Consider the following task and respond to these questions on the discussion board:
• What features of “doing mathematics” does it have?
• To what extent does it lead students to develop a relational understanding?
• To what extent does it develop mathematical proficiency?
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The Sole D’Italia Pizzeria sells small, medium, and large pizzas, The small pie is 9 inches in diameter, the medium is 12 inches in diameter and the large is 15 inches in diameter. For a small plain cheese pizza, the price is $6; for a medium it charges $9 and for a large, $12.
• Which measures should be most closely related to the prices charged- circumference, area, radius, or diameter. Why?
• Use your results to write a report on the fairness of Sole D’Italia’s pizza prices.
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This is a test to see of comments will post:-)
ReplyDeleteThe authors’ beginning ways of doing mathematics by describing the language of mathematics and the ideal environments for students doing mathematics. They then give examples of students doing mathematics using pictures, manipulative models, oral language, and real-world situations, as well as traditional symbol manipulation (Van De Walle, et al. 24). These features of doing mathematics are presented in ways that students can understand and engage in. They involve puzzles and games, hands-on activities, and problem-solving involving pictures and diagrams of situations with which they are familiar. These serve to present mathematics as an extension of their own personal experience and prior knowledge. The hands-on manipulatives also help make the problems “real” for the students by removing the abstractness often encountered in mathematics. These features address the following components of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adapted reasoning, and productive disposition (Van De Walle, et al., 24). The authors go on to show that many mathematics lessons can address all of these components within one well-designed activity. When planning lessons and activities, Piaget’s constructivist theory of leaning as well as Vygotsky’s socialcultural theory should be kept in mind (Van De Walle, et al., 21). These ideas emphasize that students must be able to actively build their own mathematical relationships and concepts from prior knowledge and experiences through engaging activities and discussions with their peers (Van De Walle, et al.20-21). The ideas represented in these theories can be seen in the authors’ examples throughout the chapter.
ReplyDelete________________________________________________________________
The Sole D’Italia Pizzeria sells small, medium, and large pizzas, The small pie is 9 inches in diameter, the medium is 12 inches in diameter and the large is 15 inches in diameter. For a small plain cheese pizza, the price is $6; for a medium it charges $9 and for a large, $12.
• Which measures should be most closely related to the prices charged- circumference, area, radius, or diameter. Why?
The area best represents the amount of pizza purchased by the customer. The circumference only deals with the length around the age of the pizza whereas the area includes all of the pizza within the circumference.
• Use your results to write a report on the fairness of Sole D'Italia's pizza prices.
A large pizza is twice the price of a small pizza therefore, to be fair, the large should have at least twice as much pizza. If not, the consumer will be better off buying two smalls. First, the radius for each pizza is determined by dividing its diameter in half. Next, the area for each pizza is determined by the formula for the area of the circle, A = πr2 using 3.14 for π. The small pizza has an area of 63.6 square inches; the medium is 113.1 square inches; and the large is 176.7 square inches. The amount of pizza per dollar for each size can now be determined by dividing each sizes’ area by its prize. The small pizza provides 10.6 square inches of pizza per dollar; the medium pizza provides 12.6 square inches of pizza per dollar; and the large pizza is 14.7 square inches of pizza per dollar. It is clear that the large pizza is the best deal as it gives you the most pizza for you money. In my opinion Sole D’ Italia’s pizza prices are fair as Sole D’ Italia gives the customers more pizza for their money if they choose to spend more.
The readings provided an understanding of how a math class should be conducted. It provides tools for teachers to promote higher level thinking in mathematics as well as creativity. The text also provides engaging activities for use with prior knowledge and scaffolding which will aid in the understanding of concepts. The teacher can use multiple approaches to aid in the development of students math skills, which will focus on a students individualized understanding. There are five strands which the textbook indicates allow for math proficiency; concepttual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition (Van de Walle, pg. 24). Reflection on these strands will ensure students develop an understanding of math as well as reach comprehension of it. Additionally, they will be able to manipulate and problem solve using different strategies in order to become proficient. The textbook also provides examples to illustrate math concepts (Van de Walle, pg. 28).
ReplyDeleteI agree with you that the readings provide fantastic information on ways that we can provide to our students that will help them use critical thinking on how to resolve mathematical problems. You are right to say that prior knowledge and scaffolding can be implemented during the mathematical activities as well different approaches to develop math skills. The ideas in the chapter seem to accomplish these goals through puzzles, games, and real-world scenarios.
ReplyDelete• What features of “doing mathematics” does it have?
ReplyDeletePatterns and Order: The pizza scenario leads the students to consider patterns and order which is part of “doing mathematics”. For example, some patterns the students could consider in this scenario are the size of the pies 9, 12, and 15 are each 3 inches greater than the next. The pattern is odd, even, and odd. The prices of the pies are $6, $9, $12. Again each is $3 greater than the next. The pattern is even, odd, even.
Language and Environment for “Doing Math”: The pizza scenario is presented in a way that allows the student to decide how to solve the problem. They are then asked to explain their reasoning. The language in this problem is not the traditional memorizing and lower level thinking words. The student is encouraged to invoke higher level thinking.
• To what extent does it lead students to develop a relational understanding?
By asking the students to write a report on the fairness of Sole D’Italia’s pizza prices the students are expected to apply their reasoning to a real world scenario.
• To what extent does it develop mathematical proficiency?
In this problem the student must be able to reason about a pattern and explain and justify their solution. This problem would require factual knowledge, procedural proficiency, and conceptual understanding as described by the NCTM Standards. In approaching this problem the student would need to design a strategy to solve, try new approaches, and be able to keep at it until solved.
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ReplyDeleteAccording to the readings, “Doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense”(P. 13). It deals with the strategies used to solve the problem, the ways in which we explore, investigate, verify, explain and develop. From the beginning of the task it leads students to develop a relational understanding by using familiar objects and experiences. Students at this point are analyzing the information and developing an explanation to the solution. According to Van de Walle, “relational understanding is -knowing what to do and why” (P. 23). In regards to the task given, it requires the five strands of Mathematical Proficiency to have an understanding of the topic. Ensuring that students get in the habit of developing a strategy, solving it, then explaining or justifying a solution will help them have a productive disposition.
ReplyDeleteThere is a variety of ways students can use math with this word problem. It has patterns with the pricing and size of the pizza, using money, and students can have an understanding of size (area or diameter). The relational understanding is brought on by the students’ possible prior knowledge of purchasing pizza. The student can relate to the problem because he or she may order pizza in their everyday life and they may have a better understanding of the problem. The text states that children may have difficulty with solving problems when they cannot translate a concept from one representation to another (pg. 27). This pizza problem can be used as a real-world situation and when students can connect with real-world situations they have a better chance of being able to understand and solve the problem. Mathematical proficiency is when one has the ability to understand mathematical concepts and operations. This word problem helps students with problem solving connected to a real-life situation, they have to be able to reason and justify their answers, communicate and represent their answers (pg. 3-4).
ReplyDeleteDeciding what size pizza to order based on size and prize is a great example of our need for math. Daily task like this gives people the chance to generate strategies that were learned during elementary school. Simple task like deciding which size pizza we need to order can be very challenging if you don’t have the knowledge to do so. Learning math strategies gives us a way to identify the problem and then come up with a solution that makes sense.
ReplyDeleteOrdering pizza is a worthwhile task that gives students a chance to explore mathematical ideas. Activities like this will give students activities that allow them to think about all the mathematical ideas that are involved in everyday life. I’ve heard friends say that they hate math and didn’t have a use for it. Everyone must embrace or recognize how math and everyday live go hand and hand. Making good decisions may be as simple as getting better at mathematical proficiency.
The Van de Walle readings really hit home with me. I am "that" student that does not like Math or have any faith in my ability to do it. I have a background that is more towards the instrumental understanding (Van de Walle, p. 23) on the Continuum of Understanding. I never realized that, nor had I thought about it that way, until I read this chapter.
ReplyDeleteThe chapter delves into what a classroom environment should be like in order for effective math learning to take place. It talks about using many different manipulatives in order to "illustrate and discover mathematical concepts". (Van de Walle, p. 27) I have seen students use counters, base ten blocks, and number lines in the Math classes I have been in. I have to say that after reading this chapter I will view them in a different way and work harder towards helping the students understand that these are more than tools for "finding the answer".
The point that the chapter made about the importance of the teacher's role as a creator a "spirit of inquiry, trust, and expectation" (Van de Walle p. 14) is especially true. I have been in classes in which the teacher had a great desire to communicate her lessons to her students and challenged them to achieve to their highest potential and she was rewarded with great results. I have also seen the other end of the spectrum as well and the results have been equal tot he effort exerted.
The most important thing that the chapter taught me is that" Mathematics is a science of concepts and processes that have a pattern of regularity and logical order. Finding and exploring this regularity or order, and then making sense of it, is what doing Mathematics is all about". (Van de Walle, p. 13) It is those patterns that I hope to gain a greater understanding of in order to be a good teacher of Math.
In the problem presented in the Module I saw that the students were using their reading and reasoning skills as well as their Math skills. They were asked to figure out what operation(s) they would use to solve the problem. The students use their prior knowledge about money counting skills to figure out that the cost of a large pizza gets you a lot for your money.
Ok so here goes! If I am understanding correctly, we are to demonstrate our understanding on the chapter by looking and the problem posed above and answer the questions
ReplyDelete• What features of “doing mathematics” does it have?
• To what extent does it lead students to develop a relational understanding?
• To what extent does it develop mathematical proficiency?
In this scenario I can see an attempt to make the students think. It uses words like related and consider which are good words to use in “doing math.” There is more than one way that a student will be able to work on this question. They can start by finding out the relationship between the pizza size and the price assigned to it, or they can start by determining which measure of the pizza is most important. They will then need to determine if the price will be related to the area radius or diameter and then asks the question “why?”. This is good because it let’s the students explain how they came up with the answer and related it back. The problem does a great job of relating to a student’s interest and prior knowledge of pizza. Most students in our classes will be very familiar with the yummy goodness! There are opportunities to build on prior knowledge of relationships between size and price and build on the those to solve a problem in the real world. The last part of the problem asks the student to make connections they haven’t already made. As previously stated the problem will help define relationships between size and values. It makes the student think about fairness in relation to price and purchase amount. Depending on the age of the student, it might connect the concept of currency and smart shopping! In all I think the problem had potential. It definitely asked the student to use a higher level of thinking than a math problem or worksheet. It gets the student to think about things, relate it to knowledge already known and possibly construct new knowledge using what they already know.
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ReplyDeleteThe task at hand can be used to address and reflect on the many concepts and approaches in mathematics. Our textbook gives an explanation of the four features of a productive classroom culture in which the students can learn from each other by doing mathematics. These four features are listed as: Ideas are the currency of the classroom, Students have autonomy with respect to the methods used to solve problems, The classroom culture exhibits an appreciation for mistakes as opportunities to learn, and The authority for reasonability and correctness lies in the logic and structure of the subject, rather than in the social status of the participants (pg 14-15). If we examine the task given we can conclude that all features can be addressed in classrooms working on the said task. There is room for active participation, each students approach to solving the problem can benefit the final outcome, mistakes made and corrected while trying to find a solution can guide students to the right answer and serve in the future scenarios, and the derivation to a well formulated answer can bring confidence to the students. Furthermore, the task leads students to develop a relational understanding to the extent that it provides one with the ability to interchangeably use the different representations of mathematical ideas described in our text as manipulative models, real-world situations, oral language, written symbols, and pictures. (p 27) The fact that the task involves a real life scenario dealing with pizzas an object you can visualize and manipulate and picture gives way to your ability to find a formula to figure out lets say the area or circumference of this object and perhaps use ratios to orally explain your reasoning. In addition, the task develops mathematical proficiency to the extent that in solving the problem it makes it important to use prior knowledge of diameter, circumference, radius, and area to find a solution and justify it as well as make connections to real life scenarios and to be confident of your answer because it makes sense particularly in this type of problem in which you are having to find out what size pizza is the most convenient and fair to purchase at the given prices.
ReplyDeleteSome features of “doing mathematics” this example has is number patterns (p 15), with the pizza sizes (9, 12, 15) and prices (6, 9, 12) each counting by 3.
ReplyDeleteBringing a real life situation, ordering pizza, to the students in a math equation assists students in understanding it as many students have either ordered or been around ordering of pizza (building new knowledge from their prior knowledge p 21). I know that I personally learn better with hands on activity or a problem where I can recall my prior knowledge, like in ordering pizza.
This can help develop mathematical proficiency by bringing students interest into classroom, into math itself. When students understand what they are learning the proficiency level increases.
Measures that should be most closely related to the prices charged needs to include the circumference because it’s the distance around the circle and the diameter because that’s the distance across the circle through the center.
The prices should be based on the size of the pizza, going up in price as it goes up in size. So a small (9 in) pizza costs $6, with this theory the large pizza would fairly be 18 inches and no more than $12. The way the prices are now, it would be more cost efficient to buy two smalls (9 in plus 9in equals 18 inches) for $6 each ($6 plus $6 equals $12) instead of a large pizza (15 in) for $12.
In response to the 3 questins posed below
ReplyDelete• What features of “doing mathematics” does it have?
• To what extent does it lead students to develop a relational understanding?
• To what extent does it develop mathematical proficiency?
Doing Mathematics will allow students to use thier mind to improve critical thinking skills and utilize key terms that are related to math in general. This will allow students to work on several ways to answer questions by seeing tha the size of the pizza and price can be a direct relation to the diamater.This in turn will lead students to a relational understanding. Using pizza to build and increase math skills will build knowledege with how size and price will work when they are using in life skills as they grow up to become productive citizens. Using real connections with thier daily lives such as ordering pizza and size contexts will allow studnets to become more comfortable with what they naturally know.
The task at hand promotes schematics. Schematics require assignment of experience and matching. Relational understanding is met when asked "Which measures should be most closely related to the prices charged- circumference, area, radius, or diameter?" Only requiring comparison of more for the money or most economical.
ReplyDeleteNo math is done. To compare or relate the scale by increments of three is not influencing connection of the dots. No stimulation of math is done. A report is proof of findings. The task seemed to be designed more to science.---CG
Interesting thoughts @Christie. Math is a science so I think that I would want to be able to prove my findings by doing the math whether it was done for me or not. I agree that a report is proof of findings, but the report should be of from person that has done the findings. I agree that the task promotes schematics, but I'm not sure if the math is done. How would the student know which pizza was for the better price or if they were all the same in comparison without either having prior knowledge of that style of pricing or doing the math to find out?
ReplyDeleteMathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to the solutions, and checking to see if your answers make sense (pg.13). As learning occurs, the networks are rearranged, added to, or otherwise modified (pg.20).One of the best ways that I feel that students learn is through socializing. Vygotsky believes that mental process exists between people in social learning settings and that from these social settings the learner moves ideas into his or her own psychological realm (pg.21) Understanding mathematics consist of both constructivist and socio cultural theories. Understanding mathematics doesn't mean that you have to know it all or not know anything; rather it means that you use your schema and add on to what you already know.
ReplyDeleteThe problem that is given about the pizza scenario evolves problem solving that can be related to a real life situation. Students also have to use prior knowledge to get a solution to the problem. They have to be able to remember the formulas that are use to get the radius, circumference, diameter and area. This problem can also be linked to patterns. Students have to write a report on the fairness of the pizza prices. This requires students to compare the sizes to the prices. Students need to use higher level thinking in order to find a solution.
AJ I agree with you that Math is a science. I also think that there was plenty of math to do in this problem. I feel like there were patterns to identify and there were relationships from Pizza size to price. The student would also need to use specific formulas for determining the actual size of the pizza. I felt like this was a good real world scenario for students to get a taste of making purchasing decisions.
ReplyDeleteI agree with you Alcala, the book provides excellent activities in which students can engage in higher level thinking. It also provides engaging activities that help students in prior knowledge and scaffolding. A great example was the example of the pizza scenario in which the students had to use the problem and use their prior knowledge from formulas to find a fair solution. This problem was a great example of using a real life scenario in math. As, you said the textbook did a great job in providing examples that help students illustrate math concepts.
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ReplyDeleteI agree with you completely. Knowing how to do math help us whenever we want to compare the price between something on sale and something not on sale. Sometimes we think that buying a product that is on sale will be a better deal than buying something listed at its normal price. And in the end we often pay more for that particular product that was on special. When problems involve our hard-earned money they acquire additional meaning. You are right about the use of real life examples when solving math problems; they definitely keep students engaged in the problem solving as well as the search for a solution.
ReplyDeleteI saw the same patterns that Katy saw in the pizza scenario and I agree that the scenario of ordering a pizza would be one that is familiar to most students. In presenting a scenario like this it would peak the interest of the students’ because who wouldn’t want to get the most pizza for the money. I have to say that I looked at this scenario from the perspective of a fourth grader. In looking at it that way I saw that the sizes of the pizzas increase by three inches and the price increase in a three dollar increments as well. In the simplest terms I believe that my kids would agree with Katy and buy two small pizzas for $12.00 and therefore get more pizza for less money. They love pizza.
ReplyDeleteGladis has a good point in that if something is not working then a different approach can be tried. The textbook talks about not just talking louder or slower, but trying a different strategy. The text also has different theorys of Piaget and constructivism or sociocultural which is the work of Vygotsky. Teachers should be able to know how their kids learn best and develop lesson plans which will meet their needs. This can also be used for grouping the students which learn in similar ways. When it comes to solving problems, students need to be challenged and given different types of strategies in order to solve to see which will work best for them. I strongly believe that all students can learn and expectations need to be set high.
ReplyDeleteKaty, I really like your ideas on setting patterns as far as using pizza sizes and prices. Teaching students number concepts is very important as well, it's not only about knowing what number is more or less than another. It's about "building on their existing knowledge" (P. 26). I also feel that children learn best when using real life situations or at least situations where children can relate to the objects in the situation. From my own experiences, I feel " hands on activities" using manipulatives have always been a great way for me learn in a classroom. I can truly say I have had great math teachers, because I have always loved math and I feel there is an easy way to solve each problem.
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ReplyDeleteAs explained in our textbook mathematics is more than completing sets of exercises or mimicking processes the teacher explains. Doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense (pg.13). The problem given can be used to tackle and reflect on different ideas and approaches in mathematics. That is, this particular problem consists of number patterns (pg.15), which allows for active participation in finding a solution to guide students to the right answer. Having the problem reflect on actual circumstances gives students room to use their prior knowledge and perhaps gain interest in solving the problem. Their interest in problem solving can guide them into a better understanding of what they are learning while additionally increasing their proficiency levels. Moreover, the problem gives students the opportunity to help define relationships between size and value and gets them to think about equality in relation to price and purchase amount.
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ReplyDeleteKaya I definitely see how this problem can make students think. I like how you pointed out that most students would be familiar with the "yummy goodies," which is very true and one as a future teacher should use this to our advantage.If the students are familiar with these goodies they may be more interested in solving the problem which can lead to a better understanding of how to get an answer. I also fee this problem was a good way to get students to relate to real life scenarios while dealing with math.
ReplyDeleteYour posts have reflected thoughtful and deep reading of the Van de Walle text. Particularly impressive are the posts which demonstrate what had been learned and apply this to the problem presented.
ReplyDeleteThis problem is rich in mathematical opportunity. One of the key ideas you will develop this semester is the critical importance of not just getting an answer from a student but taking the time to find out the thinking behind an answer.
While we should always look for opportunities to make math problems relevant to our kids' lives, there will be times when a current, real-life application cannot be made. However,we can always design work that engages students by allowing for multiple entry points when solving a problem. Many problems, such as this one, provide opportunities for us to assess our students' reasoning,number sense skills and their ability to determine relationships in numbers, all of which are skill sets they will use everyday in "real life."
Food for thought: Is there one RIGHT answer to this problem? Is there value in allowing students to "make a case" for their individual answers? What is the role of computational accuracy the pizza problem?
The best thing about problems like these are that students can relate to them, but not only that it teaches them how to be wise with their money. So not only are they learning which pizza is better for their money, but they are learning to spend their money wisely which is something they can take far beyond grade school and buying a pizza. @IbethT, I agree with your statement that math is more than just exercises. When I was in grade school that was all we did. We sat around in our cubicles and did math problems that most of the time we couldn't relate to or worse found them frustrating. I wish my math teacher had the same philosophy as this author.
ReplyDeleteI agree with AJ Moore. This exercise would help students learn to spend their money wisely. I know that I learned the basic math skills of balancing a checkbook and things like that in school, but never really understood the value of a dollar until I was older and buying food (my paycheck went quite quickly when I ate out every day as a teenager). I also agree with Ibeth, math is more than just exercise. I’m glad that we as future teachers are learning ways to teach our students beyond the old school worksheets of problems after problems.
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