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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 Brooklyn 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. Pricinpal, & 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 Bronx’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.

What Students Need Most When the Stakes Are High

Monday, June 4th, 2018

Written By: Jay Meadows, Chief Education Officer 

High stakes, end-of-year performance tasks on statewide tests have become the norm in recent years. These types of questions are designed to assess how well students can utilize their developing math skills to answer authentic—multi-step, complex problems that can be solved with a variety of strategies.

How do we prepare our students for these challenging tasks while—at the same time— ensure that we are utilizing the precious minutes in every class period and are not “teaching to the tests”? The answer lies in what we hope to accomplish in our math classrooms.

Math students in today’s changing world need to be able to able to calculate precisely and efficiently. Skill development remains a foundation in the math curriculum. However, students must also develop the ability to (1) utilize these computation skills as they work to solve complex problems, (2) reason effectively in finding efficient solutions, (3) communicate and (4) model their ideas and strategies utilizing the tools of mathematicians.

Students Need Practice

Exemplars has been creating rich and engaging performance tasks in mathematics for the purpose of instruction and assessment for the past 25 years. To master performance assessments, just as with any important skill, students need to practice. That practice time is only viable if the skills developed are relevant to their success outside of the classroom. In the 21st century, the skills needed for success include the ability to communicate effectively, collaborate within teams, and apply critical thinking to solve complex problems in new and creative ways.

By having students practice with rigorous problem-solving performance tasks, such as Exemplars, teachers can intentionally nurture these 21st-century skills while developing the math process standards NCTM has articulated as the foundation of strong mathematicians: Problem Solving, Reasoning and Proof, Communication, Connections and Representations.

Through rich problem solving, students are given the opportunities to work collaboratively in small teams, thus learning to communicate their ideas and strategies with others while listening to the explanations of their peers. Students can then bring their own ideas forward creatively, and efficiently work to solve these problems and develop persuasive arguments to explain their ideas to the whole classroom. This setting helps students develop the skills the 21st century requires.

Exemplars is the Perfect Supplement

Our collection of more than 800+ problem-solving performance tasks in mathematics present students with the opportunity to develop the skills of collaboration, communication, creativity and problem-solving.  Exemplars tasks have been classroom tested, and include student work samples that teachers and students can utilize to gain an understanding of what high-quality work actually looks like. Our material also provides standards-based assessment rubrics for teachers and students, detailed lesson planning sheets for each task, and differentiated problems. Visit exemplarslibrary.com to sign up for a free 30-day trial and gain access to our getting started materials.

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?

The four-part blog series below 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.

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

Dynamic Math Learning at The Phoenix School

Thursday, March 29th, 2018

Written By: Barbara McFall, Head of School

Extending Exemplars problem, "Park Play Area," to an irregular quadrilateral that will become a plan for one of our community’s potential park spaces … real-life math building on an Exemplars problem!

Fall is the time of year when my students begin pestering me for “real” word problems. They love solving problems that include the names of their classmates and using a collaborative experience in which they can bounce ideas off one another as they work through more complex problems. My go-to resource for problems that create mathematical thinkers of my students is Exemplars.

At The Phoenix School, for many years (first in paper form, now digital) Exemplars has been a vital component of our math program. No more memorization and regurgitation that flies in – and out! — of kids’ brains. Exemplars work solidifies concepts, which are the foundation of mathematical learning, and allows for multiple solutions as well as encouraging creative thinking. It has the flexibility that enables us to extend concepts on which we have been working … and it is so well organized that it is easy to determine which sets of problems we need to best showcase particular math concepts.

Working Exemplars problems requires students to do much more thinking than in traditional mathematical learning formats. It teaches them to analyze information and focus on what is most important in getting to the solution. Exemplars problems also encourage students to use previous knowledge to work through solutions. When students discover how mathematical systems work, they can solve almost anything.

I choose a variety of math problems at all levels to begin. Varying math topics and levels help students learn to be flexible in their thinking and to transfer learning from one topic to another with ease. To create excitement, I change the names of the people in the problems to the names of students in my class. This leads to wonderful interactions which, in turn, leads to sharing ideas and helping one another. Each student gets a folder of problems. I put them on label paper so they can stick them in their math journals, one problem at a time. I don’t specify in which order students should work the problems – leaving it open-ended makes the math more engaging, encouraging students to ask questions of each other.

Some students like to work alone, while others prefer to connect with partners to work on problems they have in common. Using Exemplars legitimizes working with partners. It teaches students to share strategies, talk to others, and collaborate. Traditionally, teachers have considered the sharing of math work to be “cheating,” but I have found the learning to be so much better for all when my students share solutions. Math at The Phoenix School often becomes a social event as we collaborate on a particularly complex challenge, especially when we turn to using materials to represent a solution. Everyone’s ideas are considered and sometimes debated with passion!

I discuss different approaches to solving problems with students and expect them to document their work using charts/graphs/tables, words, numbers, equations, pictures/diagrams, and personal reflection. This helps students see math in multiple ways, which increases the learning. Requiring visual representations helps students organize their thinking so that, as I say to them, “Your paper talks to me. I can tell exactly what you were thinking as you worked through the problem.” For my older students, I always use the more challenging form of each problem. It is easy to extend these problems into algebra since so many lend themselves to finding the pattern and taking it to the algebraic equation.

From our youngest students to our oldest, we find that Exemplars problems engage, challenge, and create mathematical thinkers at every level. Our math program is truly strengthened by the Exemplars component.

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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