Saturday, October 22, 2011
Chalk Walk: October 19, 2011
I really enjoyed doing the chalk walk activity. Doing the chalk walk with another block was fun, but I think it could have been a way to bring our block closer together if we did it with just our own block. We could have been assigned a partner through a creative method such as, pair with someone born in the same month as you. I do think we needed more time to write our directions for our partners. I was trying to describe six squares and I was only able to describe four of them. Also, I think if we had more directions on how to write the directions for our partner. For my partner, I folded the paper and and wrote the directions in each square to give a visual of what I was describing. My partner made a list of directions which were hard to follow. I think if my partner wrote his directions the way I did it would have been much more clear what he wanted. I love doing art and math was my least favorite subject when I was in school. It is a great idea to incorporate art and real world connections to math for students, especially the unmotivated students.
Tuesday, October 18, 2011
Tuesday, October 4, 2011
Reys Ch. 16
Why should measurement be included throughout the elementary mathematics curriculum?
Measurement is important for several reasons. It provides many applications to everyday life. It can be used to help learn other mathematics. It can be related to other areas of the school curriculum. It engages students in learning.
What is the measurement process?
1. Identify the attribute by comparing objects.
a. Perceptually
b. Directly
c. Indirectly Through a Reference
2. Choose a unit.
a. Nonstandard
b. Standard.
3. Compare the object to the unit.
4. Find the number of units.
a. Counting
b. Using Instruments
c. Using Formulas
5. Report the number of units
Measurement is important for several reasons. It provides many applications to everyday life. It can be used to help learn other mathematics. It can be related to other areas of the school curriculum. It engages students in learning.
What is the measurement process?
1. Identify the attribute by comparing objects.
a. Perceptually
b. Directly
c. Indirectly Through a Reference
2. Choose a unit.
a. Nonstandard
b. Standard.
3. Compare the object to the unit.
4. Find the number of units.
a. Counting
b. Using Instruments
c. Using Formulas
5. Report the number of units
Reys Ch. 5
What connections are important to aid elementary children in learning mathematics?
First ideas within math itself are connected with one another: Students who learn about fractions, decimals and percentages in isolation from one another miss an important opportunity to see the connections among these ideas. (1/4 = .25 = 25%). A second important type of connection is between the symbols and procedures of math and conceptual ideas that the symbolism represents. (3 squared = 3x3, 9). A third type of connection is between math and the real world and other school subjects/ content. As students encounter problems from real-world contexts where math is a significant part of the solution, students come to recognize and value the utility and relevance of the subject.
First ideas within math itself are connected with one another: Students who learn about fractions, decimals and percentages in isolation from one another miss an important opportunity to see the connections among these ideas. (1/4 = .25 = 25%). A second important type of connection is between the symbols and procedures of math and conceptual ideas that the symbolism represents. (3 squared = 3x3, 9). A third type of connection is between math and the real world and other school subjects/ content. As students encounter problems from real-world contexts where math is a significant part of the solution, students come to recognize and value the utility and relevance of the subject.
Tuesday, September 20, 2011
Reys Ch. 9
What prerequisites are important prior to engaging students in formal work on the four basic operations? The four prerequisites are: facility with counting, experience with variety of concrete situations, familiarity with many problem solving contexts, and experience using language to communicate mathematical ideas. Facility with counting means that as long as the student has enough time, they would be able to solve any problem with whole numbers by counting. However, there are much more efficient ways to solve problems than counting. Experience with a variety of concrete situations means that children need to have many experiences in problem solving situations to work with physical, or concrete, objects to develop and understanding about math operations. Materials also provide a link to connect each operation to real world problem solving situations. Whenever a child wants to be sure that an answer is correct, materials can be used for confirmation. Familiarity with many problem contexts means that students should view math as problem solving and having real life connections. Children need to talk about and write about math. Putting their experiences into words helps with making meaning and deeper processing. As soon as possible have students put their ideas on paper. If they are unable to write have them draw, but as soon as they are able to write they should be encouraged to write number sentences and a narrative explanation of their thinking.
Reys Ch. 8
The number system that we use has four important characteristics. The first characteristic is place value. Place value is how the position of a digit represents it's value. Base of ten is next. Base means a collection. In our system, ten is the value that determines a new collection and is represented by ten. The third characteristic is the use of zero. Zero exists to represent the absence of something. Last is the additive property. Numbers can be summed with respect to place value. These properties make the systemefficient and contribute to development of number sense. Once children understand these characteristics, the formation and interpretation of numbers becomes a natural development. Many counting and trading experiences are necessary, particularly grouping by tens. Using manipulatives, such as grouping or trading with straws or cubes are a few methods of developing place value. Place-value mats serve as a visual reminder of quantities involved and provide a bridge toward the symbolic representation of larger numbers. Establishing these bridges from the concrete to the abstract is critical in developing place value. Place value concepts are developed many years. This means that place value is not completely developed before operations are introduced, because experiences with adding, subtracting, multiplying, and dividing whole numbers develops additional competence and understanding. Also working with decimals in later grades will further extend place value concepts.Place- value concepts will be integrated and extended throughout elementary school.
Tuesday, September 13, 2011
Reys Ch. 17
What are some common misconceptions young students have about probability?
Young children often hold common misconceptions about various aspects of probability. They tend to make predictions based on their personal preference. They may also hold biases against certain numbers, such as it is harder to roll a six on a die. They may expect all outcomes in an experiment to be equally likely, or may not be surprised at all by something unlikely happening. In affect they may not be prompted to find out underlying causes for such events. It is difficult for many children to make inferences from data.
Young children often hold common misconceptions about various aspects of probability. They tend to make predictions based on their personal preference. They may also hold biases against certain numbers, such as it is harder to roll a six on a die. They may expect all outcomes in an experiment to be equally likely, or may not be surprised at all by something unlikely happening. In affect they may not be prompted to find out underlying causes for such events. It is difficult for many children to make inferences from data.
Reys Ch. 15
What types of explorations with geometry help build elementary children's spatial reasoning and visualization skills?
Using mental images helps to build elementary children's spatial reasoning and visualization skills. A classic task to help student's develop mental images is to cover up most of the shape and reveal it slowly while asking students to predict what shape is covered. Students need to work with models to help them make mental images. Have children play with shape blocks imprinting them in play dough, matching them, and tracing them. Another way to help students is to give students drawings of the faces of structures built with cubes and have them construct the building, or the other way around. There are many variations of this task. Have student's close their eyes and give them scenarios of real life shapes and different perspectives of which they can view it. From a birds viewpoint, a human, and even underground animals.
Using mental images helps to build elementary children's spatial reasoning and visualization skills. A classic task to help student's develop mental images is to cover up most of the shape and reveal it slowly while asking students to predict what shape is covered. Students need to work with models to help them make mental images. Have children play with shape blocks imprinting them in play dough, matching them, and tracing them. Another way to help students is to give students drawings of the faces of structures built with cubes and have them construct the building, or the other way around. There are many variations of this task. Have student's close their eyes and give them scenarios of real life shapes and different perspectives of which they can view it. From a birds viewpoint, a human, and even underground animals.
Tuesday, September 6, 2011
Reys Ch. 14
How do patterns help children develop algebraic thinking and ideas?
Patterns are a foundation to helping students think algebraically. Practicing analyzing, organizing and recognizing patterns is a very important prerequisite to algebra. It is important to introduce students to patterns using pictorial models. Once students are able to recognize simple repeating patterns and identify the core, prompt children to think how they can extend a pattern, fill in a missing element, and to describe a pattern. It is very important to have the students explain their thinking. Growing patterns are a little bit more complex. They may be represented verbally, with pictures, or with simple symbols, and are typically numerical patterns. With growing patterns each successive term changes by the same amount from the preceding term. When a student explains their thinking about a pattern the teacher is able to informally asses the students progress. Also, students need to be aware that there is usually more than one way to explain a pattern.
Patterns are a foundation to helping students think algebraically. Practicing analyzing, organizing and recognizing patterns is a very important prerequisite to algebra. It is important to introduce students to patterns using pictorial models. Once students are able to recognize simple repeating patterns and identify the core, prompt children to think how they can extend a pattern, fill in a missing element, and to describe a pattern. It is very important to have the students explain their thinking. Growing patterns are a little bit more complex. They may be represented verbally, with pictures, or with simple symbols, and are typically numerical patterns. With growing patterns each successive term changes by the same amount from the preceding term. When a student explains their thinking about a pattern the teacher is able to informally asses the students progress. Also, students need to be aware that there is usually more than one way to explain a pattern.
Reys Ch. 6
What does it mean to teach math through problem solving? What "signposts" for teaching guide this approach?
Teaching math through problem solving is a teaching approach in which students are confronted with problems, supported in their attempts to solve the problem, and aided to discuss and incorporate the learning that results. The primary goal is to ensure that students make sense of and remember the math they are learning so as to use them appropriately in unfamiliar situations. There are three "signposts" for teaching that guide this approach. Signpost one says to allow math to be problematic for students. It is important for students to be presented with challenging problems and to struggle. This approach allows students to think of strategies and to examine those strategies in which they use. Students math abilities will grow from having a challenge. Signpost two suggest to focus on the methods used to solve problems. Students should be encouraged to communicate their methods to each other. Students deepen their understanding by thinking about approaches and which are easier, harder, or most appropriate for the problem. Signpost three advises the teacher to tell the right information at the right times. When teachers are using a problem solving approach to help their students learn, the teacher needs to know when to let the students figure some things out for themselves. This signpost is very similar to the first one, letting math be problematic. If the teacher explains thing completely and clearly in the beginning the students may not be challenged enough to grow in their understanding. This signpost also focuses on the importance of discussions in the classroom and to teach fundamentals. Fundamentals such as, math symbols and language need to be taught first because the students will not discover these things for themselves.
Teaching math through problem solving is a teaching approach in which students are confronted with problems, supported in their attempts to solve the problem, and aided to discuss and incorporate the learning that results. The primary goal is to ensure that students make sense of and remember the math they are learning so as to use them appropriately in unfamiliar situations. There are three "signposts" for teaching that guide this approach. Signpost one says to allow math to be problematic for students. It is important for students to be presented with challenging problems and to struggle. This approach allows students to think of strategies and to examine those strategies in which they use. Students math abilities will grow from having a challenge. Signpost two suggest to focus on the methods used to solve problems. Students should be encouraged to communicate their methods to each other. Students deepen their understanding by thinking about approaches and which are easier, harder, or most appropriate for the problem. Signpost three advises the teacher to tell the right information at the right times. When teachers are using a problem solving approach to help their students learn, the teacher needs to know when to let the students figure some things out for themselves. This signpost is very similar to the first one, letting math be problematic. If the teacher explains thing completely and clearly in the beginning the students may not be challenged enough to grow in their understanding. This signpost also focuses on the importance of discussions in the classroom and to teach fundamentals. Fundamentals such as, math symbols and language need to be taught first because the students will not discover these things for themselves.
Wednesday, August 31, 2011
Vaughn Ch.15
What are some of the reasons students with learning problems have difficulty with traditional mathematics curricula?
Students with learning problems may need more time and practice to learn math facts and math computations because they often lack the "automatization" to perform math computation effectively and efficiently. Some of their difficulties may relate to understanding the problem, they may lack the computation skills to solve the problem. Some may have difficulties with motivation. Students have difficulty reading the information provided because the vocabulary is too difficult and the reading level is too high. All information in the book must be directly taught to them and the math concepts are often presented poorly. Information may be scattered and multiple concepts are introduced at one time. There are insufficient problems covering any one concept or operation and too few opportunities for application of knowledge learned. The problems are not presented in enough different situations for students to learn and transfer what they know to real life settings. Also, students already do have difficulty transferring knowledge to real problems. Students often do not have the necessary prerequisite skills mastered to move on to the next unit. In addition, students with learning problems may have trouble with traditional curricula because the organization of the text varies considerably and makes learning from the book difficult.
Students with learning problems may need more time and practice to learn math facts and math computations because they often lack the "automatization" to perform math computation effectively and efficiently. Some of their difficulties may relate to understanding the problem, they may lack the computation skills to solve the problem. Some may have difficulties with motivation. Students have difficulty reading the information provided because the vocabulary is too difficult and the reading level is too high. All information in the book must be directly taught to them and the math concepts are often presented poorly. Information may be scattered and multiple concepts are introduced at one time. There are insufficient problems covering any one concept or operation and too few opportunities for application of knowledge learned. The problems are not presented in enough different situations for students to learn and transfer what they know to real life settings. Also, students already do have difficulty transferring knowledge to real problems. Students often do not have the necessary prerequisite skills mastered to move on to the next unit. In addition, students with learning problems may have trouble with traditional curricula because the organization of the text varies considerably and makes learning from the book difficult.
Reys Ch. 10
What are some myths and facts about using calculators?
Some myths about using calculators are that they do not require thinking from students, they lower math achievement, that using a calculator always makes computations faster, and they are only useful for computations. Calculators do require that a person think of what numbers they need to compute to problem solve. For example, helping the student use process of eliminator to find common factors between numbers to further solve the problem. Calculators can also raise a student's achievement and are also useful for instructional tools. Calculators can be used as an instructional tool to facilitate search for patterns, support concept development, promote number sense, and encourage creativity and exploration. Today's calculators are able to efficiently plot graphs among other things. Also, it is not always faster to use a calculator because computations that one has memorized or that are simple enough to compute mentally will be much faster to retrieve than punching it in to a calculator. Allowing students to use calculators in the classroom can facilitate problem solving, ease the burden of long tedious computations, focus the attention on meaning, remove anxiety and provide motivation and confidence.
Some myths about using calculators are that they do not require thinking from students, they lower math achievement, that using a calculator always makes computations faster, and they are only useful for computations. Calculators do require that a person think of what numbers they need to compute to problem solve. For example, helping the student use process of eliminator to find common factors between numbers to further solve the problem. Calculators can also raise a student's achievement and are also useful for instructional tools. Calculators can be used as an instructional tool to facilitate search for patterns, support concept development, promote number sense, and encourage creativity and exploration. Today's calculators are able to efficiently plot graphs among other things. Also, it is not always faster to use a calculator because computations that one has memorized or that are simple enough to compute mentally will be much faster to retrieve than punching it in to a calculator. Allowing students to use calculators in the classroom can facilitate problem solving, ease the burden of long tedious computations, focus the attention on meaning, remove anxiety and provide motivation and confidence.
Reys Ch. 4
I feel like chapter four was a lot of information that I will not have a complete understanding of until I am able to experience doing it for myself. There were many ideas and prompts that made me think about how I can incorporate different strategies in my teaching. The chapter was very informative of all the various ways of assessment. I have heard of many of the assessment forms in my previous classes. I really think portfolios are a great idea. It is amazing for the students to see their progress and to be involved in picking what will be put in it. As a teacher I would like to incorporate portfolios into my classroom. The chapter also mentioned communication between the students, the parents or guardians, and the school administration. The portfolios are a great way to communicate with parents during conferences, with students and with the school administration about students goals and progress. Also, the chapter mentioned having students take time to reflect on themselves through self assessments. Self assessments is something I would like my students to do, and to include in their portfolio. The section about observation was very informative. I agree that observing students and recording notes on a checklist would be very helpful in monitoring student performance. I want to make it a point to create a checklist of a few important aspects to focus on for at least one student a day to observe and record notes about.
Wednesday, August 24, 2011
Ch. 3
Planning curriculum is very important and I do strongly agree that it lies at the heart of good teaching. Lesson plans should be carefully organized and developed, and enriched through activities, technology and manipulates. I am intimidated that teachers plan their entire year of curriculum before the school year begins. It is then narrowed down into units, and finally into a weekly and daily basis. I do find it comforting that we cannot expect to follow our year plan exactly, but we review the progress of our class and the students as individuals to vary the pace, reteach and revise the lessons. It is important to write out a detailed lesson plan to make predictions and to use as a way to assess your own teaching. Also, if the lesson plans are written out instead of just planned in our minds we are better able to share ideas, and receive feedback from others. I also find it comforting that the state standards and school districts provide a framework of the curriculum to be taught. Having an outline to go off of is going to be very helpful in planning an entire years worth of class activities, tests and homework. I am also hopping that when I become a teacher my colleagues will share fun and effective lesson plans with me, so I am able to use them to help my students have fun and be engaged while learning.
Ch. 2
In creating a positive learning environment, Reyes says "Make sure Students understand that they will not all learn the same things at the same time and that they will not all be equally proficient, but that everyone can indeed become proficient'. I think this is great advice and I completely agree; I wish I knew this when I was learning math. When I was learning math in elementary and middle school I always a little behind, until eventually I did fall behind and couldn't catch up to the majority. I ended up needing one on one tutoring at a learning center. Lessons not only need to be modified for students with learning disabilities or special needs, but for each students' proficiency level. This is a lot of work a teacher needs to do with a class size of 25 to 30 students, but it is very important for children to learn at their own pace and be comfortable with that. It is very important that the students feel safe and comfortable with their own progress. Part of learning and making progress comes from trial and error. Students need to feel comfortable enough to make errors. That they may not fully understand how to solve particular problems right away but it's ok. Student's need to understand that this is a natural thing, that it is different for everyone and that it is ok to make mistakes and progress at their own speed.
Ch. 1
I do agree that math is used in everyday life and that society needs all of it's members to be able to problem solve in everyday life. I do think that the basic needs of society is a starting point of knowing how advanced to educate citizens. However, we live in a fast changing world that is rapidly advancing in technology. We do not know what jobs will be available in five, ten or twenty years from now. It is important for everyone to have as much knowledge and technological experience as possible in today's age. The needs of society seem to be changing and updating every year. It is interesting to look back in past to see the pattern of what was considered the standard of basics to be taught in school. In the 1970's the minimal competency movement stressed the basics of addition, subtraction, multiplication and division which included whole numbers and fractions. Today, and even in the 1970's, advanced technology had been developed that does these calculations in a matter of seconds. I find it disappointing that only addition, subtraction, multiplication, and division were the basics of what was taught in schools. I am pleased that this has changed and society has not fallen back into the pattern of stressing only the basics, and instead we are expanding.
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