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Double Occupancy

 

A Pair

of

Singapore Style


Concept Building Reflections

 

 Similar Prisms

 

I have posited, for your consideration, a mathematical problem that allows us to explore both the volume and surface area of a rectangular prism. This activity, which prioritizes understanding before the memorization of formulas, will serve to broaden our conceptual knowledge of both volume and surface area.

In Laureate’s video, Dr. Van de Walle describes how the “base times height” formula for finding a rectangular area can be connected to a number of other geometric area formulas. The reasoning behind this is thus. If we substitute the words base for length, and height for width, it doesn’t really matter what side these labels are on. We can label the top, bottom, or side of the figure as the base. The width then becomes the height. These terms, base and height, are more negotiable than length and width when dealing with the areas of parallelograms, triangles and trapezoids. We can more easily adapt them to the unique proportions of these polygons. As we explore the strands of commonality among area formulas for rectangles, parallelograms, triangles, trapezoids, and even circles, we can extend the concepts to embrace three dimensional figures and their volumes, as well.

Once we have tweaked the nomenclature of the dimensions of the rectangle, we can attain a deeper understanding of the concept of area. No formula is necessary at this point. Our first step is to determine how many square units sit on the base, and how many layers of these units will accommodate the height. We then multiply these two numbers together for the area squared. By counting the square units contained within, we can thereby affirm the formula for the area of a rectangle to be base times height.

To begin our exercise, we are presented with a three dimensional rectangular prism that initially measures eight inches (base) by six inches (width), by two inches (height). We are then asked to determine both the volume and the surface area. Subsequently, we are directed to increase the ratio of the original base, width and height units by factors of two, three, and even more. Finally we are to reflect on the results, by discerning patterns and relationships as we observe the volume and surface area increase proportionally. We are then to utilize this knowledge to fashion a formula for finding the surface area and volume of a larger prism with a ratio of 1: n .
              I considered the first rectangular prism. A rectangular prism is a three dimensional figure with two ends and four sides. The opposite sides have the same area.  I began by drawing a three dimensional rectangular figure on graph paper.  Following this, a second drawing was made, with the prism flattened into a two dimensional shape. This enabled me to view a figure resembling a flat cross, divided into six segments. In this configuration, I was able to peruse the entire area of the surface. Reflecting on my observations, I concluded that the surface area must be the sum of the areas of all six shapes that cover the surface of the object. I also noticed these six shapes were comprised of three different sized rectangles. Each of the three had an identical counterpart located opposite to it in the original configuration.  Since each of these three areas can be found by using the base times the height formula, and each area has a matching counterpart, I can now fashion a formula for surface area: two times the width times the height (accounting for the area of two identically sized sides), plus two times the base times the height (accounting for two more identically sized sides) plus two times the base times the width (accounting for the top and bottom areas).  Since we are dealing with two dimensional areas, the answers will be expressed in square units.

Crunching the numbers, I determined that with a width of six inches, a base of eight inches, and a height of two inches, the first rectangular prism would have a resulting surface area of 24 plus 32 plus 96 which equals 152 square inches. The second prism had a width, base, and height that was twice as large as the first one. This would give us a width of twelve inches, a base of sixteen inches, and a height of four inches. The ensuing surface area was therefore 608 square inches.  The third rectangular prism, which tripled the width, base, and height of the first one, featured an eighteen inch width, a twenty four inch base, and a six inch height. Consequently, the surface area was 1368 square inches.

Volume is the measure of the amount of space inside of a rectangular prism.  It is measured in cubic units, since it can be viewed as the number of one inch cubes needed to fill the interior. Since the rectangular prism is a three dimensional figure, we must not only account for its length and width, but also its height. This third dimension is what gives it volume. Accordingly, to discover the volume, we must expand the area of a two dimensional rectangle by the factor of its height, with a resultant formula of base times height times width, expressed in the aforementioned cubic units. 

To find the volume of the first prism I used the formula: base (eight inches), times the width (six inches), times the height (two inches). The subsequent volume was 96 cubic inches. The second prism had dimensions that doubled the first. With a sixteen inch base, a twelve inch width, and a four inch height, I determined the volume to be 768 cubic inches. The third prism tripled the base, width and height of the first. This resulted in a volume of 2592 cubic inches.

Comparing my results, I did detect some patterns and relationships. Considering the surface area first, which is comprised of square inches, I noticed that when the base, height and width measurements were doubled, the area rose from 152 square inches to 608 square inches. This is a fourfold increase, or the ratio number (two) squared. When the dimensions were tripled, the area soared to 1368 square inches which is nine times as great as the original, or the ratio number (three) squared. If I were to quadruple the original numbers I would have a base of 32 inches, a width of 24 inches and a height of 8 inches. Squaring the ratio (four), I would multiply the original 152 square inch area by 16. This would give me a total area of 2432 square inches. Plugging the dimensional numbers into the formula and calculating the results yields the same answer. My hypothetical pattern has been validated. I also detected a relationship between the fact that as the dimensions were doubled, the units were expanding not in single units, but in square units. This might account for the squared ratio number.

    Turning to the volume, when the original dimensions were plugged into the base times height times width formula, the area was found to be 96 cubic inches. Doubling each dimension gave us an area of 768 cubic inches. Analyzing this, I found that by taking the cube of the ratio number (two) and subsequently multiplying this number (two times two times two, or eight) by the original area yielded the answer of 768 cubic inches. Consequently, when we tripled the original dimensions we attained an area of 2592 cubic inches, which is the same as multiplying the cube of the new ratio (three times three times three, or twenty-seven) by the original area. Reflecting on this pattern, I can hypothesize that when I quadruple the original dimensions I will discover the area by taking the cubic value of four (sixty-four) and multiplying this by the original area of 96 cubic inches. This yields an area of 6144 cubic inches which coincides with the number attained by using the formula. The rationalization for the cubic exponent for the ratio number is that the area is expanding in three dimensions as the base, height and width is doubled, tripled and quadrupled.

    In conclusion, when students develop formulas, rather than memorize them, they gain conceptual understanding of the ideas and relationships involved. Once students understand where the concepts come from, the absorption of critical ideas is facilitated. During this authentic process, they analyze and synthesize figures and shapes. They discover patterns, and then use this knowledge to make connections. By understanding the why before memorizing the what, they will no longer be limited by esoteric information that is isolated from the reasoning behind it. Rather, they will be able to transfer their mastery of the entire concept to an array of mathematical subjects. This skill remains the lynchpin of problem solving which, in any event, is the ultimate goal of mathematical education.

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Numbers and Operations (K-5)

 

    

"The formula 2 + 2 = 5 

is not without its attraction."

-Fyodor Dostoevsky

                                                                                                                     

The subject of Elementary (K-5) numbers and Operations is intriguing. This particular strand encourages the exploration of the the process of problem solving, concept development, and representation, and how each of these entities interacts with the other. Although teaching math through problem solving was its main theme, the course presentation was by no means monochromatic. In general, the material was presented on a richly textured canvas that integrated historical perspectives and international data, as well as offering the student-practitioner a wide selection of strategies, based on scientific research, in which to improve student learning (Stigler and Hiebert, 2004).Accordingly, we were allowed to indulge ourselves at this intellectual banquet by perusing well considered theories based on cutting edge expertise. Let me now expand on these reflections.

 


    According to the National Council of Teachers of Mathematics, the teaching of mathematics, in the United States must evolve from memorization of basic facts, rules and procedures to the realm of concepts, underlying ideas, and the construction of one’s own knowledge (Laureate Education, 2007a). The reason for this is simple. Traditional math has simply not worked for the overwhelming majority of students (Laureate Education, 2007a). As the global economy’s work skills suddenly metamorphose from a shopkeeper modality, with plentiful opportunities for those with weak math foundations, into a high tech, Kafkaesque landscape, employees must now be adroit at problem solving, be flexible in their approaches, and knowledgeable about technology (Laureate Education, 2007a). Unfortunately, young learners are not offered the opportunity to acquire these skills in many of today’s schools (Laureate Education, 2007a).  The NCTM has promoted an educational reform movement to reverse this trend. Their standards include conceptual understanding, procedural knowledge, flexible computation strategies, and problem solving ability. These abilities dovetail neatly with real world job requirements (Van de Walle, 2004).


    Instilling in children the confidence to correctly solve problems is another shibboleth of the NCTM (Van de Walle, 2004). This process begins with a safe learning environment. Here, the teacher, as a facilitator in this community of learners, encourages risk taking, feedback, the sharing of ideas, and continual practice.  The teacher’s other responsibilities are to clarify misconceptions, and summarize what has been learned (Van de Walle, 2004). Otherwise, they are encouraged to step back and allow the students to share their thought processes with each other (Van de Walle, 2004). Educators further inspire, as part of this communal culture, reasoning, exploring, conjecturing, hypothesizing, justifying, and, again, communicating (Bell and Bass, 2005). It is in this milieu that the concepts of math develop.


    The NCTM also emphasizes the theory that concepts and procedures should be taught in the context of problem solving. In other words children learn math by doing math (Van de Walle, 2004). Math problems presented for the students’ consideration should contain certain qualities. Firstly, important math content should be embedded in these problems. In addition, they should be relevant and engaging (Van de Walle, 2004). Moreover, they can be open ended and should provide multiple entry points, as well as entail different methods of solution (Laureate Education, 2007b). Additionally, good problems must also include modifications for students with varying skills, abilities, and learning styles (Askey, 1999). When students are building on previously learned concepts and strategies, actively looking for relationships, analyzing patterns, identifying productive strategies, justifying results and challenging the thoughts and solutions of others, they are mastering the concept (Van de Walle, 2004).


     Before the subject of concepts, strategies and theories can be expanded upon, groundwork for understanding the initial concept must be established. This foundation is important since it is in this setting that students begin their math journey. The NCTM stresses that effective lessons begin where the children are academically, not where the teacher is. Instructors should therefore begin with ideas that the students have already mastered. These ideas will then be used to create new ones (Van de Walle, 2004). In circular fashion, once mastered, this fresh knowledge will deepen the students’ understanding of their previous notions (Askey, 1999). Finally, what goes around comes around. Success in these endeavors breeds confidence; confidence, in turn, facilitates future learning. This aphorism applies equally to teachers as well as the students.


    The teacher signals the beginning of the learning cycle through direct modeling. This prerequisite step, so necessary for understanding the concept, involves an array of area, length and set models. These manipulatives, along with the accompanying counting, ensure that thinking and computing skills develop simultaneously (Van de Walle, 2004). Students unveil and absorb the concepts, as they practice their numbers. In this manner, their sense of numbers also begins to form, as they compute, compare and order quantities.  Finally, once the concept is taught, symbolic names, or the language of math, are introduced (Laureate Education, 2007c). Symbolic representation is necessary if these concrete skills are to be later applied in abstract contexts (Ball and Bass, 2005).


    Computational fluency is another aspect that must be discussed. It has been defined as using and understanding efficient and accurate methods of computing (Van de Walle, 2004). Mastery of computational fluency is evidenced by the ability to utilize problem solving approaches that include, but are not limited to, mental math, paper and pencil calculations, invented strategies, estimations with friendly numbers, and traditional algorithms (Lobato, 1993). Knowledge of place value plays a central role.


    Development of computational fluency has derivative benefits as well. Students come to view computation as more than just a means to produce answers. It is increasingly viewed as a conduit leading to deep appreciation of the number system. The resultant is that students develop a strong number sense.  Students who have number sense can break apart and combine numbers. They also understand the relative size of numbers, can identify sensible answers, and utilize this knowledge to accurately compute (Laureate Education, 2007d).


    As students drink deeply from the computational well, they will quickly learn for themselves, and in their own words, that computational fluency and math confidence are two components that inform and support each other. The more they successfully compute, the more confident they will feel. The more confident they feel, the more likely it will be that they will calculate successfully.  


    Invented strategies are a natural correlative of not only computational fluency, but concept mastery, as well. When math concepts are intertwined with strategies, they are not easily forgotten (Laureate Education, 2007d). Furthermore, the idiosyncratic nature of these approaches gives the student a vigorous sense of ownership, as well as a healthy sense of numbers (Laureate Education, 2007d). These two constituents cannot be underestimated as motivational imperatives.


    Whether invented strategies are suggested by the teacher, a peer, or the class as a whole, they must be constructed and comprehended by the student (Van de Walle, 2004). Each invented strategy must be valued by the instructor, and this approbation should be perceived by the student. Additionally, these approaches not only complement the natural reading process with their left to right format, but provide a facile entre into mental math and estimation (Van de Walle, 2004). Invented strategies include such configurations as front end methods, and adding and subtracting by tens, hundreds, and other friendly numbers. Other derivatives  include the splitting and chunking of numbers for addition, counting up to calculate subtraction, partitioning and clustering to advance multiplication, and identifying missing factor approaches to simplify division (Van de Walle, 2004).  Students initially scaffold these operations with a variety of concrete models, such as Cuisenaire rods, base ten blocks, number lines with benchmarks, and ten frame cards. Later, when they have connected the concrete to the abstract, these manipulatives are, for the most part, discouraged. Despite the celebrity that has been invested in invented strategies, traditional procedures and algorithms should not be ignored (Van de Walle, 2004). The opportune moment to connect the invented strategies with standard algorithms, however, is after the students have mastered the concept, and not before (Laureate Education, 2007).


    Reflecting on the problem solving process, (Polya, 1945), has crystallized for me, the challenges children face when attempting word problems. By utilizing the sequential steps of understanding the problem, devising a plan, carrying out the plan, and looking back, children will be able to clarify the variables and choose correct operations to crack the problem. This process brightlines the concept of organization (Polya, 1945). Since problem solving is the central goal of math education, this methodology offers the students the dual opportunity of solving problems in a logically sequential way, while developing a deeper understanding of the underlying concepts (Burris, 2005). Manipulatives, drawings, number lines with benchmarks, as well as area, length and set models, are integral appendages to the process. These scaffolds support the learners as they peel away the obfuscating scales, one by one, in order to view the underlying ideas in the light of day (Burris, 2005). The welcome resultant is accurate computation, conceptual knowledge with a long shelf life. Additionally, this systematization offers an inductive, discovery oriented path, to move the child from the general to the specific (Van de Walle, 2007).


    Assessing the learning that has taken place is no less complex than the problem solving process. (Van de Walle, 2007). Rubrics can be developed that evaluate such skills as conceptual understanding, adeptness with procedures, and computational accuracy (Van de Walle, 2007). The problem solving itself, with students explaining the why and how of their solutions, creates ongoing assessment data that can be used to make instructional decisions, help individual students, and communicate with parents (Van de Walle, 2004).


    Mathematics is not a collection of isolated topics. In order to deepen our understanding of this discipline we must make connections not only to previously learned material, but to other subjects and the world at large, as well (Laureate Education, 2007e). The opportunity for students to experience math in other contexts is important. Math is used extensively in science, social studies, medicine, and business (National Council of Teachers of Mathematics, 2000). It therefore behooves teachers to include math in many cross curricular activities, in order to impress upon students the broad-based nature, importance, and relevance of the discipline (Fuson, 2002). Real world connections are integral, as well. Mathematics makes more sense when real world connections are made (Laureate Education, 2007e). These verisimilitudinous connections motivate students, facilitate understanding, and provide a purpose for learning (Laureate Education, 2007e). Accordingly, presenting new material on a canvas that integrates meaningful, real-life experiences can expedite and enhance the learning process (Laureate Education, 2007). In this context, students will experience math, not just memorize a list of procedures (Burns, 2007). Furthermore, they will no longer wonder why they are learning a particular concept. Real life applications teach them where and when they will use the skills they are now mastering (Willis and Checkley, 1996).


    In conclusion, I have sharpened my own discernment of the pedagogical framework promoted by the NCTM by responding, on a twice-weekly basis, to prompts that stressed these abovementioned concepts, and much more. In choosing to read, view, and reflect upon the resources offered, I have come to see the interrelated aspects of mathematics in a more encompassing light. I have also realized that elementary math is not a superficial discipline, as we have been led to believe, but a complex dynamic incorporating a myriad of interrelated concepts (Askey, 1999). Accordingly, it is an intricate subject that demands deep understanding, and careful, time consuming preparation by the teacher (Askey, 1999). Only by thoroughly understanding the number and operation involved, and mastering the concepts behind the skills I plan to teach, will I feel confident in my abilities to teach this material to my students. In this manner I plan to teach, as well as imbue, in my students, a thirst for math knowledge, so necessary for success in the real world.

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References


Askey, R. (1999, Fall). Knowing and teaching elementary mathematics. American Educator,
23
(3). Retrieved November 4, 2004, from the American Federation of Teachers
Web site http://www.aft.org/pubs-reports/american_educator/fall99/index.html


Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G.
Martin, & D. Schifter (Eds.),

Burns, M. (2007). About teaching mathematics. (3rd ed.). Sausalito CA: Math Solutions
Publications.

Burris, A. C. (2005). A five-step problem-solving process. In Understanding the math you teach:
Content and methods for prekindergarten through grade 4 (pp. 24–32). Upper  
SaddleRiver, NJ: Merrill/Prentice Hall.

Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J.
Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles
and standards for school mathematics (chap. 6, pp. 68–94). Reston, VA: National
Council of Teachers of Mathematics.

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Two. "Mathematics in  
the  United States” Baltimore: Trafton, P and McIntosh, R.  

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Three. "Problem Solving,
Part Two”, Baltimore: McIntosh, R   

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Four. "Representation”
Baltimore: Sheer, J..

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Six, "Number Sense”   
Baltimore: Trafton, P.

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Nine, “Connections.”
Baltimore: Steele, D.

Lobato, J. E. (1993, February). Making connections with estimation. Arithmetic Teacher, 40(6),
347–351.

National Council of Teachers of Mathematics. (2000). Connections. In Principles and standards
for school mathematics. Retrieved February 15, 2005, from
http://my.nctm.org/standards/document/chapter3/conn.htm

Polya, G. (1945). How to solve it: a new aspect mathematical method. London, England:
Penguin Books Ltd


Stigler, J. W., & Hiebert, J. (2004, February). Improving mathematics teaching. Educational
Leadership, 61(5), 12–17.

Van de Walle, J. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.

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;

 

 

 

"Any time Detroit scores more than 100 points and holds the other team below 100 points they almost always win."

-Doug Collins

 

 

 

Consider these:

 

 

A. Devon, Matt and Brandon went for a walk in Montana, which is grizzly bear country.  About 1 mile into their journey they came to a deep, wide river. There was no bridge. They didn’t have a boat or raft or any materials to build one. None could swim or float. How did they cross the river?

 

B. A man pushes his car up to a hotel. He pays the owner of the hotel $200. Then he pushes the car away. What's going on? 

 

C. Destiny lives in Yellowstone National Park. She walked out the back door of a farmhouse on Thursday afternoon and found a man’s pipe, a scarf and 3 lumps of coal lying on the wet grass. They weren’t there on Tuesday. No one had been near the house in weeks. Where did the objects come from?

 

 

The Answers:

 

 

A. The boys went on their hike in the middle of winter. The river was frozen so they merely walked across.

B. The man is playing Monopoly.

C. The weather got warmer and Destiny’s snowman melted.

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"Think beyond the bun!!"

 

d

 

Theodore Geisel Library, San Diego

 

 

 

 

World's Tallest Building

Khalifa Tower, Dubai

 

Form Follows Function

 

 

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