As I recently worked on correlating our materials to the newly released Common Core State Standards, I was surprised to find many of the topics we commonly associate with a high school geometry course placed throughout the standards for elementary grades.
I was surprised, but not shocked. In fact, I’ve believed for a long time that much of the geometry traditionally reserved for high school is prime material for middle school students. As long as the pedagogical approach supports students’ truly exploring geometry—through activities that allow them to draw, measure, construct, explore, and reason—it’s always seemed to me that topics such as congruence, similarity, transformations, properties of shapes, and classification systems are completely appropriate in middle school.
Take this Dynamic Geometry activity as an example. Students can see the properties of shapes through the way they behave as their vertices are dragged, and suddenly relationships between members of the quadrilateral family don’t seem so abstract.
This is one of our many middle school geometry activities which I expected would correlate strongly to the middle school standards in the Common Core. But they didn’t… because those geometry standards are placed in the 4th and 5th grades!
Now I’m not an expert on elementary education—in my 14 years of teaching I taught mostly high school students and some middle school students—but I do have a daughter who is now starting high school. So I tried to imagine her and her friends back when they were between 8 and 10 years old dealing with these standards:
Grade 3 (Geometry Standard 1) —
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
Grade 4 (Geometry Standard 2) —
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
Grade 5 (Geometry Standards 3 and 4) —
Classify two-dimensional figures into categories based on their properties.
• Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
• Classify two-dimensional figures in a hierarchy based on properties.
At first, I thought to myself “This is crazy,” and joked with a colleague that our high school geometry book nails a 5th grade standard. But the more I immersed myself into the Common Core, the more I began to appreciate the substantive transformation of math education they propose. These are well thought out learning trajectories that combine age-appropriate pedagogical approaches with rigorous mathematics at younger ages. As the standards quoted above demonstrate, there is a conscious development of fundamental concepts over multiple years, not the usual snapshot of apparently random concepts listed for each grade level.
The trick then is in how to teach these standards at this level. Just taking the old high school geometry approach and dropping it down six or so grades would be pure torture for those little learners. But the Common Core emphasizes standards for mathematical practice, such as using appropriate tools strategically, both as overarching goals and as an integral component of specific content standards. To clarify, consider these students as they explain constructions they have made with patty paper.
The students in this video are in high school, but it’s not a great a stretch for me to envision younger students working with these same tools to identify and describe geometric relationships, and I’d be interested in hearing experiences from elementary and middle school teachers on the ways in which they teach these topics.
The key to success in adopting the Common Core is to not only change the grade level of when specific topics are taught, but also to thoroughly reevaluate the methods used at those grade levels, and to ensure that students are given the learning experiences that will deliver on the promise that these new standards engender.