Major Implemented Strategies
My students will be able to apply the strategies listed below during classroom discussions as well as later on in their math careers. I will introduce the 3 main strategies that include Polya’s Problem Solving Methods, Three Reads and Math Talks. I will discuss the effectiveness of all and intertwine them to give students a structure that will help them be successful in a discussion-based environment. I will highlight the advantages of each and describe how they build off of one another to create a culture for rich dialogue in my classroom. I hope that they will be able to apply the discussion techniques in other classrooms at Pierce.
Strategy of Polya’s Problem Solving Methods
Students need a tool that can be applied in many junctures in their math careers. This is essential to my unit’s success. I will reference the model frequently in classroom discussion and in other units later on in the year. My students need some sort of roadmap that they can apply not only for this unit, but throughout the progression of all units covered. My students are in need of an overall structure to help them organize their problem solving efforts. For this, I will probably eventually want to use Polya’s 4-step structure.3
The problems sets will require them to read carefully and analyze what is to be done. These can be problems with somewhat complicated wording, but with relatively modest mathematical requirements. Understand The Problem can be a serious roadblock that will hinder problem access. It is the first step.
Then, when the problems presented are a little more involved, perhaps two steps rather than one step, they will have to figure out what they should do. This second step is Devise A Plan.
When they become skillful at devising a plan, I will present problems for which the calculations are somewhat more demanding with harder numbers (fractions or all variables), so they will have to work harder to Carry Out The Plan.
Then, after I have discussed a considerable variety of problems, I will select some to compare, discussing how they are similar and how they are different, and perhaps sorting them into groups of problems that are mathematically similar, although they have different scenarios. That is, I will have them Reflect, something that they are not used to doing. This step will allow my students to answer the following questions. Is the answer reasonable? Can I use another method and get the same answer? It is in this fourth step that I believe the most learning eventually will take place. I usually will have students keep a journal to help organize their thoughts and reflections post-solving. It is important to have a reading and writing dialogue with students with limited capabilities in the language arts. It helps engage learners who may not be fully motivated to talk in class. This way, students can have discourse in different ways. The written portion is another way in which students can communicate and is an important aspect to a discussion-based classroom.
Figure 3. Flowchart of Polya’s Problem Solving Steps
This is the perfect strategy to help my students develop a problem solving nature. My students have not yet developed a lot of stamina for persevering through a problem. Polya’s method will allow students entry into the problem and give them a roadmap to follow. The most important piece I want my students to gain from method is the area that links reflection to understanding. As my students get more experience with problem solving, reflection can lead to a broader understanding of the topic. Eventually, students will see similarities and differences between problems. Furthermore, this will lead to insight into why certain approaches are more productive than others. Instead of not engaging with the material or question at hand, they will have a jumping off point to get started and go through the steps. Once problems get more complex in format and wording, students will need to figure out a clear direction to go. They will require a patient and gradual practice to overcome the building of complexity. My problem bank has been designed with this in mind.
3 Reads Strategy
A strategy that I will continue to use alongside Polya’s Four-Step Method will be the Three Reads Protocol.4 The Three Read Strategy is another tool that students can rely on once they have been doing problems with me for a while. This can be used alongside Polya. This protocol focuses on SMP 1, where students have to persevere through longer problems. It helps support their sense making and their ability to engage in the solution process. The Three Reads Protocol is a specific approach to Polya’s step 1, understand the problem and will continue into step 2, devising a plan.
This strategy involves students reading a text three times and pulling out specific information each read. It is important to introduce this idea with a higher level of cognitive demand task. During the first read of the problem, students are asked to either talk about or write what the problem is. It can be done through a variety of strategies such as: Think, Pair, Share or just whole group sharing. This is very similar to Polya’s understand the problem. The second read, students are asked to pick out all of the quantities involved in the problem. The students are asked what they mean, how do they relate to each other. Here, I am usually scribing, meaning, writing on the blackboard student responses when they are picking out the numbers and the units. Finally, the last read will revolve around asking the students to combine all the different quantities into mathematical statements. What equations that fit the scenario can you think of? This will allow students to be wrong or partially correct in a positive setting. This will relax the environment because the focus is on understanding rather than the solution.
Math Talks
As in all my classes, I will continue to use the Math Talk principles from Intentional Talk.5 There are four major ones to keep in mind.
Clearly defined goals are a must. Every talk will have a goal, to be accomplished by its end. For each talk, I will need to pay attention to what my students’ specific needs are and how the talk will address and ultimately meet that need.
Students should have some sort of universal format when sharing and engaging with other classmates. My favorite one is ‘Agree/Disagree with ideas and not the person’ because this helps increase a respectful culture. Ideas and thoughts can be very personal and disagreement may lead to hurt feelings that hinder the learning process and slow growth. Additional Resource 1 lists some of the norms that I try to foster in my classroom. I find myself revisiting them frequently during instruction.
The third principle is to set students up for the cultivation of their own ideas. This is not an easy process to develop in my classroom because I have to refrain from over-scaffolding when students struggle.
Lastly, I make clear that all ideas are important to the learning process and that brains matter. I sometimes have told my classes that it is more important to advance the conversation than to be correct, and that their fellow students will be grateful for making mistakes for them. This level of respect and care throughout develops whole citizens. This is aligned with my school mission and vision for our students. Cultural norms help foster critical thinkers who engage in a respectful manner while developing love and compassion for learning.
I use Math Talks year after year, and maintain the same protocol. It makes it easier on me when I loop with the same students the following year. Also, I can have a seamless transition into my new school this upcoming year with the protocols already figured out in advance. Although the style or ideas change from talk to talk, I need to stay consistent in my delivery. Some ideas for a protocol are located in Additional Resources 2. I usually limit talks to 15 minutes. However there is no reason to end a talk early. If there are wonderful ideas on display and there is value in the material being discussed, I firmly believe that talks should organically continue their own pace until they reach a natural ending. I try to make time for talks one or two days a week.
To start a math talk, I give a prompt, and then allow for 1-2 minutes of think time. This is silent in nature and I want to save all the discussion for when it is appropriate. My hope is that during the think time students will be applying all strategies intended to allow for access and develop a structure to organize a classroom dialogue. If a student is combining both Polya’s Method and the Three Reads strategy effectively, then they should have productive think times and be able to find multiple ways to solve the problem presented. When it is time for whole class sharing I act as scribe for my students. I want to make sure that I record the name of the student who contributes each idea. That student owns that idea and it is crucial for students to see where that idea came from. This generates another growth measure for cultural capital and emphasizes that students’ ideas are the primary focus.
Below are the problem sets and phrases that will be discussed during our talks. I will outline at the beginning of each operation how talks can be used. Furthermore, I will explain how am I going to use the format of math talks to increase their awareness that many different phrases can be translated to the same expression. Overall, students should be able to carefully read each expression and produce the same end result. I will include in the Additional Resources 3 some questions that I plan to use to help facilitate deeper conversation.
I plan to introduce Math Talks in a very specific manner. I will practice the method, depending on the class familiarity with the discussion technique. I will start off using the Four 4’s problem as an initial talk for this unit. The Four 4’s problem is to make expressions using four number 4s, with as many different values as possible, using the standard signs for arithmetic expressions in any configuration that is legal. For example, 4 + 4 + 4 + 4 evaluates to 16, while 4 + 4 – 4/4 evaluates to 7. This is particularly useful so that I can start to see where students struggle. I have to keep in mind that it’s OK to give hints along the way. I can support student thinking by showing one or two examples. My favorite is 4,444. This solution is reached by pushing the numbers together. I will also give recognition to different ways to obtain the same value.
This takes careful thinking on the part of the students because they have to create as many numbers as they can, using any combination of operations, but only the number four. A total of four of them will be used and the students describe how they ended with their end number. This might show the students that they can be as creative as they want when generating the expressions, as long as they respect the grammar of arithmetic. An important aspect of this problem is to raise sensitivity to and appreciation of the grammar of mathematical formulas, and the use of symbols, and especially, the order of operations. Further comment is located in the activity list.
The next series of talks will include English to algebra problems, and algebra to English problems. The problems below are categorized by operation and will vary in the form in which they are delivered. I envision using the sets as a basis to start fluid discussions.
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