Math Methods
Identify how you would develop the following number ideas with children: group recognition
Group recognition is a continuation of number conservation. (Ex. one line with 6 marbles, a circle of 6 marbles, and two rows of three marble each). Now ask if any of the groups has more. A student might count each group to figure it out. We are working towards the students being able to see a group and instantly see "how many". So, the next step would be teaching them different groupings of numbers so they can recognize them without having to count them.
Recognize the five content standards described in chapter 3 of the PSSM document. (Jaci)
Number & Operation Algebra Geometry Measurement Data Analysis & Probability
Distinguish between the following set ideas. (Rachel, green handout)
Equal sets- when both sets have the same elements Empty or null set- set containing no elements Subset- every element of set A is also an element of set B Union- the union of two sets A and B is the set of all elements that are either in A or B, or in both A and B Equivalent sets- two sets are equivalent if their elements can be placed in one-to-one correspondence Disjoint sets- two sets are disjoint if they have no elements in common Intersection- set of all elements that are both A and B One-to one correspondence-
Recognize the six principles described in chapter 2 of the PSSM document and be able to describe how the principles may influence your teaching of elementary mathematics. You may find it helpful to connect this knowledge with the reading you did in chapter 3 of your text. For example, while the discussion on the Teaching Principle references the areas listed below, your text discusses them in a little more depth.
Equity: "Excellence in mathematics education requires equity—high expectations and strong support for all students." -All students must have the opportunity to learn Mathematics -not necessarily identical instruction, but rather the accommodations needed to succeed -interwoven with the other principles -"Equity requires high expectations and worthwhile opportunities for all." -"Equity requires accommodating differences to help everyone learn mathematics" -"Equity requires resources and support for all classrooms and all students" Curriculum: "A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades." -"A mathematics curriculum should be coherent." -"should focus on important mathematics" -"should be well articulated across the grades" Teaching: "Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well." -"Effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies. -"Effective teaching requires a challenging and supportive classroom learning environment." -"Effective teaching requires continually seeking improvement." Learning: "Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge." -"Learning mathematics with understanding is essential" -"Students can learn mathematics with understanding" Assessment: "Assessment should support the learning of important mathematics and furnish useful information to both teachers and students." -"Assessment should enhance students' learning" -"Assessment is a valuable tool for making instructional decisions" Technology: "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning." -"Technology enhances mathematics learning" -"Students' engagement with, and ownership of, abstract mathematical ideas can be fostered through technology. " -"Technology supports effective mathematics teaching" -(use as a tool) -"Technology influences what mathematics is taught"
Describe the two ideas that are foundational to understanding place value. (Johannah p. 163)
Explicit grouping or trading rules are defined and consistently followed: This provides a constant reminder of the importance of grouping by tens to place value. (Example: 1 can be traded for 10 tenths) The position of a digit determines the number being represented: The position of a number represents completely different quantities. (Children should experience the role of zero in place value early and often)
Identify how you would develop the following number ideas with children: Number conservation
First, note that this is something that is hard for students first-second grade and younger. Begin by having a row of two different things and asking if each row has the same amount. Spread out a row so that it is longer, but still has the same amount. Ask the child again if each row has the same amount. If they do not recognize that they still have the same amount, the child has not reached full understanding of number conservation.
Identify how you would develop the following number ideas with children: pattern recognition
(Think of what we did in math the other day with the shape manipulatives - find one difference, now find two differences, etc.) Start this at a young age, but have a simple pattern. This could be a pattern that you make on a bracelet and the kids copy the pattern exactly. The next level would be having students find the next one in the pattern. After children are able to figure out what the next thing in the pattern would be, have them extend a pattern. Lastly, have them making their own patterns.
Describe the four principles upon which the counting process is based. (Maddie) (Pg. 144)
Each object to be counted must be assigned one and only one number name. a one-to-one correspondence between each object and the number name The number-name list must be used in a fixed order every time a group of objects is counted. 1, 2, 3... stable-order rule: start counting using "one" and continue in specific order The order in which the objects are counted doesn't matter. order-relevance rule: the child can start with any object and count them in any order The last number name used gives the number of the objects. regardless of which object is counted first or in which order the objects were counted, the last object counted will always be the "answer" or number of objects counted in total
Read and write numbers in bases other than 10. (Maddie)
Example (in base 6): Block: 216; Flat: 36; Long: 6; Unit: 1 How to write it in Base 6: Block: 6^3; Flat: 6^2; Long: 6^1; Unit: 6^0 (I don't know how to type powers on the computer)
How can a teacher teach through problem-solving? Be able to discuss types of problems that a teacher might use to accomplish this goal. (Alissa, pg 111-116)
Teachers should allow math to be problematic for students. Give them time to wrestle with a problem, solving it in their own way based on former knowledge, instead of showing them how to do it. Teachers should focus on the methods used to solve the problems. There are many ways to solve any given problem, and students are likely to come up with different ones. Give them time to talk with each other about the methods they used and why they worked. Teachers should tell the right things at the right times. Leaving students floundering with a problem that is too difficult with not make them successful. Teachers should tell students about mathematical notations and technical definitions. Teachers also should share alternative methods that the students did not come up with on their own. Finally teachers should highlight the big ideas that come up in problem solving discussion. The text listed these types of problems (Pg. 113-114): Problems that ask students to represent a mathematical idea in various ways Problems that ask students to investigate a numeric or geometric concept Problems that ask students to estimate or to decide on the degree of accuracy required or apply math to practical situations Problems that ask students to conceptualize very large or very small numbers Problems that ask students to use logic, to reason, to strategize, to test conjectures, or to gauge the reasonableness of information Problems that ask students to perform multiple steps or use more than one strategy Open ended problems
Identify how you would develop the following number ideas with children:Classification
This can be done with or without numbers, but should begin at a young age and progress to be more difficult as children get older. I would first start with one concept (separate boys from girls). Classifying is being able to find characteristics of an object. For example, I would classify myself as a female, brunette, El. Ed. major, etc. As the children begin to understand the idea of finding characteristics, I would start adding a level of difficulty (find more than one classification - this group is girls, and these three have blue eyes). The students are finding things that fit together in a group.
Recognize the five mathematical processes described in chapter 3 of the PSSM document and be able to describe why each process should be included in the elementary mathematics curriculum? What are the implications for your teaching of elementary mathematics?
... Problem Solving - It is critical for all people to be good problem solvers. Problem solving is foundations to mathematics and something that you do in everyday life. As a teacher it is important to chose worthwhile problems for the class. We also need to reiterate how a student solved a problem. We are to label the strategy they used to solve the problem and then ask if any other students used a different strategy. Another thing that we are to make sure of is that students understand how to solve the problem. We are to teach and use a variety of appropriate strategies to solve problems and make sure that the students reflect and think about why a strategy did or didn't work. Reasoning & Proof - (Alissa, I did my paper on this) Reasoning and proof are the background of mathematical concepts and procedures. If children do not learn reasoning and proof, they are simply memorizing their math, and it will neither motivate them nor help them to make connections and use math in the real world. Further, as they learn more and more complex concepts, they will be unable to build on prior knowledge because they won't understand why simple procedures work. Communication - Connections - Representations -
Describe appropriate methods and materials for teaching place value and numeration
In developing place value and establishing number names, it is far better to skip beyond the teens and start with the larger numbers. It is important that children have experiences thinking of numbers in various ways. Concrete physical models→ semi-concrete organizational models→ symbolic representational models Position of the digit determines it's value. Explicit trading/grouping rules are defined and consistently followed. Tens & Ones - It might use money (2 dimes and 5 pennies) or another model, such as bean sticks. With the bean sticks, 25 could be represented several different ways (see diagram on pg. 166). Hundred Charts Darts Game (as played in class during stations) The base-ten blocks, together with the place-value mat, as shown in Figure 8-8 (on pg. 167), can be used to model the additive property and illustrate expanded notation for 123.
Describe at least four strategies for managing your classroom that will allow you to support problem solving learning for your students. For one of these strategies be able to describe in detail how you would carry it out in your classroom (Kendra, pg.112).
Knowledge - Students must learn to make connections between new problems and problems they have solved in the past. They must learn to recognize underlying structural similarities among problems and choose the appropriate approach to solving each problem. (Incorporating prior knowledge into every lesson or concept being taught in the classroom. Include the following aspects into your classroom: word walls, student work, examples of problems/solutions, brainstorming, valuing students' work, etc.) Beliefs and Affects - Students' problem-solving abilities often correlate strongly with their attitudes, their level of self-confidence, and their beliefs about themselves as problem solvers. Teachers must show they believe ALL students can be good problem solvers and encourage them to develop their own strategies. (Continually encourage students in your classroom, appreciate different paths to a destination, appreciate all cultures and value all ethnicities, create classroom community, etc) Control - It is extremely important for students to learn to monitor and control their own thinking about problem solving. Good problems solvers spend a good time up front, making sure they understand the problem, and at the end, looking back to see what they did, analyzing how they can modify or improve their solution, then thinking about other problems it is similar to. (Incorporate reflection time after every lesson, encourage students to share how they solved the assigned problems - esp. if they used a different problem solving method) Sociocultural Factors - The atmosphere of the classroom should encourage students to use and further develop the problem-solving strategies that they have already developed naturally through experiences outside the classroom. To use problems effectively, teachers need to consider the time, planning aids, resources, technology, and classroom management. (Bring in guest speakers, be sure to include different ethnic names in problems, include technology)
Describe symptoms of a lack of conservation of number. Why is this an important early number idea? How might a teacher help a child develop an understanding of conservation of number? (Kendra, p.140-141 & class discussion)
Lack of Conservation is when a child does not understand that two glasses of water can have the same amount of water even in one is a tall glass and the other is a wide glass. Another example is if a teacher takes two balls of clay that are the same size and shows them to the child; then the teacher flattens one ball and the child claims the flattened one must be bigger. Conservation is important in early number ideas because it allows the child to look at two sets of six marbles and know they are equal because the quantity does not change, just the arrangement of the marbles. The child also knows he or she does not need to recount the number of objects if the teacher "mixes them up". A teacher could help a child learn this through doing many different activities (like the clay, mixing up objects, or the water) and then showing the child that the only difference is the way the object is arranged, but the quantity is the same. A teacher could do this through pouring the liquid into the same sized glasses, rolling the clay ball again, or recounting all the "mixed" items to show the results are still the same.
Describe why "looking back" is an important step to include in your instructional planning for problem solving. Describe at least three approaches that you could use in teaching the looking back step of problem solving. (Julie).
Looking back is important because students can see the final answer to the problem and the steps they took to get there. It allows them to review their own thought processes while problem solving and develops their metacognitive abilities (thinking about one's own thinking). Looking back and talking through the processes helps students become better problem solvers. Looking back at the problem: Students should generalize after solving a problem, which means they make connections between other problems and recognize the similarities/differences. This helps them see how the details of a problem are different, but the structure is similar. Looking back at the answer: Students should look back to see if their answer makes sense. Looking back at the solution process: Students use different strategies to solve problems. They should look back and consider each step of their problem solving process so that they know why they got their answer, rather than just focusing on the answer itself. They learn the relationships in the problem solving technique instead of the numbers.
Define an open-ended problem and give at least two examples. Why might a teacher use open-ended problems in teaching problem solving? (Jaci) (Pg. 114-115)
Open-ended problems are problems that can have more than one correct answer. With open-ended problems, the answer depends on the approach taken (of course, the answer must be reasonable). Different students approach open-ended problems in very different ways, so such problems are ideal for ensuring that students at all levels can experience some measure of success. Open-ended questions are especially appropriate for cooperative group work, in which case they should be followed with a class discussion in which the mathematical ideas and planning skills involved are explored and students get a chance to clarify their thinking and validate their decisions. Teachers may use open-ended problems while teaching problem solving skills because open-ended problems push students to think outside of their box and search for multiple strategies to solving, or viewing, a given problem. Examples of Open-Ended Problems: Randy has $13.00 to spend at his favorite restaurant. He wants to order one main dish, two side dishes, and one dessert. He knows he will spend $1.50 on video games while he waits for his order. Find three different meals that Randy could choose. Show your calculations and explain how you thought about that problem. Using the given triangles, order them according to your own definition of "biggest", such as tallest, widest, thickest, etc, and be able to defend your own answer in whole-group discussions. (See page 115 for more details!)
Identify the four properties that make our number system efficient. (Johannah p.162)
Place Value: the position of a digit represents its value Base of Ten: ten is the value that determines a new collection and is represented by 10 Use of Zero: A symbol for zero exists and allows us to represent symbolically the absence of something Additive property: Numbers can be written in expanded notation and summed with respect to place value
Distinguish between the following counting strategies. How are these strategies related to the understanding of number ideas? (Rachel, pp.144-148)
Rote counting- the number names are known but the number sequence is unknown Point counting- Rational counting- correct sequence with correct correspondence Counting back- correct number as counting backwards from a particular point Skip counting- correct number names while counting by 2s, 3s, 4s, etc. Counting on- correct number names and can start at any number and begin counting
Identify how you would develop the following number ideas with children: seriation
Seriation skills develop with each individual child. This is being able to arrange objects by size. First, children have to know what large and small means. Then they need to be able to compare objects. First, children start off with two to three objects. This one object is smaller than this object. As they start to master this skill, they can look at objects and say this object is smaller than this object, but larger than that other object, etc.
Identify how you would develop the following number ideas with children: Set inclusion
Sets, subsets, etc. - note the term sheet that we received with the different concepts). Children start off being able to put things into one set (ex. the whole class). They then can separate things and create another set based off of one thing (ex. our whole class, ECE majors from El. Ed. majors). The ECE majors and the El. Ed. majors can all be their own set, or they could all be combined if you are looking at the set of the whole class.
Discuss at least three problem solving strategies identified in your text. What are the important components of each strategy? What would you need to stress with children as you taught each strategy? (Jenna) (Pages 119-126)
You should always encourage students to generate their own ideas about how to approach a new situation Act it out: Helps children visualize what is involved in the problem. When you teach this strategy you should stress that the objects used do not have to be the real thing (For example, real money is not needed to act out a problem involving coins, only something labeled 5 cents or 25 cents). Also stress that they need to focus their attention on the actions and not the objects. Earlier grades can actually act it out usually whereas older grades may need to simply "play the movie" in their heads to visualize it. Make a drawing or diagram: Helps you depict the relationships among the different pieces of information in a problem in a way that makes those more apparent. Stress to the children that there is no need to draw detailed pictures. Instead encourage the children to draw only what is essential to represent the problem. (Example: circles can represent students, and a square can represent the classroom, etc.) Look for a pattern Construct a table: Helps children discover a pattern and identify missing information. Efficient way to classify and order large amounts of information. Provides a record of what's been tried so that the children need not to retrace nonproductive paths. Important to stress/ make sure that the children learn how to construct a table from scratch (not just depending on reading one to find an answer). They need to determine for themselves what form the table should have. You can have the students do this by having them collect information, organize it, and report it. Guess and check Work backward Solve a similar but simpler problem
Describe how each of the following pre-number concepts contributes to the development of meaningful counting and number sense. (Maddie) (Pg. 136; 142; 139; 141)
classification it is a prerequisite to any meaningful number work before children can count, they must know what to count, and classification helps identify what is to be counted comparisons essential in developing number awareness students must be able to discriminate between important and irrelevant attributes must be familiar with descriptions such as more than, less than, and as many as patterns mathematics is the study of patterns four ways that they could be used to develop mathematical ideas: copying a pattern finding the next one extending a pattern making their own patterns group recognition (like on a dice) before even starting school, most children can identify quantities of three things or less by inspection alone without the use of counting techniques this is important because: it saves time it is the forerunner of some powerful number ideas it helps develop more sophisticated counting skills it accelerates the development of addition and subtraction
Describe what it means for students to construct their mathematical knowledge. What underlying tenets of constructivism need to be considered in planning mathematics instruction? (See additional information in Chapter 2 of your text)
construct their mathematical knowledge by: (How Children Learn Mathematics powerpoint) actively creating or inventing knowledge they do not passively receive knowledge reflecting on their own physical and mental actions by engaging in the social process of dialogue and discussion with themselves and others Underlying tenets (p. 24 of the textbook) several characteristic and identifiable stages of thinking exist, and children progress through these stages as they grow and mature learners are actively involved in the learning process learning proceeds from the concrete to the abstract learners need opportunities for talking about/communicating their ideas with others
Describe Polya's four-stage model of problem solving. How would you help students to carry out each step of the model? (Maddie) (Pg. 119)
provides a general picture of how to move through the process of solving a problem The steps are not discrete; it is not always necessary to perform every step 1. Understanding the problem. reading the problem and looking for possible answers 2. Devise a plan for solving it. thinking through which strategy would make the most sense to use 3. Carry out your plan. work through the strategy that you choose and make sure to include all the information given in the problem 4. Look back to examine your solution. work backwards and make sure that your solution is correct