3.4+Process+Standards

Problem Solving What is an exercise?
 * step-by-step
 * surface thinking
 * quick and easy
 * comfortable
 * doesn’t always encourage understanding
 * follow by example
 * could encourage retention (procedural)
 * plug and chug
 * a way to stay in “math shape”
 * oftentimes isolated or segmented

What is a Problem?
 * Thinking through a question.
 * Not using step-by-step procedure, but experimenting with the question.
 * Something that challenges the regular thinking process.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Encourages understanding

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">What is a non-problem?
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Observations
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Statements
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Opinions
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Ideas
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Discussions
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">“No one correct answer”
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Theorems
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Facts
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Axioms
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">...

<span style="background-color: #ffffff; color: #0000ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Reasoning and Proof-- 6-8

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">1. “Recognize Reasoning and Proof as Fundamental Aspects of Mathematics” <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">It is very important for students to realize that understanding why certain things happen in mathematics is crucial in their ability to learn. In the article, it states that <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">“Reasoning and proof should be a consistent part of students’ mathematical experience in prekindergarten through grade 12 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">” (1-3). That way, students will understand why things occur and thus, will be able to figure out any problem or situation dealing with those mathematical concepts. If they don’t ever learn the reasoning and proof of certain ideas, they will never fully understand it and won’t be able to use those ideas in the future. The way I think of it is like building a house. Sure, you can put some walls up with a roof on top and call it good, but if you don’t build the foundation right and make a steady base, then the house will fall down.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">2. “Make and Investigate Mathematical Conjectures” <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Another important aspect of learning as middle school students is the ability to <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">form conjectures <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">and elaborate upon them. Rather than just telling a student what happens in a problem, the teacher should ask questions to lead their students to discover the concept themselves. “To make conjectures, students need multiple opportunities and rich, engaging contexts for learning” (57-1). This means the teacher should give students problems and ask questions that grab their attention, but also don’t spell out how to figure the problem out. <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The students should be able to form their conjectures, then express them using different materials, symbols, drawings, and tools. <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> That way they can explain their thought process to other students, and they can explore their conjectures together.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">3. “Develop and evaluate mathematical arguments and proofs” <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Being able to <span style="background-color: transparent; color: #ff00ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">give reason <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> to mathematical concepts develops over time; while early elementary students may look at thinking that there is one “left out” in odd numbers, older students will draw on mathematical results (57-2). Teachers can help students form argument by asking <span style="background-color: transparent; color: #ff00ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">helpful questions <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> -- does this always work? why? (57-2). <span style="background-color: transparent; color: #ff00ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Classroom discussion <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> between students helps them form prove techniques (57-3). For example, a student who claims all numbers bigger than 2 are even could be proved wrong with a (or many) counterexample. The format of proofs does not matter so much, rather the concepts that students unravel (57-4).

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">4. “Select and use various types of reasoning and methods of proof” <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The way students reason should develop over time; younger students will draw on what they already know (6+6 is 12, so 6+7 is 6+6+1=13) while older students need to be introduced to a <span style="background-color: transparent; color: #ff00ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">variety of proof techniques <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> (58-1). Younger students also like to use trial and error methods in solving a problem, but as the get older this technique should <span style="background-color: transparent; color: #ff00ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">become more systematic <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">. Page 58 offers a “proof by cases” that a 5th grader came up with-- although the mathematics is simple, this type of proof technique is useful (as we even used it in 3140).

<span style="background-color: #ffffff; color: #0000ff; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Standards for School Mathematics: Communication

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Communication is a vital part of mathematics and mathematics education. Through the sharing of ideas students can see them become objects of reflection, refinement, discussion, and amendment. The process of communication can also help with clarifying understanding and building a meaning/permanent presence for ideas. As the section states (60-1); “ <span style="background-color: transparent; color: #38761d; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing. Listening to others' explanations gives students opportunities to develop their own understandings. Conversations in which mathematical ideas are explored from multiple perspectives help the participants sharpen their thinking and make connections. <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">”

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The communication standard from chapter 3 of Principles and Standards for School Mathematics states that instructional programs from prekindergarten through grade 12 should enable all students to do the following four activities:


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">organize and consolidate their mathematical thinking through communication;
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">analyze and evaluate the mathematical thinking and strategies of others;
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">use the language of mathematics to express mathematical ideas precisely.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; vertical-align: baseline;">__Organizing and consolidating mathematical thinking through communication:__

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Through communication students can i <span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">ncrease their insight into mathematical thinking <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">. This can be achieved by presenting methods for solving problems, justifying their reasoning to others, or by formulating a question about an issue that is giving them problems. Communication also <span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">plays a large part in reflection <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">, whether it is from a teacher asking thought provoking questions or a student consolidating their work on paper, communication can cause students to rethink and reflect on their reasoning.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; vertical-align: baseline;">__Communicating mathematical thinking coherently and clearly to peers, teachers, & others:__

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Communication can also be <span style="background-color: transparent; color: #cc0000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">a way to give students the opportunity to test their ideas in a public setting allowing classmates to reflect upon them <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">. In order to build a classroom community that encourages students to freely express their ideas, teachers must keep in mind the age of their students and carefully model questions that can help clarify expectations for student work. <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Written communication should also be emphasized <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;"> in the classroom in much the same way, by describing ideas informally and formally on paper students are given a chance to expand on their ideas.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; vertical-align: baseline;">__Analyzing and evaluating the mathematical thinking and strategies of others:__

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;"> Sometimes students see things in different ways, therefore the students can <span style="background-color: transparent; color: #351c75; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">benefit each other by talking through some problems <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">. One students might know how to solve a word problem once they get the equation, and another student might know how to get the equation. As a teacher, putting these kids together would probably be a good idea. Students have to <span style="background-color: transparent; color: #351c75; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">learn how to question each other and think critically <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; vertical-align: baseline;">__Using the language of mathematics to express mathematical ideas precisely:__

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;"> Students learn at a young age that there are words they might know in everyday language that have a different meaning in math. In the middle school years, students <span style="background-color: transparent; color: #b46504; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">should be able to understand the role of math definitions and should be able to incorporate them into their work <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">. However, there should be no rush in being good at formal math language because students need to understand why they are learning and knowing these definitions.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 16px; text-decoration: none; vertical-align: baseline;">Technology gives other opportunities and challenges for the development of mathematical language. Student will benefit from comparing normal expressions with those used in spreadsheets or calculators.

<span style="background-color: #ffffff; color: #0000ff; font-family: Arial; font-size: 20px; text-decoration: none; vertical-align: baseline;">Connections <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Students should be able to: <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">1. <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">R <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">[|ecognize and use connections] <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> among mathematical ideas;

<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">2. U <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">[|nderstand how mathematical ideas interconnect] <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> and build on one another to produce a coherent whole;

<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">3. R <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">[|ecognize and apply mathematics] <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> in contexts outside of mathematics

<span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Math is an integrated field of study <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> (paragraph 2), so students need to be able to make connections from topic to topic, so they can have a deeper understanding of each subject, and the field as a whole. Teachers are the key to having students view math as a <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">set of connections, versus a set of isolated concepts and skills <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> (paragraph 3). Using the phrase “How is (topic) connected to our discussion last week about (topic)?” is a good way to emphasize the importance of making connections (paragraph 3). Students should be <span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> encouraged and expected to ask themselves how the current topic is related to a previous topic <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">This understanding of math as a whole compared to math as a set of isolated concepts, leads to students being able to do complex problems, because they can use other ideas easier, due to the connections from one topic to another.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Students should do math problems that are given in a context, likereal world examples, so they can relate the problems to something that impacts them. <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Real world examples allow students to connect more with a problem, so they are more willing to work at a harder problem. <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> A good example of a real world problem is teaching them different things about banking, like interest rates and how to calculate a total with interest rates and principal amounts. The more connections there are for the student, the better the student can apply concepts to real-world problems or situations.

<span style="background-color: #ffffff; color: #0000ff; font-family: Times New Roman; font-size: 20px; text-decoration: none; vertical-align: baseline;">Representation Standard for Grades 6-8

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Representations, such as physical objects, drawings, charts, graphs, and symbols, exist everywhere in the middle-grades mathematics curriculum proposed by the NCTM.

<span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Students should be able to:


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">create and use representations to organize, record, and communicate mathematical ideas;
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">select, apply, and translate among mathematical representations to solve problems;
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">use representations to model and interpret physical, social, and mathematical phenomena.

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Students may use <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">standard and nonstandard forms of representation <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;"> to solve a problem. The power of representation is that it <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">blends visual and numerical information. <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;"> This is especially helpful to those students who are visual learners. Visual representations can facilitate the relationship of representations, tables, and graphs with anything seemingly non-mathematical or mathematical. Representations <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">allow students to organize data in order to visualize <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;"> the patterns or irregularities. The use of graphing calculators and computer software can aid the student in creating the visual representations, instead of creating these by hand.

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Having students arrange their ideas with visual representations will help them make sense of what is going on as well as make it easier to remember. These representations <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">do not always have to be standard <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">; as long as the representations make sense to the student then the representation has done it’s job. However, standard forms of representations, such as number lines or graphs, are important for students learn because this will help them <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">communicate their mathematical ideas with others.

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Teachers must help students to <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">utilize different types of representations <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">. This will assist students by listening, questioning, and making a sincere effort to understand what the student is trying to <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">communicate via a representation <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">. It is important that students build confidence and competence in creating their own representations when needed. Students must also be <span style="background-color: #f4cccc; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">exposed to how other students use representations <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;"> and recognize the characteristics that make a representation flexible, appropriate, and useful.