Problem Solving and the Common Core

Teaching Strategies

Interactive Student Notebooks

Interactive Student Notebooks (ISNs) are an important part of my math curriculum and are utilized for both note-taking during direct instruction and individual student practice. The right side of the student created notebook is occupied by teacher-determined content, such as a foldable graphic organizer of the rules of exponents. The left side of the journal is reserved for student created content such as reflection on in-class activities or answers to practice problems.

The ISNs are an excellent tool for organization, guiding students to follow various protocols and guidelines as they build the content in their journal throughout the year. In addition to organization, the journals address students’ multiple intelligences. For example, creating or inserting graphs, tables and other images will be beneficial for visual learners. Finally, ISNs can also be used to assess student learning. Teachers can monitor student progress on the left side of the journal while students can self-assess and reflect as they look through the content of the right side of the journal. Overall, the journals allow students the ability to “internalize and personalize the content provided” leading to more ownership over the material covered.⁶

Number Talks

A number talk is a quick 10-15 minute exercise that asks students to practice mental computation around various skills. It is meant to be repeated on a regular basis with set routines and is not intended to necessarily line up concurrently with the ongoing curriculum. In number talks, “students are asked to communicate their thinking when presenting and justifying solutions to problems they solve mentally”. The process is intended to create more flexible, efficient and accurate mental problem solving. In my classroom, I have utilized this structure to touch on various review concepts or skills such as equivalent expressions or order of operations without needing to deviate entirely from the curriculum as scheduled.⁷

Kate Kinsella Precision Partnering

Dr. Kate Kinsella, a professor at San Francisco State University, specializes in ELD teaching strategies and has developed a protocol for effective partnered conversation in any classroom. Titled “Precision Partnering” this practice centers around four actions for students to employ in any partner conversation stated simply as “Look” (at your partner), “Lean” (in towards your partner), “Lower” (your voice) and “Listen” (carefully to your partner). These actions along with varied strategies for eliciting student responses and holding students accountable for each conversation serves to help create effective and productive partnered conversations. I use these strategies regularly in the classroom to ensure that students are on task and engaging with their classmates during number talks or other class discussions.⁸

Manipulatives

A major component of this unit is a hands on and manipulative heavy approach to teaching equations. Although it is essential that students develop the capacity to solve equations symbolically it has been found that “many students can benefit from working with physical problems that can be symbolized mathematically”.⁹ By combining the two strategies, students will develop a more intrinsic understanding of the essential knowledge needed for increasingly difficult algebraic reasoning.

Pan Balance

I will employ a visual model with a balance scale to best physically demonstrate the concept of the properties of equality to students. For example, to demonstrate the addition / subtraction property of equality I will put two unknown but equal weights on a balance scale, students will see the scale is balanced and not tilted to one side or the other.

Students will quickly see that if an additional weight is added to one side, the balance will become off center, teetering to one side.

If that same weight is added to the other side, however, the balance will again be even. The same visual representation can be employed for the multiplication and division properties of equality by doubling or tripling the weight in one pan, for example, and doing the same in the other.

Hands on Equations

One manipulative that I will employ is called “Hands on Equations.” This approach uses a scale (similar in theory to the previous pan balance scale activity) and symbolic representations of variables and negatives. Quite a bit of research supports this approach. One such study found that “the isomorphism between the object itself and the mathematical notions implied allows students to form a mental image of the operations that they have to apply. They are able to reactivate this self-evident image at any moment. ”¹⁰ While this method will be most useful and relevant for simple and more straight forward problems, the process will solidify students’ foundational understanding of the content, allowing them to develop mastery of more complex problems. This tool will additionally be useful in scaffolding the material. Students who absorb the content quickly can move on to a strictly symbolic representation of the problems assigned, whereas students still struggling with the concepts can continue to use the physical, hands on model.

I will use this tool extensively when discussing the essential knowledge and strategies for developing algebraic reasoning and equations. To start, it will be integral in discussing inverse operations. Through the model, students utilize inverse operations to simplify equations. For example with the equation 2x = x + 4 the scale would be set up as follows with two pawns on one side and one pawn on and a four counter on the other side:

The first step students will take to solve this is to move all the pawns (variables) to the left side. To do this, students add a dark colored pawn to each side. The white pawn represents a positive x, while the dark pawn represents a negative x. Adding the dark pawn to both sides represents a “zeroing out” of the white pawn. Symbolically this would be equivalent to x + -x = 0.

Since the white pawn has been “zeroed out” it can now be removed from the right side of the equation and the solution can be clearly seen.

We can see now that each pawn is worth 4. Through exercises similar to this one, students will become familiar with the process of using inverse operations to eliminate terms from the equation. This model is additionally helpful for reinforcing the property of equality as it shows students that the equation maintains equality when the dark pawn is added to both sides.

Examples of equations that result in either no soluyion or infinitely many solutions, can also be demonstrated with the Hands on Equations balance scale to give students a physical representation of the equations. The equation 2x+3=2x+6, for example will reduce down to 3=6, students will easily be able to see on the balance scale that the 3 cube is not equal to the 6 cube, and therefore no value of x could make that equation true. Similarly, an equation with infinitely many solutions could also be represented on the balance scale. 2x+3=2x+3, for example, would reduce down to 3=3. From this result a discussion would follow about what values for x would make this equation true, upon trying a few potential solutions student should discover that any number substituted for x would yield the same result. This, in turn, should lead them to the conclusion that there are infinitely many solutions for this equation.

Bar Models

The model method was developed in Singapore in response to students’ reliance on key words and other superficial clues to solve word problems. The model method, often referred to as bar models, “puts the focus back on the relationships and actions presented in the problem, and helps students choose both the operations and sequence of steps that are needed to solve a problem”.¹¹ In an algebraic context, the bar models can be used to help students visualize a problem in a similar way to the Hands on Equations balance scale. Two bars are set equal to each other, divided into various parts that represent different values in the equation. Through a process of elimination, a physical representation of the properties of equality, students can visualize the process of solving an equation. Consider the following problem as an example:

Jimmy bought two bags of assorted candies each with the same number of candies in each. He also bought two individual candies. If Jimmy had a total of 24 candies, how many were in each bag?

Although this is not drawn to scale, it is clear that the two from the first bar and two from the second bar can be eliminated because the two bars will remain equal if we remove the same amount from each.

We can now see that the two x’s are worth 6 total. Or one x is worth three, x = 3.

This model can help students create a visual representation of the information presented in a problem. The bar model can also be very useful with more complex algebraic problems, take the following as an example.

Al and Bob lift weights to gain for football. Al weighs 195 pounds. Bob weighs 205 pounds. Al wants to gain three pounds a week, Bob wants to gain two pounds a week. How long will it be until they weigh the same?

Set up the known information in two bars. Let w represent the number of weeks gaining weight.

Al

Bob

What the bar model above makes apparent is that even though Al starts off weighing less, he is gaining more per week. By setting up the bar model with the w’s being equivalent, it also becomes apparent that w and 10 are equivalent. It will therefore take 10 weeks for their weight to be the same.