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Standards-based assessment and Instruction


Archive for the ‘Math’ 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.

How PS 12, the Dr. Jacqueline Peek-Davis School, Took Student Math Scores “Ahead of the Pack”

Tuesday, December 11th, 2018

Exemplars changed my teaching,” says one teacher at Brooklyn’s PS 12, the Dr. Jacqueline Peek-Davis School. And it changed the school’s learning outcomes, too: So substantial were its improvements in math scale scores that it made New York City’s top 10 list of positive percentage changes for 2017.

Why do the school’s educators attribute that to Exemplars? Aldine Finnikin-Charles, a special education teacher at Brooklyn’s PS 12, credits the problem-solving procedure students learn as they work through our tasks. “That’s the best thing for them,” she says. “Exemplars changed the way my students look at problems and how they unpack them.” Challenging students with complex multi-step tasks, Problem Solving for the 21st Century: Built for the Common Core gives them a robust framework for developing solutions — so they’re well equipped to tackle such problems on standardized tests.

LaToya Garcia, assistant principal and former teacher at PS 12, cites the practice students get through regular classroom use. “Students in previous years had struggled with the test’s open response questions and the writing that was required with the math,” she says. That changed with Exemplars, since every task challenges students to communicate their thinking. “Exemplars really gives them the practice and support that they need,” says LaToya, “not only to be able to solve the problem, but also to solve it using multiple strategies, critically think about how they could expand the problem, and explain their thinking in a precise way.”

Nyree Dixon, the school’s principal during the 2017-2018 school year, says one of the keys to PS 12’s improvements has been Exemplars’ impact on student self-assessment. With Exemplars standards-based rubric to support them, she says, “Our children are not waiting for the teacher. They’re not waiting for ‘authority’ or a supervisor. They’re assessing their work themselves, and aligning it to the rubric.” As they do this, they develop critical skills of self-assessment and metacognition, which puts them, Nyree says simply, “ahead of the pack” — and enjoying a real improvement in learning outcomes, too.

We thank the Brooklyn PS 12 team for making us part of their school’s top 10 story, and for sharing their thoughts about how Exemplars helps their students succeed. 

A Mindset Shift in Problem Solving

Monday, October 1st, 2018

By: LaToya Garcia, Asst. Principal, & Kimberly Naidu, Third Grade Teacher, at P.S. 12, Brooklyn, NY

Exemplars Problem Solving Procedure has demonstrated that students can be very creative when they are allowed to choose their own methods for solving mathematical problems. Many educators, we believe, focus on enhancing mathematics skills to meet academic expectations, but fail to promote more critical thinking. Without the ability to inquire while thinking, students will not learn to make decisions.

At our school, Exemplars has given students the opportunity to learn from one another and provide opportunities to use multiple solution strategies for the same problem. The Problem Solving Procedure has supported students in understanding a problem, identifying a strategy to use, solve the problem, communicate findings in words, and trying to solve a problem in another way.

Exemplars has also helped to prepare our teachers for a more cognitively guided instruction. Our goal was to modify our current curriculum to help students become mathematically literate citizens. We realized that problem solving wasn’t about knowing the solution to the problem, it was about understanding that knowledge. In order to enhance learning, our teachers use a student’s prior knowledge, transfer that information to new situations and organize their students’ knowledge using model-based problems from the Exemplars Library. Teachers are able to plan activities to help students build their mathematical knowledge through problem solving and reflect on the process of mathematical problem solving. Our teachers find that students connect different meanings, interpretations and relationships to the four operations in math.

Our training with Exemplars helped to develop a more differentiated learning environment as well. We now integrate mathematics across our curriculum. For example, teachers use more math read-alouds in the classroom as a visual interpretation of mathematical problems. Teachers modify problems from the Exemplars Library by simply changing the character’s name to one students recognized from a literacy lesson. Students are also provided with manipulatives to show math situations. Manipulatives and other mathematical tools are made apart of our daily routines. Materials are made accessible in math centers and students are encouraged to self-select resources when needed. Our entire school community understands that our students need to improve in the area of communication. Students practice using math language to explain what they did while solving a problem and represent their math thinking through drawings and diagrams.

Exemplars has helped to guide and organize our students’ thought processes–a skill that can be transferred to all content areas. Throughout the school, students have begun to recognize the positive impact of a problem-solving strategy guided by specific steps. A third-grade student Michael C. commented, “Exemplars is fun. It helps you to do math and writing. While you’re doing Exemplars, you can also work on your writing.” Another third-grade student Desayee S. said, “What I like about Exemplars math is that it helps me to practice using different strategies for math, like diagrams, models and tables.”

Additionally, Exemplars has led to an increase in student-to-student conversation in students who had previously shied away from mathematics and students who are currently learning English as a second language.

As the year continues and students become even more experienced Exemplars math problem-solvers, we anticipate an improvement in students’ critical thinking and problem-solving skills. Students have already begun to develop a love for mathematics and are learning to persevere until they arrive at their answer.



10 Things Educators Love About Exemplars, Plus 1 Big Bonus

Monday, October 1st, 2018

We asked a powerhouse teaching team to share their Exemplars experience. In an inspirational conversation, a panel of educators at the Brooklyn’s PS 12 offered ten amazing ways that using Exemplars has transformed their school.

1. It encourages students to be flexible and creative in their approach to solving problems. “My students love the freedom Exemplars tasks give them,” says Kimberly Naidu, a third-grade teacher. “They’re never forced to use a specific strategy. They can create one or use one they’ve learned before.”

2. It builds students’ math vocabulary. Aldine Finnikin-Charles, an experienced special education teacher, says, “My students are always using math nicknames,” says Aldine, “so the tasks that require them to use at least two math vocabulary terms really help them a lot.”

3. It helps students become capable peer and self-assessors. PS 12 teachers ask students to consider their own work alongside Exemplars anchor papers. “It really helps them get a good understanding of how to meet the standard,” says Kimberly. “They see exactly what needs to be done.”

4. It spurs students to demonstrate their understanding. They learn to communicate their ideas clearly and effectively. Plus, it’s fun, says Cherron Knight, who teaches grades 4 and 5: “Our students love showing their strategies. They can always argue, ‘Why did you use that strategy?’ They love that part, explaining what they think.”

5. It motivates students to make new mathematical connections. “When kids bring their work up to me, I say, ‘Go back and see what else you can do. What’s a different way of showing it?’ They find that enjoyable,” says Kimberly, “because they can always do it another way.”

6. It lets students practice a process for solving problems — so they can internalize it and apply it to every task. “That’s the greatest thing,” says Aldine, “the best thing for them. They are able to become reflective thinkers.”

7. It gives students a clear understanding of what kind of work meets the standards. Kimberly says, “They really want to become Experts,” and because of the Exemplars rubric, “they know exactly what needs to be done to achieve that level.”

8. It empowers students as peer and self-assessors. With what Principal Nyree Dixon calls “these amazing rubrics” that are at the heart of Problem Solving for the 21st Century, “they’re not waiting for the teacher,” she says. “They are not waiting for quote-unquote ‘authority.’ They are assessing their work themselves,” and experiencing the benefits.

9. It inspires students to grow as problem solvers — and as leaders, too. When kids solve Exemplars tasks in a problem-solving classroom, they become the experts. And they become skilled collaborators and advocates, Nyree says. “They’re learning to work together and disagree or agree,” she explains. “They’re learning to take a stance and defend their responses and answers.”

10. It helps educators establish a classroom culture of problem solving. In a lot of schools, this constitutes a major shift in thinking. “We went through growing pains. We still do,” Nyree says. “We have to keep our minds open and be willing and ready to learn.” And we’re there to help them every step of the way. After all, every teacher in every school should be able to say, as Nyree does, “We do not ever, ever, ever lower our expectations.”

The PS 12 team reported another benefit, too: Their students made a dramatic jump in math scores on standardized tests, an improvement they attribute to Exemplars — but more importantly, students made enormous strides forward as problem solvers.

“I think Exemplars did a really, really good job in helping the students to think critically about the math that they were learning,” says LaToya Garcia, Assistant Principal. “Exemplars really gave them the practice and support that they needed to be able to not only solve the problem but also to solve it using multiple strategies, critically think about how they could expand the problem, and explain their thinking in a precise way.”

Our thanks to the team at PS 12 for sharing their insights. We’re thrilled to be a part of your students’ success.

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.)


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