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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.

Posted in Education | No Comments »

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!

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Posted in Education, Math | No Comments »

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!

Posted in Common Core, Education, Math | 1 Comment »

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.”

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Posted in Common Core, Education, Math, Professional Development, Resources | 1 Comment »

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.

Posted in Common Core, Education, Math | No Comments »

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.

 

 

Posted in Common Core, Education | 1 Comment »

Reflections on Partnering with Exemplars

Tuesday, June 26th, 2012

Written By: Leslie Koske, Curriculum Specialist, Ginnings Elementary School, TX

In this piece, Leslie Koske shares her experiences with her RTI group and Exemplars.

Response to Intervention (RTI) begins with both high-quality instruction and universal screening tests for all students to determine levels of learning competency. Intensive interventions in small group settings are then provided to support students in need of assistance with mathematics learning. Student responses to intervention are regularly measured to determine whether students are making adequate progress within the three-tier model.

Beyond the “bare facts” approach, the use of a well-designed mathematical performance task like those developed by “Exemplars” may reveal how well a student has grasped and applied the math concept in an intervention or lesson(s). The performance task rubric is critical in providing the intervention team with information as to how to help the student continue to increase problem-solving thought patterns. It also provides the interventionist and other school personnel with data that can be used to place students in groups within the three tiers of RTI instruction.

While common skill assessments can identify and direct remediation of math weaknesses, it is a leap of faith to move the student into the arena of open-ended problem solving. Unlike a student armed with the tools of math facts and basic computation skills plus adequate reading skills, the RTI student may be undertaking a complex task with minimal skills in all areas.

So, with heart in hand, we begin to delve into the world of creative problem solving with tons of scaffolding to keep the students engaged and afloat.

METHODOLOGY

First, we approached Exemplars not as a “math problem” (immediate defeat), but as a “math story” full of fun. Students begin by analyzing the meat of the text with verbs and action, armed with their best reading strategies (seeking main idea, keeping summary and inference with character and plot in mind) and using the famous five W’s: who, what, when, where, and why. “Who is this story about?” “What do we know?” “What are we looking for?” “Why and when did this happen?” “Can we predict what will happen next?”

We chart out information from the Exemplars math problem on a four-quadrant chart loosely referred to as “UPS Check” model borrowed from Polya’s work: Understand, Plan, Solve, Check.1 This framework supports the organization of a complex math problem by directing the student to “chunking” the parts: understand and paraphrase the question, set up a solution plan (t-chart, number line, picture, labels, etc.), actually solve the question, then evaluate and justify the answer. This method is often a group project with four students, each one taking a fourth of the quadrant. A weaker student may need to copy the problem and ask for help reading it, while students with other strengths will tackle the “plan, solve and justify” quadrants.

Believe me, just understanding where to begin is a major and very risky undertaking for the struggling student. We usually work in pairs or small groups in order to spur ideas. We also incorporate another problem-solving strategy called “RUBIES” in the “understand” quadrant that is a problem-solving acronym we borrowed from the science people: Read and Reread, Underline to understand the question, Bracket information, Identify key Elements. This is yet another support to clarify deeply connected math embedded within fictional text.

RTI students need many structures to support and verify their thinking as they investigate possible solutions. I provide “wipe boards” to sketch out solutions, because mistakes can be wiped away without fuss and muss. Students select from a variety of manipulatives to give physical evidence to their thinking. I also feel it is comforting to begin the process with a whole-group experience as the teacher and students plunge into analytical thinking together using “wait time” (be quiet and wait for students to ponder) and “think aloud” (model thought processes out loud so students don’t think teachers were born with answer keys in their heads) and other “active listening” strategies to demonstrate the process of true problem solving as being a walk-in-the-dark to new ideas and not a quick answer. Additionally, I give great attention to modeling different approaches to problem solving and relish using the student work that you [Exemplars] provide to show students the many ways that a solution can be discovered. During this time, we discuss the process: Working backwards, we make a table or chart, find a pattern, and use simpler numbers and so on until students no longer need this structure.

What follows is an example of scaffolding the integration of a well-known perimeter investigation with a similar Exemplars math problem.

INTEGRATED “SPAGHETTI AND MEATBALLS FOR ALL” WITH “SEATS AND TABLES”

ENGAGE: Read Spaghetti and Meatballs for All by Marilyn Burns 2 to the students. Use color tiles to model the various table arrays to find different seating arrangements as the teacher reads.

EXPLORE: Students will color the models of their tiles on centimeter paper and draw conclusions as to the effect of the dimensions of the arrays on the number of people at the table.

(This is an introductory activity with all the same shapes.)

EXPLAIN: Teacher asks students to reflect on the table arrangements and the number of people per table. Does the length or width of the array effect the seating? Are there hidden sides? Develop definitions for perimeter, square, rectangle, array, sides, and edge. (For ESL students, a pre-teaching of vocabulary for this lesson is recommended.)

ELABORATE: Exemplars Task: Seats and Tables (click to download task)

“You are in charge of setting up a classroom with 20 places for people to sit. You can use any number of tables and any combination of 3 kinds of tables. A hexagon-shaped table has 6 places. A square table has 4 places and so does a rhombus shaped table. How would you set up your tables so that 20 people have a place to sit?” Show how many people can sit at each of the tables and how do you know there are places for 20 people.”

  • You may use pattern blocks.
  • Pretend the paper is a miniature room.
  • You need exactly 20 places.
  • Provide: graph paper, colored pencils

REAL-WORLD CONTEXT: We have four different kinds of tables in our room (rectangle, hexagon, circle and small rectangle private office). During lunch and work time, there are specific numbers of people allowed at each table. This creates social strains and naturally gets kids talking about the classroom set up on a daily basis. They initiate their own discussions of how to maximize their contact with people or minimize it with others. I decided to introduce this problem because it is a familiar topic for them and they seem interested in solving their own classroom seating issues.

WHAT THE STUDENTS DID: The students took the shapes and tried various arrangements to get to 20. They had a hard time remembering to match sides — not vertices — when making their arrangements. Students traced the shapes and really experimented with all kinds of structures.

Some students lost the questions and went to 20 pieces — not 20 sides or “seats.” They did not really relate to the shapes of the tables in the classroom and needed redirection to relate this activity to the real-life situation around them.

To solve the problem, students used the shapes of tables in the classroom. They traced and counted sides, and then added a different shape (triangle, for instance) to reach 20 seats.

Some students placed numbers at each angle instead of at the sides. They added two squares are eight and then added up the squares and hexagons (8 + 12 = 20). Some students multiplied the tables, which represent the same amount of people (5 x 4 = 20) and used equations and counting to add them up.

Students tried to use all the shapes and changed their minds when the numbers did not count up to 20. Some students traced shapes that were a correct solution, but were not able to write an equation and/or the numbers.

EVALUATE: Using Exemplars rubric categories and Task-Specific Assessment Notes, the student’s work is evaluated.

At this time our efforts are modest as we venture into the waters of true explorers of math thought and away from canned textbook algorithms. I believe our partnership with Exemplars is rock solid and can only lead to mind-expanding experiences through the wonder of thoughtful questioning.

References:

1. Polya, George. 1945. How to Solve It. Princeton: Princeton University Press.

2. Burns, Marilyn. 1997. Spaghetti and Meatballs for All. City: Scholastic Press, Inc.

Posted in Education, Math | No Comments »

#2 Tips for Planning Successful Problem-Based Learning in Your Math Classroom

Thursday, May 10th, 2012

Written By: Julia Watson, Ph.D., Exemplars Consultant and Gifted and Talented Specialist

In her last post, Dr. Julia Watson provided an overview of Problem-Based Learning (PBL). In this segment, she offers suggestions on how teachers might go about incorporating this approach into their classrooms. You can access her first post here.

Where to begin?

(1) Think of your students, of their ages, maturity levels, and their interests. What school-level project possibilities might exist, just outside the classroom door? What local (community) issues or priorities could be integrated as a math challenge?

(2) Begin to map out your ideas in two sections, with Section I preceding Section II:

  •  Section I: Addresses preparation by a teacher/team.
  •  Section II: Deals with implementation for the unit in the classroom, either as a whole class or for a small team who may need this type of challenge.

What does it look like?

The following is an example of a possible PBL “experience” based on a local news article:

A recent proposal is being considered that may partially remediate Elm City’s budgetary crisis. At the last city council session, members suggested reducing the city debt by not funding the animal shelter for the next fiscal year. This recommendation is one of the possible cuts mentioned concerning the city budget. The suggestion caused immediate concern and debate among the citizens who attended the meeting. The suggestion has been tabled until the next meeting, in one month.

What are the steps?

(1) Define important questions for the unit.

  • What are the economic costs of having an animal shelter each year?
  • What is the impact on the city if there is no animal shelter to house stray and/or abandoned animals?
  • What information is needed and how can this information be presented to the city council so they are able to make a well-informed decision about the animal shelter?
  • What are some alternative solutions and costs for our community in order to provide for stray and abandoned animals?

(2) Select standards and learning outcomes to be developed during the PBL experience.

  • Based on your state standards and local curriculum, what mathematical possibilities exist within this problem?

(3) Integrate across content areas, making connections.

  • Brainstorm possible content area connections, network with other teachers/instructors to connect the learning possibilities.

(4) Define a possible problem statement.

  • What information about the possible closing of the animal shelter should we present to the city council so that they can make an informed decision that will be amenable to the community?

(5) Design assessments for the unit.

  •  Identify ongoing “check points” and formal/informal measures.

(6) Determine length of unit.

  •  Think of time frame and need to “be ready” for next session/end of unit. Set timeline, possibly working backwards.

 How do I implement the PBL experience with students?

(1) Meet the problem.

Sunflower City in Colorado has determined that in order to meet the city’s requirement for debt reduction by 20% the city has proposed a number of items to be cut from the budget. One of these is the support funding for the local animal shelter. Without this additional funding, the animal shelter cannot stay open. Concerned fourth graders have decided to present at the next city council meeting. The fourth graders job is to convince the council members to review alternative solutions and to convey the impact closing the shelter will have on the community. The meeting will be held next month.

(2) Construct “Know/Need to Know” statements.

  • We know Sunflower City’s proposal to cut the deficit.
  • We know some of the animal shelter needs.
  • We know students’ perceptions about what happens to abandon animals.
  • What are the operational costs for the animal shelter per month and year?
  • What are the donation and adoption monies received by the shelter per month and year?
  • What are the consequences for the closure for the city, the citizens, and the community?
  • How does this impact the city as a community?
  • What are alternative solutions?
  • What donations or sponsors can support the shelter?
  • What are the needs of the animals if they are left to roam the city untended?
  • What are the opinions of the community members for a place to house abandoned animals in the city?

To answer our “Need to Know” statements we need to:

  • Create a KWL chart.
  • Complete a cause and effect graphic organizer.
  • Determine flexible groups to research various solutions.
  • Obtain resources from multiple areas.
  • Create a financial fact sheet.
  • Create a list of questions for the animal shelter.

 (3) Class defines the problem statement.

How might we present accurate information regarding the effects on the community of not having a way to care for homeless animals in our city so that the council can make an informed decision about the management of the animal shelter?

(4) Students gather information.

Activity I: Making Connections.

  • Economics: Review alternative sources of funding including sponsors and grants and economic impacts on the community of the loss of the shelter.
  • Civics: Learn decision-making process of the council and gather multiple perspectives of the community.
  • Science: Investigate animal needs, such as food, water and shelter. What are the impacts if needs are not met?

Activity II: Financial Analysis Assignment. Each subgroup receives a financial fact sheet that answers the following financial questions:

  • How many animals are there on an average each month?
  • How many of each kind?
  • What does it cost to feed them?
  • What are the personnel and building costs?
  • What additional monies are donated and where are they from?
  • What are the costs of running the shelter per month and annually?
  • What would be the funds needed to replace the budget cut by the city?

Students use this information to create a balance sheet of funds coming into the shelter and shelter expenses per month. Next, they determine the costs annually. They calculate the income after the 20% reduction from the city and determine how much funding the shelter will lose if the cut occurs.

(5) Students share information with class.

(6) Students generate possible solutions to the problem.

(7) Students determine best-fit solution.

  • Which solution can be best prepared in time to present to the council?
  • Which solution will present the researched information most clearly to the council?
  • Which solution will most impact the decision-making process of the council?

(8) Solution is presented.

(9) As a class, debrief the PBL experience.

Discuss the effectiveness of the final presentation that was given to the city council and other members of the school and/or community. Think about:

  • What happened as a result of the presentation?
  • Were those the results you expected?
  • What are some other things you can do to support the shelter at this time?

Incorporating the 21st century skills of critical thinking, innovation and collaboration will empower our students to face challenges and problems, using their mathematical knowledge and skills. As educators, it is our task to help students develop skills to work with others to solve these future problems. Problem-based learning provides an effective instructional strategy for this purpose … and who knows, maybe Train A will never overtake Train B anyway …

What are some PBL opportunities that you have found effective with your students?

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#1 Tips for Planning Successful Problem-Based Learning in Your Math Classroom

Monday, April 30th, 2012

Written By: Julia Watson, Ph.D., Exemplars Consultant and Gifted and Talented Specialist

In this blog, Dr. Julia Watson provides insight on Problem-Based Learning (PBL) and offers suggestions on how teachers might go about incorporating PBL into their classrooms. It is published as a two-part series. The first segment features background information on PBL and its benefits on student learning, while the second will focus on steps for implementation.

If Train A leaves the station at 9:53 a.m. and is traveling at 32 mph, when will Train B that leaves the same station, traveling at 40 mph, overtake Train A?

Do any of you recall this mathematical challenge from your former education days? As a student, I found myself wondering “Who cares?”  After all, there was no train where I lived …

What does this imply for problem-based learning? I hope the answer is apparent– providing students with realistic, intriguing, open-ended tasks can engage those (of us) who don’t care about Train A OR Train B.

What is Problem-Based Learning (PBL)?

PBL is a focused, experiential learning opportunity organized around the investigation and resolution of messy, real-world problems that “leads to relevant and connected learning for students.” (Illinois Mathematics and Science Academy 2011)

Why use PBL?

Research indicates:

  • When students are asked to respond to situations that arouse their curiosity they display improved academic performance. (Engel and Randall 2009)
  • Challenging students to use higher-level thinking to solve complex problems is “superior when it comes to long-term retention, skill development and satisfaction of students and facilitators.” (Strobel and VanBarneveld, 2009 44)

What are the roles of students in PBL?

Students participating in PBL become:

  • Engaged by problematic situations.
  • Active learners.
  • Higher-level thinkers.
  • Self-directed learners and problem solvers (learning how to learn).
  • Decision-makers about the nature and structure of their own learning.

(Adapted from Barell 2010 and Torp and Sage 2002)

What are the roles of facilitators in PBL?

Facilitators engaged in PBL become:

  • Co-investigators who model using the problem-solving process.
  • Designers of learning experiences who determine the desired outcomes of the experience.
  • Coaches who ask questions.
  • Monitors of the learning experience who provide feedback to students.

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Review of Exemplars

Friday, March 16th, 2012

Written by: Ellen and Alan Alquist, IB/PYP Consultants, learn2inquire.com

Introduction:

Ellen and Alan Alquist have worked for years with International Schools, principally in the Middle East. Ellen has also been a consultant to International Schools in Europe and Africa. Her focus has been on the International Baccalaureate, Primary Years Program (PYP), Understanding by Design (UbD), mathematics and science. In this short blog, Ellen and Alan describe why and how they use Exemplars with these programs.

When teachers consider lesson planning, they typically think of the assessments last. This is natural, because the end of a unit of study typically culminates with the “big test,” or “final exam.” This misconception, however, causes inefficient and ineffective instruction. Assessment should be considered first. Careful consideration of summative assessments is fundamental to effective instructional planning. This is especially important when designing instructional units for inquiry-based learning or “Understanding by Design.” Making assessment the primary focus causes the teacher to concentrate on exactly what it is she or he wants their students to be able to do and understand as a result of instruction.

Assessments such as those designed by Exemplars provide a practical and rigorous way of doing this. The tasks are grounded in real-world experiences, and include a rubric and anchor papers with which to evaluate student responses. Materials are also aligned with national, IB and Common Core standards.

In our experience oversees, we use Exemplars tasks as a way of focusing teachers on assessment first when working with them on implementing the PYP (Primary Years Program) or UbD (Understanding by Design). After selecting an Exemplars task suitable for evaluating students’ mastery of the central ideas or enduring understandings in a particular unit, the teachers did the assessment themselves. They then marked their papers against the anchor papers. This led to a discussion among the teachers about the appropriateness of the particular assessment for the unit of instruction in question. When there was consensus that the assessment matched the goals of the unit, the task was adopted as a summative assessment. Occasionally modifications were made to be more culturally appropriate, depending on the country in which the tasks were being used. The numbers in a mathematics assessment were also tweaked from time to time to achieve a more curricular alignment.

After giving the test, teachers used the anchor papers to evaluate the students’ performance. The overall results were discussed by the teachers and used to inform planning for future teaching. It was often decided that the learning engagements or lessons needed to be altered. In many cases the Exemplars task was very suitable, but the teachers realized that the initial instruction had not adequately prepared their students for success. Use of Exemplars tasks had a dual benefit. Not only was student thinking revealed in greater clarity, but also teachers were able to be more reflective about their practice.

Many of the teachers we worked with reported that their students were genuinely engaged and enjoyed working on the assessments. This differs sharply with the typical student reaction to end-of-unit tests. In contrast to standard testing practices that assess products, the tasks set forth by Exemplars provide teachers with a way to examine the processes of student thinking. This is particularly important because research shows that teachers using alternative assessments focused on process are more disposed to be flexible and responsive to their students’ learning needs. (Spinelli 2001) Teachers who routinely use process-oriented assessments tend to continue expanding their practices with a greater range of choices and strategies. (Larrivee 2000) Teachers, staff developers and administrators seeking ways to transform and improve instruction will find Exemplars an invaluable resource.

Notes

Spinelli C. G. 2001. “Interactive teaching strategies and authentic curriculum and assessment: A model for effective classroom instruction.” Hong Kong Special Education Forum 4(1): 3–12.

Larrivee, B. 2000. “Transforming teaching practice: Becoming the critically reflective teacher.” Reflective Practice 1(3), 293–308.

Posted in Education, International, Math | No Comments »

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