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:
- 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.