Editor’s Note: The Standards for Mathematical Practice, which are included in the Kentucky Academic Standards for Mathematics, describe the expertise that educators should strive to develop in their students. Students should be provided multiple opportunities to demonstrate proficiency of these standards. These standards are often referred to as the 8 mathematical practices.
By Brooke Powers
I hope that in the weeks since the first part of this series was published – “Incorporating the math practice standards for students” – you have had an opportunity to incorporate some of those math routines in your classroom in order to engage students in the 8 mathematical practices.
In the long stretches of school between breaks like we are currently in, I find the use of these methods and routines brings structures and focus to my classroom on a daily basis that the kids and I both need. The practices and routines I covered in my last column –are some of my favorites. But I find the ones shared here are equally important in getting students to be problems solvers and mathematicians.
Make sense of problems and persevere in solving them.
One of my student’s favorite strategies is when we incorporate problems from Open Middle into class. Open Middle gives students an opportunity to engage their brain in a problem with multiple solutions and entry points. It encourages students to look at math problems as more than just something to solve and instead as something to reason and think about from multiple viewpoints.
After time for individual thinking and small group sharing, we share out student thoughts as a class. I reward the students who thought outside of the box and came up with unique solutions. The problems on the website are sorted by grade level and domain, so it is so easy to find one with which to start your class.
Here is an example of one of the many problems found at Open Middle.
Directions: Using the numbers 1 to 9 at most once each time, fill in the blanks to make the equality true:
Attend to precision.
“Find the Blub Day” may not be the most electric of all the resources I have shared, but probably brings some of the greatest opportunities for critical thinking we have in class.
There isn’t a specific website or resource I use for this activity. I use formative data from my classroom to identify an area of weakness for my students and capitalize on this as an opportunity for them to do error analysis.
When students enter the room on Find the Flub Days, I have an incorrectly completed problem on the board with errors the students have typically made. The student’s job is to find all the errors and then work the problem correctly on their desk or paper.
After the initial time to think and work, students share their findings with a shoulder partner before we have a whole class discussion about the errors and the correct response. This activity helps strengthen a student’s ability to find mistakes in their own work and encourages them to have a growth mindset about their own mistakes.
Look for and express regularity in repeated reasoning.
Counting circle is the number routine in our classroom that takes up the most time, because it generally leads to rich discussion about ways to think about numbers flexibly.
For this math practice, I begin with a blank number line on the board and then give the students a random starting value and an amount I would like them to count by. The amount of challenge in these assignments vary based on the course and the time of year.
At the beginning of the year, we focus on things like starting at 1, but counting up by 2. That’s harder than it seems. By this time of year, we are starting at 5 and counting up by 2-8/9. While I move around the room selecting students to say the next number, I am scribing every answer – correct and incorrect – on our class number line.
If a student makes an error, I go ahead and record the error on the number line. The next student sometimes compensates for the mistake in their answer and sometimes they miss it, but I keep scribing the whole time.
Once our number line is complete, we go through and correct errors as a class and talk about ways we could have thought more flexibly in order to make the mental math easier. In the case of 2-8/9 – which we just completed in class – students shared that some of them thought it was easier to add 2 first and then add the 8/9 as needed, while others added 3 and then subtracted 1/9.
Below is a picture of counting circle from my class. In this example, we started at zero and were counting up by 1-7/8. Students share strategies for determining which numbers are correct and which ones are incorrect. The incorrect answers, as determined by the class are crossed out as the mathematical discourse progresses.
Use appropriate tools strategically.
This is perhaps the most difficult math practice to write about, but the most important for developing students into true mathematicians. As I tell my students, mathematicians in the real world use whatever tools are at their disposal to solve problems and think critically. We work throughout the year on using all of the mathematical tools we can to solve problems in class.
Students should be given opportunities to choose and use the tools strategically and not be handed the appropriate tool each time. However, in order to accurately incorporate this standard for math practice into your class, students must be taught when to use and how to use a variety of tools and then they will develop the ability to select which tool is appropriate to use for a given a task.
I try to develop this in my classroom through the use of take home tool kits. As students are introduced to different manipulatives and tools – such as paper algebra tiles, colored paper, index cards and rulers – they add them to their tool kit. When they are doing independent work later, they have the knowledge and experience to be able to select which tools are most appropriate to use to solve a given problem.
My classroom routine to intentionally incorporate the 8 math practices continues to develop and transform my instructional practice all the time. Although I have grown from those early days of ignoring them altogether, I still have a long way to go until I feel I am using them to their full potential daily. However, the use of these routines and activities certainly has accelerated my student’s potential as mathematicians and problem solvers.
Brooke Powers is an education blogger and 7th-grade teacher at Beaumont Middle School (Fayette County). You can read more about providing students a real-world and engaging math experience on her blog by clicking here.