1. Process Standards for Mathematics in Action!
Building Background for Educators

2. Project Information
This presentation hopes to answer two questions for educators:
How do you integrate the Process Standards for Mathematics into instruction of daily lessons in conjunction with the content?
What does this integration look like for teachers and for students?
With the implementation of Indiana’s CCR Academic Standards, many educators are leaving out the implementation of the Process Standards for Mathematics. The content standards are important, but the PS are an equal asset to an effective math classroom. This project hopes to help teachers develop a better understanding of the process standards and how they can integrate them into their daily lessons by providing resources for professional development. What we want teachers to understand is that the PS don’t guide your lesson, your content guides your lesson, but the PS need to be a natural part of the process. It is not something that you do on Fridays or a special day, it happens each and every day in every math lesson.

3. Process Standards for Mathematics
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
These are the 8 process standards for mathematics as outlined in Indiana’s Academic Standards. Unlike the content standards which identify a specific grade-level, these standards should be integrated into content instruction at all grades K-12. These can be defined as “processes and proficiencies with longstanding importance in mathematics education.” This is what we want students to take away believing about what mathematics is as a field. Too often, students develop beliefs and dispositions about mathematics that suggests it is not applicable to the real-world or that mathematics is mostly about learning rules without understanding. The Process Standards emphasize the importance of highlighting mathematics as a field that is focused on problem solving, understanding, justifying, and representing quantitative thinking. We want students to leave their K-12 mathematics experiences with a realistic view about mathematics as a field that focuses on inquiry and knowing know they can apply their thinking.
One of the things to make clear is that these practices are not content-specific. Unlike many content standards, you probably will not be able to spend a few days on “constructing viable arguments” and expect students to have mastered this practice. Rather, these practices should be inherent in your daily instruction and sustained throughout the year. Every lesson should ideally address one or more of these standards, and you should revisit the standards regularly.

4. The Indiana Department of Education and others have suggested that the process standards be paired to emphasize important connections between them. Problem solving and precision are overarching ideas, and the others can be grouped to focus on reasoning and explaining, modeling and using tools, and seeing structure and generalizing. We will be discussing each of the eight standards in greater depth throughout this webinar.

5. Make sense of problems and persevere in solving them
Process Standard#1
Make sense of problems and persevere in solving them
Process Standard#1: Make sense of problems and persevere in solving them.

6. What is a problem?
A problem is defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution method.
Hiebert et al., 1997
Before we look at this practice more closely, it’s important to understand what a problem truly is.
According to this definition: A problem is defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution method.
This may not be how you’ve thought about problems. But it’s important when thinking about this first mathematical practice, that you focus on this, or a similar, definition.
As teachers, we often want to help students solve problems by providing TOO much assistance. Keep in mind that if we remove all obstacles to problem solving for our students, we are not supporting the perseverance aspect of this Mathematical Practice and we thus deny students the opportunity to make sense of the mathematics.

7. Explain the meaning of a problem
Make sense of problems and persevere in solving them Students should be able to…
Explain the meaning of a problem
Describe possible approaches to a solution
Consider similar problems to gain insights
Use concrete objects or illustrations to think about and solve problems
Monitor and evaluate their progress and change strategy if needed
Check their answers using a different method
What should students be doing while engaged in this process?
They should explain the meaning of a problem and describe possible approaches to a solution.
They should ask themselves: Is this problem similar to one I solved before? If so, what approach did I use then that might work now?
They should use resources such as manipulatives, or other concrete objects, and drawings to help them think about the problem.
They should monitor and evaluate their progress. Do they recognize when the method they are using isn’t working? They should change strategy if needed.
And finally, they should check their answers using a different method. If students check their answer using the same approach they used to solve the problem, the student may conclude that their solution is correct when it’s not. But if they check their answer using a different method, they are apt to find the errors in their reasoning and eventually come to the correct solution.
All of these ideas help students to not only make sense of problems, but to persevere in solving them.

8. Make sense of problems and persevere in solving them Grade 5
There are 228 players in the softball league. How many 12-member teams can be formed if each player is placed on exactly one team?
Choosing a good task or problem is critical to engaging students in this process standard.
Here is an example of a problem that might be given to 5th graders.
This problem was chosen because the context would be interesting to many students, it would challenge them, it builds on work they’ve done in 4th grade (where they divided by one-digit numbers) and it supports the use of multiple strategies. Students might use repeated subtraction, they might rely on their knowledge of multiplication facts, or they might use some type of manipulative, like base-ten blocks.

9. Make sense of problems and persevere in solving them Kindergarten
I have 5 things on my plate.
Some are peas. Some are carrots.
How many of each could I have?
How many peas? How many carrots?
Adapted from Investigations in Number, Data, and Space
Here is a problem for kindergarten.
I chose this example because the problem addresses meaningful mathematics. What do I mean by meaningful mathematics? I am referring to mathematics that will propel or move students forward in their mathematical knowledge at an appropriate level. And through the use of manipulatives and written work, the problem will reveal information about their understanding of how to decompose numbers. To decompose a number, means to break it down into two or more parts. In this case, they are decomposing the number 5 into two parts.

10. Make sense of problems and persevere in solving them
Teachers should…
Provide Good Problems
To determine whether it’s a good problem, ask yourself:
Is the problem interesting to students?
Does the problem involve meaningful mathematics?
Does the problem provide an opportunity for students to apply and extend mathematics they are learning or have learned?
Is the problem challenging for students?
Does the problem support the use of multiple strategies?
Will students’ interactions with the problem reveal information about students’ mathematics understanding?
To help students meet this standard, teachers need to.
Provide good problems. This is one of the most important decisions you will make for your students. You may be thinking, “How do I know if it’s a good problem or not?” Here are some questions you can ask yourself to help make that determination.
{Read the questions}
Let me add some clarification to the last question. What we mean by students’ interactions with the problem are such things as: students’ work, their discussions, and the processes and strategies they used to solve the problem. You want to be able to look at student work and listen to their discussions and gain information about their understanding of the mathematics in the task.
I referenced this list in the two examples I showed you – one for 5th grade and one for kindergarten. But keep in mind that you are the best person to answer these questions for tasks you choose because you know your students better than anyone.

11. Teachers should… Provide access to appropriate tools/resources
Make sense of problems and persevere in solving them
Teachers should…
Facilitate student engagement in the problem-solving process
Provide access to appropriate tools/resources
Encourage discussion and questions
Support students to
Make sense of the problem
Make connections
Check reasonableness of solutions
In addition to choosing a good problem, you should facilitate engagement in the problem-solving process. You can do this by having a variety of tools and resources available for them to use and by encouraging questions. One question from a student can lead to a very interesting class discussion.
And finally, you want to support students as they try to make sense of the problem, help them make connections to similar problems they’ve solved and ensure that students always check the reasonableness of their solutions. We all know of instances in which students come up with a solution that makes no sense in the context of the original problem. One way to help students is by having them estimate the solution before they even start solving the problem.

12. Process Standard#6 Attend to precision.
The Process Standard of attending to precision relies heavily on the mathematical norms established in the classroom. Developing this mathematical practice is dependent upon students realization of appropriate use of informal and formal mathematical language, procedures, and processes as they engage with mathematical problems.

13. Attend to Precision Students should be able to…
Communicate precisely to others
State meaning of symbols and definitions and use them appropriately
Specify units of measure
Calculate accurately and efficiently
This standard is comprised of four sub-skills: communicating precisely to others, stating the meaning of symbols and definitions and using them appropriately, specifying units of measure, and calculating accurately and efficiently. Attending to precision focuses on the need for both teachers and students to communicate precisely and correctly. Precise mathematical communication could involve understanding and appropriately using symbols, definitions, and units of measure. This practice requires of students to be accurate and appropriate in the use of calculations, keeping in mind that the level of appropriateness is often determined by the context of a problem.

14. Attend to Precision Grade 3
José ate ½ of a pizza.
Ella ate ½ of another pizza.
José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Who is right? Support your answer with words and pictures.
Students in a third grade class were asked to respond to the question shown here (read question). This question is discussed in more detail as an example of Process Standard 3. The class discussion leads to an insightful exchange about the importance of the recognizing the whole in context when comparing what seem to be equivalent fractions. As part of the discussion, the teacher models the process standard of attending to precision by introducing terms and their respective definitions for numerator and denominator, as well as recognizing the importance of the context when determining the appropriate quantities to compare.
Student B Response
Student A Response

15. Attend to Precision Grade 6.
Student A Response
I found a pattern of adding five toothpicks for each hexagon. Since 21 toothpicks are in 4 hexagons, I know that five more makes 26. I could also multiply 5 by 6 and subtract 4.
Students in a sixth grade class were asked to respond to the question shown here (read question). This question is discussed in more detail as an example of Process Standard 3. Through orchestrating a classroom discussion that highlights these two solutions, the teacher models the Process Standard of attending to precision as she asks each student to explain and support his or her solution strategy. This discussion leads to an insightful exchange that produces three equivalent equations to represent the pattern. The teacher takes the opportunity to record the equivalent equations and discuss the meaning and uses of the algebraic symbols involved in each.
Student B Response
How many toothpicks are needed to make 4 hexagons? 5 hexagons? 10 hexagons? n hexagons?
To find the number of toothpicks in 5 hexagons, add five to 21 (the number in 4 hexagons).

16. Attend to Precision Teachers Should…
Model appropriate use of mathematics vocabulary, symbols, and explanations
Provide opportunities for students to share their thinking
Prepare students for further study
The teacher’s role in helping students develop this process standard of attending to precision includes modeling the appropriate use of mathematics vocabulary, symbols, and explanations; providing opportunities for students to share their thinking, and preparing students for further study. First, teachers help students attend to the importance of precision by modeling appropriate precision when discussing solutions to problems. This modeling often comes into play as student strategies are being recorded for the class. Oftentimes as strategies are shared by students, informal language is used to describe their process. It is important for teachers to stress the importance of the use of mathematically correct vocabulary and symbolism as these strategies are recorded. Providing opportunities for students to share their thinking engages the entire class in determining mathematical precision. Other students, as well as the teacher, in developing proper language to describe mathematical precision. Finally, teachers should think ahead by using appropriate language that supports the further study of mathematics. Statements such as “multiplication makes bigger and division makes smaller” hold true for whole numbers, but often lead to misconceptions in the further study of integers, fractions and decimals.

17. Reason Abstractly and Quantitatively.
Process Standard#2
Reason Abstractly and Quantitatively.
The Process standard of reasoning abstractly and quantitatively relates to students’ sense-making skills. Students make sense of mathematical situations by carefully thinking through ideas, by taking into account examples, counterexamples, and other alternatives, and by forming and reflecting upon questions, conjectures, and hypotheses. The ability to reason abstractly and quantitatively forms the foundation for student engagement in all areas and levels of mathematics.

18. Reason Abstractly and Quantitatively Students should be able to…
Make sense of quantities and their relationships in problem situations
Contextualize and decontextualize
Create a coherent representation of the problem at hand
This Process Standard is comprised of three sub-skills: making sense of quantities and their relationships in problem situations, contextualizing and decontextualizing mathematical situations, and creating a coherent representation of the problem at hand. Reasoning about quantities and their meanings is of particular importance in the learning of mathematics. Understanding and sense making around quantities include the abstract understanding of quantities expressed through symbols, as well as the understanding of quantities expressed in context. This Process Standard is firmly based in number sense and stresses the importance of the development of students’ sense-making skills in both concrete and abstract representation of quantities.

19. Reason Abstractly and Quantitatively Grade 1
Decontextualize: Write a number sentence to represent this story – There are 12 girls and 13 boys in Mrs. Johnson’s class? How many students are in Mrs. Johnson’s altogether? =☐
These examples from first grade illustrate the basic ideas of contextualizing and decontextualizing simple mathematical situations. In the first, students are asked to reason abstractly about an additive mathematical situation by decontextualizing a mathematical story. In the second example students are required to ponder the underlying meaning of the operation of subtraction.
Liam had 15 pencils. He gave 9 pencils to his classmates. How many pencils does Liam have left?
Contextualize:
Write a story that could be represented by the number sentence 15 – 9 = ?

20. Reason Abstractly and Quantitatively Key Words are not the Keys to Mathematical Reasoning
Oftentimes students are encouraged to look for key words in problems that are associated with mathematical operations. These examples illustrate the consequences of relying solely upon the key word approach to problem solving. Mathematics education researchers used the birthday problem shown here in their research on children’s mathematical problem solving. (read problem) In the study, 10 percent of the participating kindergartners and first graders attempted to answer the question posed by either adding or subtracting the two numbers involved even though sufficient information is not presented. The percentage rose to 30% for the 2nd grade participants. The results show that even 45% of fifth graders attempt to solve the problem. The study results highlight the importance of developing the ability to reason through mathematical situations, including the ability to make sense of the situation while decontextualizing it.

21. Reason Abstractly and Quantitatively Key Words are not the Keys to Mathematical Reasoning
Bernard had some coins. Samuel gave him 6 more. Now Bernard has 14 coins. How many coins did Bernard have to begin with?
This example illustrates the importance of students making sense of the context. The key word “more” is associated with the act of summing and generally indicates an unknown result. However, in this problem the initial value is unknown rather than the result. Students relying on the keyword approach to problem solving run the risk of choosing to find the sum of 6 and 14 rather than the difference.

22. Reason Abstractly and Quantitatively Grade 6
3 pineapples
1 serving = ½ pineapple
Given the information in the box above, write a mathematics word problem for which 3 ÷ ½ would be the method of solution.
This sixth grade example is from the National Assessment of Educational Progress (NAEP) and requires students to reason about the division of fractions, a concept that is difficult for many children and adults. Here, students are asked to contextualize a mathematical situation involving fractional parts of a whole consisting of three objects. Creating a context such as “How many people will receive servings of ½ of a pineapple if the restaurant has a total of 3 pineapples?” around this mathematical situation provides students the opportunity to reason deeply and conceptually about division involving fractions.

23. Reason Abstractly and Quantitatively Teachers Should…
Provide opportunities for students to
Express interpretations about number
Apply relationships between numbers
Recognize magnitude of numbers
Compute
Make decisions involving numbers
Solve problems
The teacher’s role in helping students develop the process standard of reasoning abstractly and quantitatively includes providing opportunities for students to strategically reason about number quantities and their applications. First, teachers provide distinct opportunities for students to develop number sense to form a foundation for mathematical reasoning. Providing students the opportunities for developing a deep number sense includes empowering students to express interpretations about number by relating quantities in abstract ways with the application of the quantities in context. For example, the numbers 5, 17, 32 can be related to each other by comparing which sets are greater or less than the others, or considered in context as 5th place, 17 ounces, or age 32. Providing students with the opportunity to apply relationships between numbers supports student number sense by promoting mathematical efficiency. For example, knowing the relationship between the numbers 3 and 9 allows students to replace 1/3 with 3/9 in the problem 1/3 + 4/9 to efficiently solve. Students also need opportunities to recognize the magnitude of numbers in context in order to make sense of their world. For example, children may consider 93 dollars to be a large sum of money, but 93 grains of rice would be a very small serving of food. Providing students with the opportunity to use multiple computational strategies, as well as multiple representations of number facilitates a development of deep number sense. Generalizing about computation based on individual cases is important. For example, realizing that a number multiplied by two is the same as doubling the number or that multiplication does not always make the product bigger are big conceptual ideas. Providing students opportunities to make decisions involving numbers in their own lives is paramount to the development of strong number sense. Opportunities to translate information into algebraic representations of contextual problems to use in the solving of the problem results in increased number sense.

24. Reason Abstractly and Quantitatively Teachers Should…
Draw students’ attention to numbers and their applications
Encourage discussion that promotes reasoning
In the classroom, it is important for teachers to draw students’ attention to numbers and their applications by considering numbers to be a source of exploration for students. Teachers should encourage classroom discourse that promotes going beyond the answer. Through the teacher’s use of probing questions, students thinking can be stretched beyond the answer to the underlying mathematical connections between the quantities being considered.

25. Construct viable arguments and critique the reasoning of others.
Process Standard #3
Construct viable arguments and critique the reasoning of others.
The process standard of constructing viable arguments and critiquing the reasoning of others relates to students’ analysis of various mathematical situations. Students make conjectures and arguments in support of mathematical situations while listening to and judging the reasonableness of the mathematical claims made by others. The ability to construct viable arguments and critique the reasoning of others forms the foundation for student mathematical communication.

26. Compare two plausible arguments
Construct Viable Arguments and Critique the Reasoning of Others Students should be able to…
Understand and use stated assumptions, definitions, and previous results
Analyze situations
Justify conclusions
Compare two plausible arguments
Determine domain to which argument applies
This process standard is comprised of five sub-skills: understanding and using stated assumptions, definition, and previous results, analyzing situations, justifying conclusions, comparing two plausible arguments, and determining the domains to which an argument applies. Constructing viable arguments and critiquing the reasoning of others focuses on the process of solving problems and provides students with the opportunity to describe how they solved problems and how their problem-solving strategies connect with strategies of others.

27. Construct Viable Arguments and Critique the Reasoning of Others Grade 3
José ate ½ of a pizza.
Ella ate ½ of another pizza.
José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Who is right? Support your answer with words and pictures.
Students in a third grade class were asked to respond to the question shown here (read question). Student A drew an illustration showing how Jose could be right, while student B supported Ella’s conclusion. Alone, these responses are appropriate, but do little to build the understanding of the importance of congruent wholes when comparing fractions. Through orchestrating a classroom discussion that highlights these two solutions, the teacher asks each student to explain and support their position. She engages other students in the class by asking them to restate each position in their own words. During this exchange a few students question student A and student B about their thinking as they were responding to the question. This leads to an insightful discussion about the importance of the recognizing the whole in context when comparing what seem to be equivalent fractions. The teacher then extends the problem by asking students if they can create an example in which Ella would have eaten more pizza than Jose.
Student B Response
Student A Response

28. How many toothpicks are needed to make 4 hexagons? 5 hexagons?
Construct Viable Arguments and Critique the Reasoning of Others Grade 6 .
Student A Response
I found a pattern of adding five toothpicks for each hexagon. Since 21 toothpicks are in 4 hexagons, I know that five more makes 26. I could also multiply 5 by 6 and subtract 4.
Students in a sixth grade class were asked to respond to the question shown here (read question). After students answer the initial questions, the teacher realizes that students are having a difficult time generalizing an equation that describes the pattern in determining the number of toothpicks in n hexagons. She also notices that Student A is close to forming a generalization, but the student does not recognize the generalization. The teacher decides to ask Student A and Student B to share their work to find the number of toothpicks in five hexagons with the class. Through orchestrating a classroom discussion that highlights these two solutions, the teacher asks each student to explain and support their position. She engages other students in the class by asking them to restate each position in their own words. During this exchange other in the class recognize that Student B’s solution may help them think about the general equation of the pattern. This discussion leads to an insightful exchange that produces three equivalent equations to represent the pattern.
Student B Response
How many toothpicks are needed to make 4 hexagons?
5 hexagons?
10 hexagons?
n hexagons?
To find the number of toothpicks in 5 hexagons, add five to 21 (the number in 4 hexagons).

29. Establish supportive social norms
Construct Viable Arguments and Critique the Reasoning of Others Teachers Should…
Establish supportive social norms
Provide opportunities for students to make and evaluate conjectures
Facilitate meaningful discussions of mathematics
The teacher’s role in helping students develop the process standard of constructing viable arguments and critiquing the reasoning of others includes establishing supportive social norms in the classroom, providing opportunities for students to make and evaluate conjectures, and facilitating meaningful discussions of mathematics. First, teachers can work towards establishing supportive social norms in the classroom by modeling appropriate behaviors when discussing solutions to problems. By sharing several correct solutions to the same problem, teachers can emphasize the importance of the reasoning behind each solution, resulting in a focus on the process. Teachers can provide opportunities for students to make and evaluate conjectures by asking students to consider alternative solution paths for problems, as well as extending problems to other cases (as in the pizza problem shared earlier). Teachers facilitate meaningful discussion of mathematics by either providing opportunities for students to discuss and question each other’s mathematical thinking or by asking probing questions to help mathematical thinking emerge when student discussion is stalled.

30. Model with Mathematics.
Process Standard#4
Model with Mathematics.
This process standard is often misrepresented, mainly because of the word “model.” Some believe this word either means to show something or to represent it using a physical model, but that is not the sole intent of the practice.

31. Model with Mathematics Students should be able to…
Apply mathematics to solve problems in everyday life
Make assumptions and approximations to simplify a problem
Identify important quantities and use tools to map their relationships
Reflect on the reasonableness of their answer based on the context of the problem
The most important aspect of this process standard is application. Can students apply mathematics to real-world contexts?
When they are solving these real-world problems, are they attending to what assumptions they are making and the impact of these assumptions on their solution?
Do students have opportunities to identify the quantities they need and use tools to represent their thinking?
Finally, do students, think about whether their answer is reasonable?
The knot that ties all these ideas together is the importance of solving authentic, real-world problems that require mathematics.

32. Model with Mathematics Grade 1
Real-World Scenario: David had $37. His grandpa gave him some money for his birthday. Now he has $63. How much money did David’s grandpa give him?
Symbolic Model:
37 + ☐ = 63
Real-World Scenario:
David had some stickers. He gave 37 to Susan. Now he has 26 stickers. How many stickers did David have before?
For example, the real-world scenario and symbolic models shown here are common in first grade. Students’ first encounter with models usual involve the solving of early word problems. Their mathematical models may be in the form of an equation like the ones given. Students should attend to the operations and order of numbers in their models.
As students progress through the grades, the ways they model and represent situations will change as students learn more mathematics.
Symbolic Model:
☐ - 37 = 26

33. Model with Mathematics Grades 4-5
Whole number division: 25 ÷ 4 =
If 25 students are going on a field trip, and each vehicle can hold 4 students, how many vehicles are needed?
25 ÷ 4 ≈ 7
If 25 pennies are divided equally among 4 kids, how many pennies will each kid receive?
25 ÷ 4 = 6 R1
If Candace shares 25 candy bars equally among 4 people, how many candy bars will each person get?
25 ÷ 4 = 6 1/4
If John cuts a 25-meter board into 4 equal pieces, how long will each of the pieces be?
25 ÷ 4 = 6.25
Whole number division is an important content focus in grades 4 and 5. Consider the division problem 25 divided by 4. What’s the answer. Well, depending on the context of a real-world problem, we make assumptions about how to report the answer – with a remainder, as a decimal, as a fraction, or by rounding. Some textbooks ask students to use one of the methods without helping students to think about why. Students should be given opportunities to reflect on what a reasonable answer is for a given context. Part of modeling is understanding what assumptions are being made when solving a problem and what answers are reasonable.

34. Model with Mathematics Grade 8
Given data from students’ math scores and absences, make a scatterplot. Draw a line of best fit, paying attention to the closeness of the data points on either side of the line.
In grade 8, students’ begin to analyze data involving 2-variables. Data modeling might include creating scatterplot and line to fit the data. This question can easily be extended to have students make estimates using the line of best fit. For example, you might have students estimate the score on the test for a student that missed 2 class sessions. This problem requires students to produce multiple representations. However, it also engages students in a problem in a familiar context.

35. Modes of Representation
One way we model mathematics is by representing mathematics in multiple ways. In early grades, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed.

36. Model with Mathematics Teachers Should…
Provide opportunities for students to solve real-world problems
Focus students’ attention on sense-making and reasonableness of results
Have students develop real-life contexts to support mathematical expressions
To help students meet this standard, teachers need to
Provide opportunities to solve real-world problems. The more authentic these problems are, the better. Even within your own school or community there are many ways you can connect the mathematical work of your students.
Focus on sense-making and reasonableness of results. Although students can often develop ingenious ways of solving problems, you want students to be able to consider whether their solutions are reasonable and appropriate given the constraints of the context.
Finally, we often have students solve word problems. One of the ways to strengthen students’ ability to model is by having them develop the context or word problem. For example, you might have students write a word problem for 4/5 – 1/2. Students must be careful that they are attending to the whole when writing such problems.

37. Use Appropriate Tools Strategically
Process Standard #5
Use Appropriate Tools Strategically
The nature of mathematics facilitates the use of a variety of tools for teaching and learning. In fact, some math content, like measurement require the students to use appropriate tools.

38. Take a look at this picture
Take a look at this picture? Not using the right tool can often create increase the burden of doing mathematics, which in turn, can cause students to develop negative impressions of mathematics.

39. What is a tool?
So, what do we mean by tools? Of course, tools in mathematics would include the traditional tools we’ve used in math classes, like using a protractor to measure angles or compass to construct arcs, but it also means technological tools, like calculators and computer software for geometry and/or data analysis. Manipulatives can also be considered tools, such as pattern blocks or base ten blocks. Finally, the word “tool” can be extended to include conceptual tools, such as different types of graphs or common formulas and algorithms that are used to accomplish a task. Even estimation is considered an important mathematical tool needed to solve problems. Thus, “tools” can represent a wide variety of mathematical devices, both physical and conceptual.

40. Use appropriate tools strategically Students should be able to…
Consider a variety of tools and choose the appropriate tool to support their problem solving
Use estimation to detect possible errors
Use technology to help visualize, explore, and compare information
Knowing HOW to use mathematical tools is only part of this mathematical practice. In additional, students need to know when a tool is appropriate to use and how to use them strategically in their work.

41. Use appropriate tools strategically Kindergarten
Decomposing Numbers
In kindergarten, students learn to decompose numbers. Students should use many tools, such as two-color counters, ten frames, unifix cubes, and part-part whole representations to represent these decompositions. Students should be familiar with these different tools and understand how they can use them to help them construct their arguments about different ways numbers can be decomposed.

42. Use appropriate tools strategically Grade 4
Determine how many unique pairs of Main Dish and Drinks can be made.
3 x 3,
because of the Multiplication Counting Principle
In this problem, students were asked to determine how many unique pairs of main dishes and drinks can be made. To solve this task, students may use a variety of conceptual tools, like a tree diagram, a table, or the multiplication counting principle. Students should be aware of the different choice of tools they have in solving such problems.
Some teachers like to always select the types of tools that students use, but it’s also important that students develop the ability to choose appropriate tools to express their thinking.

43. Use appropriate tools strategically Grade 7
Triangle Inequality Theorem
Students in 7th grade are expected to be able to construct geometric figures with given constraints or to provide reasoning if a figure cannot be constructed. Technological tools, like Geometer’s Sketchpad, can be useful to aid students in creating constructions and arguments. For example, this figure was used to justify why the triangle inequality theorem holds true, showing that you cannot create a triangle with side lengths 6 cm, 3 cm, and 2 cm.
In evaluating the instructional value of a technological tool, it’s important to focus on whether the tool provides a means to further the students’ mathematical thinking by giving new insights.

44. Use appropriate tools strategically Teachers Should…
Provide students with access to appropriate tools
Facilitate students’ selection of tools
Help students become aware of the power of tools
To ensure this process is met, teachers need to
Provide students with access to appropriate tools.
Facilitate the selection of tools. Remember that many tools have multiple purposes and can address different content areas. For example, two-color counter are useful for teaching topics in whole number operations, fractions, and probability.
Help students become aware of the power of tools. Remember that tools should not be used simply for the sake of using tools, but rather the tools should support the students’ ability to engage and develop mathematically.

45. Look for and make use of structure
Process Standard#7
Look for and make use of structure
Process Standard #7:
Look for and make use of structure.
Students are familiar with structure in their daily lives. The routines they follow on a daily or weekly basis provide structure for their time. There’s also structure in mathematics that students should look for and make use of. And just as with structure in our lives, structure in mathematics helps us know what to expect. In fact, a major contribution to the beauty of mathematics is its structure.

46. Look for and make use of structure Students should be able to…
Explain mathematical patterns or structure
Shift perspective and see things as single objects or as composed of several objects
Explain why and when properties of operations are true in a context
For this process standard, students should explain the patterns and structure they see in mathematics.
They should shift perspective and view things as a single object or as composed of several objects.
And finally, students should explain why and when the properties of operations are true? Do they know, for example, when the commutative property holds? Do they know why it works for multiplication?

47. Look for and make use of structure 3rd grade
7 x 8 = ?
I know that
7 x 7 is 49. And one more 7 makes 56.
So 8 x 7 = 56.
Students in third grade are expected to apply properties of operations and mental math strategies to multiply and divide. [3.OA.5]
Here are two different ways students might multiply 7 x 8.
The student on the left used an array to help visualize the problem and to decompose 7 into By using the structure of the array, the student used two easier multiplication facts to find the product of 7 and 8.
The student on the right used a related number fact (7 x 7) to help him find the product. He decomposed the number 8 into
Both students looked for and made use of structure to solve the multiplication problem.
Then = 56

48. Look for and make use of structure 5th grade
When we multiply a number by 10, the value of each digit becomes 10 times larger.
Students in 5th grade are asked to examine the pattern in the number of zeros that appear in the product when multiplying a whole number by powers of 10.
Often students learn a “rule” for multiplying by 10, 100, and so on. If you multiply by 10, add one zero. If you multiply by 100, add two zeroes. But they don’t really understand why the rule works.
This figure illustrates that when one multiplies by 10, the value of each digit in the original number becomes 10 times larger. Here you see 10 groups of 34 toothpicks. Looking top to bottom, you can see that each group of 10 toothpicks in the 34 becomes 10 tens or 100 toothpicks. Each single toothpick in the 34 becomes a group of 10 toothpicks. Looking across the bottom of the slide, one can see that after multiplying 10 x 34, we now have 3 hundreds and 4 tens or 340. The result is that each digit shifted one place to the left. THAT is why we add a zero on the end of 34!
The structure of our place value system helps us explain why this rule works.
Each digit moves one place to the left.

49. Look for and make use of structure 7th grade
Find the area of the trapezoid shown below using the formulas for rectangles and triangles.
Derive the formula for area of a circle using what you know about the area of a parallelogram.
Another domain in the common core standards is geometry. The study of geometry relies a lot on the structure of objects. [7.G.6. and 7.G.4]
Here are two area problems. In the first example, students are asked to find the area of a trapezoid, using only formulas for rectangles and triangles. In order to complete this task, students need to shift perspective, and see the trapezoid as composed of smaller shapes.
In the second example, students must use the structure of the circle to decompose it into wedges that can be rearranged into the shape of a parallelogram. Then using prior knowledge of how to find the area of a parallelogram, they can derive the formula for area of a circle.

50. Look for and make use of structure Teachers should…
Draw students’ attention to the structure in mathematics.
Provide examples that are conducive for exploring structure
Engage students in exploring patterns in numbers.
Patterns can be represented both visually and numerically and are a good context for recognizing structure
Help students make use of structure.
Problems with a certain structure can be solved in a similar manner.
To help students meet this standard, teachers need to
Draw students’ attention to the structure in mathematics. To do this, you need to provide appropriate examples that will encourage students to notice the structure and allow them time to talk about and share their findings.
There are lots of patterns in numbers, and these patterns can be represented in different ways. They provide one context for recognizing structure in mathematics.
And finally, help students make use of structure. Noticing structure is only the start. Being able to apply that structure to new situations will help students solve problems efficiently and effectively.

51. Look for and express regularity in repeated reasoning
Process Standard #8
Look for and express regularity in repeated reasoning

52. Look for and express regularity in repeated reasoning Students should be able to…
Notice if calculations are repeated and use information to solve problems
Use and justify the use of general methods or shortcuts
Self-assess to see whether a strategy makes sense as they work, checking for reasonableness prior to getting the answer
When engaged in this process, students move beyond simply solving problems to finding ways to generalize the methods they use and to determine shortcuts for those procedures.
Students should notice if calculations are repeated and use that information to solve problems.
They should use AND justify the use of general methods or shortcuts. The word AND is important here. Often students can use a method or shortcut, but don’t understand why it works.
And finally, do students use metacognitive skills? In other words, do they think about their thinking? In particular, do they ask themselves if a strategy makes sense as they work? Do they check for reasonableness? These are important skills that will help students notice the regularity in their reasoning.

53. Regularity and repeated reasoning in mathematics is useful
but in mathematics, we're talking about regularity in reasoning, which means the ways in which students generalize methods or justify shortcuts
Let’s look at some examples….

54. Look for and express regularity in repeated reasoning 3rd grade
Commutative Property for Multiplication
For example: 3 x 6 = 6 x 3
An array demonstrates the concept:
In third grade students investigate whether order matters when multiplying two numbers. In the example shown, we can see that it is not immediately obvious that 3 groups of 6 is the same as 6 groups of 3. Initially, students may need to count to verify this.
After a few, similar examples, students conjecture that the order doesn’t matter when multiplying.
The commutative property for multiplication can be further illustrated through the use of arrays in which the number of rows and the number of columns are interchanged.
4 rows of 3 or 4 x rows of 4 or 3 x 4

55. Look for and express regularity in repeated reasoning 5th grade
Angelo has 4 pounds of peanuts. He wants to give each of his friends 1/5 of a pound. How many friends can receive 1/5 of a pound of peanuts?
In fifth grade, students divide whole numbers by unit fractions.
Here’s an example: Angelo has 4 pounds of peanuts. He wants to give each of his friends 1/5 of a pound. How many friends can receive 1/5 of a pound of peanuts?
A diagram like the one shown can help students to solve this problem.
Students may explain the solution like this: “Since there are 5 fifths in one whole, 5 friends can share one pound. There are 4 pounds, so 20 friends can each receive 1/5 of a pound of peanuts.”
After more problems like this one, students conjecture that the solution can be found by multiplying the number of wholes by the number of parts in each whole. This thinking eventually leads to the shortcut we all know and use: “invert and multiply.”
[5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.]

56. Look for and express regularity in repeated reasoning 8th grade
Transformations
In 8th grade students study transformations such as reflections and rotations.
After completing several problems, students may notice that when an object is reflected across the y axis, the x-coordinate of the image is the opposite of the x-coordinate of the pre-image. And the y-coordinate stays the same.
In a similar manner, when a figure is rotated 180 degrees about the origin, students notice that each coordinate of the image is the opposite of its pre-image.
When students look for and recognize this regularity, transformations become easier to perform.
[8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.]
Reflection across the y-axis
1800 Rotation about the Origin

57. Look for and express regularity in repeated reasoning
Teachers should…
Avoid Teaching Shortcuts Before Students Develop Understanding of Important Concepts
Provide opportunities to look for the regularity in the calculations.
Scaffold Examples to Highlight Regularity in Repeated Reasoning
Use multiple examples to help students move from seeing the repeated reasoning of a single example to being able to build a general method.
Establish Expectations for Students, and Share Conjectures About General Methods
Establish norms in which students are expected to make and share conjectures related to what they notice
To help students meet this standard, teachers need to
Avoid teaching shortcuts before students develop understanding of important concepts. When students can make sense of a process or method, they take ownership of it, understand why it works, and are less likely to use it incorrectly or inappropriately. The only way this will happen is if the conceptual understanding comes first.
Teachers should scaffold the examples they use so that students move from seeing the repeated reasoning of a single example to being able to build a general method. Asking students to describe the processes they use and look for repetition in those processes provides the scaffolding needed for students to make sense of the method.
And finally, it’s important to establish classroom norms for student expectations. Students should be expected to make and share conjectures related to what they notice.

58. Reference:
Our primary reference for this webinar has been the book series: Common Core Mathematics in a PLC at Work, co-published by Solution Tree and the National Council of Teachers of Mathematics.

59. Heather Baker hbaker@doe.in.gov 317-518-4577
Questions
Heather Baker