Standards-based assessment and Instruction

# Archive for the ‘Education’ Category

## Supporting the TEKS Mathematical Processes with Exemplars Performance Tasks and Rubric at the Kindergarten Level

Tuesday, February 9th, 2016

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

#### Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 1 of a six-part series that features a problem-solving task linked to a Unit of Study for the Math TEKS and a student’s solution in grades K–5. Evidence of all seven Mathematical Process Standards will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress towards being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Processes is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with scoring rationales that demonstrate the alignment between the Exemplars assessment rubric and the Math TEKS can be insightful for educators. Anchor papers and scoring rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Processes are embedded in the scoring rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in-hand while reviewing a piece of student work.

#### Blog 1: Observations at the Kindergarten Level

The first anchor paper and set of assessment rationales we’ll review in this series is taken from a Kindergarten student’s solution for the task, “Boots.” You will notice that the teacher has “scribed” the student’s oral explanation. This method allows teachers to fully capture the mathematical reasoning of early writers.

“Boots” is one of a number of Exemplars tasks aligned to the Counting and Cardinality Unit designed by Exemplars for the new Math TEKS. This task could be used toward the end of the learning time allocated to this Unit. Prior to “Boots” being given, students have already completed a number of tasks with questions that state, “How many ears?”, “How many shoes?”, “How many balloons?”, etc. “Boots” gives students an opportunity to bring a stronger understanding of the concept how many to their solution.

Five students wear boots to go outside for recess. When the students come in from recess they must put all boots on a rubber mat to dry. The teacher counts seven boots on the mat. The teacher thinks some boots are missing. Is the teacher correct? Show and tell how you know.

The Counting and Cardinality Unit involves understanding numbers and how they are used to name quantities and to answer questions, such as:

• How many balls is the clown juggling?
• Do you have enough cups for each member of your group to have one?

The TEKS standards covered in this Unit include K.2 Numbers and Operations:

• K.2A count forward and backward to at least 20 with and without objects.
• K.2B read, write, and represent whole numbers from 0 to at least 20 with and without objects or pictures.
• K.2C count a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order.
• K.2D recognize instantly the quantity of a small group of objects in organized and random arrangements.

Mathematical Process Standards: K.1A, K.1B, K.1C, K.1D, K.1E and K.1G

## Great Minds Don’t Always Think Alike

Friday, January 15th, 2016

Written By: Jaclyn Mazzone, Fifth Grade Special Education Teacher at P.S. 94 in Brooklyn, NY

Working in special education, I help students with special needs as well as other students who struggle with math. One of the most beneficial features of Exemplars is the ability to differentiate easily for struggling learners. Some need just a little extra support through small group instruction. The more accessible version of Exemplars tasks is perfect for them!

The more accessible versions present the same problem-solving elements as the grade-level task and the more challenging task, but in a simplified form: they’re less wordy, involve simpler numbers, and require fewer steps to complete. At the same time, they’re still demanding enough that they challenge each student to think more deeply to determine what the question is asking, decide what information they need to know, choose a strategy to solve the problem, complete the mathematical computations, and show their work.

### Repetition Is Key

Specifically in my class, I’ve noticed repetition is key. My students complete Exemplars at least once a week, so they’ve been able to build a routine. A colored version of the Problem Solving Procedure is provided for each student. It has become automatic for them to carry out the first two steps of the Procedure independently. They start out by reading the problem twice; the first time they just read it, and the second time they annotate, highlight/locate important information in the problem, and underline the question. Once they’ve determined what the problem is asking them to do, they then move on to writing their “I have …” and “I will …” statements independently. Next, as a group we read the entire problem together — their third time to read it — ensuring that the important information has been located, underlined, or highlighted, and that they understand what the problem is asking them to figure out. Many of my students are English Language Learners so language is also sometimes an issue for them. Therefore, we carefully break apart the math tasks sentence by sentence. We address any unknown vocabulary and I encourage them to use their prior knowledge to connect to the problem.

### Small-Group Discussion and Scaffolding

Then, the students engage in a small-group discussion and discuss what the problem is about. They brainstorm different strategies that could be used to solve the problem and discuss why those strategies would work. Then I step in to scaffold and guide them to set up their chosen strategy correctly, whether it be a table, number line, diagram etc. Once I’ve supported them in setting up their chosen strategy and answered any questions they might have, they work with their small group to continue solving the problem. The students know to show all of their mathematical thinking on paper — this expectation has been instilled since day one of their work with Exemplars. As they work, I circulate around the room to provide one-on-one support where necessary.

After I allow approximately 20-25 minutes for them to complete the problem with their peers, we come back together as a group and discuss the different ways we solved it. The students then move on to the next step of the Problem Solving Procedure by making mathematical connections. I encourage them to try to extend the problem, write what they notice in their work, or try to find a pattern. Because connections are still rather challenging for them, I link the task to our current unit of study or math topics we have previously worked with to help them make those connections.

### The Importance of Tools

Tools are important resource for my students. Working with learners in a small group, I am able to accommodate their different learning styles. Depending on the task, I use manipulatives such as connecting cubes, fraction bars, unit cubes, or sometimes even a basic multiplication table chart to further support their needs.

### Peer Assessment

My students also peer-assess each other’s work. They swap papers with a partner and use the Exemplars student-friendly rubric to score their partner’s work. Since the student-friendly rubric addresses the same criteria as the teacher rubric in a simpler form, this practice has made them very familiar with the four levels of achievement and what’s required to become a Practitioner (meet the standard).

Peer assessment has also built up their self-confidence. As they assess each other’s work, I ask them to find a “Glow” statement — something their classmate did well — and a “Grow” statement — an area for improvement. In doing this, they notice that other students have challenges, too, and might even make the same mistakes as they do. Peer assessment has not always been an easy task for them, but through repetition and teacher feedback, the students have picked up on sophisticated math language to use when providing helpful Glow/Grow comments to peers.

###### An example of a student’s peer-assessment comments using the “Glow and Grow” strategy.

Exemplars has been a very beneficial tool in my classroom to reinforce the mathematical concepts we learn throughout the year. Through careful scaffolding and support, my students have learned to persevere when problem solving, show all of their mathematical thinking, use different strategies to achieve the same answer, and provide peer feedback. My students consistently aim to achieve a level of Practitioner and some even aim for Expert. Exemplars has become a weekly activity in my classroom. The students truly find it enjoyable!

###### Jaclyn Mazzone’s Biography

Jaclyn Mazzone is a fifth-grade special education teacher at P.S. 94 in Sunset Park, Brooklyn. She has a master’s degree in special education from Touro College and a bachelor’s degree in childhood education from the College of Staten Island.

## Exemplars in the Classroom: “They Want to Become Experts.”

Tuesday, October 27th, 2015

Written By: Danielle Descarfino, Fifth Grade Teacher at P.S. 94 in Brooklyn

#### Getting Started

From the beginning of the school year, I used Exemplars problem-solving tasks regularly to create routines that have helped my fifth grade students grow and succeed. Following the Problem-Solving Procedure is a central part of this.

Although each task is different, the procedure helps kids internalize a framework for approaching a problem. I provided each student with his or her own color copy (in a sheet protector for safe keeping.) Each time we begin an Exemplars task, the students take out their Problem-Solving Procedures and refer to it. I also have a poster-sized version prominently displayed in the classroom, which I hold up and point to while guiding and facilitating tasks.

#### Building Background Knowledge

My class is made up of English Language Learners and former English Language Learners, so I anticipate that reading and understanding the problem may be especially challenging for them. We read the problem together, I ask questions to activate their background knowledge, and I often provide pictures that help them visualize the problem.

For example, we recently completed “A New Aquarium,” a 5.MD.C.5a task involving volume. We had been working on this math concept for only a few days and this was our first volume Exemplars task. Before reading the problem, I displayed a photo of an aquarium on the Smart Board and discussed the following questions with the class:

• What is an aquarium?
• What type of solid is this aquarium?
• How could you figure out how much space this aquarium takes up? What steps would you take?

Although many students initially were not familiar with the word “aquarium,” after this discussion, they understood that an aquarium is a fish tank and a rectangular prism, which meant that we would be calculating its volume to find out how much space it takes up. Using visual aids and background questions to ensure that students understand the situation in the problem has been very helpful when completing Exemplars with English Language Learners.

#### Differentiation

We always utilize the differentiated Exemplars tasks. Students are aware of which problem-solving group they are in and know where to sit when it is time for an Exemplars task. One group gets the More Accessible Version; they are guided through the problem as they work with the Special Education teacher at a kidney-shaped table. The other two groups receive the Grade Level and More Challenging versions and sit with their groups in desk clusters, like a team of problem solvers.

For the Grade Level and More Challenging groups, we discuss background information, read the problem out loud, annotate it, and write our “I have to find …” statements. Then the students go on to work with their groups to complete the task while the teacher takes on the role of a facilitator, conferring with groups. Students share ideas, address misconceptions, and explain their mathematical reasoning to one another as they solve.

#### Motivating Students

I love hanging Exemplars tasks on bulletin boards. I think it’s useful for students to look at the page and see all of the different ways their classmates organize and express their mathematical thinking through equations, representations, and writing.

From day one, I have made it clear that it is expected that their finished work clearly communicate their problem-solving steps to the reader. Not only should the students make an effort to write neatly, but they should also organize their problem-solving steps on the page in a way that makes sense. Sometimes if a student is not showing all of their steps or it is unclear, I’ll say, “I am confused. When I look at your paper, I don’t understand the steps you took to solve the problem.” When the students have the understanding that a goal is to communicate their math thinking to a reader, it helps them create a higher-quality finished product.

Another great way to motivate students is through mathematical connections. I have given a strong emphasis to connections, as I initially noticed that once students solve the problem, they feel like they are done! This is not the case, because noticing mathematical connections, patterns, and alternate strategies really helps students understand mathematics on a deeper level and practice critical thinking skills.

To help them stretch their thinking, I discourage students from writing “boring” connections, like “This number is greater than that number” or “John ate the least amount of pizza.” Instead, I encourage them to use mathematical language, create a second representation, show steps to solving with alternate strategies, convert fractions/decimals/percents, or extend the problem by adding to the story in the original problem. Once they get the hang of it, they start being more creative, going above and beyond to make more complex math connections. During the volume unit, I taught students how to use grid paper to make scaled models of rectangular prisms. When completing these tasks, many students decided to build models to represent the rectangular prisms in the task and attach them to make 3-D Exemplars. They looked great, and the students loved making them!

#### Peer Assessing

At the beginning of the year, I explained each portion of the Exemplars rubric to the students. The rubrics are very student-friendly and I find that they inspire students to want to become Experts.

Each time I assess Exemplars, I use the rubric along with a sticky note full of feedback. The sticky note always contains one “Glow,” something the student did well, and one “Grow,” something the student could improve upon. At the beginning of the year, I let the students know that when they become more comfortable with Exemplars, they would learn how to peer assess. After a few months, I told the students that they were ready to peer assess one another’s work. They were so excited! This made them feel proud that they had reached a new level of expertise in problem solving and feel empowered that they were now trusted to assess a classmate’s work.

To peer assess, they do exactly as the teacher has done all year: complete the student rubric and use a sticky note to write “Glow and Grow” feedback. An example of this can be seen below. From the start, I was so impressed at how well the students were able to assess one another’s work with Exemplars. I found that regularly providing students with written feedback and referring to the rubric when expressing expectations is a great way to model peer-assessment. Furthermore, the experience of assessing Exemplars helps students get new ideas from their classmates and become more aware of how their own work will be graded.

(More Accessible Version)

Joseph has a new rectangular aquarium. The aquarium has a length of four feet, a width of two feet, and a height of two feet. What is the volume of Joseph’s new aquarium? An aquarium holds one inch in length of fish for each twelve square inches of the area of the base of the aquarium. Joseph can buy fish in two different sizes—about three inches in length or about five inches in length. About how many three-inch fish can Joseph put in the new aquarium? About how many five-inch fish can Joseph put in the new aquarium? Show all your mathematical thinking.

Danielle’s Biography

Danielle Descarfino is a fifth grade teacher at P.S. 94 in Sunset Park, Brooklyn. She graduated from Fordham University with a Masters of Science in Teaching English to Speakers of Other Languages. Danielle grew up in Tappan, New York, and currently lives in Brooklyn. She was inspired to become a teacher after spending time as a volunteer teaching English at an orphanage and community center in Salvador, Brazil.

## Preparing for the New Math TEKS: Using Rubrics to Guide Teachers and Students

Tuesday, October 6th, 2015

By: Ross Brewer, Ph.D., Exemplars President

As you begin preparing your staff to focus on the new math TEKS this year, rubrics should play a key role in terms of helping your teachers and students achieve success with the new standards.

#### What are rubrics?

A rubric is a guide used for assessing student work. It consists of criteria that describe what is being assessed as well as different levels of performance.

Rubrics do three things:

1. The criteria in a rubric tell us what is considered important enough to assess.
2. The levels of performance in a rubric allow us to determine work that meets the standard and that which does not.
3. The levels of performance in a rubric also allow us to distinguish between different levels of student achievement among the set criteria.

#### Why should teachers use them?

The assessment shifts in the new math TEKS pose challenges for many students. The use of rubrics allow teachers to more easily identify these areas and address them.

For Consistency. Rubrics help teachers consistently assess students from problem to problem and with other teachers through a common lens. As a result, both teachers and students have a much better sense of where students stand with regard to meeting the standards.

To Guide Instruction. Because rubrics focus on different dimensions of performance, teachers gain important, more precise information about how they need to adjust their teaching and learning activities to close the gap between a student’s performance and meeting the standard.

To Guide Feedback. Similarly, the criteria of the rubric guides teachers in the kind of feedback they offer students in order to help them improve performance. Here are four guiding questions that teachers can use as part of this process:

• What do we know the student knows?
• What are they ready to learn?
• What do they need to practice?
• What do they need to be retaught?

#### How do students benefit?

Rubrics provide students with important information about what is expected and what kind of work meets the standard. Rubrics allow students to self-assess as they work on and complete problems. Meeting the standard becomes a process in which students can understand where they have been, where they are now and where they need to go. A rubric is a guide for this journey rather than a blind walk through an assessment maze.

Important research shows that teaching students to be strong self-assessors and peer-assessors are among the most effective educational interventions that teachers can take. If students know what is expected and how to assess their effort as they complete their work, they will perform at much higher levels than students who do not have this knowledge. Similarly, if students assess one another’s work they learn from each other as they describe and discuss their solutions. Research indicates that lower performing students benefit the most from these strategies.

#### Rubrics to Support the New Math TEKS.

Exemplars assessment rubric criteria reflect the TEKS Mathematical Process Standards and parallel the NCTM Process Standards. Exemplars rubric consists of four performance levels (Novice, Apprentice, Practitioner (meets standard) and Expert) and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation).

Our rubrics are a free resource. To help teachers see the connection between our assessment rubric and the TEKS Mathematical Process Standards, we have developed the following document: Math Exemplars: A Perfect Complement for the TEKS Mathematical Process Standards aligns each of the Process Standards to the corresponding sections of the Exemplars assessment rubric.

It’s never too young to start.

Students can begin to self-assess in Kindergarten. At Exemplars, we encourage younger students to start by using a simple thumbs up, thumbs sideways, thumbs down assessment as seen in the video at the bottom of the page.

Our most popular student rubric is the Exemplars Jigsaw Rubric. This rubric has visual and  verbal descriptions of each criterion in the Exemplars Standard Rubric along with the four levels of performance. Using this rubric, students are able to:

• Self-monitor.
• Self-correct.
• Use feedback to guide their learning process.

#### How to introduce rubrics into the classroom.

In order for students to fully understand the rubric that is being used to assess their performance, they need to be introduced to the general concept first. Teachers often begin this process by developing rubrics with students that do not address a specific content area. Instead, they create rubrics around classroom management, playground behavior, homework, lunchroom behavior, following criteria with a substitute teacher, etc. For specific tips and examples, click here.

After building a number of rubrics with students, a teacher can introduce the Exemplars assessment rubric. To do this effectively, we suggest that teachers discuss the various criteria and levels of performance with their class. Once this has been done,  a piece of student work can be put on an overhead. Then, using our assessment rubric, ask students to assess it. Let them discuss any difference in opinion so they may better understand each criterion and the four performance levels. Going through this process helps students develop a solid understanding of what an assessment guide is and allows them to focus on the set criteria and performance levels.

Deidre Greer, professor at Columbus State University, works with students at a Title I elementary school in Georgia. Greer states that in her experience,

The Exemplars tasks have proven to be engaging for our Title I students. Use of the student-scoring rubric helps students understand exactly what is expected of them as they solve problems. This knowledge then carries over to other mathematics tasks.

At Exemplars, we believe that rubrics are an effective tool for teachers and students alike. In order to be successful with the learning expectations set forth by the new math TEKS, it is important for students to understand what is required of them and for teachers to be on the same “assessment” page. Rubrics can help.

## Understanding Mathematical Connections at the Fifth Grade Level

Tuesday, August 18th, 2015

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

This blog is the final post 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.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of today’s mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

As part of the other blogs in this series, we reviewed solutions from a first grade student and third grade student to observe how they successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in their work.

#### Blog 4: Mathematical Connections at the Fifth Grade Level

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?

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 at the Third Grade Level

Wednesday, August 5th, 2015

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

This blog is Part 3 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.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classrooms to help their students become more proficient in making mathematical connections.

As part of the second blog, we reviewed a first grade solution and how this student successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in his or her work.

#### Blog 3: Mathematical Connections at the Third Grade Level

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?

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 First Grade Level

Monday, July 20th, 2015

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

###### Summer Blog Series Overview:

This blog represents Part 2 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.

In the previous blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum.

We also reviewed the Exemplars rubric and offered the following strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

1. Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct.
2. Encourage students to continue their representations.
3. Explore the rich formal language of mathematics.
4. Incorporate inquiry into the problem-solving process.
5. Encourage self- and peer-assessment opportunities in your classroom.

#### Blog 2: Mathematical Connections at the First Grade Level

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.

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

Wednesday, June 24th, 2015

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. As part of this series, student work will be examined at Grades 1, 3 and 5.

#### 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.

(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 Grades 3 and 5 will follow in subsequent posts of this series.)

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

Tuesday, March 3rd, 2015

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.

## Why A Focused Mathematics Curriculum Matters and How Exemplars Can Help Texas Educators

Monday, November 3rd, 2014

Written By: Dinah Chancellor, Exemplars Math Consultant

Prior to 2006, many states—including Texas—had a math curriculum that was perceived to be “an inch deep and a mile wide.” Teachers were required to teach a large number of math skills that spiraled from grade to grade and seemed both disconnected and fragmented. When Texas’ own Cathy Seeley became President of the National Council of Teachers of Mathematics (NCTM), she determined that a more focused mathematics curriculum that was built around fewer “big ideas” would give students and teachers the luxury of time—time to plumb the depths of major math concepts, and time to form a foundation of connected mathematical understandings.

Therefore, in 2006 NCTM published the Curriculum Focal Points—A Quest for Coherence. The Texas response to the Curriculum Focal Points was the new state assessment program—STAAR—the State of Texas Assessment of Academic Readiness taken by students in grades 3-8. STAAR focuses on fewer skills at each grade level and it is expected that these skills will be taught at greater depth. When the new math TEKS were written, released in April 2012 and implemented in the fall of 2014, the writing teams focused on fewer skills at each grade level. Teachers are expected to address these skills and understandings by teaching rich lessons in which students make critical connections between foundational big ideas in mathematics. Because of the need to teach a focused mathematics curriculum, it does not make sense to teach each of the new math TEKS in isolation.

To assist Texas educators in achieving this goal, Exemplars latest K–5 product, Problem Solving for the TEKS, groups the individual math TEKS student expectations into rich Units of Study. Four or more instructional tasks/formative assessments and one or more summative assessment is provided to address the big mathematical ideas within each Unit. Tasks are meant to supplement a school or district’s existing curriculum. Teachers may choose to use all or only a few of the instructional tasks/formative assessments in a Unit. The summative assessments include anchor papers that exemplify each of the performance levels in the Exemplars Rubric—Novice, Apprentice, Practitioner (meets the standard) and Expert.

###### A Look at a Sample Unit

The Place Value Unit represents one of eleven Units in the third grade. The math TEKS covered in this Unit include: 3.2A, 3.2B, 3.2C, 3.2D. These math TEKS were grouped together to provide a cohesive Unit that enables 3rd grade students to understand the Properties of Place Value and to apply this understanding to compare and order whole numbers. See the full list of Units of Study for K–5.

##### How Might a Teacher Use the Tasks in This Unit?

As the Place Value Unit progresses, a teacher may want to use one of the instructional tasks to teach students the expectations of the Exemplars Assessment Rubric. A lesson using the task “Tables for a Party” may include the following steps:

•  Whole Group: Read the task together and ask students to underline the question, identify important information in the problem and summarize the task by restating what the question is asking them to do. Example: I need to find out how many tables need to be set up for 34 students with no more than 10 students sitting at each table. On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student understand the problem?
•  Small Group: Ask students to work together, think of a plan, and write it down. Example: I will draw a diagram of tables with students sitting at them. At this point, students will implement their plan to solve the problem.

Example:

(Refer to the task Planning Sheet for additional examples of solution strategies.)

Students will check their plan to make sure it works and put a box around their answer.

Example:

On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student have a plan? Does the student get the correct answer? It is also scored in the category for Reasoning and Proof—Does the student show a systematic implementation of the plan?

Small Group: Ask students to polish their papers—

• Explain your plan and how it solved the problem.
• Create a representation—such as a diagram with a key, use a model (such as manipulatives), use a table, use a number line.
• Use mathematical vocabulary and/or symbolic representation.

On the Exemplars Rubric this step is scored in the category for Communication—Does the student use at least two mathematical vocabulary words, at least two correct symbolic representations or one of each? It is also scored in the category for Representation—Is the representation correct and appropriate to the solution?

Small Group: Finally, make a connection—

• Make an observation.
• Identify and describe a pattern.
• Make a comparison between this task and other tasks. Explain how the math is similar.
• Identify a rule.
• Create a hypothesis or conjecture to test.
• Solve the problem using a different strategy to prove the original solution is correct.
• Recreate the problem and show a different solution.

On the Exemplars Rubric, this step is scored in category for Connections—Does the student include a mathematically relevant connection? Making connections requires students to look at their solutions and reflect.

###### Using Anchor Papers & Scoring Rationales

Anchor papers and assessment rationales are provided with every summative assessment task. These problem-solving tasks are given at the end of a Unit of Study to assess students’ understanding. A summative assessment must represent a student’s total independent solution. One Hundred Miles is the summative assessment for the grade 3 Place Value Unit.

Anchor papers and scoring rationales provide a great way to show both teachers (in professional development sessions) and students the expectations of the Exemplars Rubric; i.e. What a Practitioner (meets the standard) piece of student work looks like. Analyzing Exemplars anchor paper solutions and rationales at the Practitioner and Expert levels help students polish their own work and measure their own progress toward a specific goal. Analyzing the Novice and Apprentice samples can help identify for students where the work falls short of the goal and specifically how the papers could improve.

To view other sample tasks and anchor papers for grades K–5, you can sign up for a free 30-day Trial for Problem Solving for the TEKS.

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