# Math

## Unraveling the Common Thread of Big Ideas in Geometry and Measurement

By Dr. Michele Douglass

Student achievement in the areas of measurement and geometry has been lacking for years, as evidenced by TIMSS data as well as state-to-state student achievement data. Two contributing factors are that textbooks typically spend less time developing these concepts and teachers often don’t spend the instructional time that is needed for them.

Now is the time to address this deficiency in mathematical understanding by beginning to understand the Common Core State Standards (CCSS).

The CCSS bring about many changes from state to state. One of the biggest adjustments is thinking about the topic of geometry as a separate entity from measurement or data and statistics. As evidenced in the standards, statistics is not introduced as a topic until students have had years of working with data, especially as it relates to geometry and measurement.

So, what really is the difference between geometry and measurement, and how do we support in-depth student learning of these concepts?

• Measurement helps us describe shapes by quantifying their properties. Angular measures, in particular, play a significant role in the properties of shapes.
• Geometry is the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

Mark Driscoll describes the importance of having a geometric habit of mind that involves: reasoning with relationships, generalizing geometric ideas, investigating invariants, and balancing exploration and reflection to improve student achievement (Driscoll, 2007, Fostering Geometric Thinking).

These categories of thinking are certainly one way to think about geometry and its connection to measurement. However, when you look across the domains of measurement and data, along with geometry, one might see a blurring of these ideas. It isn’t these categories that will bring clarity to instruction or build deeper knowledge in students. We need to change instruction to include the connections among and between the standards in terms of common big ideas that thread through all the standards.

The big ideas that are important to use as you design your instruction are as follows:

• Visualization
• Properties that define relationships
• Dimensions and measurement
• Problem solving

Visualization. This is where everything begins. We ask students at an early age to visualize the differences in shapes to sort and categorize them. Categorizing shapes visually leads to the connection of seeing and learning properties. For example, think about quadrilaterals. Students can sort these shapes into parallelograms, or more specifically to see the differences and similarities among squares, rhombi, and rectangles, which connect to the learning of properties of shapes.

Visualization also connects to building meaning of congruence and finding congruent shapes within other shapes, which connects to symmetry and geometric transformations. Think about how a shape might be transformed when you see it appear in a real-life situation. Not only does the shape need to be recognized, but you must also be able to accurately identify the properties of that shape. Think specifically about Pythagorean Theorem. What happens if the right angle of the triangle is actually in the upper right corner of the triangle? This different orientation requires changes in the visualization students have about the location of the sides or legs of the triangle versus the hypotenuse.

Finally, visualization connects to the measurement of perimeter, area, volume, and surface area. Here one must see that, even when a triangle has been rotated and shown in a different orientation, there is still a base and a height, even if the base isn’t in its traditional location – the length at the bottom of the triangle. Furthermore, visualization is a basis for building meaning of the relationship between finding the area of a trapezoid and finding the area of a triangle. One must be able to visualize the triangles that exist in a trapezoid to make meaning of this relationship.

This same type of visualization must occur to build meaning in finding the surface area of three-dimensional shapes. One must be able to visualize all the faces of the three-dimensional shape when it is drawn on a two-dimensional surface in order to find the area of each of the faces. Likewise, it takes visualization to see the two-dimensional shapes that exist within a specific plane that intersect a three-dimensional shape. We need students to have visualized the shapes such as the circle or ellipse prior to learning their specific properties and transformations that occur when graphing.

During your planning, think about how you are providing opportunities for students to visualize the object from multiple vantage points. How do they need to see the shape? Do they need to be able to decompose the shape? Is there reason to make sense of how the object is composed of various other shapes? Do they need to see a shape within an object to build connections to understand a specific relationship or property? These are some examples of questions you should be asking yourself as you are planning with visualization in mind.

Properties That Define Relationships. We often think about our own geometry experience and remember proof, proof, proof. Yes, we must know properties, but what we often miss is that properties define specific characteristics, and many properties connect to one another due to specific relationships. For many students, the learning of properties is extremely challenging because the list of properties is so long, and the relationship of these properties is not explicitly taught.

For example, think about learning the properties of a triangle. This topic begins in third grade and extends into high school. If we think about the relationship of the properties to the triangle, it provides students with a hierarchy as well as an organization for learning the properties. Think about the properties of angles of a triangle. First, angles can be acute, obtuse, or right. Likewise, a triangle can be classified by these same words. The relationship is that these words describe properties of the types of angles that exist within a triangle.

We can also look at the properties that exist for the lengths of the sides of the triangle. The length of the sides of a triangle defines it as being scalene, isosceles, or equilateral. Some of these same words can be used to describe the properties of a trapezoid. When a trapezoid has one right angle, it is a right trapezoid. When the two nonparallel sides are congruent, we name the trapezoid an isosceles trapezoid. As we move up in grades, additional properties of the lengths of the sides of a triangle are discovered and learned, including Pythagorean Theorem as well as trigonometry functions. The use of these two sets of relationships is defined by the type of angles that are present within the triangle.

While you plan your instruction of a geometry or measurement standard, you want to extend beyond visualization to think about properties and relationships. You should consider these questions focused around the big idea of the properties and relationships that exist. What are the properties that you are teaching that link to the concept? How do these properties show a relationship? How does the property connect to another shape? Another formula? Think about the connection between the properties and the relationships so that students are not seeing every property as a new  entity, but rather as an extension of a relationship they already know.

Dimensions and Measurement. Dimensions and measurement is where the properties that we just discussed become quantified. Measurement is an issue of understanding the attribute that is being quantified. Attributes are measured in various ways and with different tools. For example, length is an attribute that is often measured in feet or inches using a ruler as a tool, whereas a table has the additional attribute of the space it takes up in a room. Space is the attribute of volume, which could be measured using cubic inches or cubic feet.

The dimensions of an object dictate the type of attributes that can be measured. For example, a line is in a plane, and when two lines intersect, you have two attributes that you could measure: length of a segment or angle measure. However, if you think back to the table, which is three-dimensional, you have multiple attributes. You could measure angles, lengths, area, volume, weight, and more. Each of these measurements is determined using specific tools such as a protractor for angles, rulers for lengths, and scales for weight.

Measurement has another aspect to it in the concept of conversions. The idea that is often lost in the act of unit conversions is making meaning of the inverse relationship that exists between the measurement and the unit of measure. More specifically, think about measuring the length of your bedroom using inches vs. feet. The length could be 120 inches or 144 inches versus 10 feet or 12 feet long. Notice how the quantity of units decreases. Many times students get confused if they only are paying attention to the total measurement without considering the size of the unit. Good instruction involves building meaning of the relationship that exists between a measurement and the unit size.

Since we are now measuring with a unit that is greater in length, it takes less iterations of that unit to match the length of the object. This is the inverse relationship – a greater unit of measure takes less iterations of the unit. Notice how understanding this relationship is critical in building meaning of dimensions and measurements.

Measurement also connects to some properties that we learn are specific measurements, such as a right angle or complementary angles. If you think back to the dialogue about properties, you may recall the number of properties that exist based on the angle measures of a shape. For example, when you have a regular polygon, you know all the angles are congruent. Vertical angles are two angles formed by intersecting lines, and they are congruent. A square has four right angles. While we often think of measurement as area or perimeter, let’s not forget the impact angle measures have within geometry and measurement.

As you are planning to teach a specific standard, you should be asking questions about measurement as well. Consider some of these questions:  What is the attribute that is being measured? Is one unit of measure better to use than another? Which tool would I use to make this measurement? How can I help students understand the inverse relationship that exists between the measure and the unit size? How does the measurement link to the properties we know about the shape?

Problem Solving. Most likely, the hardest part of geometry is the problem solving. This is hard for many reasons, but at the top of the list is that students don’t have enough opportunities to see how the visualization, properties, and measurement connect to everyday life. For this reason, problem solving must be a focus of every lesson. From classifying shapes to solving for a missing angle, students need multiple opportunities to connect the learning to everyday life, as this is most often the way the concept is tested.

Student achievement will not improve if the instructional approach is not enhanced to meet the demands of the Common Core State Standards. You can make a difference for students by investigating the standards to see the connections described in this blog.

As you begin with a standard, go back to each of the sections above and use the questions posed as a starting point for your planning. Design your instruction to place emphasis on how visualization, properties and relationships, and dimensions and measurement can support students in learning the intended content.  Use these connections to support students to build a greater understanding of geometry and measurement.

Michele Douglass, Ph.D., is the president of MD School Solutions Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as professional development and technology programs.

## Professional Development Must Engage Math Teachers in the “Big Picture”

By Dr. John Woodward

We understand the issue more and more every day. For years, we’ve been told that our students don’t stack up in math when compared with their peers in other countries. Our performance isn’t that bad at the fourth grade, but TIMSS and PISA data clearly show significant comparative declines as our students end eighth and tenth grade. One of many interpretations of these data is that math at the intermediate and middle grades is an exceedingly weak link in our educational system.

Were that not enough, the link between mathematical competence and success in the workplace is becoming ever clearer as the economy slowly emerges from a deep recession. A recent and fascinating issue of the Atlantic Monthly (Davidson, 2012) provides a lucid account of the extraordinary gaps in knowledge between highly successful manufacturing workers and their less-skilled counterparts who are employed, at least for now, on the same factory floor.  The former possess increasing amounts of quantitative knowledge, while the latter live in fear of automation or outsourcing. Success in math at the middle grades, which is obviously fundamental to success in high school and beyond, is a cornerstone for securing the future for American students.

Standards such as the Common Core are one way to renew our commitment to raising mathematical performance. Yet the challenges are significant, as evidenced in a recent survey of school districts from around the country (Center on Education Policy, 2011). Most districts agreed that the Common Core State Standards are more rigorous than most state standards and that, if implemented well, they will improve student math skills. Yet respondents also felt that new curricular materials, as well as fundamental changes in instruction, would be needed.

The Need for Professional Development

Every business organization, including school districts, wants to hire “turnkey” employees. These are teachers who can hit the ground running and deliver instruction at a high level. Yet with changing standards and what we know about how long it takes any professional to develop a high level of skills, this desire is unrealistic. The hope for turnkeys also puts aside the millions of teachers who already work in our schools. Again, the international message is clear and consistent: high-achieving countries hire the best candidates they can, but they continue their professional development through many years of employment (Akiba & LeTendre, 2009; McKenzie & Company, 2007). We need to adopt this thinking if we have any hope of raising the math performance of our students in today’s schools.

There are distinct features to high-quality professional development in mathematics for today’s teachers. First, it is crucial that teachers understand the concepts they are teaching. Some would argue that this means extensive refresher courses in college-level mathematics, most of which are taught in a traditional, symbolic fashion.

Learning more formal mathematics can possibly help some teachers, but it is an unlikely solution for most. Also, there is little guarantee that any of this kind of professional development transfers to the classroom. Instead, teachers need vivid demonstrations of key concepts (or “big ideas”) as well as opportunities to engage in learning activities that promote the kinds of instruction advocated in the Mathematical Practices component of the Common Core. Teachers – and their students – need opportunities to analyze, discuss, and reason about concepts. They also need to solve the kinds of problems that promote strategic thinking and persistence. Naturally, how to integrate thoughtful skills practice is also part of the picture.

Teachers also need to see the “big picture” within the different strands of mathematics. For example, they need to see how rational numbers develop in complexity over grades 3 through 7. This kind of connected understanding of a strand helps teachers see how the big ideas link together, how what was taught at a previous grade level needs to be reviewed, and how what students do at one grade level is important for the next grade level.

Vivid examples of classroom practice are also critical. How do I use fraction bars effectively? How do I orchestrate a classroom discussion with an eye toward students who do not normally participate? How do I assist students when they get stuck grappling with rich mathematical problems? Well-designed video examples can go a long way to improve practice, and they are something teachers can return to again and again.

Finally, teachers need a tremendous amount of assistance when it comes to instructional planning. Linking the contents of a district’s math adoption to Common Core State Standards is challenging in itself. Even more, creating opportunities within a unit of instruction for students to engage in mathematics at a high level is new to many teachers. It is easy to skip this kind of instruction, particularly if it is a new kind of classroom practice. Teachers need guided assistance doing this as well as developing a variety of assessments that tap into the kind of thinking we want today’s students to do in math.

There is good news.  We can provide the kind of professional development our teachers need. Our challenge is to accept the fact that this kind of work is an unavoidable feature of today’s successful school systems.

Dr. Woodward is a professor and dean of the School of Education at the University of Puget Sound. In a project funded by the U.S. Department of Education, he worked with the REACH Institute on a collaborative five-year program that examined teaching methods for helping students in grades 4-8 with disabilities succeed in standard-based instruction. Dr. Woodward is coauthor of the TransMath mastery-based intervention solution for middle and high school students and a lead trainer for NUMBERS math professional development.

References

Akiba, M., & LeTendre, G. (2009). Improving teacher quality: The U.S. teacher workforce in a global context. New York: Teachers College Press.

Center on Education Policy. (2011, September). Common core state standards: Progress and challenges in school districts’ implementation. Washington, DC: Center on Education Policy.

Davidson, A. (2012, January/February). Making it in America. The Atlantic Monthly. Retrieved January 26, 2012 from http://www.theatlantic.com/magazine/archive/2012/01/making-it-in-america/8844/

McKenzie & Company. (2007). How the world’s best-performing countries come out on top.  Retrieved January 26, 2012 from http://mckinseyonsociety.com/downloads/reports/Education/Worlds_School_Systems_Final.pdf

## Be a Paid Education Blogger! Enter Sopris Learning’s Blog Contest

By Sopris Learning

Sopris Learning is looking for passionate educators to share their views with an online community of colleagues through our EdView360 blog. Enter by blogging about your choice of three given topics and submitting of a short video explaining why we should hire YOU! The public will vote, and the winner will write for EdView360 at \$100 per blog! Click Here for contest details.

## Strategies to Maximize Math Instruction

By Dr. Michele Douglass

Multiple researchers discuss the best practices to use to maximize student achievement in mathematics. The good news is that the authors share many of the same big ideas.

One thing that stands out is a strong focus on procedures that has existed for years in the elementary grades. This emphasis is linked to the language and testing of our state standards, but lacks problem-solving development and the foundations of knowledge needed for higher-level mathematics.

The strategies that exist across multiple authors for improving mathematics achievement include:

1. Making number sense a part of everyday instruction
2. Focusing all content around problem solving
3. Using communication every day
4. Supporting learning with the use of tools and representation for all levels of students

1. Make NUMBER SENSE a part of every day’s lesson.

Number sense doesn’t build in a single chapter or topic; it builds over time. By providing students opportunities every day to build number sense, we help them become better problem solvers. They also learn to think about the size of the number while they learn to work with numbers in flexible ways.

Building number sense also develops estimation skills. We all know that in the real world, we rarely as adults pull out a pen and paper. Rather, we think about the numbers to estimate solutions. Think about how you solve the problem 15 x 16 without a pencil or paper. What about 45% of 250? We must support students to think about numbers in multiple ways so they don’t have to rely on an algorithm or the calculator on their cell phone.

Making this happen isn’t as hard as it seems. You can incorporate number sense into your warm-ups by figuring out the number sense that scaffolds into your lesson. Use a timer, as it’s easy to make number sense an entire lesson. Set the timer for 10 minutes. If you are working on exponents, your number sense might be on multiplying repeated factors to see how students group the factors to find the product. If your lesson is on multiplication, your number sense might be on multiplying numbers by 10 or 100. You might estimate your age in seconds or the height of 1,000 or 1 million pennies. You know the set of number sense topics that are critical at your grade level. Use these specific topics as the basis for your number sense problems. Some days, you might do a single problem, and as students learn methods for thinking about numbers, they will be able to do more than one problem.

2. Integrate more PROBLEM SOLVING.

Students learn new skills through the process of solving problems such as learning facts. Problem solving is a great way of connecting conceptual knowledge with procedural knowledge. While we often think as adults that the problem-solving problems are the hardest, children often need the context of a problem to connect the meaning within a procedure.

To begin with problem solving, choose problems that are open-ended, allowing for multiple ways to arrive at a solution. Problems need to allow students to make and test conjectures. They should foster creativity while either using formulas or connecting procedures to concepts. Many times you can find a problem in your textbook that you can turn into an open-ended problem. For example, turn a simple area problem into a comparison of two sets of dimensions and add a context. Which has the greater area and why? Ask students to justify their answers in more than one way.

3. COMMUNICATE, Communicate, communicate.

The one thing to remember about communication is that you can’t communicate either in written or oral formats without having the knowledge to express the idea in a coherent manner. To support students in communicating, begin with a safe learning environment.

Through communication, students are orally processing what it is they think they know. However, communication also gives the instructor the opportunity to be aware of how students are thinking about a concept or set of symbols or even a definition. For example, if all you ask is the answer to 2 to the 4th, you might not realize that the student is simply multiplying 2 times 4. When you ask students to go beyond giving you an answer, you learn whether they grasped that 2 to the 4th is the same as 2 times 2 times 2 times 2.

Oral and written communication in mathematics also supports language learners and students who struggle with language development in a content area. By speaking and communicating, students are building language skills and specific math academic language.

One way to begin with communication is to ask the question “Why?” And don’t just ask this when a student provides an incorrect solution. Asking “Why?” all the time makes students rethink the solution and the steps they used to arrive at the solution. Asking “Why?” to a student whose solution is correct enables you to hear the student’s process. If the process is accurate, the student is providing teaching to the class. However, there are many times when a student is getting the correct answer for the wrong reasons. If we never ask “Why?” then the student is being set up for making continuous errors.

4. Use TOOLS and REPRESENTATIONS.

Tools and representations help students build relationships among numbers, construct knowledge and meaning of concepts and ideas, and make connections between concepts and connecting procedures. Tools and representations have also been found to help maintain a positive attitude about mathematics, as they support the sense making of mathematics. Mathematics is abstract even from the nature of the symbols. There isn’t anything concrete about the way you write the number 5 to know that it represents 5 things. This must be learned. Tools provide methods for solving problems by allowing space to organize, think, reason, and test ideas.

As you begin with tools and representations, refer to your text and capitalize on the representations used by the authors. Supplement to use a variety of items. As you use different manipulatives, help students transfer understanding by showing a representation of the manipulative on paper. For example, if you use place-value blocks, you might teach students to draw a square to represent the 100 block but to use a line for the 10 rod and a dot for the unit pieces.

It takes time upfront, but the time we spend reviewing could be minimized if we spend more time supporting students in building concepts and strategies through these methods.

Michele Douglass, Ph.D., is the president of MD School Solutions Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as professional development and technology programs.

## It Takes a Village

By Sopris Learning

It takes a village to raise a child, and it takes a community to teach one. At Sopris, one of our key objectives for the 2011–2012 school year is to build a community of educators online that reaches across borders and boardrooms to explore the issues that are important to increasing student achievement—across the board.

With most of the country adopting Common Core State Standards, we find ourselves in an environment where educators are putting progress above politics and agreeing to work together to establish common ground on which to build—or rebuild—a successful K–12 education system. Districts and states across the nation have used the principles of response to intervention (RtI) to create their own multitier systems of supports (MTSS) for increasing student outcomes.

Within this framework of unity, we believe that educators can learn from one another’s successes with evidence-based academic and behavioral interventions. Educators and administrators are reaching out to one another on Facebook, Twitter, and other social media outlets to find professional support and practical strategies that translate into success in the classroom.

In addition to connecting educators, researchers, and authors on Facebook and Twitter, Sopris is launching a blog that will provide a forum for discussion around today’s education issues titled EdView360. We will hear first from Sopris’ own Stevan Kukic, Ph.D., about the “end of an ARRA” and whether this economic boost has actually stimulated positive, sustainable change. Kukic is a past president of the National Association of State Directors of Special Education (NASDSE), formerly served as Utah’s state director of At-Risk and Special Education Services, and has been instrumental in MTSS efforts across the country.

We hope you will tune in to EdView360 to hear from Kukic and other education leaders who will share their opinions, experiences, frustrations, inspirations, and big-picture insights on a variety of topics that are important to you and, ultimately, your students.

We look forward to hearing your viewpoints as well and fostering an open forum of communication and collaboration toward a common goal—empowering all students to rise to their full learning potential. You work hard 365 days a year, and we hope that your efforts come full circle this fall! Best wishes for a successful school year!

Written by Kathy Lee Strickland, marketing editor for Sopris, a member of Cambium Learning Group. She earned her Bachelor of Journalism from the University of Missouri-Columbia, has worked as a newspaper and magazine editor, and has taught at the high school and adjunct university levels.