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Standards-based assessment and Instruction


Archive for the ‘Common Core’ Category

Performance Tasks: What’s the Point?

Tuesday, April 23rd, 2019

Written by: Jay Meadows, Exemplars Chief Education Officer & Cornelis de Groot, Ph.D., Exemplars Secondary Math Editor

Twenty-five years ago, Dr. Ross Brewer, founder of Exemplars, sat in a warm farmhouse in upstate New York with Grant Wiggins and Jay McTighe and asked a very important question, “How do we determine if students can utilize the mathematics they are learning in the classroom to address and solve the complex problems they eventually will be asked to address in the “real world?” To answer this question, these leaders of education decided to employ performance tasks with the explicit purpose of putting students in the authentic role of the problem solver.

Authentic Tasks and the Skills It Takes to Solve Them

Utilizing mathematics to solve real-world or authentic tasks requires more than the foundational skills of arithmetic and calculation. Authentic tasks require the knowledgeable utilization of a combination of several mathematical concepts. To solve these tasks, a person or team must decide on a strategy, choosing which math skills and tools to utilize and in what order. Once the team has determined their strategy and arrived at a solution, they must then design a clear explanation of their solution path, demonstrate and explain their thinking and articulate the reasonableness of their solution. In essence, students must learn to develop a persuasive argument and use precise mathematical language to provide clear mathematical evidence that supports their thinking.

These problem-solving skills are not innate to most students.

The Role of Intentional Practice

To help students prepare for the journey of becoming great problem solvers, teachers must clearly explain the purpose of solving these complex tasks so students can understand why they are being asked to work on tasks that can be more challenging then they are used to.

Solving complex problems requires intentional practice. Success with performance tasks can take time. Specific stages of student understanding can be scaffolded to work methodically towards strong products. Persistent success is not going to be found if we only give performance tasks in isolation. The trajectory to mastery of problem solving can provide a rich supplement to any curriculum. Spending time having students working in teams to develop potential solution paths and develop persuasive arguments can help develop several skills which are fundamentals of the 21st Century: Collaboration, Communication, Critical Thinking, Creativity and Problem Solving.

Why Are Performance Tasks Important?

Why do we ask students to attempt performance tasks? Creating opportunities for students to practice authentic problem-solving skills within their classrooms, in safe and supportive environments, will provide them incredible opportunities to learn to create real solutions to real problems. In this way they can develop the ability to solve the problems of the 21st Century.

Performance tasks ask students to do far more than calculations. Rich performance tasks ask students to adapt and apply their developing knowledge and understanding of mathematics, to take risks and explore possible strategies, to persevere while being flexible with their mathematical skills and understandings in their efforts to become deep mathematical thinkers.

The Importance of Marinating

In this type of learning environment, the initial instructional priority is helping students to clearly understand the task they are working to solve. This goes beyond simply highlighting keywords. Instead, we must ask students to talk with each other about the task: What are they being asked to solve, what strategies have they learned in the past that may connect to this task and help them to find a solution? What do they know and wonder from the task? This time spent marinating in the task is a foundational skill for the great problem solvers throughout history. Often the most important time spent in solving a task is the time spent carefully looking at and contemplating a task. Additional time realizing how a new task connects with something the students have done in the past can provide a strategy for getting started. “How does my prior knowledge possibly connect with this new task?”

Great problem solvers spend more time marinating in a task then in any other phase of the problem-solving process.

As students work this spring to succeed on end-of-the-year performance tasks, remember these are powerful tools for preparing them to be ready to use the math concepts and skills we spend years helping them develop. These high-level expectations will help develop a generation of great problem solvers.

Exemplars Problem Solving in Our Schools

Monday, April 22nd, 2019

Written by: Phil Sanders, Elementary Math Supervisor, Plainville Community Schools, CT

For the last three years, Plainville Community School teachers have dedicated themselves to the use of the Exemplars Problem-Solving Program. There are many reasons for this. First, it is an excellent, Common Core standard aligned, comprehensive problem-solving program. Teachers from Kindergarten to grade 5 comment most often about how it really works for their students. Principals also agree that they have seen tremendous growth in students’ ability to understand a problem, develop a plan to solve a problem and create models to show their thinking about how they solved the problem.

The Challenge

Plainville has adopted a rather unique approach to tackling the issue of students being successful on pencil-and-paper math work but falling down on problem solving. Three years ago we realized that our students were not understanding what the problem-solving tasks were asking them to do, and students often failed to develop a well-thought-out plan to solve the tasks. When examining student work, teachers saw students picking numbers out of the problem and running them through the latest algorithm. When asked to discuss their thinking, students responded with confusion or lack of understanding.

A Three-Fold Solution

Exemplars has provided us with the vehicle to tackle these seemingly insurmountable issues. The program provides standard-aligned problems that lend themselves to students being successful. Our approach was three-fold: Understanding was our focus the first year, Communication the second year and Accuracy the third year.

  1. Teachers, through their district-wide PLCs, developed slideshows based on the problems, helping to create contextual understanding of what the problem was discussing. This helped activate the UDL aspect of instruction, allowing all students to gain a foothold for their understanding.
  2. Through small group work and modeling of different problem-solving strategies, students next developed their proficiency at drawing models to communicate their thinking. All this information came directly from the Exemplars Preliminary Planning Sheets, which teachers found to be crucial to understanding all aspects of the problem.
  3. Last, we focused on accuracy and found that students were ahead of the curve. Because of their developed understanding and ability to use models, students’ overall accuracy increased tremendously.

One of the ancillary benefits to the Exemplars program is that we found students were able to have either small group (Turn and Talk) or whole group discussions about what they have learned and were able to defend their findings using evidence stemming from the Exemplars problem they had completed.

Teachers find the copious amount of information included in the Exemplars program to be extremely helpful. We have incorporated Exemplars Summative problems as end-of-topic assessments, and in grades 3–5 we have seen growth of our SBAC scores. We attribute these gains to our work with the Exemplars Problem-Solving Program.

Productive Discourse and Student Choice with Exemplars

Monday, April 22nd, 2019

Written by: Brendan Scribner, 4th Grade Teacher at the Bernice A. Ray School

For the past 21 years, I’ve enjoyed the benefits of using Exemplars math problems within my classroom. The Exemplars problems are aligned to my curriculum, promote productive discourse and enable student choice. The efficient and user-friendly website allows me to spend less time planning, and more time focusing on student sense-making, intentional sharing of student thinking, and allowing for students to debate their ideas.

Laying a Foundation for Discourse

Within my 4th grade classroom, we have established ground rules for productive student discourse. It’s important to note that this foundation is built beginning on day one of the year and slowly cultivated every day. These norms help to provide a base in which we function as a learning classroom. Students must agree with and embody the norms. Students must learn and use the talk moves. We strive to have a classroom that from an outsider perspective looks, how may I say this politely, chaotic. I strive to have my students engaged in discourse related to academic talk for a minimum of 50% of all class work. It’s important that students have ownership of the talk, and I benefit greatly as the teacher listening to the discourse and planning my moves based upon the talk in the room. With a talk structure in place, this is where the Exemplars problems shine.

Classroom Norms

Student Owned Talk Moves

The Exemplars Routine

In order to support a talk rich classroom, I need to feed the learning. The Exemplars problems are rich and engaging tasks that my students eagerly unpack. With each new unit of study, I follow an instructional routine that allows students to acquire knowledge and skills in a predictable manner.

Prior to a unit of study, I visit the Exemplars Library website and review all tasks within the standard my class will be practicing. I use the summative task as my pre/post assessment for each unit. The website features underlying mathematical concepts, possible problem-solving strategies, mathematical language and symbolic notation, a planning sheet, possible solutions, and connections. These resources are user-friendly, and help to efficiently allow you as the teacher to begin to anticipate student work that will be produced for the problems presented. It is imperative that you as the teacher solve the problem on your own, and if possible, with your teaching team as well. Unpacking the problem on your own, and with colleagues, will better prepare you for guiding discourse in your classroom.

After reviewing tasks, and administering the summative as a pre-assessment, I collate all the “more challenging versions” into one condensed handout. For a recent unit: Arrays, Factors, and Multiplicative Comparison, we focused on standards 4.OA.B.4, 4.OA.A.2. As such, we used Dog Years, Feeding Lizards and Frogs, A Jumping Good Time, Making Cakes, Boxes for Mini Muffins, The Fitness Center, Hot Dogs and Buns for Friends, Marching Ants, Posters, Puppet Shows, and Snacks on the Playground. I gathered all the problems listed for these standards within one student handout. Students have a choice of which problems they attempt to solve. Students must solve at least four, then select one of the four to present to the class. To prepare the students I engage them in unpacking the grade level version of Dog Years, using a gradual release of the problem to promote contextual understanding, student engagement, and discourse.

Here is an example of how this release occurs:

Task: Dog Years

It is said that dogs age seven years for every “people year.” Mason’s dog, Shep, was born on Mason’s eighth birthday. When Mason was nine years old, Shep was seven dog years old. If Shep is fifty-six dog years old, how old is Mason? Show all your mathematical thinking.

  •  Slide1: It is said that dogs age seven years for every “people year.”
    • What do you notice?
    • Tell me 3 things about dog or people years.
  •  Slide 2:  Mason’s dog, Shep, was born on Mason’s eighth birthday.
    • Now what do you notice?
    • Are the ages of Shep and Mason a lot or a little different?
  • Slide 3: When Mason was nine years old, Shep was seven dog years old. In dog years, how old will Shep be on Mason’s twelfth birthday?
    • What do you wonder?
    • What’s the solution?
    • Estimate, how many more/less?

Scribed class anchor chart

Using a gradual release supported by classroom norms and talk moves allows for a rich and vibrant launch into problem solving. Students apply their understandings and have time to listen to all class discourse.

As the facilitator of this conversation, I scribe a class anchor chart solution for the problem. This typically includes all of the possible solutions offered on the Exemplars site. This anchor is critical as it models how we want each student to show their thinking within their math journals for every problem opportunity.

As our discussion reaches a class consensus on a solution, I then have students begin working on the Exemplars problems for the unit of study. Students will work on problems in math class each day for a part of each class session. The first two class sessions are independent work time. During the remaining class sessions, students partner with peers to engage in shared thinking time. This includes a review of work they have done alone, as well as some time working on the same problem. Again, talk moves are leveraged during partner work. After 5–7 class periods, we will begin the process of sharing our thinking.

Strategy Share Planner

In order to engage our classroom in a thoughtful and comprehensive review of the Exemplars problems, I use a strategy share planner. Students have worked toward completing at least four (many students complete all) of the problems, and now select one to share with the class. The share time often comes outside of math class. We have used snack and lunchtime effectively to ensure that we come to closure on our problems and hear from all students equitably.


We methodically review all problems solved by the class sequentially through the handout. If more than one student chose a problem, we take turns sharing each solution. This usually means we have sustained, in-depth discussions about most of the problems within the standard. This engaged sense-making opportunity allows all to share thinking, and ultimately affords us rich debates about problem-solving strategies used by all students.

Students Create Their Own Problems

A particular draw for students is the creation of their own original problem. As a result of our process of discourse, gradual release of the anchor problem, student choice, and sharing, students are typically very ready to craft their own story problems. Students write their problem and then solve them. I encourage them to use the Exemplars problem as a template. My students love to transfer their understanding and apply their developing ideas to a context that is part of their everyday life.

Using the Exemplars platform of problems has enriched my teaching practice. The resource-rich website offers readily available story problems that engage students in the practice of making sense of the world around them and appreciate that math really is everywhere. I have noticed that the intentional gradual release of problems, shared ownership of talk moves, embedded classrooms norms, and use of a strategic planner have enhanced my students’ enjoyment and success with problem-solving. I look forward to continuing to make sense of Exemplars problems with my students for many more years.

Understanding Mathematical Connections at the First Grade Level

Thursday, March 29th, 2018

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

In today’s post, we’ll look at a first-grade student’s solution for the task, “Pictures on the Wall.” This anchor paper demonstrates the criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the Exemplars assessment rubric. It also shows a solution that goes beyond arithmetic calculation and provides the evidence that a student can reflect on and apply mathematical connections. The beauty of mathematical connections is that they often begin with the other four rubric criteria. In other words, the Exemplars rubric provides multiple opportunities for a student to connect mathematically!

In this piece of student work, you’ll also notice that the teacher has “scribed” the student’s oral explanation. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

This blog will offer tips for the type of instructional support a teacher may provide during this learning time as well as the type of support students may give each other. Teacher support may range from offering direct instruction to determining if a student independently included mathematical connections in her or his solution. After reading this post, give the task a try in your own classroom along with the Exemplars rubric. You may view other Exemplars tasks here.

First Grade Task: Pictures on the Wall

There are sixteen pictures on a wall. The art teacher wants to take all the pictures off the wall to put up new pictures. The art teacher takes seven pictures off the wall. How many more pictures does the art teacher have to take off the wall? Show all your mathematical thinking.

Common Core Alignments

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

Understanding Mathematical Connections at the Third Grade Level

Thursday, March 29th, 2018

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

In today’s post, we’ll look at a third-grade student’s solution for the task “Bracelets to Sell.” This task is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 3.OA.3. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these in your classroom along with the sample task and the Exemplars assessment rubric. How many mathematical connections can your students come up with?

3rd Grade Task: Bracelets to Sell

Kathy has thirty-six bracelets to sell in her store. Kathy wants to display the bracelets in rows on a shelf. Kathy wants to have the same number of bracelets in each row. What are four different ways Kathy can display the bracelets in rows on the shelf? Each bracelet costs three dollars. If Kathy sells all the bracelets, how much money will she make? Show all of your mathematical thinking.

 Common Core Alignments

  • Content Standard 3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6

Understanding Mathematical Connections at the Fifth Grade Level

Thursday, March 29th, 2018

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

In today’s post, we’ll look at a fifth grade student’s solution for the task “Seashells for Lydia.” This task is one of a number of Exemplars tasks aligned to the Number and Operations in Base Ten standard 5.NBT.2. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these along with the task and the Exemplars assessment rubric in your classroom. How many mathematical connections can your students come up with?

5th Grade Task: Seashells for Lydia

Lydia started collecting seashells when she was five years old. At age seven, Lydia had 12(10)2 seashells. At age nine, Lydia had 24(10)2 seashells. At age eleven, Lydia had 48(10)2 seashells. Lydia wants to collect 75(10)3 seashells. Lydia continues to collect seashells at the same rate. How old will Lydia be when she has 75(10)3 seashells? Show all of your mathematical thinking.

Common Core Alignments

  • Content Standard 5.NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6, MP7

Understanding Mathematical Connections

Thursday, March 29th, 2018

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

What is a mathematical connection? Why are mathematical connections important? Why are they considered part of the Exemplars rubric criteria? And how can I encourage my students to become more independent in making mathematical connections?

This blog represents Part 1 of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students. We find many students enjoy making connections once they learn how to reflect and question effectively.

A Brief Introduction to the Exemplars Rubric

The Exemplars assessment rubric allows teachers to examine student work against a set of analytic assessment criteria to determine where the student is performing in relationship to each of these criteria. Teachers use this tool to evaluate their students’ problem-solving abilities.

The Exemplars assessment rubric is designed to identify what is important, define what meets the standard and distinguish between different levels of student performance. The rubric consists of four performance levels — Novice, Apprentice, Practitioner (meets the standard) and Expert — and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation). Our rubric criteria reflect the Common Core Standards for Mathematical Practice and parallel the National Council of Teachers of Mathematics (NCTM) Process Standards.

The Importance of Mathematical Connections

Exemplars refers to connections as “mathematically relevant observations that students make about their problem-solving solutions.” Connections require students to look at their solutions and reflect. What a student notices in her or his solution links to current or prior learning, helps that student discover new learning and relates the solution mathematically to one’s own world. A student is considered proficient in meeting this rubric criterion when “mathematical connections or observations are recognized that link both the mathematics and the situation in the task.”

NCTM defines mathematical connections in Principals and Standards for School Mathematics as the ability to “recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.” (64)

The Common Core State Standards for Mathematics (CCSSM) support the need for students to make mathematical connections in problem solving. Reference to this can be found in the following Standards for Mathematical Practice:

  • MP3: Construct viable arguments and critique the reasoning of others. “… They justify their conclusions, communicate them to others, and respond to the arguments of others.”
  • MP4: Model with mathematics. “… They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”
  • MP6: Attend to precision. “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose … They are careful about specifying the units of measure and labeling axes … They calculate accurately and efficiently express numerical answers with a degree of precision appropriate …”
  • MP7: Look for and make use of structure. “Mathematically proficient students look closely to discern a pattern or structure …”
  • MP8: Look for and express regularity in repeated reasoning. “… They continually evaluate the reasonableness of their intermediate results.”

The CCSSM also state, “The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word ‘understand’ are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations …” (Common Core Standards Initiative, 2015)

When students apply the criteria of the Exemplars rubric, they understand that their solution is more than just stating an answer. Part of that solution is taking time to reflect on their work and make a mathematical connection to share.

What Can Teachers Do to Help Students Make Mathematically Relevant Connections?

When students begin to explore mathematical connections, teachers should take the lead by providing formative assessment tasks that introduce new learning opportunities and provide practice, so they may become independent problem solvers. As part of this process, teachers will want to focus on five key areas to help students develop an understanding of mathematical connections.

(1) Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct. When students demonstrate an additional or new strategy or representation in solving a problem, a mathematical connection is made. The Common Core includes a variety of representations students can apply to solve a problem and justify their thinking. Examples include manipulatives, models, five and ten frames, diagrams, keys, number lines, tally charts, tables, charts, arrays, picture graphs, bar graphs, linear graphs, graphs with coordinates, area/visual models, set models, linear models and line plots. By practicing these different approaches, students will begin to create new strategies and representations that are accurate and appropriate to their grade level. This in turn opens the door for them to use a second or even third representation to show their thinking in a new way or to justify and support that their answer(s) is correct.

Using formative problem-solving tasks to introduce and practice new strategies and representations is part of the problem-solving process. Teachers should provide formal instruction so that students may grow to independently determine and construct strategies or representations that match the task they are given. An example of this can be seen in the primary grades when many teachers introduce representations in the following order: manipulative/model, to diagram (including a key when students are ready), to five/ten frames, to tally charts, to tables, to number lines. This order allows students to move from the most concrete to the more abstract representations.

(2) Encourage students to continue their representations. Mathematical connections may be made when students continue a representation beyond the correct answer. Examples of this can be seen when a table or linear graph is continued from seven days to 14 days or when two more cats are added to a diagram of 10 cats to discover how many total ears a dozen cats would have. Another example includes adding supplemental information to a chart such as a column for decimals in a table that already has a column indicating the fractional data. In this case, the student extends his or her thinking to incorporate other mathematics to solve the task. It is important to note that connections must be relevant to the task at hand. In order to meet the standard, a connection must link the math in the task to the situation in the task.

(3) Explore the rich formal language of mathematics. Mathematical connections may be made as students begin to use the formal language of mathematics and its connection to their representations, calculations and solutions. Mathematical connections can be seen in the following examples: two books is called a pair; 12 papers is a dozen, the pattern is a multiple of 10; 13 is a prime number so 13 balls can’t be equally placed in two buckets; and the triangle formed is isosceles. The input and output on a table can also help students generalize a rule with defined variables. Students will quickly learn that making connections promotes math communication (formal terms and symbols) and that using math communication promotes connections. Again, these connections must link the math in the task to the situation that has been presented.

(4) Incorporate inquiry into the problem-solving process. Asking students to clarify, explain, support a part of their solution to a math partner, the whole class, or a teacher, not only helps develop independent problem solvers but also leads to more math connections. In your discussions, use verbs from Depth of Knowledge 2 (identify, interpret, state important information/cues, compare, relate, make an observation, show) and from Depth of Knowledge 3 (construct, formulate, verify, explain math phenomena, hypothesize, differentiate, revise). By asking students questions that provide them the opportunity to show and share what they know, connections become a natural part of their solutions.

Instead of asking, “Do you see a pattern in your table?” say, “Did you notice anything about the numbers in each column in your table?” Try asking a primary student, “I know you have a cat. Would you like your cat to join the cats in your problem?” “What new numbers are you using?” “I heard you tell Maria that all the numbers in your second column are even. Can you help me understand why they are all even numbers?” Every time a student provides you with a correct answer to your or another student’s inquiry, stop and say, “Thank you for explaining/showing/sharing your thinking. You just made a mathematical connection about your problem.” If you hear a student make a mathematical connection outside of class, stop and comment, “You just made a math connection!” Some examples of these student connections may include, “Look, we are lined up as girl, boy, girl, boy, girl, boy for lunch. That is a pattern,” “In four more days it is my birthday,” “Art class is in 15 minutes because we always go to art at 10 o’clock,” “We can have an equal number of kids at each table because four times six equals 24,” “My dad says we have to drive 45 miles per hour because that is the speed limit, so I think I can write each student as ‘per student’” or “I think I can state all the decimals on my table as fractions.”

(5) Encourage self- and peer-assessment opportunities in your classroom. Encourage students to self-assess their problem-solving solutions either independently, with a math partner or with the support of their teacher. The more opportunity students have to use the criteria of the Exemplars assessment rubric to evaluate their work, the more independent they become in forming their solutions, which will include making mathematically relevant connections.

Exploring Authentic Examples of Mathematical Connections

In the next blog post of this series, we’ll look at a problem-solving task and student solution from Grade 1 to observe how mathematical connections have been effectively incorporated. We’ll also explore the type of support a teacher may provide during this learning time as well as the type of support students may give each other. (Solutions from Grade 3 and Grade 5 will follow in subsequent posts of this series.)


A Problem-Solving Lab to Support the Math Practices

Monday, October 31st, 2016

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

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

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

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

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

7 Things I’ve Learned on My Journey to Implementing Problem Solving in the Classroom

Monday, October 3rd, 2016

Written By: Suzanne Hood, Instructional Coach, Georgia

I’ve always believed in the power of students to use their own childlike curiosity to problem solve. These problem-solving experiences happen for our students naturally, through the math they use in cooking, playing games and playing with toys, among other things. Problem solving is a life-long skill all mathematicians use. The true power of a mathematician is the ability to see math in all situations and solve problems using a toolbox of proven strategies.

While I believe that students are innate problem solvers, I also believe that learned algorithmic thinking corrupts a child’s natural ability to problem solve and discourages perseverance. Although I have met many teachers who share my belief that problem solving should be the focus of the math, many struggle to create this culture in their classroom.

This is becoming more apparent—and the stakes of ignoring problem solving much higher—as we approach testing season. The classrooms that will likely fall behind in this new era are those who insist on teaching math through algorithmic thinking. Conversely, I am convinced that teachers who use problem solving to teach math, supported by materials like Exemplars, will have students who score proficiently on the state assessment and are more prepared for success beyond the classroom.

So how can teachers help their classrooms make this critical transition to problem solving? My personal story of transformation, which began after participating in one of Exemplars’ Summer Institutes, offers a path forward. This was when I realized two important things: first, I needed to work on my own personal proficiency in teaching problem solving. And second, I wasn’t alone; veteran teachers confessed their frustration in teaching problem solving, and many admitted their backgrounds did not include support in how to instruct students through the problem-solving process. Here are seven things I’ve learned on my journey to becoming an educator fully committed to teaching mathematics through a problem-solving approach.

1. Nurture a community of trust.

Based on my experience as a Mathematical Instructional Coach in Georgia, I believe it is essential to nurture relationships and establish a community of trust between teachers, so that discussions are authentic and all voices are included. Trust is a prerequisite for being able to assess the strengths, weaknesses and gaps of teacher readiness in the classroom. Only when teachers feel they are in an environment where they can share their knowledge, their doubts and their pedagogical weaknesses, will they be able to feel comfortable.

2. Establish a baseline of teacher readiness.

Evaluating teacher readiness and needs and getting them on the same page is an important first step. How can you get teacher teams to have collegial conversations when everyone has a totally different math background? Do all teachers even want a problem-solving classroom? Do they know what that means? Asking these questions can be illuminating, albeit tough. As such, using universally agreed-upon protocols such as those from the National School Reform Facility can establish a baseline to work from, encourage collaboration, and support an atmosphere of trust.

3. Assess student work so you can see where the gaps are.

One way to assess teacher acuity and readiness in teaching problem solving is by assessing student work using an Exemplars task. Here’s how it worked for me: At the first Professional Learning session, I asked teachers to bring classroom samples from their most recent classroom Exemplars task. As a community, we agreed to facilitate the discussion with the protocol Atlas – Learning From Student Work. As I observed teachers at the meeting, I noticed that while some teachers were proud to display their samples, others pretended to forget their samples or chose to stick their student work in their tote bag. As we used the Exemplars standards-based rubric to score our samples, it became clear that our understanding of the skills needed to meet the standards did not align. The journey began; teachers began to talk about problem solving.

4. As a team, align your mathematical beliefs towards problem solving.

When we began, we knew we shared some foundational mathematical beliefs. We also knew that we needed to solidify a shared understanding of how a mathematics culture transfers knowledge from the teacher to the student. We used the Math Framework (a document listing all the mathematical beliefs of the faculty) as a tool to target instructional strengths and weaknesses. As a team, we revised the document to build cohesion and a shared understanding of our beliefs. Next, I had the team read a book rooted in Vygotsky’s constructivist theory to increase our group’s understanding of the problem-solving trajectory. Because we had been working hard to build an atmosphere of trust, teachers felt safe sharing their struggles and personal hardships with teaching problem solving. We discovered that we shared similar experiences, and that we all believed our students would be successful at any problem if we just taught them the necessary skill set. The student samples, however, told a different story.

5. Create simple tools to help teachers and students internalize the standards and assess their progress.

At our next meeting, we reviewed Exemplars student work samples and discovered a misconception: we thought we knew how to teach problem solving, but we were actually teaching skills in isolation. Why? Quite simply, it turns out that many teachers lacked background knowledge about the Standards of Problem Solving. To facilitate the understanding of the standards, I created posters with clear icons for each standard. These anchor charts would support teachers and students. It worked. Now, teachers could explain each standard. Each classroom in our building displayed the posters. It was a great reference for both students and teachers. We made a replica of the posters into a small book that students put in folders for their own reference. Students used the folders as portfolios to track their problem-solving progress, and created data notebooks to reflect on their growth and set goals for their next Exemplars task. Using data notebooks empowered kids to self-reflect on their own progress.

6. Hold individual meetings with students to track progress and set goals.

Currently, I am encouraging teachers to hold one-on-one Exemplars conferences with their students. Individual conferences support differentiated instruction, meet students where they are, and set goals for the next problem-solving task. Although this approach makes some teachers uneasy at first, they become more confident over time. Allowing other teachers or coaches to observe and co-teach the process can lead to greater transparency and effect change in teacher practice.

7. You may not get the teacher of the year award, but you’ll still be changing students’ lives.

At the beginning of my career, I thought Oprah would call me to announce my Disney Teacher of the Year Award. While this hasn’t happened yet, I do have countless memories of the sparkle in a child’s eye when he or she announces, “I get it!” I believe I have the responsibility to show up every day prepared to change the lives of children and equip them with the skills to be life-long mathematicians. Exemplars provides the problem-solving tools necessary to guide teaching and build capacity for each child’s mathematical journey.

Preparing our Kids for Success Beyond the Classroom

Friday, March 25th, 2016

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 consistently strong standards from state to state. In the past, many standards have lacked focus and coherence, giving 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, analyze quantitative data and to communicate verbally and in writing.

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

In recent years, many 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.

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 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 to 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.

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