Did you first think to discuss in your solution why you are not just adding four with every square? Did you first try to think of the singular form of the word vertices? Initial thinking is not a linear activity, especially in mathematical problem solving. Yet, the result of problem solving—the written solution—often looks like a linear, step-by-step procedure.
Good problem solvers brainstorm different thoughts and ideas when first presented with a problem, and these may or may not be useful. Problem solvers can use a graphic organizer to record random information but not process it. A student can later reflect upon usefulness of the information and ideas. If the information and ideas help the student make relationships between concepts, then they are essential.
It also gives every student a starting point for the problem-solving process. Figure 1: Four Corners and a diamond mathematics graphic organizer. Figure 1 depicts the four corners and a diamond graphic organizer. This graphic organizer was modified from the four squares writing graphic organizer described by Gould and Gould The four square writing method is a formulaic writing approach, originally designed to teach essay writing to children in a five paragraph, step-by-step approach.
The graphic organizer portion of the method specifically assists students with prewriting and organizing. We saw beneficial problem-solving aspects in the graphic organizer portion of this writing method for mathematics. Actually, the form in Figure 1 does not have to be given to the students each time.
Figure 2 shows how students, using a blank piece of paper, make the four corners and a diamond graphic organizer template. The teachers reported that students later e. Figure 2: Four Corners and a diamond folding template. In terms of objectives, it does not. Obviously, the four corners and a diamond graphic organizer is designed to help students understand the problem, devise a plan, carry out the plan, and look back Polya, It is the implementation process, how students form their response, that is the important aspect of the four corners and a diamond graphic organizer Zollman, a.
The pictorial orientation allows students to record their ideas in whatever order they occur. If students first think of the unit for their final answer, then this is recorded in the fifth, bottom-right area. This idea the unit , then, is not needed in the short-term memory because a reminder is recorded. If students first think of a possible procedure for their answer, this is recorded in the third, upper-right area. The four corners and a diamond graphic organizer allows, and even encourages, students to use their problem-solving strategies in a non-hierarchical order.
A student can work in one area of the organizer and later work a different area. It also shows that completing a problem-solving response has several different, but related, aspects. Students do not begin writing a response until some information or ideas are in all five areas. The four corners and a diamond graphic organizer especially encourages students to begin working on a problem before they have an identified solution method.
As in the four square writing method, the students then organize and edit their thoughts by writing their solution in the traditional linear response, using connecting phrases and adding details and relationships. The steps for the open response write-up are as follows: 1 state the problem; 2 list the given information; 3 explain methods for solving the problem; 4 identify mathematical work procedures; and 5 specify the final answer and conclusions. The graphic portion of the organizer allows all students to fill in parts of the solution process.
Further, teachers quickly can identify where students are confused when solving a problem by simply examining the graphic organizer. The teacher should model proper use of the four corners and a diamond graphic organizer and have students work in groups when introducing this tool. Working in groups allows students to see that many problems can be worked in more than one way and that different people start in different places when solving a problem.
In their small-group discussions, students identify relationships between the areas in the graphic organizer and among the various solutions. Four corners and a diamond provides students with a logical framework for writing about problem-solving tasks.
Graphic organizers can benefit students when they take standardized state mathematics assessments, specifically open-response problem-solving items. Most states use a scoring rubric for these types of items. In Illinois, for example, the scoring rubric has three categories: mathematical knowledge, strategic knowledge, and explanation Illinois State Board of Education, Higher-ability students sometimes skip steps in their explanations. The four corners and a diamond graphic organizer helps each type of student produce a more complete response in each of the three categories and, thus, receive a higher score.
Nine middle school teachers decided to use the open-response mathematics questions as the focus of their action research on the effects of using graphic organizers. Teachers administered pre- and post-tests with their students to see if using the four corners and a diamond graphic organizer impacted their performance. The graphic organizer helped students coordinate various parts of mathematical problem solving: a What is the question? Zollman, a; b. The teachers found the use of graphic organizers in mathematical problem solving to be very efficient and effective for all levels of students.
The teachers saw that their lower-ability students, who normally would not have attempted problems, had now written partial solutions. The organizer appeared to help average-ability students organize thinking strategies and help high-ability students improve their problem-solving communication skills Zollman, b. Students now had an efficient and familiar method for writing and communicating their thinking in a logical argument.
The samples of student work in Figures 3 and 4 are from an open-response squares and vertices problem before and after the use of graphic organizers in the classroom. Sample 1 shows the work of a student who was presented the problem before becoming familiar with the four corners and a diamond graphic organizer.
This work shows a misunderstanding of the problem, limited strategy, and no explanation. While it is not a perfect response, understanding, organization, development, and reflection are all strongly represented on the graphic organizer. However, without any explanation, the teacher cannot know what strategy, if any, the student was attempting.
Again, this work shows a misunderstanding of the problem, limited strategy, and no explanation. For mathematical knowledge, the formula is well explained in words, not as an algebraic expression. This would be acceptable on state assessments, as the problem did not specifically ask for an algebraic expression. Sample 5 is the post-test work of a higher-ability student. The drawings also suggest that the student feels a sense of ownership of and satisfaction with the solution and probably finished the problem with plenty of time to spare.
We hoped the students in our action research study would improve their problem solving with an instructional intervention from pre-test to post-test; however, no single instructional method directly affects learning.
Rather, instruction is one of many factors that may influence learning. Others include the curriculum, the student, the class, and the teacher. The crucial factor in the effectiveness of any instructional method is how it is implemented. If four corners and a diamond graphic organizer is used as a linear, systematic procedure to teach problem solving, it will succeed sporadically.
In fact, any direct teaching about problem solving is likely to have intermittent success. However, students may remain uncertain about where to start a problem, confused by essential versus non-essential information, or unaware how to communicate important steps and reflections in their solutions. We found that graphic organizers aid students in all three of these areas. Allowing students to first use their own thinking—and then reflect, revise, and re-organize their knowledge, strategies, and communication—helps them improve their problem-solving abilities.
Initially, teaching about problem solving as a hierarchy of procedural steps is neither efficient nor effective. Our results confirm other studies that found teaching via problem solving is the key instructional process Lester, As our work suggests, effective reading and writing strategies like graphic organizers may have crossover effects in mathematics for students of all ability levels. We found that four corners and a diamond, when properly used, was an extremely useful instructional method in the middle grades mathematics classroom.
Our instructional approach helped students construct content knowledge and strategic knowledge and, we contend, it also improved their mathematical communication skills. Extensions The author shows how graphic organizers that are typically used to help students organize their thoughts while writing in ELA can also be used to help them think through problem-solving tasks in mathematics.
For example, a simple subtraction problems could be expressed as:. This is the underlying procedure or schema students are being asked to use. Once they have a list of schema for different mathematical operations addition, multiplication and so on , they can take turns to apply them to an unfamiliar word problem and see which one fits.
Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper. Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number. Show students how to make an educated guess and then plug this answer back into the original problem. To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared.
Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:. Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem.
This will allow students to keep track of their thoughts and pick up errors before they reach a final solution. Checking work as you go is another crucial self-monitoring strategy for math learners. Model it to them with think aloud questions such as:. But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks before arriving at a final answer.
Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. Show students how to backtrack through their working out to find the exact point where they made a mistake. Read up on how to set a problem solving and reasoning activity or explore Mathseeds and Mathletics, our award winning online math programs.
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