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 21^st 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?

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.

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.

Join Katie Dello Stritto, Assistant Principal at P.S. 58, as she discusses the implementation of Exemplars for problem solving.  In this video, she chats with Ross Brewer, Exemplars President, at an Exemplars Institute about the Carroll School’s approach and teacher support structure.  They discuss the work of teacher vertical teams assessing together as well as the outcome of Exemplars use in the classrooms.

Life is a series of things we outgrow … our shoes, our clothing, hairstyles, and ultimately, even definitions of ourselves. Teaching is no exception. Are ideas about teachers outgrown as well? What are the visual symbols of a teacher?

We’ve all received our share of apples, quaint schoolhouse ornaments and sticky notes with pencils and books. But is that how we see ourselves in the 21^st century as we lay aside our traditional grade book and log on to our computer assessment program, for example? And now, even that old apple has turned into Apple i-pad textbooks!

Responding to Karen Ahern a teacher from Haverhill, MA they set out to rebrand us and our mission. NPR’s Studio 360 recently took on the challenge and asked a design group, Hyperakt, to formulate a new set of visuals related to who we are as educators and the universe we inhabit. It is imaginative, playful and thought provoking … just like school should be!

The visuals and background to the piece can be found http://www.studio360.org/2012/jan/20/teacher-redesign-revealed/slideshow/. I hope you enjoy it!

Have a student identified as gifted in mathematics? Hmmm, what to do … ? Give more problems? Jump to the next book?

Because each learner comes to his/her school experience at a differing level of strength and need, a Response to Intervention (RtI) approach is suggested. Using data to identify levels of service, a continuum of programming options must be matched to student needs in order to most appropriately reach mathematically promising students. For some students, in-class differentiation can meet student needs. In this case, the suggested modifications included with Exemplars tasks make it easy for teachers to provide mathematical challenges for a wide range of learning needs. In situations where a student is one or more grade levels above his/her assigned math class, more intensive programming options must be considered.

Dr. L.J. Sheffield, author of Extending the Challenge in Mathematics (Corwin Press, 2003), describes a continuum of mathematical thinking along which one can place students in a relationship to their level of mathematical thinking; the continuum ranges from non-mathematical thinkers to the most sophisticated mathematical thinkers:

innnumeraters  > doers  > computers > consumers > problem solvers > problem posers > creators

In order to raise mathematical thinking to the highest level, Dr. Sheffield has identified five components necessary for profound mathematical problem solving; students must be able to relate, investigate, create, evaluate and communicate mathematical ideas.

Programming for mathematically promising students should include a combination of enrichment and acceleration, as these students are capable of processing math concepts with less practice and at a faster rate than other students. Instruction should be at an accelerated pace and the curriculum should have increased challenge and rigor focusing on the depth and complexity within mathematics. The tasks within Exemplars can be used in place of the regular curriculum (more depth and complexity) or as an extension of the current curriculum as the tasks have been aligned to NCTM and Common Core standards.

The concept of cluster grouping should be applied to meet the needs of gifted students. The ongoing support of  “mental mates/peers” can help students as they face and work through challenges. Together, students can wrestle with and discuss complex problems to develop a greater understanding of mathematical concepts. Cluster grouping can support adjustment to the increased demands of complex problems and students can be encouraged to share their results and thinking with students who “speak the same language.”

Student enrichment can be provided through the addition of higher-level problem-solving activities and application of mathematics to real-life problem-solving situations. Look for problem-based learning opportunities in local news media or in everyday life and surroundings. For example, in problems concerning:

• water resources
• wildlife population management
• transportation routes and traffic pattern
• design of buildings
• parks or playgrounds

there is an opportunity to develop public opinion polls and surveys. These can be used for modeling and statistical analysis as well as to develop the concepts of geometry. Math activities should apply information from other content areas to increase student interest and involvement. 

Teachers of mathematically promising students need to know the content of mathematics and understand how their students develop mathematical concepts over time. Teachers should serve as facilitators to help students develop the necessary skills to discuss their thoughts about the concepts they used to solve the problem and how these concepts relate to other curricular areas. Strong teachers of mathematics provide students with activities that allow students to show what they really know about mathematics. Problems that have multiple entry points, multiple pathways to solutions and/or multiple solutions should be commonplace.

In mathematics classrooms, teachers must create a “culture of opportunity” where students have the time for and feel free to explore the wonder and beauty of mathematics. In this environment, students expect to be challenged and are encouraged to develop their own problem-solving strategies. Additionally, students need opportunities to learn how to evaluate and critique the solutions of others in a collaborative way. Such classrooms require discussions among students regarding their solutions to problems; it takes a savvy teacher to know when to step in to clarify or to ask students to share solutions that enhance and clarify understanding for others. Rubrics and anchor papers from Exemplars can assist students in more accurately rating and improving performance in order to increase achievement and success.

Making Math Meaningful through high-end problems and challenges for gifted learners using Exemplars is an exciting opportunity for students as well as teachers of mathematics.

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

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

There are eight Standards of Mathematical Practice.

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

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

• Math Exemplars: A Perfect Complement for the Common Core aligns each of the Standards of Mathematical Practice to the corresponding sections of the Exemplars assessment rubric.

• Common Core Standards of Mathematical Practice and the Practitioner Level of the Exemplars Math Assessment Rubric provides much more detail. It aligns the language within each Standard of Mathematical Practice to the Practitioner level of the Exemplars assessment rubric.

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

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

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

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

Successful formative assessment includes:

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

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

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

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

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

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

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

Through our web site and blog, useful material and other resources will be made available. Some will be new from Exemplars and some will come from educators that have successfully implemented innovative approaches in their schools. We are also developing a video library to highlight strategies that have worked in the classroom. Exemplars’ blog is designed for educators to communicate and share ideas with each other.  We welcome your questions and comments at any time.

We look forward to hearing from you!