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Concreteness Fading: Concrete, Representational, Abstract (CRA) Approach
Concreteness Fading: Concrete, Representational, Abstract (CRA) Approach
"Some know this idea as concreteness fading, while others have called this progression concrete, representational, abstract (CRA). In either case, the big idea is the same. Start with concrete manipulatives, progress to drawing those representations and finally, represent the mathematical thinking abstractly through symbolic notation."
This idea will be used as a foundation for our class as we look a student learning progressions, and how teachers build student conceptual understanding.
The problem solving techniques outlined below require that students can build visual models, which is one step in the concreteness fading learning model.
Making Math Moments Matter With The Concreteness Fading Model.pdfpolya problem solving techniques.pdfPolya's Problem Solving Techniques
Polya's Problem Solving Techniques
Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our lives where we encounter problems—not just math. Although the method appears to be a straightforward method where you start at Step 1, and then go through Steps 2, 3, and 4, the reality is that you will often need to go back and forth through the four steps until you have solved and reflected on a problem.
https://www.opepp.org/lesson/hsdm-unit7-tool-for-field/#:~:text=Nearly%20100%20years%20ago%2C%20a,classic%20method%20for%20solving%20problems.Qualities of a Math Drawing
Qualities of a Math Drawing
How Can We Use Strip Diagrams and Other Math Drawings? Visual representations can often help us make sense of a problem, formulate a solution strategy, and explain a line of reasoning. Simple drawings that show relationships between quantities and are quick and easy to make can be especially helpful. We call such drawings math drawings. Math drawings should be as simple as possible and include only those details that are relevant to solving the problem. One type of math drawing is the strip diagram, also called a tape diagram. Strip diagrams use lengths of rectangular strips to represent quantities. Because strip diagrams use lengths, they connect readily with number lines. Strip diagrams can help students formulate equations, including algebraic equations. Strip diagrams are used throughout this book.
Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 44). Pearson Education. Kindle Edition.Singapore Math Strategies – Drawing Bar Models
Singapore Math Strategies – Drawing Bar Models
Drawing bar models is a powerful tool to help students break down information and visualize math problems. Bar modeling also helps students improve mathematical fluency and number sense.
1. Read the problem.
Start by reading the problem carefully sentence by sentence.
2. Determine the variables of who and what.
As you are reading the problem, identify the who and what of the problem. Underline the who and what.
3. Draw the unit bars.
Do your best to draw the unit bar or bars based on the problem.
4. Re-read the problem and make sure that the unit bars match the information.
You can adjust the unit bar/ bars to make sure that the bar models accurately reflect the information.
5. Place the question mark.
Look at your bar model. Based on the problem, what are they asking? Now, place a question mark on the bar model.
6. Compute the problem.
Compute the problem. If it is a multi-step problem, all steps should be shown. All working should be clearly shown too. This way if the student gets it wrong, we can refer to the steps and the working.
7. Write a complete sentence that answers the question.
Writing a sentence ensures that students know how to respond to the question. Sometimes students have trouble comprehending the problem so this step is important to make sure the student understands the problem and communicates effectively.
https://singaporemathlearningcenter.com/singapore-math/drawing-bar-models-singapore-math
Reciprocal Teaching In Math
Reciprocal Teaching In Math
Developed from research on reading conducted by Annemarie Palincsar and Laura Klenk at the University of Michigan and Ann Brown at the University of Illinois at Urbana-Champaign (1984). It was adapted for math in approximately 2009. Variations on this method a popular today.
Reciprocal teaching refers to an instructional activity in which students become the teacher in small groups. Teachers model, then help students learn to guide group discussions using the strategies: predicting, clarifying, visualizing, calculating, and summarizing. Once students have learned the strategies, they take turns assuming the role of teacher in leading a dialogue about math problems.
https://www.pghschools.org/cms/lib/PA01000449/Centricity/Domain/1247/austrailia%20recip%20teaching.pdfGood Explanations in Math
Good Explanations in Math
While an oral explanation helps you develop your solution to a problem, written explanations push you to polish, refine, and clarify your ideas. This is as true in mathematical writing as in any other kind of writing, and it is true at all levels. You should write explanations of your solutions to problems, and your students should write explanations of their solutions, too. Some elementary school teachers have successfully integrated mathematics and writing in their classrooms and use writing to help their students develop their understanding of mathematics.
Like any kind of writing, it takes work and practice to write good mathematical explanations. When you solve a problem, do not attempt to write the final draft of your solution right from the start. Use scratch paper to work on the problem and collect your ideas. Then, write your solution as part of the looking back stage of problem solving. Think of your explanations as an essay. As with any essay that aims to convince, what counts is not only factual correctness but also persuasiveness, explanatory power, and clarity of expression. In mathematics, we persuade by giving a thorough, logical argument, in which chains of logical deductions are strung together connecting the starting assumptions to the desired conclusion.
Good mathematical explanations are thorough. They should not have gaps that require leaps of faith. On the other hand, a good explanation should not belabor points that are well known to the audience or not central to the explanation. For example, if your solution contains the calculation 356 ÷ 7, a college-level explanation need not describe how the calculation is carried out, except when it is necessary for the solution. Unless your instructor tells you otherwise, assume that you are writing your explanations for your classmates.
The box above lists characteristics of good mathematical explanations. When you write an explanation, check whether it has these characteristics. The more you work at writing explanations, and the more you ponder and analyze what makes good explanations, the better you will write explanations, and the better you will understand the mathematics involved. Note that the solutions to the practice exercises in each section provide you with many examples of the kinds of explanations you should learn to write.
Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 47-48). Pearson Education. Kindle Edition.Information in this section corresponds to: 2.1 Solving Problems and Explaining Solutions: Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 42-48). Pearson Education.
Information in this section corresponds to: 2.1 Solving Problems and Explaining Solutions: Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 42-48). Pearson Education.
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