  Brynjolfsson and McAfee offer this stark summation of current technological trends:

Technological progress is going to leave behind some people, perhaps even a lot of people, as it races ahead. … there’s never been a better time to be a worker with special skills or the right education, because these people can use technology to create and capture value. However, there’s never been a worse time to be a worker with only ‘ordinary’ skills and abilities to offer, because computers, robots and other digital technologies are acquiring these skills and abilities at an extraordinary rate. (The Second Machine Age, p11)

Unfortunately, as the most recent international reports make clear, while American students have made incremental improvements on international tests of problem solving, the position of the United States continues to slip as other nations advance more rapidly.

One of the reasons for weak student performance on international tests has historically been the absence of a unified set of standards across all 50 states. The mishmash of state standards, in addition to lacking focus and coherence, have given prominence to simple skills that are easily measured while minimizing problem solving and communication, skills employers identify as being important.

Why is problem solving so important? At a mathematical level, problem-solving skills are critical to the development of understanding more advanced mathematics and the ability to perform other complex tasks. This in turn creates the foundation to solve problems in the real world. Indeed, among the essential employee skills identified by employers are the ability to solve problems, process information and analyze quantitative data and to communicate verbally and in writing.

Thankfully, we are in the midst of a transformation whose aim is to close our country’s global competitiveness gap and prepare our children for a global economy.

In recent years states have moved to address weaknesses in problem solving and the associated process skills. The Common Core State Standards (CCSS) is the most wide-ranging effort, and have been adopted by 45 states.

In addition to the Content Standards, the CCSS give at least equal importance to a set of process standards, the Standards for Mathematical Practice. These eight process standards describe ways in which students are expected to engage with the content. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. They emphasize the problem solving, reasoning, analytical and communication skills and are given equal prominence at each grade level along with the Content Standards. Even states that are not participating in the CCSS are prioritizing sets of process standards.

How Exemplars Supports Problem Solving

So how does Exemplars tackle the problem-solving imperative facing today’s teachers and students? We were founded more than 20 years ago with a single mission: to engage students’ interests and develop their abilities to problem-solve in today’s world. From the beginning, the focus of our mathematics material has been on the following process standards: problem solving, reasoning, communication, representation and connections. Exemplars tasks are designed to help teachers instruct students in mathematical problem solving and allow students to demonstrate their understanding of problem solving.

Our latest K–5 material, Problem Solving for the Common Core, offers teachers a supplemental resource to help develop their students’ problem-solving and critical-thinking skills. Our real-world tasks, rubrics and anchor papers are designed to encourage:

  • Students’ problem-solving abilities
  • Students’ use of representations and making the link between the problem and the underlying mathematics
  • Students’ ability to communicate mathematical thinking and provide reasoning and proof to justify their answer or approach
  • Students’ application of appropriate mathematical language and notation
  • Students’ self-assessment skills
  • Formative assessments, which allow teachers to understand how their students are doing and to adjust their instruction to improve performance
  • Engaging summative assessments, which allow teachers to evaluate if their students have met the standard

In short, problem solving is at the core of everything we create at Exemplars. You can try our new material with your students by signing up for a free 30-day trial or by downloading sample tasks. Let us know what you think.

A Common Core “Must Read” Paper

Friday, November 30th, 2012

By: Ross Brewer, Ph.D., Exemplars President

Jay McTighe and Grant Wiggins have written a “must read” paper – “From Common Core Standards to Curriculum: Five Big Ideas,” in which they offer key ideas to guide the work of transforming the Common Core Standards to a functioning curriculum in a school or district. The paper highlights some of the important misconceptions that readers bring to the Common Core and focuses on important processes that will lead schools and districts to creating an effective curriculum that actually embraces the Common Core Standards.

You may download a copy of this important paper here.

Using Exemplars for School Improvement

Tuesday, October 30th, 2012

By Tammy Krejcarek, Assistant Principal, Virginia

Several years ago, I was hired as a math specialist to help a school entering school improvement status. Our math scores were the lowest in the county, and our disparity gap was over 30 points. One of the first changes we made was to implement Exemplars. I began by training my teachers on how to lead students in math talk while sharing their various strategies for solving the problems. I found the anchor papers were a great place to start when introducing Exemplars to students, as they promote that rich discussion you are looking for. The built-in rubrics teach students how to self-evaluate their progress from one Exemplars task to another. The rubrics also offer teachers a tool for providing timely and meaningful feedback to students. One of the key benefits of Exemplars is that students don’t have to get to the correct answer in order to be successful or to stretch their thinking. The dynamics of students sharing and discussing their thought processes with one another is what’s so invaluable — it is NOT always about the answer; it’s about the process. Exemplars allows students to explore strategies without the pressure of getting a “bad” grade.

We started doing Exemplars school wide every Friday during math time. Depending on the focus, we worked as a whole group, in smaller groups, as partners and sometimes individually. Our students quickly learned from one another how to represent their thinking with pictures AND words, as well as how to create tables and diagrams. These strategies helped students organize information into a format they could understand without being overwhelmed. Because Exemplars tasks are based on real-world situations, they provided relevant and engaging context from which the students could make meaning. The organization of the tasks into strands and concepts made it easy for teachers to correlate Exemplars with their mathematics lessons. The tiered levels allowed for differentiated instruction to ensure the success of ALL students.

That first year we implemented the program, our math scores increased by over 30 points to well above passing while decreasing the disparity gap to within 10 points. That was five years ago, and Exemplars is still thriving. Based on this success, most schools in the district are now implementing Exemplars into their mathematics program.

This year, I am at a new district that is excited to begin using Exemplars in their buildings. With the increasing rigorous demand of high-stakes testing, Exemplars is a “must-have” component to any mathematics program. I have been in education for over 18 years and have seen programs come and go. Exemplars is one of the few initiatives that has proved effective time and time again!

Exemplars New Math Samples, K-8

Friday, September 7th, 2012

Our real-world performance tasks are differentiated and aligned to the Common Core Content Standards as well as to the Standards for Mathematical Practice. The anchor papers have been assessed using the Exemplars rubric.

New Samples:

Please share these with your colleagues. We suggest trying our problem-solving tasks in your school and discussing the student work as a team to see how your students approach them.

Have a great start to the school year!

Using Anchor Papers to Help Teachers and Students Understand the Common Core

Wednesday, August 22nd, 2012

By: Ross Brewer, Ph.D., Exemplars President

Assessing what our students know and are able to do, where they stand with regard to meeting the standards, and how teaching and learning activities might be improved are among the most common uses for evaluating student work. Key to this is creating sets of anchor papers. With the new standards and learning expectations outlined in the Common Core, anchor papers can be a useful tool for helping your teachers and students see and understand what meeting the new standards will “look” like in their classrooms.

What are anchor papers?

Anchor papers are examples of student work at different levels of performance that, along with rubrics, guide formative and summative assessments. Schools and districts can either build their own collections of anchor papers over time or reference examples like those provided by Exemplars.

How can they help?

In addition to identifying where students are in terms of meeting a particular standard, anchor papers can be examined as a way to understand the learning opportunities we are, and are not, giving our students. These can also be used to train school and district assessment teams as well as evaluate how accurately and consistently teachers are assessing students. One way to do this is to ask teachers to assess previously assessed work and compare their scores to the “approved” scores. There are guides and protocols for these types of activities, which are, no doubt, the most important uses of student work. For specific examples and to learn more, visit the Looking at Student Work Web site.

Becca Lindahl, formerly the School Improvement Coordinator for the Diocese of Des Moines Catholic Schools, describes her diocesan’s professional development “scoring” days in the following manner:

Our diocesan’s grades four and eight scoring days are some of the best professional learning we do. Teachers, with their scorers’ hats on, learn about students’ math thinking. At the end of the day, we turn back into teachers and discuss what the data is telling us and how we can perhaps make instructional decisions from the data.

This technique can be used with teachers, schools and districts.

There are many effective ways to use anchor papers.

What does meeting the standard look like at my grade level?

Written standards and rubrics define these expectations, but student work samples help make them concrete. Having teachers analyze student work from several grade levels can answer the question “Where did my students come from and where are they going?” An example of this can be seen in the Exemplars task, Marshmallow Peeps, which provides student work samples from grades: two, four, six and at the high school level.

 This technique can be used with teachers, schools and districts.

Solving problems and studying previously solved problems.

A report published by the U.S. Department of Education titled Organizing Instruction and Study to Improve Student Learning states that students learn more by alternating between studying problems that have already been solved and solving their own problems, as opposed to just solving problems. (NCER 2007-2004, U.S. Department of Education, available online from the Institute of Education Sciences)

A large number of laboratory experiments and a smaller number of classroom examples have demonstrated that students learn more by alternating between studying examples of worked-out problem solutions and solving similar problems on their own than they do when just given problems to solve on their own. (9)

According to the report, using anchor papers with students addresses two classroom challenges. It saves time, as fewer problems need to be worked out, and eases the burden of assessing additional work. It also tackles the shortage of good problem-solving material that is available.

This technique can be used by teachers and students.

Teaching students to self- and peer-assess: using anchor papers as a tool.

In an earlier blog, we discussed research that showed the power of student self- and peer-assessment. Anchor papers may be used to help students learn to be successful self- and peer-assessors. After your teachers have introduced the assessment rubric to students, try putting a piece of anonymous student work on the overhead. Ask students to solve the original task (making sure they understand the solution). Then, using the assessment rubric ask students to assess the piece and share their analysis once everyone has finished. As they discuss various perspectives, students learn what work meets the standard and what work doesn’t. A great deal is also learned about problem solving.

To further extend this exercise, you could ask students how they might improve upon weaker samples so that they meet the standard. Teachers can also take work that meets the standard and ask students how they would turn it into work that exceeds the standard. By doing this, students will learn what meeting and exceeding the standard looks like.

This technique can be used by teachers and students.

Providing guidelines for students.

Anchor papers can provide students with examples of the kind of work their teachers expect. Ask your teachers make copies of student work samples for a set of problems. Include anchor papers that don’t quite meet the standard as well as work that meets and exceeds the standard. Have them discuss these pieces and link each of the solutions to the parts of the rubric that are applicable. Doing so will enable students to have a much clearer understanding of the work that is expected.

This technique can be used by teachers and students.

Making use of errors.

By highlighting errors in anchor papers, teachers can create learning opportunities for their students. In Japanese classrooms teachers use errors in student work as a teaching opportunity, whereas in American classrooms this is rarely done. In the U.S., teachers tend to continue polling students in search of the correct solution, generally ignoring errors.

Discussing errors helps to clarify misunderstandings, encourage argument and justification, and involve students in the exciting quest of assessing the strengths and weaknesses of the various alternative solutions that have been proposed. The Learning Gap (Summit Books, 1992) p. 191

 This technique can be used by teachers and students.

Anchor papers to support the Common Core.

The essence of the anchor paper is to provide an accurate picture of what student work looks like at various performance levels with regard to a specific standard. Working with real student samples can help both teachers and students visualize the new learning expectations set forth by the Common Core.

Over time, your teachers can work together to build collections of student work. Exemplars also offers a large library of problem-solving tasks that are aligned to the Common Core. Each of our performance tasks include annotated anchor papers that correspond to the four levels of our assessment rubric. These are a great resource that schools and districts can use to get started.

To learn more about our performance material or view sample tasks with anchor papers select from these grade levels K–2, 3–5, 6–8 and scroll down to the links in the “Task-Specific Assessment Notes.”

Preparing for the Common Core: Using Rubrics to Guide Teachers and Students

Tuesday, August 14th, 2012

By: Ross Brewer, Ph.D., Exemplars President

As you begin preparing your staff to integrate the Common Core this year, rubrics should play a key role in terms of helping your teachers and students achieve success with the new standards.

 What are rubrics?

A rubric is a guide used for assessing student work. It consists of criteria that describe what is being assessed as well as different levels of performance.

Rubrics do three things:

  1. The criteria in a rubric tell us what is considered important enough to assess.
  2. The levels of performance in a rubric allow us to determine work that meets the standard and that which does not.
  3. The levels of performance in a rubric also allow us to distinguish between different levels of student achievement among the set criteria.

Why should teachers use them?

The Common Core assessment shifts will pose challenges for many students. The use of rubrics will allow teachers to more easily identify these areas and address them.

For Consistency. Rubrics help teachers consistently assess students from problem to problem and with other teachers through a common lens. As a result, both teachers and students have a much better sense of where students stand with regard to meeting the standards.

 To Guide Instruction. Because rubrics focus on different dimensions of performance, teachers gain important, more precise information about how they need to adjust their teaching and learning activities to close the gap between a student’s performance and meeting the standard.

To Guide Feedback. Similarly, the criteria of the rubric guides teachers in the kind of feedback they offer students in order to help them improve performance. Here are four guiding questions that teachers can use as part of this process:

  • What do we know the student knows?
  • What are they ready to learn?
  • What do they need to practice?
  • What do they need to be retaught?

How do students benefit?

Rubrics provide students with important information about what is expected and what kind of work meets the standard. Rubrics allow students to self-assess as they work on and complete problems. Meeting the standard becomes a process in which students can understand where they have been, where they are now and where they will be. A rubric is a guide for this journey rather than a blind walk though an assessment maze.

Important research shows that teaching students to be strong self-assessors and peer-assessors are among the most effective educational interventions that teachers can take. If students know what is expected and how to assess their effort as they complete their work, they will perform at much higher levels than students who do not have this knowledge. Similarly, if students assess one another’s work they learn from each other as they describe and discuss their solutions. Research indicates that lower performing students benefit the most from these strategies.

Rubrics to Support the Common Core.

Exemplars rubrics can provide a valuable bridge for staff transitioning to the new standards.

Our rubric criteria reflect the Common Core Standards for Mathematical Practice and parallel the NCTM Process standards. Exemplars rubric consists of four performance levels (Novice, Apprentice, Practitioner (meets standard) and Expert) and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation).

Our rubrics are a free resource. To help teachers see the connection between our assessment rubric and the eight Standards for Mathematical Practice, we have developed two alignment documents:

Which alignment one uses will depend on the intended purpose of the user.

It’s never too young to start.

Students can begin to self-assess in Kindergarten. At Exemplars, we encourage younger students to start by using a simple thumbs up, thumbs sideways, thumbs down assessment as seen in the video at the bottom of the page.

Our most popular student rubric is the Exemplars Jigsaw Rubric. This rubric has visual and  verbal descriptions of each criterion in the Exemplars Standard Rubric along with the four levels of performance. Using this rubric, students are able to:

  • Self-monitor.
  • Self-correct.
  • Use feedback to guide their learning process.

How to introduce rubrics into the classroom.

In order for students to fully understand the rubric that is being used to assess their performance, they need to be introduced to the general concept first. Teachers often begin this process by developing rubrics with students that do not address a specific content area. Instead, they create rubrics around classroom management, playground behavior, homework, lunchroom behavior, following criteria with a substitute teacher, etc. For specific tips and examples, click here.

After building a number of rubrics with students, a teacher can introduce the Exemplars assessment rubric. To do this effectively, we suggest that teachers discuss the various criteria and levels of performance with their class. Once this has been done,  a piece of student work can be put on an overhead. Then, using our assessment rubric, ask students to assess it. Let them discuss any difference in opinion so they may better understand each criterion and the four performance levels. Going through this process helps students develop a solid understanding of what an assessment guide is and allows them to focus on the set criteria and performance levels.

Deidre Greer, professor at Columbus State University, works with students at a Title I elementary school in Georgia. Greer states that in her experience,

The Exemplars tasks have proven to be engaging for our Title I students. Use of the student-scoring rubric helps students understand exactly what is expected of them as they solve problems. This knowledge then carries over to other mathematics tasks.

At Exemplars, we believe that rubrics are an effective tool for teachers and students alike. In order to be successful with the new learning expectations set forth by the Common Core, it is important for students to understand what is required of them and for teachers to be on the same “assessment” page. Rubrics can help.

To learn more about Exemplars rubrics and to view additional samples, click here.

Mathematical Practice and Problem Solving: Preparing Your Teachers for Common Core

Thursday, July 26th, 2012

By: Ross Brewer, Ph.D., Exemplars President

The Common Core State Standards – Mathematics is divided into two parts: Content Standards, and Standards for Mathematical Practice. A major focus of the Standards for Mathematical Practice is on using problem solving to reinforce important concepts and skills and to demonstrate a student’s mathematical understanding.

To fully prepare for the implementation of the Common Core, teachers must have an understanding of what problem solving is, why it is important and how to go about implementing it. For many, the successful teaching of problem solving will require real pedagogical shifts. What do teachers need to know?

To help answer this question and prepare your staff, you might turn to findings in the recent report, Improving Mathematical Problem Solving in Grades 4 Through 8, published in May 2012 under the aegis of the What Works Clearinghouse (NCEE 2012-4055, U.S. Department of Education, available online from the Institute of Education Sciences). This report provides educators with “specific, evidence-based recommendations that address the challenge of improving mathematical problem solving.”

In the Introduction, the panel that authored the report makes the following points:

  •  Problem solving is important.

“Students who develop proficiency in mathematical problem solving early are better prepared for advanced mathematics and other complex problem-solving tasks.” The panel recommends that problem solving be part of each curricular unit.

  •  Instruction in problem solving should begin in the earliest grades.

“Problem solving involves reasoning and analysis, argument construction, and the development of innovative strategies. These should be included throughout the curriculum and begin in kindergarten.”

  •  The teaching of problem solving should not be isolated.

“… instead, it can serve to support and enrich the learning of mathematics concepts and notation.”

  • Despite its importance, problem solving is given short shrift in most classrooms.

To address these points and improve the teaching of problem solving, the panel offers five recommendations.

Recommendation 1

Prepare problems and use them in whole-class instruction.

In selecting or creating problems, it is critical that the language used in the problem and the context of the problem are not barriers to a student’s being able to solve the problem. The same is true for a student’s understanding of the mathematical content necessary to solve the problem.

Recommendation 2

Assist students in monitoring and reflecting on the problem-solving process.

“Students learn mathematics and solve problems better when they monitor their thinking and problem-solving steps as they solve problems.”

Recommendation 3

Teach students how to use visual representations.

Students who learn to visually represent the mathematical information in problems prior to writing an equation are more effective at problem solving.

Recommendation 4

Expose students to multiple problem-solving strategies.

Students who are taught multiple strategies approach problems with “greater ease and flexibility.”

 Recommendation 5

 Help students recognize and articulate mathematical concepts and notation.

When students have a strong understanding of mathematical concepts and notation, they are better able to recognize the mathematics present in the problem, extend their understanding to new problems, and explore various options when solving problems. Building from students’ prior knowledge of mathematical concepts and notation is instrumental in developing problem-solving skills.

The panel also identifies two specific “roadblocks” to implementing these recommendations:

Roadblock 1

“Traditional textbooks often do not provide students rich experiences in problem solving. Textbooks are dominated by sets of problems that are not cognitively demanding …”

Exemplars was started precisely to meet this need — to provide the rich problem-solving tasks that teachers and students lacked in traditional texts.

Roadblock 2

Lack of time/opportunity to do problem solving in the classroom.

The panel notes that in addition to spending time solving problems, research shows that students benefit by studying already solved problems.

Exemplars annotated anchor papers help meet this need.

As president and founder of Exemplars, it is validating to see the fundamental elements of our material affirmed in this rich research-based report. So much of what is discussed is at the core of what Exemplars math material is all about and has been since we began publishing 19 years ago:

  • The importance of success with problem solving
  • The critical role formative assessment plays in the classroom
  • Students’ use of representations in making the link between the problem and the underlying mathematics
  • Students’ ability to communicate their thinking
  • Students’ application of appropriate mathematical language and notation
  • Helping teachers instruct students in mathematical understanding and allowing students to demonstrate that understanding.

We believe all of these factors should play a critical role in instruction, assessment and professional development.

As teachers are asked to implement more problem solving in their classrooms in support of the Common Core Standards for Mathematical Practice, Exemplars math tasks provide a valuable resource. The tasks are also an effective tool for staff development.

To view samples of our current material and the respective alignments to the Common Core, click here: K–2, 3–5, 6–8.

 

 

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