Part 1 of 8, Strategies for Integrating the Mathematical Practices into Instruction
By Dr. Michele Douglass
This year’s work has been deeply concentrated on the implementation of the Common Core State Standards. As I work with school sites as they are planning and designing lessons, we end up in conversations about the mathematical practices.
As we focus on the key shifts of Focus, Coherence, and Rigor, lesson planning adjusts to how to use the mathematical practices as a way to address these shifts. It isn’t easy to do when most of us have had our own instruction delivered to us through very traditional approaches that use one procedural method.
This series of blogs will examine each practice individually with special attention to implementation. The first practice is Reasoning Abstractly and Quantitatively.
The practice of Reasoning Abstractly and Quantitatively asks students to make sense of quantities and their relationships in problem situations. Students should have the ability to decontextualize (to abstract a given situation and represent it symbolically and be able to manipulate these symbols) and the ability to contextualize (to connect the symbols back to the references in the problem that each represent). This practice also asks students to reason with quantities using habits that create reasonable representations of problems—considering the unit of measure(s) being used versus just computing; and knowing and flexibly using different properties of operations and objects.
Now the big question is what does this look like in a classroom? You may be saying, “Yah, right? I can do this, but my students aren’t even close to doing these things as a habit!”
Getting students to create this as a habit is dependent on the types of activities or lessons you plan. It requires that students have opportunities to think in this way. I have found that there are many wonderful websites to support my own thinking, which I have included in the references below.
Here are a couple of problems that I have used with students this year to provide them with this opportunity.
Example 1: Without using the standard algorithm, what is the quotient of 25 divided by 0.05?
There are several things that might happen as students reason with these numbers. You might find that they contextualize the numbers to think about money. If they do this, they are working to find out how many nickels are in $25. You might also find that some students relate this problem to multiplication. In fact, some students might say, “I know 5 x 5 = 25.” From here, students might get stuck. If this is the case, think about this sequence of problems for students to use to help reason about the quantities (each one could be connected to the context of a measurement if you think about a length that is 25 meters long):
- 25 divided by 1 = (If you take 1-meter steps, how many would you take?)
- 25 divided by 0.1 = (If you take 1-decimeter steps, how many would you take?)
- 25 divided by 0.01 = (If you take 1-centimeter steps, how many would you take?)
When students are reasoning about quantities, they can use context to make sense of the idea that when you divide by a value that is smaller than the whole unit, it takes a lot of them. In other words, the quotient results in a larger value than the dividend.
Example 2: Your principal bought 8 tacos and 5 burritos at Taco Town for $13.27 on Monday. On Tuesday, he went back and bought 6 tacos and 7 burritos for $14.47. The store didn’t charge any taxes due to a special they were having. How much does each teacher need to pay for a taco? What about a burrito?
Instead of just jumping into this problem, think about all the reasoning that could take place. You might begin with writing 8t + 5b = 13.27 on the board. Ask students what this represents. Be specific in asking why 8t is used versus 8b? Ask what t is representing. Follow this by asking students to write an equation for Tuesday. Ask students to show two different ways they could figure out the cost of a taco or a burrito. They might choose to do graphs of the equations, t-charts connected to trial and error, or an algebraic approach such as elimination. Be sure to connect the thinking of the method of choice back to the context of the problem.
Example 3: You had 25 coins in a jar. Your dad put some coins into the jar over the weekend but didn’t tell you how many he added. When you recount the coins, there are now 37 in the jar. How many coins did your dad add to the jar?
Here is an example for the lower grades. If students are reasoning with quantities, they might not even write an equation. They might count on as a strategy to answer this question. Another strategy could be using a Hundreds Chart. Starting at 25, you move down one row (add 10) to 35 and then to the right 2 squares (add 2) to end at 37. What would you expect students to draw to illustrate this problem? You could write their equations on the board to ask which one(s) could represent the problem and why?
25 + ⃝ = 37 37 – ⃝ = 25
37 = 25 + ⃝ 37 – 25 = ⃝
The important aspect to be thinking about is how you create the lesson that is not just about writing and solving equations. Rather, you are slowing down and pausing to ask students to reason with quantities using various strategies and methods. Reasoning needs to connect quantities to the context of the problem.
As important as it is to find appropriate problems to use in lessons allowing students opportunities to reason abstractly and quantitatively, it is equally critical to ask questions of students versus moving on as the clock is ticking. Here are some questions to use as the teacher to push students to reason abstractly and quantitatively:
- What does the number ____ represent in the problem?
- How can you represent the problem with symbols and numbers?
- Create a representation of the problem.
- How does your picture/model represent the problem? Where are the _____ in your picture?
- Have you solved a problem like this before?
- How did you know to break the numbers apart in that way?
- Why does that strategy make sense to you?
- What properties might you use to find a solution?
- Could you use another property or operation to solve this problem? Why or why not?
References:
Common Core State Standards Initiative Website: http://www.corestandards.org/Math/
Implementing the Mathematical Practice Standards Website: http://mathpractices.edc.org/
Inside Mathematics Website: http://www.insidemathematics.org/index.php/mathematical-practice-standards
Mathematics Assessment Resource Service Website: http://map.mathshell.org/materials/index.php
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.
