Are my students able to add and subtract within 100? Can they read a histogram or multiply mixed numbers? These are the types of standards-based questions that we often ask ourselves as teachers. In a world of high-stakes testing and accountability, it is easy to focus instruction primarily on content standards. They are, after all, the skills by which students are most commonly assessed. Practice standards, such as NCTM’s Process Standards or the Common Core’s Mathematical Practices, are less concrete, so they are more easily either ignored or reduced to “checklist thinking” – post them on a wall, give students a problem, and then ask students which process or practice they used. Check. Done!
When used appropriately though, practice standards can be an excellent vehicle for both improving content understanding and increasing student engagement. To use practice standards effectively, they cannot be afterthoughts or add-ons at the end of a lesson. Planning lessons with the practice standards in mind from the outset helps frame your lessons in a way which is far more engaging and impactful. Although the application of a practice standard may not be directly assessed, purposeful engagement with the practice standards leads to a deeper understanding of concrete content.
Let’s explore this with CCSS Mathematical Practice 1: Make sense of problems and persevere in solving. Give your students some word problems and you can check MP1 off the list, right? Not quite – rote problem solving is not the intention. It is easy to oversimplify the meaning of the practice standards. In the case of MP1, the key phrases are “make sense” and “persevere.” A very common teaching approach is to provide direct instruction, lead a guided group practice, and then give students a series of similar problems to which to apply the same solution path.
In such a case students are not problem solving or “making sense.” They are simply following a prescribed solution path. In this scenario, perseverance means slogging through a batch of look-alike problems and maybe a challenge problem. True perseverance is about tackling novel problems that require the application of previously learned content in a new way. It’s about willingness to risk “getting it wrong” or willingness to backtrack and try a different approach. Repetitive, “same-context, different numbers” problems do not encourage perseverance and discourage risk-taking.
Educator Magdalene Lampert proposes flipping this method of teaching. In her approach, students individually wrestle with a unique problem related to a new content or skill. They then come together in small groups to discuss ideas and approaches. The lesson culminates in a group sharing of solution paths and findings, which provides opportunities for discovering and understanding underlying mathematical processes. Hmmmm… that sounds a lot like CCSS Mathematical Practice 8: Look for and express regularity in repeated reasoning. Many of the practice standards are strongly interrelated; when you integrate a particular practice into a lesson, others will naturally lend themselves to the work as well.
Let’s see how this can work with actual content. Suppose you want students to understand how to find the area of figures composed of rectangles. Students have explored area by tiling rectangles with unit squares and counting units squares on a grid. They have discovered (hopefully through open-ended problem solving) that multiplying length by width gives the area of a rectangle. Now present them with a diagram of an L-shaped room and ask them to determine the area.
Some students may subdivide the figure into unit squares and count. Some may multiply the longest length by the longest width (12 x 11). It can be difficult to watch students go down a “wrong” path like this, but group discussion helps them compare, contrast, reject, and/or build off one another’s ideas. A student who sees the 12 x 11 approach may object to the fact that part of the 12 x 11 rectangle is missing. He or she may then subtract the area of the “missing part”: 12 x 11 – 9 x 3. While not the standard method, this is a perfectly acceptable, mathematically-sound strategy. Through individual exploration and small group discussion, the idea of subdivision into two smaller rectangles will invariably come up.
During the whole class debrief, the relative strengths and weaknesses of each approach can be discussed and debated. Hey, isn’t that CCSS Mathematical Practice 3: Construct viable arguments and critique the reasoning of others? Lampert’s novel problems approach makes explaining and justifying mathematical reasoning a cornerstone of the mathematics classroom.
Through this process, students learn that multiple approaches are viable and acceptable, as long as they are clear and defensible. This encourages them to use mathematical language authentically to document and defend their thinking. Students who are allowed to wrestle with problems like this are no longer passive participants stuck on trying to figure out “what the teacher wants.” Instead, they are engaged in formulating and testing ideas both individually and with others. Such open-ended opportunity builds resilient problem solvers, who are willing to wrestle with novel problems and are confident in their ability to apply their mathematical understanding to tackle unique challenges.
How can you use a process-first approach to encourage rich problem solving in your classroom?