K-3 Mathematics Curriculum Processes and Attitudes Framework with
Parts and Pieces

Concepts or Big Ideas	Outcomes	Activity Sequence	Evaluation
Problem Solving	Problem Solving	Problem Solving	Problem Solving
A problem solving heuristic is a series of generalized steps that is helpful to think about when solving problems.	Begins and follows through solving problems with a heuristic. Identify a problem in different contexts. Use a heuristic (generalized pattern/strategy) to solve problems. Use metagcognition during the process. Identify the words in a problem that describe the mathematical relationships and operations. Accurately explain a problem in their own words. Identify information needed to solve the problem. Identify unneeded information in a problem. Select a strategy to solve the problem. Try a different strategy when one appears to be at an impass. Solve the problem. Try to solve the problem in a different manner to gain confidence in your answer. Reflect on what was learned and how it might be used in other contexts. the process and the accuracy of the solution. Share the process, strategies used (successful and unsuccessful), attitudes, and solutions.	Develop a generalized pattern/strategy (heuristic) to solve problems.	Use a heuristic when confronted with new mathematical situations that related to their lives.
Problems can be solved with different strategies. Strategies for solving problems are: Work a simpler problem Trial and error, guess and check Work backwards Use smaller numbers Systematic steps Look for patterns Physically acting - out the problem Use a graphical representation - model, drawing picture or diagram Problems can be solved with models. Organize data to look for patterns sequence, chart, table, making a graph, Venn diagrams Process of elimination or process of identification Write an open sentence Use algebraic reasoning	Apply and adapt a variety of appropriate strategies to solve problems; Use manipulatives to represent objects and actions in the problem. Act out the problem. Use models as a strategy to think. Draw pictures as a strategy to think through a problem. Break a problem into two related problems and solve the original problem in two steps: one for each problem. Use trial and error, guess and check. Recognize and describe patterns. Categorize information to find relationships and patterns that will assist reasoning and proof.	How many pockets are there on the indoor clothes of the students	Do students come up with their own strategies for solving problems, or do they expect others to tell them what to do? What do their strategies reveal about their mathematical understanding? Do students understand that there are different strategies for solving problems? Do they articulate their strategies and try to understand other students' strategies? How do students use materials to model a mathematical situation to find solutions? How do students keep track of and record their work? Is it difficult for them to talk, draw, and write about their work? Do they solve mathematical problems in ways that make sense to them? How do they work with peers cooperatively, participate in whole class discussion and share ideas, materials, creations? Learn form the thinking of others How do they work independently? Rely on their own thinking When given choices what do they choose? The same, different, stay in one place, move comfortably, move around a lot...
Reflection is Metacognition helps to solve problems.	Monitor and reflect on the process of mathematical problem solving Use self talk, group discussion, to talk through a problem and problem solving process to reflect on all the decisions that are possible to better insure an accurate solution.
The more problems I solve (persistence) the easier it is to solve problems and use mathematics. Following a heuristic is helpful to think about what you know or have done and what you need to find out or do.	Build new mathematical knowledge through problem solving;	Using provided story problems, students will high light the necessary information and cross out extraneous information.
Reasoning and Proof	Reasoning and Proof	Reasoning and Proof	Reasoning and Proof
Every idea in mathematics has to be proved (skeptical) before it can be used as a mathematical idea. For a proof to be valid every change or every step has to have a reason that justifies its accuracy.	Recognize reasoning and proof as fundamental aspects of mathematics; Use data to support a mathematics idea. Asks others to explain their reasoning Desires to explain their understanding, reasoning, and procedures. Rejects solutions that have insufficient evidence and lack reasonable explanations.
A conjecture is a conclusions based on insufficient evidence. Conjectures are ideas used to reason on evidence (which provides no doubt) to a conclusion.	Make and investigate mathematical conjectures; Makes generalized statements from specific examples.
A mathematical arguement that is used to prove a conjecture must have a valid reason for every change (or every step) from what is in question to something that is know or has previously been proven mathematically.	Develop and evaluate mathematical arguments and proofs; Explains how a problem was solved. Interpretes and explains other students' solutions as accurate, inaccurate, or not sure. Looks for addition examples for justification. Sove problems in different ways to justify accuracy. Recognize that a proof has to justify every possible example. Uses concrete representations to justify all examples of an argument and proof.
Mathematical ideas can be proven by manipulating a representation of the conjecture. Mathematical ideas can be proven by manipulation of symbols based on previously proved manipulations.	Select and use various types of reasoning and methods of proof from the following: Uses concrete representations to justify all examples of an argument and proof. Uses models to demonstrate and justify reasoning. Use symbols to represent all possibilities when proving a conjecture.		.
Communication	Communication	Communication	Communication
Mathematical ideas can be communicated. Written and spoken words, pictures, manipulatives, symbols, and movements help communication. Charts and graphs can be used to see relationships. Different ways to communicate mathematically iclude speaking, writing and drawing, use manipulative objects, acting it out, pictures, and symbols.	Organize and consolidate their mathematical thinking through communication; Seeks new vocabulary, because of the power it provides by representing a grouping of specific ideas. Speaks, manipulates, draws, and writes to collect, organize, chart, and interpret mathematical ideas.
Mathematical ideas can be communicated more accurately by matching the kind of communication to the intended audience. Mathematical ideas can be communicated in virtually every medium known. Mathematics is most often communicated orally, in writing, and through graphic representations. The closer the communcation is to accurately describing the mathematical ideas, the more likely the communication will be understood.	Communicate their mathematical thinking coherently and clearly to peers, teachers, and others; Writes, draws, uses manipulatives, and animates objects to represent externally their internal understanding.		Acts out problems when solving. Develops methods for recording numerical information: pictures, words, and/or numbers
People think in different ways so it is important to be open minded to understand what other people mean mathematically.	Analyze and evaluate the mathematical thinking and strategies of others; Uses other peoples oral and written ideas.
I can use mathematical language or ideas to communicate better my ideas to other people. When other people use mathematical ideas that I understand, it helps me understand what they mean better.	Use the language of mathematics to express mathematical ideas precisely Uses mathematical concepts, representations, vocabulary, and, symbols to explain parts of data, whole sets of data, mathematical processes, proceudures, examples, and solutions orally, in writing, pictures, using models, symbols, and acting/demonstrating real world problems.		Doesn't attemt to; Attempts to; Accurately; uses mathematical concepts, representations, vocabulary, and, symbols to explain parts of data, whole sets of data, mathematical processes, proceudures, examples, and solutions orally, in writing, pictures, using models, symbols, and acting/demonstrating real world problems.
Connections	Connections	Connections	Connections
Connecting ideas helps understanding, problem solving, and memory. Mathematical ideas can be connected to other mathematical ideas. .	Recognize and use connections among mathematical ideas; Use existing ideas to create new mathematical ideas. Connects one mathematical idea to another (.25 = 1/4 = 25 cents).		Recognize numbers in their environment (clock...) Relate ordinal and cardinality to each other. Compose and decompose numbers. Connect join, separate, part-part-whole, equivalent or balance to addition and subtraction. Make comparisons between geometry, numbers, and measurement.
One mathematical idea can be conected to another mathematical idea through a relatationship. As more relationships are created they in turn can be connected to other ideas to discover new ways of understanding the world. No matter how many ideas are brought together they are all connected in one way or more through all of the different relationship as well as through the processes of mathematics that was used to create them and to join them all as mathematical ideas.	Understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
Mathematics can be used to describe objects and ideas in the world. Mathematical ideas can be connected to the real world.	Recognize and apply mathematics in contexts outside of mathematics Solves mathematics problems outside of mathematics class. Solves real world problems.		.
Representation	Representation	Representation	Representation
Thinking of something related to a mathematical problem or idea and imagining, sketching, or symbolizing what it is and what might happen to it can help understand it, solve the problem, and provide confidence in the solution(s).	Create and use representations to organize, record, and communicate mathematical ideas; Model number values with concrete objects and pictures Use models to represent and understand quantitative relationships such as more, less, same, different Collect and organize data with oral, written, graphical, or physical models. Identify circles, squares triangles, rectangles ... Recognize spatial concepts of left/right, above/below, over/under, near/far ...		Represent number values with concrete objects, 100 chart, pictures Use models to represent and understand quantitative relationships Represent and animate addition and subtraction with objects, pictures,( joining,equalizing, seperating) ... (if we take the two bears on this table and put them with the two bears on that table, how many bears will be on that table?)
There are many ways of representing mathematical ideas and it is important to match a representation to an event or problem to be accurate.	Select, apply, and translate among mathematical representations to solve problems; Create charts and graphs.
Representations are useful in many ways of looking at and understanding the world.	Use representations to model and interpret physical, social, and mathematical phenomena.		.
Attitudes	Attitudes	Attitudes	Attitudes
I can use math to explain my world. The more I think about mathematics the more I understand. Asking questions is a good way to learn. Mathematics is a powerful tool that can be used to describe and prediict everyday occurances. The strategy that people use to attempt to solve a problem is a much bigger factor for determining success than luck. I can communicate mathematical ideas in a variety of ways. I have the ability to solve problems. I can recognize patterns. I can create solutions. Sometimes you have to think for awhile before an idea comes to you or you can understand an idea (perserverence, effort, open minded).	Able to solve mathematical problems. Recognzes patterns. Use mathematical ideas to describe or predict events in his or her life. Attribute his or her success to being open-minded, objective, focusing on the task, creating alternate possible strategies and solutions, persevering till a solution is achieved, and having self-efficacy to being able to select appropriate strategies, and to solve problems. (open-minded, objective, focused, creative, perserverence, self-efficacy, reasonable effort.)		Do students explore materials? Initiate their own ideas? Observe others? Follow prompts or suggestion? Desires to explain their understanding, reasoning, and procedures. Are persistent - follow through on an idea to completion or begin one construction and abandon it for another. Desire to show and justify solutions. Switch materials Work alone Work with others Share materials Join together two projects to make one Like to use mathematical manipulatives. Are confident that they are correct
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