Archive for February, 2014

Our good friend, Dr. Ted Coe, had the following article published on


Why Arizona needs Common Core standards

Math professor: It’s all about how we think and learn

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By Ted CoeMy TurnTue Feb 25, 2014 12:54 PM

In a recent surveyconducted by Expect More Arizona, 43 percent of the state’s likely voters said they favor Arizona’s College and Career Ready (Common Core) Standards, which define the knowledge and skills necessary for K-12 students to succeed in college and careers.

After being given a short description of the standards, support rose to 71 percent.

Yet just last week, the state Senate Education Committee passed bills to abolish or allow schools to opt out of these standards and literally revert back to the level of standards from the last century. This is not the direction Arizona should be moving.

To understand why, you must first understand three key areas of mathematics teaching and learning: ways of doing, ways of thinking and habits of thinking.

Ways of doing: I have been a math educator for 20 years. I have seen how our culture views mathematics simply as ways of doing. Consider the typical math classroom experience: A teacher explains a particular mathematical process and students complete the process 30, 40 and sometimes up to 100 times.

The next day, the teacher introduces another mathematical process, perhaps unrelated to the previous one. Students collect these pieces for up to five months and the system rewards those who are able to memorize the greatest number of processes.

Think about it: When you finished the last of your mathematics classes, did you see math as a collection of random pieces or as a completed puzzle?

Ways of thinking: For too many, math ends up being merely a collection of random pieces, when in reality the pieces should always fit together. After all, mathematics is the one academic subject that makes logical sense.

Unfortunately, many lose the notion that what we do in math should follow from how we think about math. Our ways of doing need to develop from ways of thinking.

Arizona’s College and Career Ready Standards set a new expectation in this regard because they are built on progressions of thinking. That is, the standards emphasize not only the mathematics processes but also conceptual understandings that are consistent across the grades.

Math instruction should make sense both day-to-day and year-to-year. Teachers who use clever shortcuts to supplant thinking do their students a disservice in the long run.

Habits of thinking: It is important to encourage good mathematical habits of thinking. Arizona’s College and Career Ready Standards will develop mathematical practices such as problem-solving, persevering, reasoning abstractly, constructing arguments and attending to precision. The standards will require our students to think about and apply their math skills to real-world contexts.

Educating our state’s future leaders has never been more important. Arizona’s College and Career Ready Standards provide a framework that will ensure our youth get the quality education needed to prepare for college and the workplace.

In the realm of mathematics, that means doing more than helping students ace an exam. Rather, it’s about encouraging and supporting ways of doing, grounded in ways of thinking, while developing essential mathematical habits.

Dr. Ted Coe is an assistant dean at Grand Canyon University, a member of the Partnership for Assessment of Readiness for College and Careers Math Operational Working Group and a co-director of the Arizona Mathematics Partnership, an $8.7 million National Science Foundation grant to promote excellence in middle school math and improve student achievement.


The Common Core State Standards Initiative is a state-led effort coordinated by the National Governors Association Center for Best Practices and the Council of Chief State School Officers. To date, forty-five states have adopted the Common Core Standards and are working to develop new assessments to measure student success based on these new standards.

Content standards have been a part of educational reform at the local, state, or national level for many years. However, the Common Core State Standards in Mathematics offer a new and exciting focus that has not been seen before – Standards for Mathematical Practices. These mathematical practices focus on the ways of thinking and habits of thinking necessary for students to understand mathematical ideas that can be used as they prepare for college and career.

The philosophy of the getrealmath team is in sync with the Common Core Standards for Mathematical Practices. Let’s look at the eight mathematical practices and see how our  approach can support the development of these practices.

Common Core Standards for Mathematical Practices

  1. Make sense of problems and persevere in solving them.
    Solving problems is at the heart of our  approach. Students can better make sense of problems that are based on real-world situations. As students work to solve these problems, they learn to persevere. Perseverance is best learned by, well, persevering! Initially, perseverance may mean that a student sticks to it for just a minute or two. As students find success solving interesting problems that are connected to motivational contexts, they are apt to stick to a problem for longer periods of time before giving up.
  2. Reason abstractly and quantitatively.
    Rather than to only mimic procedures, students using activities based on real-world contexts must engage in reasoning. In fact, the activities often ask students to explain their thinking, to make connections, and to justify their arguments. These kinds of activities will help students to reason abstractly and quantitatively. Also, the activities make appropriate use of mathematical symbolism to help students move from the concrete to abstract.
  3. Construct viable arguments and critique the reasoning of others.
    When working to solve real problems, students often need to construct an argument defending their thinking, their problem solving approach, or their conclusions. This is a skill that will be beneficial for preparing students for college and career. All of the activities can be easily incorporated into group work and/or classroom discussions and instructors are encouraged to do so. When working together with a partner or a team, students will be afforded the opportunity to listen to the reasoning of others and to provide feedback. Learning to provide constructive criticism of the reasoning of another is a valuable skill that can be developed while solving real-world problems.
  4. Model with mathematics.
    The Common Core Standards for Mathematical Practices states, “mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” Teaching with a focus on real-world activities provide students the opportunity to develop this problem solving ability.
  5. Use appropriate tools strategically.
    These tools include paper and pencil, calculators, Internet resources, measuring devices, and even mental images. Students need to learn not only how to use these tools but also when to use these tools appropriately. Some computations can and should be performed mentally (e.g. 6 x 8 = 48) while others can be done efficiently, in the context of a real-world problem situation, using a calculator. In some problem situations, students may need to conduct research to learn more about the situation as they make sense of the problem. Collecting data, making measurements, using concrete tools to model a situation are real-world problem solving strategies.
  6. Attend to precision.
    Certainly, computing accurately is part of attending to precision. Another important part of attending to precision is helping students to develop precise explanations and descriptions of their thinking, their conclusions, and their strategies. As students construct viable arguments, write down their conclusions, or orally present their thinking, we can help them to do so precisely. Describing the mathematics as it connects to a real-world situation can be helpful in developing this precision.
  7. Look for and make use of structure.
    One of the goals of our approach is to help students to develop a conceptual understanding of the mathematical ideas that they are learning. For example, the idea of a linear function is strongly connected to the idea of a constant rate of change. As students work to make sense of linear functions in a real-world context, they can better see the structure of the mathematics and make important conceptual connections.
  8. Look for and express regularity in repeated reasoning.
    Mathematics is often described as the science of patterns. Students who learn to identify patterns can develop a profound understanding of the mathematical ideas. While working in real-world contexts, students are often encouraged to practice reasoning about the solution methods so that the ideas can be developed in a profound way.

The Common Core Standards initiative is committed to assessing students’ development of these mathematical practices. That is, student success on standardized tests at the state level will soon require that students work to develop them. Using the Make it Real Learning Activities, students will be led through a development of the above-mentioned mathematical practices while working in real-world contexts.