Appendix A:

Coding and the Standards for Mathematical Practice

This appendix will show how you can support coding through the application of computational thinking within the Common Core State Standard Mathematical Practices. Each standard will be mapped to the corresponding ISTE Standards for Students. Individual standards will also be broken down to show the connections to the ISTE and Computer Science Teacher Association’s (CSTA) computational thinking vocabulary. Tables for each standard detail how K–5 students progress through the standards using computational thinking.

Standard 1

CCSS.Math.Practice.MP1: Make sense of problems and persevere in solving them.

Corresponding ISTE Standards for Students: Computational Thinker 5b, 5c.

You will find this mathematical practice standard in every math problem every day. It means that students must first understand the problem, then find an organized way to attack it, and finally, work until it is done. Let’s decompose the standard and see how it relates to computational thinking. Vocabulary that relates to computational thinking has been bolded in the following standard description.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway, rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

Mathematically proficient students can explain the correspondence between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.

Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can identify correspondences between different approaches.

ISTE/CSTA Computational Thinking Vocabulary

Abstraction: Meaning of a problem, meaning of a solution

Problem Decomposition: Entry points, plan a solution pathway

Data Collection: Analyze givens, constraints, relationships, and goals

Data Analysis: Consider analogous problems, try special cases and simpler forms, gain insight into its solution, ask “Does this make sense?”

Data Representation: Explain correspondence between equations, verbal descriptions, tables, and graphs, or draw diagrams of important features and relationships, graph data; search for regularity or trends

How to Teach It

Teachers should give students tough problems and wait them out. Allowing wait time for yourself and your students will be the toughest obstacle to overcome with this standard. Work for growth in thinking and the “aha” moments. Teachers will know they are on the right track when math becomes about the process and not about the one right answer. The key is to lead with questions and not with your chalk, pencil, or smartboard pen.

Begin by encouraging students to estimate a solution prior to starting the task. Ask questions and provide manipulatives. Manipulatives are not only for K–2 teachers; they should be used by all K–5 teachers as a way to allow for students to productively struggle without lowering the cognitive demand for the task. Then have students share solutions and strategies, while having them intentionally make connections.

K–5 Student Progression for Standard 1

Kindergarten

Students begin to build an understanding: “doing math” is actually solving a problem. They wrestle with the meaning of the problem by solving through trial and error.

First Grade

Students realize that “doing math” involves solving a problem. They begin to look for the meaning of the problem before they begin. They are beginning to identify critical information needed to solve problems.

Second Grade

Students look for meaning in the problem and will begin to plan out a problem-solving approach. They can locate critical information needed to solve the problem. They are beginning to identify when they need to make changes in their solution pathways to arrive at a reasonable answer.

Grades 3–5

Students can provide a reasonable estimate prior to solving a problem. Students can identify when they need to make changes in their solution pathways to arrive at a reasonable answer. Students will look for meaning and listen to the strategies of others for meaning, as well as errors in thinking. They will try different approaches and look for the most efficient way to solve problems. At this point, students should be using another method to check and justify the validity of their answers.

Standard 2

CCSS.Math.Practice.MP2: Reason abstractly and quantitatively.

Corresponding ISTE Standards for Students: Computational Thinker 5c.

This is probably the hardest of standards for those without a mathematical degree. There is a reason why most elementary teachers do not major in mathematics. That said, this is also one of the most important concepts for teachers and parents to understand. It boils down to making sure that student not only understand the standard algorithm for math, but also what the algorithm means.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: 1) the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and 2) the ability to contextualize—to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing the flexibility using different properties of operations and objects.

ISTE/CSTA Computational Thinking Vocabulary

Abstraction: Represent it symbolically (e.g., with the written numeral “8”) and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents (e.g., referents = 8 physical objects).

How to Teach It

Having students draw representations of a problem, while working with the manipulatives will provide students with the means to figure out what to do with the data themselves. Allow students to move from the manipulatives (concrete), to symbolic drawings (representations), to the algorithms (abstract).

Young students progress with the knowledge that five objects equal the number five (5). From there they begin to use manipulatives and explore fact families. If fact families are only done with the abstract numbers, the students never solidify their ability to reason abstractly. For students to truly understand the abstract, they must be able to apply the concepts to their referents (manipulatives) first. Teachers can help engage students in this mathematical practice by asking questions such as:

Just like reading uses the read-aloud to demonstrate teacher metacognitive thinking, teachers can use the think-aloud in math to demonstrate their own computational thinking in action.

K–5 Student Progression for Standard 2

Kindergarten

Students begin to recognize that numbers represent a specific quantity of something. They also connect the quantity to the written symbol.

First Grade

Students begin to compose and decompose quantities and construct written equations and fact families. For example:

4 + 5 = 9

5 + 4 = 9

9 − 5 = 4

9 − 4 = 5

Second Grade

Students solidify understanding and use properties of operations, and can relate addition and subtraction to the length of an object.

Third Grade

Students can connect quantity to written symbols. They use this knowledge to create a logical representation of problems, considering both the quantities and the units needed to represent those quantities.

Grades 4–5

Students’ understanding moves from whole numbers to fractions and decimals. Students can write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts.

Standard 3

CCSS.Math.Practice.MP3: Construct viable arguments and critique the reasoning of others.

Corresponding ISTE Student Standard: Computational Thinker 5.

It is important that students are able to talk about math using mathematical language to support or refute the work of others. Communication and collaboration are two key digital age skills that have been identified in a variety of content specific domains. If we want our students to acquire these skills, then we must intentionally model, teach, and assess them on these skills.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases and can recognize and use counter examples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

ISTE/CSTA Computational Thinking Vocabulary

Data Analysis: Use stated assumptions, definitions, and previously established results in constructing arguments; distinguish correct logic or reasoning from that which is flawed

Problem Decomposition: Able to analyze situations by breaking them into cases

Algorithms & Procedures: Logical progression of statements; construct arguments

Simulation: Justify their conclusions, communicate them to others, and respond to the arguments of others

How to Teach It

Teachers should post mathematical vocabulary and make students use it. Teacher practices have always included questioning, but students also need to be provided with sentence frames in order to understand the correct context with which to use these words.

K–5 Student Progression for Standard 3

Kindergarten

Students begin to develop mathematical communication skills by referring to objects and drawings, and asking questions such as, “How did you get that?” Students also explain their own thinking and listen to the thinking of others, as well as respond back with the help of simple sentence frames.

Grades 1–2

Students expand their communication to include error analysis in identifying when someone’s thinking may not be correct. They communicate their own thinking and ask questions to determine if others’ explanations make sense. Students start asking, “How did you get that? Explain your thinking. Why is that true?”

Third Grade

Students work on refining their thinking, but should still be using concrete manipulatives and representational drawings to support their thinking. Students ask questions and are strengthening how they respond to others’ thinking.

Grades 4–5

Students should be moving toward the use of standard algorithms to demonstrate their thinking; however, they still use correct verbal explanations that demonstrate cohesive quantitative reasoning. They should refine their thinking by understanding the strategies of others and be able to explain which strategy is best for them and why.

Standard 4

CCSS.Math.Practice.MP4: Model with mathematics.

Corresponding ISTE Student Standard: Computational Thinker 5c.

Understanding the word around us is essential for adult success. Creating representations and models of processes is how we interact with our surroundings. Young children are continually taking input from their surroundings and trying to “make sense of it.”

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.

Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important criteria in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. 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.

ISTE/CSTA Computational Thinking Vocabulary

Abstraction: Simplify a complicated situation

Automation: Map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas

Simulation: Can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace

How to Teach It

With the implementation of the reading workshop, K–5 teachers are familiar with the read-aloud, in which teachers model their metacognitive thinking so that students can see and hear what good readers do. Similarly with mathematics, teachers can model and connect different representations and processes that students can use to solve tasks by using a think-aloud.

K–5 Student Progression for Standard 4

Grades K–2

In grades K–2, students experiment with representing problems using objects, pictures, numbers, words, charts, lists, and even acting the problem out. Students need time to create connections between the way in which they represent problems and how others represent problems. Allowing time for students to make their thinking visible and explain how they arrived at their answer will help them increase the strategies they can use. Students should be expected to use a variety of strategies.

Grades 3–4

Students will expand upon their skills by evaluating the strategies and processes they used versus those that others used. Students will begin to explore similarities and differences between the strategies and processes used.

Fifth Grade

Students will solidify their thinking on strategies and processes by making decisions on the most efficient way to solve problems, stretching their thinking to realize that efficiency can be influenced by the perspective of the author/creator.

Standard 5

CCSS.Math.Practice.MP5: Use appropriate tools strategically.

Corresponding ISTE Student Standard: Computational Thinker 5b.

Understanding the word around us is essential for adult success. Young children are continually taking input from their surroundings and trying to “make sense of it.” Learning what tools help in solving mathematical problems correctly and efficiently will help our student to better understand and find success in our world.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator.

They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technology tools to explore and deepen their understanding of concepts.

ISTE/CSTA Computational Thinking Vocabulary

Automation: Consider the available tools when solving a mathematical problem; tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, and/or a spreadsheet.

How to Teach It

Don’t tell students what tool to use; leaving the decision-making process in the hands of the student will possibly cause frustration, which is what teachers should want. This will lead to discussions about what tool worked best and why. Teachers should provide a variety of tools so that students can make choices and discover what works best for them.

K–5 Student Progression for Standard 5

Kindergarten

Students begin to consider available tools (including estimation). For example, students may use linking cubes to represent two quantities to compare them side by side.

First Grade

Students expand their knowledge of available tools. For example, they may first estimate and then use linking cubes, colored chips, or a ten-frame to model an addition problem.

Second Grade

Students move into the representational stage by adding the use of graphic organizers. For example, students may use a drawing of a ten-frame or tally marks, as well as concrete items.

Grades 3–5

Students will expand upon their tools with graph paper, rulers, calculators, protractors, compasses, t-charts, Venn diagrams, fraction tiles, self-created graphic organizers, and so on.

Standard 6

CCSS.Math.Practice.MP6: Attend to precision.

Corresponding ISTE Student Standard: Computational Thinker 5b.

How many teachers have opened an item to be assembled, only to discover that the directions in the box could not be easily followed? Being able to help our students explain themselves so that they can be understood is an essential component for helping them to be career and/or college ready.

Our current system has a fundamental flaw (Reinhart, 2000) when teachers stand in front of the class demonstrating and explaining. It would behoove teachers to change their thinking of good teaching from “One who explains things so well that students understand” to “One who gets students to explain things so well that they

can be understood.” This again suggests that teachers must create a learning environment that values risk-taking and making mistakes (Boaler & Dweck, 2016).

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, including 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 context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school, they have learned to examine claims and make explicit use of definitions.

ISTE/CSTA Computational Thinking Vocabulary

Data Collection: Use clear definitions.

Data Analysis: State the meaning.

Data Representation: Communicate precisely.

How to Teach It

Ask good questions by focusing on process questions, rather than product questions. With pressures in the amount of content that teachers need to cover, they often resort to asking things like, “When you add 3 + 5, that will equal …?” and they wait for the students to respond back, “8.” These questions are much faster than asking students a process question such as, “If you have three objects and five objects, how many are there all together?”

K–5 Student Progression for Standard 6

Kindergarten

When students say, “I don’t get it,” they will begin to understand that they need to be clear and precise in the language they use in explaining what they “don’t get.”

Grades 1–2

Students will eliminate the use of “I don’t get it” by replacing it with clear and precise language. Teachers will continue to strengthen clear and precise mathematical language by having students explain their thinking in front of their peers. Teachers will ask clarifying questions so that students can explain their thinking.

Grades 3–5

Students will expand upon their use of clear and precise language by specifying units of measure, meaning of symbols, and appropriate labels for diagrams and graphs they create.

Standard 7

CCSS.Math.Practice.MP7: Look for and make use of structure.

Corresponding ISTE Student Standard: Computational Thinker 5d.

Seeing things from multiple perspectives is a common theme throughout the Common Core State Standards in a variety of subject areas. Finding patterns and repeated reasoning helps students create connections to help solve more complex problems. For example, when students learn about fact families, they come to understand that addition and subtraction are really about parts that make up a whole. In 8 − 3 = 5, 8 − 5 = 3, 3 + 5 = 8, and 5 + 3 = 8, the number 8 represents the whole, and 3 and 5 make up the parts.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three plus seven more is the same amount as seven plus three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see that 7 × 8 equals the well remembered (7 × 5) + (7 × 3) in preparation for learning the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.

ISTE/CSTA Computational Thinking Vocabulary

Data Analysis: Discern a pattern or structure.

Parallelization: See complicated things as single objects or as being composed of several objects.

How to Teach It

Help students find patterns and repeated reasoning that can be built upon and lead to solving more complex problems. As students get older and gain more experiences, they will learn more strategies and get better at breaking apart complex problems into pieces so they can apply strategies they know to solve each piece.

K–5 Student Progression for Standard 7

Kindergarten

Students begin to discern number patterns and/or structure. For example, students might recognize that all “teen” numbers start with a 1 and end with numbers 0-9. They also begin to recognize that 2 + 3 = 5 and 3 + 2 = 5.

First Grade

Students recognize the communitive property of addition (9 + 5 = 14 and 5 + 9 = 14). They also work with recognizing groups of five and ten within a group of numbers being added or subtracted. For example (4 + 6 + 4) = (10 + 4) = 14 or (9 + 11) = (29 − 10) − 1 = (19 − 1) = 18.

Second Grade

Students begin using the patterns and structure to develop fluency (mental math strategies). Using patterns and strategies such as making ten, fact families, and doubles helps students develop this fluency with numbers.

Third Grade

Students look closely to discover patterns and structures that involve strategies to multiply and divide. They begin with repeated addition and repeated subtraction, but move on to begin using the commutative and distributive properties to develop efficiency and fluency.

Fourth Grade

Students look closely to discover patterns and structures that explain calculations, such as the partial products model. They relate representations of counting problems, such as tree diagrams and arrays, to the multiplication principal of counting. They will begin generating number and shape patterns that follow a given rule.

Fifth Grade

Students look closely to discover patterns and structures that use previously mastered properties of operations as strategies to add, subject, multiply, and divide with whole numbers and begin applying the strategies to fractions and decimals.

Standard 8

CCSS.Math.Practice.MP8: Look for and express regularity in repeated reasoning.

Corresponding ISTE Student Standards: Computational Thinker 5a, 5d.

Help students see the “big picture” of mathematics! Being able to generalize strategies, procedures, and processes will help students be able to solve real-world problems of any type in any context. For example, if students need to determine the surface area of a cylinder, they should not have to memorize the formula. Students should first understand that a cylinder is made up of a rectangle and two circles. In breaking down the object into its pieces and using prior knowledge of the area of rectangles and circles, students should be allowed to explore how their previous knowledge applies to finding the surface area of this new shape.

Mathematically proficient students notice if calculations are repeated and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of a slope as they repeatedly check whether points are on the line through (1,2) with slope 3, middle school students might abstract the equation (y-2) / (x-1) = 3.

As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their immediate results.

ISTE/CSTA Computational Thinking Vocabulary

Algorithms & Procedures: Calculations are repeated; solve a problem.

Automation: Look both for general methods and for shortcuts.

How to Teach It

Help students see the “big picture” of mathematics. Students should understand that the surface area of a cylinder is made up of a rectangle and two circles. Going further, they will need to understand that the rectangle width is the same as the height measurement in the original formula. More importantly the student will need to understand that the length of the rectangle is the distance around the circle (circumference). To find the circumference of the circle, one must take the radius × 2 (diameter) multiplied by 3.14. Hopefully students also understand that represents the number of times the length of the radius will wrap around the entire circle. For example, if the diameter (the radius multiplied by 2) can be represented by a string 4 inches long, then the 4-inch string will wrap about the circle 3.14 times. That is a lot of understanding to be done! Khan Academy has a great video explanation of cylinder volume and surface area (khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/v/cylinder-volume-and-surface-area).

K–5 Student Progression for Standard 8

Kindergarten

Students notice repetitive actions in counting and computation. For example, students notice that the next number in a counting sequence is one more and, when counting by tens, that the next number in the sequence is ten more.

First Grade

Students begin to understand place value as they add and subtract numbers. They look for the use of tens and multiples of ten when adding and subtracting larger numbers. Students continually ask themselves, “Does this make sense?”

Second Grade

Students will use rounding strategies to round up or round down. Adjustments to compensate for the rounding will then occur. Teachers should intently look for this strategy and allow for mistakes as students work through the process.

Third Grade

Students begin to look for shortcut methods to build upon what they already know. For example, if a student is asked to find the product of 7 × 9, they might decompose the 7 into a 5 and a 2. They can then work with 5 × 9 and 2 × 9 to arrive at 45 + 18 = 63.

Fourth Grade

Students will notice repetitive actions in computation to make generalizations. Students will use models to explain calculations and understand how algorithms work, as well as create their own algorithms (for example, the use of visual fraction models to create equivalent fractions).

Fifth Grade

Students will apply all these concepts to new domains when working with fractions and decimals. Exploration and understanding of area models for multiplication and division will help students verbally explain why basic algorithms work. Formulas will be understood through the decomposition of their parts.