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A Problem-Solving Lab to Support the Math Practices

Friday, October 31st, 2014

Written By: Donna Krachenfels & Debra Sander, Teachers from PS 54

The school administrators at PS 54 had a vision to create a math laboratory based on the eight Standards of Mathematical Practice. The idea was to create a setting in which students could focus on multi-step problem solving.

The Exemplars program has given our students many opportunities to build and strengthen their problem-solving skills. Students were also able to strengthen their close reading skills as they reread problems multiple times to identify and think about the relevant information necessary to find a solution. Collaboration allowed students to become confident in their problem-solving skills and increased their abilities to construct viable arguments as they defended their solutions and critiqued the solutions of their classmates. Students were not afraid to take risks as they tried different representations and strategies to solve problems. As a result of the Exemplars math program, our students became more confident and more independent problem solvers.

The math laboratory is in its second year at PS 54. Last year, our data saw increased math scores for the classes that participated in the problem-solving lab. This year, the trend continued and all general education students passed the state math exam.

Special thanks goes to Exemplars professional development consultant Deb Armitage for all of her help and support. She is a true math educator!

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

Thursday, August 28th, 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 5 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 5: Observations at the Fourth Grade Level

The fifth anchor paper and set of rationales we’ll review in this series is taken from a fourth grade student’s solution for the task “Sharing Muffins.” This task is one of a number of Exemplars tasks aligned to the Numbers and Operations–Fractions Standard 4.NF.3c.

“Sharing Muffins” would be used toward the end of the learning time allocated to this standard. This task provides fourth graders with an opportunity to apply different strategies to determine the number of muffins needed for each of nine friends to have one and one-third muffins. In solving this task, there are a variety of strategies for students to consider. Some examples include using actual muffins to model one and one-third muffins per friend or diagramming the muffins using a table, tally chart or number line. In their solutions, students may replace each mixed number with an equivalent fraction. Addition, subtraction and multiplication of fractions may also be used.

Fourth Grade Task: Sharing Muffins

Nine friends are going to equally share some muffins. Each muffin is the same size. Each friend gets one and one-third muffins. How many muffins did the nine friends equally share? Show all your mathematical thinking.

Common Core Task Alignments

  •  Content Standard 4.NF.3c: Add and subtract mixed numbers with like denominators e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

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

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.”

Linking the Common Core Standards of Mathematical Practice to Exemplars Rubric

Wednesday, December 14th, 2011

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

If your school or district is preparing for the integration of the Common Core State Standards (CCSS) into its mathematics curriculum, Exemplars materials are a great bridge. Our problem-solving tasks, rubrics and anchor papers can help with the transition and aid in preparing your staff.

As many of you know, the CCSS for Mathematics are divided into two parts, the “Content Standards” and the “Standards of Mathematical Practice.” The Standards of Mathematical Practice describe the ways “student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years.” (p.8)

There are eight Standards of Mathematical Practice.

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

To help teachers see the connection between the Exemplars Standard Rubric and the Common Core, we have created the following alignment documents:

Which alignment one uses will depend on the intended purpose of the user. Exemplars mathematics tasks are also aligned to the CCSS Content Standards. These alignments can be found on our web site.

We hope that by making these resources available, you will see the natural fit between Exemplars and the CCSS.

How is your school or district preparing for the Common Core in math?

Formative Assessment Tools

Thursday, December 1st, 2011

What are some of the strategies that you use in your classrooms to foster formative assessment?

Effective use of formative assessment in the classroom is one the most powerful ways to improve student achievement. Research shows that the improvement in performance is dramatic.

Successful formative assessment includes:

  1. Asking meaningful questions, increasing the wait time for student answers and having rich follow-up activities that extend student thinking. (13)
  2. Providing meaningful feedback to students on what was done well, what needs improvement and offering guidance on how to make improvements.
  3. Ensuring that students have a clear understanding of the standards and are taught the skills of peer- and self-assessment. (15)

Exemplars rubrics can be a useful tool in implementing some of these strategies. Many students are able to successfully internalize standards that are reflected in rubrics.  In addition to our Assessment Rubrics, we have developed a number of student rubrics that can be used by children when they are very young. Our “Jigsaw Rubric” combines both verbal and visual components that make each element of the Exemplars Standard Math Rubric explicit for students. While this rubric was initially developed for primary students, it is popular with middle school and even high school teachers. Other examples of student rubrics for math and science can be seen here.

Students can be introduced to rubrics at a very young age. For tips and suggestion on how to do this, refer to our article, “Introducing Rubrics to Students.” There are also several sample introductory rubrics available on our web site. While we do make these examples available, it is important for your students to first develop their own rubric before exposing them to these. Through this process, students learn what a rubric is and how to use it. A student favorite is the “Chocolate Chip Cookie Rubric.” Another favorite asks students to develop a rubric for assessing running shoes.

In our classroom modeling workshops, a “Thumbs Up – Thumbs Down Rubric” is used with very young students. To see this in action, click on the video link at the bottom of this post.

Teachers can also use Exemplars anchor papers to help students learn how to better use the rubric to assess their own work as well as that of their peers. Additionally, anchor papers can be used to help students visualize what work meets the standard and what work doesn’t.

In addition to our rubrics and anchor papers, Exemplars offers many other formative assessment tools for both teachers and students, such as questioning guides.

Paul Black, C. H., Clare Lee, Bethan Marshall, and Dylan Wiliam (2004). “Working Inside the Black Box: Assessment for Learning in the Classroom”. Phi Delta Kappan: 9-21.

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