In designing the new state standards, educators believed in developing eight key mathematical practices that would reflect the overall goal of developing a higher level of mathematical thinking in all students across the United States. These eight ideas are called the Standards for Mathematical Practices. When working with your students on math concepts, it is vital that they are exposed to learning experiences that encourage the use of these eight practices to develop their highest level of mathematical success.

Information below from: http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.PDF

Information below from: http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.PDF

## 1. Make Sense of Problems and Persevere in Solving Them

Picture this: Your student is working on a math question and is getting extremely frustrated. Tears come and anger escalates. Sound familiar? Unfortunately this situation is all too common in math. This is why the first mathematical practice for students is to make sense of problems and persevere in solving them. The California Department of Education begins the definition of this mathematical practice by saying, "Mathematically proficient students start by explaining to themselves the meaning of a problem and look for entry points to its solution."

How to implement this practice:

How to implement this practice:

- Start each math question by asking your student (or having your student ask themselves) "What does this problem want me to do?" and then follow that question up with "What choices do I have to begin solving this problem?"
- Provide students with a way to express their confusion in a safe environment and come alongside them to ask guiding questions to lead them towards a path that will eventually result in the desired mathematical solution.
- Encourage diligence. Encourage patience. Encourage creativity.

## 2. Reason Abstractly and Quantitatively

Consider the inequality x+3< 5. Many middle school students can solve for x and get the "correct" answer. However, this second mathematical practice asks students to go deeper than simply getting a correct answer. Students are asked to actively reason about their mathematics when working with a problem. In the example listed above, a student would need to explain that x<2 signifies that any value of x that is less than the number 2 will make the inequality statement true.

How to implement this practice:

How to implement this practice:

- Ask students to explain why their answer is logical and what their answer means in the context of the situation.
- Ask students to consider a different approach to the math problem to see if it would result in a viable solution to the problem. Why does that new approach work or not work?

## 3. Construct Viable Arguments and Critique the Reasoning of Others

Mathematically proficient students are able to explain their work, justify their steps, and justify their concluding answer. They can apply and identify mathematical steps and processes in solving a question. In addition, mathematically proficient students are able to explain the work of others and justify the steps that someone else used. They can analyze and make educated guesses on why another mathematician solved the problem in the way that they did. Most importantly, mathematically proficient students are able to explain whether their answer or another student's answer makes sense.

How to implement this practice:

How to implement this practice:

- Ask students to explain how they got their answer. Allow them to explain the procedural steps they took to solve the problem.
- Ask students to explain why their answer makes sense.
- Give students opportunities to evaluate the work of other students and see if they are able to explain how that student got their answer and if their answer makes sense. (Consider using the following website: http://mathmistakes.org/category/elementary-school/)

## 4. Model with Mathematics

Too often, math appears to be a jumbled up series of rules and numbers that make no sense to students even after they have derived an answer. By encouraging students to model their math using drawings, pictures, bar diagrams, charts, scales, etc. you are allowing your student to give math a representable identity. This is not just for elementary students! Middle and High School students can represent their math in creative ways using counting blocks, scales, and even cartoons! Encourage your student to be creative in modeling their math.

How to implement this practice:

How to implement this practice:

- Give students time to work on a math problem in a hands-on way before showing them the numerical (procedural) way of solving the problem.
- Challenge students to represent a particular math problem without using numbers! It sounds silly, but students get very creative and are very good at doing this!
- Use visual models side-by-side with numerical procedures to demonstrate new concepts for students.

## 5. Use Appropriate Tools Strategically

How many times has a student asked you, "Can I use a calculator?" Often times, teachers do not want students to use a calculator because they want to ensure that a student understands

How to implement this practice:

*what*the calculator is actually*doing*and*calculating!*Calculators need to be viewed as a type of tool for students to use when appropriate for various situations. However, there are many other mathematical tools for students to use to help them with math. Some of them are digital resources, and others of them are simply different approaches to math. This fifth mathematical practice encourages students to fill a metaphorical tool bag with math strategies and options for solving problems.How to implement this practice:

- Keep a giant poster throughout the year of tools and ideas that students have learned to help them solve problems. Refer to this poster when students come across a problem that is challenging to them.
- Ensure that students understand the function of a tool and
*what* that tool is actually doing before solely relying on it for an answer. (Graphing calculators, calculators, and protractors are the most commonly misunderstood tools but there are many others!)

## 6. Attend to Precision

Did you ever get marked down on a quiz or test because you forgot a label? This sixth mathematical practice is essentially

How to implement this practice:

*why*that happened! Mathematically proficient students are precise in their procedures and answers. Including (and deeply understanding) labels is key to success in math. If a student is finding the volume of a cube and uses the label "square centimeters", they are not demonstrating an accurate knowledge that volume is a three-dimensional measurement of space and needs a "cubic" label. While this may seem a bit extreme, we want to encourage students to always be thinking about their math and*what*their math represents. Precision is a natural side-effect of a deep mathematical understanding.How to implement this practice:

- By encouraging your students to make sense of the problems they are working with and the numbers they are generating, students should be naturally inclined to also be precise. If the precision is missing, encourage students by asking them about the process of their work and
*what*their numbers represent. Begin by having them add labels to their calculations.

## 7. Look for and Make Use of Structure

Has your student ever been working on a math problem and had a "light bulb moment"? You may recognize the moment as the instant when their face lights up and they suddenly truly

How to implement this practice:

*understand*something. This is most likely the result of this seventh mathematical practice happening. When a student truly understands something, it is because they have found a connection to something else they already knew. They found a similarity, a pattern, or a type of familiar structure that they can compare to something else in their life. This is exactly what students should be experiencing in math as well. Students should be looking for and identifying patterns that they are able to see in math across grade levels and across concepts.How to implement this practice:

- Ask students to find similarities between new concepts and previously mastered concepts.
- Encourage students to apply previously known strategies to solving new types of problems.
- Congratulate students when they naturally find a comparison between two different concepts!

## 8. Look for and Express Regularity in Repeated Reasoning

Everyone loves shortcuts. They make our lives faster and easier! Shortcuts in math are no different! When a student is able to identify a repeating pattern, translate that pattern into a mathematical rule, and apply that rule to other math problems, they have found a shortcut and ultimately have demonstrated the eighth mathematical practice!

How to implement this practice:

How to implement this practice:

- Encourage students to look for patterns that they can use in math.
- When a pattern is discovered, explore it with your student and try it in different math scenarios. See if the pattern works always, sometimes, or never and figure out why. Great conversations can be had!