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.
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