In the newly adopted Texas Essential Knowledge and Skills (TEKS) standard 1c for each grade level states, “The student is expected to select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.” And this standard is stated in the Common Core State Standards under MP5 saying, “Use appropriate tools strategically.”
The term “tools” is used loosely here and doesn’t necessarily mean just those tools that you can manipulate with your hands but also encompasses the tools that are in your mind and built from strategies for problem solving.
In my classroom I always started by teaching my students the problem solving strategies that I wanted them to use. These strategies of course changed over time but in the end they were pretty much the same overall with a different focus on things that I wanted to see out of them in their work. Was I helping them by doing this? Was I hindering their thinking by doing this? The point could be made both ways but I also was open to letting students solve their problems in a different manner as long as they explained their thinking behind it. Communication was KEY!
Many of the abstract tools that we need to teach our students are DANG HARD! While it may take time to teach perseverance through a problem situation it is possible. Exploration through mathematical thinking and building conjectures- again totally possible although they take time. Now how about teaching students to develop strategies based on prior learning to help them solve current problem situations. WHOA, back up the truck there! You want me to teach my students to bring something out of the depths of their brains and use that to build on what we are currently learning? Yep, I sure do! Example time…
Around fourth grade students begin to learn how to add fractions with the same denominator. During this process they are introduced to fractions strips, part/whole relationships, composing and decomposing fractions, unit fractions and more. This is all part of building a solid fraction foundation. Now, look back at what they have been introduced to… fractions strips are a manipulative that allows them to “connect” their fractional pieces. We have been teaching students to build groups using manipulatives since Kindergarten. Part and Whole Relationships start when are working with Composing and Decomposing numbers back in Kindgergarten as well. Students learn the number bonds that create a larger number. Unit fractions… now this stuff is newer… students are to learn that anytime you have ONE PART of a fraction, it is a unit fraction and therefore teaches you what the smallest form of that fraction can be.
Now how can we build upon that? Well the next thing that students need to be able to do is add fractions with unlike denominators. To get to this step they must have an understanding of what we have already taught them as well as common mulitples of numbers. You can see in the examples provided that because the student was understanding of how to use bar models to represent fractions they were able to easily connect what the equivalent fractions were.
Not only has the student built on prior knowledge (a tool), they have also made conjectures (a tool), used bar models (a tool), and communicated their thinking (a tool). All of these TOOLS were used to solve one problem. Teaching your students different ways to fill their toolbox with strategies that will help them break down a problem is more than just a standard, it is a critical attribute to fulfill their understanding.
I can’t wait until next time
where I will be talking about choosing the right strategies to solve problems. I will discuss the different methods that I have used and taught with over the years and pros and cons of each so that you can decide what is best for your classroom.