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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.

 

 

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.

#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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