Looking at Common Core Through the Eight Mathematical Practices

As we move further into the school year, the implementation of Common Core Standards continues throughout states, school districts, and classrooms. With the actualization of the standards varying from classroom to classroom and access to sample standardized test questions being limited, it may seem difficult to know how to help prepare our students for what they will encounter in the classroom and on homework and tests. As the Education Department strives to find the best ways to help you assist your students in this time of change, we’d like to provide a comprehensive reference on the framework of the Common Core Standards for Mathematics. Use this guide to deepen your understanding of the new Standards’ intentions as we witness the reality. Parents are hearing these ideas from school and are seeing attempts at implementing them in their children’s homework. Understanding the Standards and being able to point to similar themes in Mathnasium curriculum will help you convince parents that you are the answer to their children’s math struggles!

The Standards’ framework has three major aspects that provide guidance for student support:

• Focus
• Coherence
• Rigor

Focus

The Common Core Standards have reduced the number of topics and skills that students should have awareness and mastery of at each grade level. This should allow them to focus on a greater depth of understanding of essential grade-level topics, rather than trying to race through as many as possible in a single school year.

Coherence

As we’ve heard Larry say, mathematics isn’t a bunch of disparate parts that stand alone. Rather, mathematics is a fabric of deeply connected strands that form a cohesive whole. The Standards are designed around the idea of coherent progressions from grade to grade. It is critical that the “strands” of learning are connected across grades so that students can build new understanding onto strong foundations gained in previous grades.

Rigor

According to the Common Core Standards, in order for curriculum and teaching practices to have rigor, they should lead to a genuine understanding of appropriate grade-level math concepts, rather than making math more difficult by introducing advanced concepts at lower grade levels. Per the Standards, this includes intensely pursuing conceptual understanding, procedural fluency, as well as application through problem solving at each grade level.

In our goal to teach children in a way that makes sense to them, our curriculum and Assessment process are designed to focus a child’s efforts on key foundational skills by evaluating what students know and don’t know yet and creating a customized Learning Plan. The Mathnasium Hour is designed to help ensure students work on filling in missing foundational skills as well as addressing grade-level topics at each session.

Mathnasium Assessments and the resulting Learning Plans provide coherence across grade levels. We help students build on numerical fluency, the ability to recall basic number facts effortlessly for addition, subtraction, multiplication, and division. This leads to computational fluency, the ability to know when and how to use standard and non-standard algorithms to perform arithmetic and algebraic computations. We use Mathnasium constructs to link ideas across multiple grade levels.

When a child determines where a stack of 2-inch blocks and 3-inch block will meet, they are laying the foundation for the idea that the LCM of 2 and 3 is 6. With that knowledge, they can then know that while you cannot add “halves” and “thirds” you can give them the same name of “sixths” and then add them.

In addition to the Assessment strands, these coherent ideas form the backbone of the Mathnasium Method.

What we term mastery refers to the number sense and conceptual understanding that we build in children by helping them understand why things work and how they know their answer is correct. This looks a lot like what Common Core calls rigor. Through the process of combining written work and discussion directed by skilled Instructors, students develop a deeper understanding of mathematics and of their own thinking processes. Students progress at their own pace and only move on to the next level after they have demonstrated mastery of skills and concepts in their current Learning Plan.

Mathnasium and the Eight Standards For Mathematical Practice

In support of the above framework, Common Core has included eight Standards for Mathematical Practice in addition to the more traditional content standards. Striving to achieve these standards drives most of the instructional and assessment challenges that are currently being experienced in the transition. Mathnasium’s curriculum and instructional methods support children as they work to achieve success through the 8 Mathematical Practices. (You will see them referenced in Common Core materials as MP.1 – MP.8)

1. Make sense of problems and persevere in solving them.
Students are able to explain what the problem means first, and then begin looking for solutions. They consider the relationships, restrictions, goals, and similar problems or simpler forms. They plan a course of action, solve the problem, and then ask themselves if the answer makes sense.

Problem solving, critical thinking, awareness of one’s own thought process, and the ability to clearly and concisely explain one’s own reasoning have always been parts of every student’s customized Mathnasium program. At every step, there is an intense focus on understanding and mastery. Download curriculum samples here ^[1].

2. Reason abstractly and quantitatively.
This means students are able to do two things:

• Decontextualize – generalize a situation, represent it symbolically, and manipulate the representing symbols as though they have a life of their own
• Contextualize – at any time during the manipulation process, pause as needed in order to refer back to the meaning behind the symbols

At Mathnasium, we help students learn how to analyze problems so that they can move fluidly from a specific problem to equation and formula representations and back again from the formula and generalization to its applications in specific situations. Download curriculum samples here ^[2].

3. Construct viable argument and critique the reasoning of others.
Students have the ability to understand their own assumptions, definitions, and conclusions, as well as those of others. Then, they can use that understanding to construct a logical progression of statements to critique, explain, and justify their conclusions to others, as well as respond to others’ arguments.

This Metacognition is a key part of the Mathnasium Teaching Method. Making sure that students are able to explain their thinking, how they arrived at their answers, and why their conclusions are correct is a regular learning center process. This is done both in written form within specific curriculum pages, and verbally with Instructors. Download curriculum samples here ^[3].

4. Model with mathematics.
This means students apply the math they know as they solve problems arising in everyday life, society, and the workplace.

During the part of the Mathnasium Hour designated for Workout Books, students work on curriculum designed to help them develop critical thinking and problem solving skills by solving non-routine problems that don’t require the use of standard algorithms. As appropriate, students also spend part of the Mathnasium Hour working on homework problems that cover a wide range of applications. Download curriculum samples here ^[4].

5. Use appropriate tools strategically.
This means students consider available tools when solving mathematical problems. These tools might include pencil and paper; concrete models; a ruler; a protractor; a calculator; a mathematical table or chart; or any appropriate tool.

DeskTools and Manipulatives are key aspects of the Mathnasium Method. The Mathnasium curriculum is designed with icons that guide Instructors and students so they know when it’s appropriate to use written, verbal, and visual models or equations and, when necessary, physical manipulatives when problem solving. Download curriculum samples here ^[5].

6. Attend to precision.
According to the standards, students should try to communicate precisely with others. They should attempt to use clear definitions in discussions with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem’s context.

At Mathnasium, we believe that students should experience math through the five modes of learning: mental, verbal, visual, tactile, and written. It is important that students are able to precisely and accurately show and explain their thinking and solutions—both in their written work as well as in their dialogue with Instructors. Download curriculum samples here ^[6].

7. Look for and make use of structure.
Whether students are working on computation, problem solving, or conceptual understanding, they should be able to look for patterns and underlying structure to help them figure out the answers. They can break down complex problems and see them as being composed of several simpler problems or processes. Download curriculum samples here ^[7].

8. Look for and express regularity in repeated reasoning.
This means that students look for repeated calculations, ways to generalize, or opportunities to use appropriate shortcuts when problem solving.

At Mathnasium, once students gain fundamental understanding of concepts and computation, we take them a step further to extend their knowledge. This helps them to look for ways to generalize problems, make use of work they have already done or, when appropriate, use shortcuts based on sound mathematical understanding to help them gain efficiency when doing math problems. Download curriculum samples here ^[8].

As students transition from one Assessment level to the next, the curriculum helps them apply the eight Standards for Mathematical Practice at deeper and more sophisticated levels.

Next in the series… Common Core Assessments and some tools to help your students prepare.