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Parallax View


Teaching Geometry in the Elementary Grades

 

 

 

Over a half-century ago, Pierre van Hiele and Dieke van Hiele-Geldof, mathematics teachers from the Netherlands, devised a five-level hierarchy of geometric thought (Laureate Education, 2007). This method provided a framework for students to showcase their level of geometric understanding (Laureate Education, 2007). Today, the Van Hiele theory has become one of the most influential factors in the American geometry curriculum (Van de Walle, 2007). The National Council of Teachers of Mathematics (NCTM) lends supports to the Dutch approach in their “Curriculum and Evaluation Standards.” They echo that the development of geometric ideas advances through a hierarchy of stages (National Council of Teachers of Mathematics, 1989).


The van Hiele methodology promotes five sequential levels of geometrical reasoning (Van de Walle, 2007).  Progress from one level to the next is more dependent on the richness and quantity of mathematical experiences than on chronological age (Laureate Education, 2007). Additionally, the use of physical materials and drawings are de rigueur at the first few levels.
Central to their theory is the observation and discernment of the sophistication of students’ thinking, as they engage in geometric tasks (Van de Walle, 2007). As a result of these assessments, the instructor can implement teacher guided, inquiry-based activities, in which the pedagogy matches the child’s level of thought. Accordingly, during these activities, it is imperative that teachers use geometric terminology that is in synch with the students’ levels of thought. Without this proper correlation, the students and teacher will be in two different geometric places, figuratively speaking, and consequently, the children’s learning curve will be seriously compromised (Malloy, 1999). The overarching, dual objectives of this five stage approach are thus: mastery over the present level’s concept, as well as preparing the child to advance to the next hierarchical stage.


At the Visualization level, students generally recognize figures by appearance alone, oftentimes comparing them to an example. At this juncture, the manipulative in front of them represents the concept. As such, they cannot relate it to the universal idea of the shape (Laureate Education, 2007). In addition, transformations such as rotating and flipping will cause the shape to lose its identity (Malloy, 1999). Consequently, students may think that a rotated square is a diamond and not a square because it looks different from their visual image of square. The neophytes’ vocabulary is also at the minimalist stage. Standard nomenclature is rare. Hence, a parallelogram may be a “slanty rectangle” or an angle, “the arms of a clock.” Other attributes of shapes may be described as “fat” or “skinny” (Van de Walle, 2007).
As one can readily perceive, the objects of thought at this level are individual, concrete shapes. In order to reach the products of thought for Level 0, classes of shapes that seem to look alike, activities at this level should involve lots of sorting and classifying (Van de Walle, 2007).


Seeing how shapes are alike and different, and placing those in the appropriate groups should therefore be the primary focus. Moreover, the students should decide on how to sort rather than the teacher (Van de Walle, 2007). This allows the child to perform the exercise with ideas they understand and lends to them a certain amount of ownership.


At the beginning of this Visualization stage, the children’s vocabulary may include references such as “they look alike or different” or “a rectangle is wider than a square” (Van de Walle, 2007). During the course of the activities some standard terminology will become known, such as rectangles, squares, triangles, cylinders. Properties of these shapes, such as the number of sides, parallel sides, symmetry, right angles, and congruence, are included in the exercise, but only in an informal, observational manner (Van de Walle, 2007).  It is therefore important that when these properties are introduced, students should be challenged to integrate these features as they classify shapes. By listening to the type of attributes they use and discuss in their sorting the teacher will be able to discern which properties are familiar to them, and which are not (Van de Walle, 2007). 


Students working at the Analysis stage, or Level 1, are capable of considering all shapes within a class, rather than an individual shape (Van de Walle, 2007). In order for a student to reach this level, he needs to classify shapes according to properties, such as the number of sides, congruence, perpendicular lines, and symmetry. He should also be able to classify many shapes with these properties, not just the ones he is manipulating (Laureate Education, 2007).
My assigned task this week was to design an activity to help my selected learner advance from his current van Hiele level to the next. As part of my discussion, I concluded that Mateo was operating at the Visualization stage, or Level 0. My reasoning for this placement is threefold. First, he was sorting the three types of quadrilaterals into sets and subsets according to their appearance. Secondly, his total lack of content- related vocabulary brightlined the earliness of his geometric experience. Finally, it was obvious that Mateo was concentrating concretely on the shapes before him. As such, he gave no indication of an ability to relate them to a wider population of forms. The following activities will help Mateo as he progresses from the Visualization stage to the Analysis level.


Activity One
Place a selection of three dimensional figures on a table. Call on a volunteer to group the objects according to shape. If a miscue develops, the classmates can help with hints. Rearrange the shapes and let another student attempt to sort them.


Activity Two
Another productive activity to help students become familiar with the properties of shapes is for them to discover real life representatives of those shapes. Their next assignment is to explore the classroom for these figures. File cabinets and the sides of the teacher’s desk can represent rectangles. So too can the flag. Squares can be found in the shape of the floor tiles. Circles can be found around the rim of the wastepaper basket and the clock, etc.  Afterwards they can share their discoveries with the rest of the class.


Activity Three
Children arbitrarily choose two shapes. The objective is to find something that is alike and something that is different about them. The group selects one shape and places it in view of the others. Their task is to find all the other shapes that are like this model. They share their sorting rules and show examples. Afterwards, they can write in their journals about their shapes and how they conformed to the rule. Later, they will share their findings with the rest of the class (Van de Walle, 2007).


Activity Four
Place models of three dimensional shapes in a large bag. Write the shapes’ names on index cards. Put the bag on a table and place the cards next to the bag, face down. The class forms two lines, A and B. The first student in line A picks up the top card and reads it to the other team members. This participant then reaches into the bag and, without looking, tries to locate the shape. He/she takes the shape out and identifies it by name. A correct response wins a point. The shape is then returned to the bag for the next child’s turn. The team with the most points wins.


Activity Five
Give each child a length of yarn, a piece of cardboard, and several push pins. Have them create shapes on the cardboard and write in their journals about the properties of each shape. As each new shape is created by adding one or two push pins, have the student explain his/her thinking by describing the change in properties (Hatfield,  Edwards and Bitter, 1997).


Activity Six
The teacher holds up a large card with a delineated shape on it. He/she identifies the shape and the properties of the shape.  The children then copy the shape on their geoboard.  The teacher draws the shape on an overhead geoboard and rotates it. The objective is to see if the children still call it by its name. The process is then repeated with a second card, and so on. As the resulting relationships are explored, the educator places the discovered properties on a chart. He/ she writes the names of the shape, the number of sides, the number of angles, etc. (Hatfield,  Edwards and Bitter, 1997).


In conclusion, in order to help students move from Level 0 to Level 1, students should be challenged to test ideas about shapes from a variety of examples.  In general, students should be challenged to see if observations made about a particular shape apply to other shapes of a similar kind. Once the students become aware of these relationships, they should then be prepared to explain their reasoning. Following informal evaluations, such as observing the results of their efforts and listening attentively as they summarize and reflect upon what they have learned, the teacher may decide that it is time to challenge them with more complex assignments in order to advance them to the next level.


References


Laureate Education, Inc, (Producer). (2007). [Motion picture].Program 5 “Development of 
Geometric Thinking” Baltimore: Van de Walle , J.
Hatfield, M., Edwards, N., & Bitter, G. (1997). Mathematics methods for elementary and middle
school teachers. Boston, MA: Allyn and Bacon.       
Malloy, C. E. (1999, October). Perimeter and area through the Van Hiele model. Mathematics  
Teaching in the Middle School, 5(2), 87–90.
National Council of Teachers of Mathematics.(1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author.
Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.

 

Measurement has been defined as the comparison between something being measured and a unit (measuring device) which has the same attribute or quality. (Van DeWalle, 2007). Recent studies suggest that measurement is one of the most difficult concepts for students to master (Van De Walle, 2007). The National Council of Teachers of Mathematics (NCTM) has found that at least some of this obtuseness is due to the teachers’ overemphasizing worksheets and pictures. These materials have a tendency to isolate formulas and procedures, not only from conceptual understanding, but from other related topics across the mathematical spectrum (Laureate Education, 2007).  The NCTM has strongly indicates that the solution to this lies in providing the students with multiple opportunities to engage in hands-on, concept developing activities. In this manner they will see the reasoning behind the formulas, and be able to connect them to other related concepts (Van De Walle, 2007). 


In the course of developing a conceptual knowledge base of what it means to measure, the children can incorporate a three step sequential process for measurement (Laureate Education, 2007). The first step is to identify the attribute to be measured. The second step is to select a unit with the same attribute. The third step is to compare the measuring tape to various dimensions (Laureate Education, 2007).


For this week’s artifact I will utilize this process as I measure two attributes of a soda can. One attribute is the volume and the other is the surface area. My only materials will be a measuring tape (in inches) and a calculator. By comparing the unit (measuring tape) with the object (the soda can), I will determine the numerical value of the attributes (Laureate Education, 2007).


A soda can is a right circular cylinder (Burns, 2007). As such its sides are perpendicular to the base and it has three dimensions. The volume is the space inside the can. My first concern is finding the area of the base, which is a circle. After determining the size of this area, my next step is to imagine myself covering this base with a layer of substance. The height will then tell me how much of the layering to emplace on the first one to cover the entire interior of the can. Therefore, when I find the area of the base (circle) and multiply this by the height, I will be able to determine the volume. As a result of this visualization, I can derive a formula for volume: B x h where B equals the area of the base and h is the height of the can. Since we are dealing with a three dimensional cylinder, the answer will be in cubic units. Using a measuring tape for the measuring unit, I discover that the can is approximately 4.75 inches high and the radius of the base is about 1.25 inches.


I will begin my calculations by finding the area of the base. This is a circle so it will involve pi and radius or diameter. The formula for area of the base is: pi times the radius squared. This would be 3.14 x 1.25 x 1.25, which equals 4.90. To obtain the volume of the soda can, I now have to multiply the area of the base (4.90) by the height (4.75). This equaled 23.28 cubic inches. This is the volume of the soda can.


For my second attribute to be measured I have chosen the surface area of the can. This answer will be expressed in square units. The surface area is the total of all the areas on the can’s surface. In other words, the area that would be covered if the surface areas were peeled off the can and laid flat.  This area will therefore be the sum of area of the top, the area of the bottom and the area of the side. The area of the side can best be imagined by emplacing an imaginary label on it. Consider that the label is completely covering the entire side of the can with no overlap. Therefore, when it is removed, it becomes a rectangle. The width of this rectangle is the circumference of the can and its length the height of the can. So if I measure the circumference (approximately 7.85 inches) and consider this the width of the rectangle and multiply this width by the length or height (4.75) I will find the area of the side (37.28 square inches). I know from my volume problem that the area of the bottom is 4.90 inches. Doubling this to account for the top and adding it to the area of the side, I can find the total surface area of the soda can (47.08 square inches). From my calculations, I can fashion a formula for the surface area of a circular cylinder: 2 pi times the radius squared (area of the top and bottom) plus 2 pi times the radius (circumference of the can or the width of the rectangular label) times the height.


One of the obstacles I first encountered in this exercise was visualizing the can with all of its component parts. I eventually found that by breaking the can into its various constituents and drawing each one, I was able to detect similarities to other geometric shapes, concepts, and formulas to help solve the problem at hand (Van De Walle, 2007). 


I initially contemplated choosing the attribute of how many fluid ounces the can contained. However, I decided that this basic concept would not relate to many other math topics and unwrapping the understanding behind it would provide less discovery then the attributes I subsequently selected.


In conclusion, this week’s artifact involved a concrete experience. Activities such as this one will support me in understanding the reasoning behind the formulas. Accordingly, it will not only help me to remember the formulas, but concomitantly deepen my comprehension of the measurement concept (Kribs-Zaleta and Bradshaw, 2003). By utilizing this conceptual knowledge, connecting it to other mathematical topics, and using it to develop various discovery based activities, I will be sure that the students are exposed to conceptually-based problem solving strategies.  After this foundation has been established in a meaningful way, the students are then no longer required to memorize formulas as isolated pieces of mathematic facts, but can derive them from what they know about the subject (Van De Walle, 2007). Such is one of the myriad advantages of teaching for conceptual understanding. 

 

This week’s application has posited, for our 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 (Van de Walle, 2007).


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 (Laureate Education, 2007). 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 (Laureate Education, 2007). 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 (Van de Walle, 2007).


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. (Van de Walle, 2007). 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 (Laureate Education, 2007).
I considered the first rectangular prism. A rectangular prism is a three dimensional figure with two ends and four sides (Kajander, 2007). The opposite sides have the same area.  I began by drawing a three dimensional rectangular figure on graph paper (Laureate Education, 2007).  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 (Kajander, 2007).  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 (Tent, 2001). 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 (Van de Walle, 2007).

References


Burns, M. (2007). About teaching mathematics. (3rd ed.). Sausalito CA: Math Solutions
Publications.
Kribs-Zaleta, C. M., & Bradshaw, D. (2003, March). A case of units. Teaching Children
Mathematics, 9(7), 397–399.
Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program One, “Introduction to
Measurement” Baltimore: DeWalle, J.
Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching

 

                     This week’s application involves a hands-on activity, i.e., utilizing pattern blocks to create tessellations. In describing the tessellation process, we are to note patterns, as well as describe common characteristics among the shapes employed in the tiling.


A tessellation is a special type of design that consists of a repeated pattern of at least three interlocking shapes. Most importantly, these formations should have no gaps or overlying dimensions. In addition, every vertex, or point where the shapes meet, must have the same configuration (Hatfield, Edwards, & Bitter, 1997). This dynamic is mandatory since the polygons of the tessellation must cover the plane at each vertex. Therefore, lest there be a gap due to a shortfall of angle degrees, or an overlap due to a surfeit, the sum of the interior angles must equal 360 degrees.
The tessellating, or tiling pattern is created by replicating the original shape and moving it into its interlocking position by means of transformations (flips, slides and turns) (Van de Walle, 2007).
There are different types of tessellations. Regular tessellations are made up of one type of regular polygons (which have equal sides and equal angles).  Our particular regular designs will therefore consist of all squares, all equilateral triangles or all hexagons. A semi regular tessellation is made up of two different regular polygons, again identically arranged at every vertex point (Laureate Education, 2007).
Pattern blocks are an excellent medium to explore tessellations. In addition to the regular polygons enumerated above, my set also contained the trapezoid and rhombus. Since all quadrilaterals can tessellate, I felt confident that these two figurations will fit in well with “the group”. 


My initial tessellating attempt consisted of connecting ten congruent, yellow hexagons. My choice of movement in forming straight horizontal rows was primarily slides. It was only when I created the alternating row that I used a rotation movement to affect the interlocking aspect. The end product resembled a honeycomb and a perusal of the tiling indicated that there were no gaps or overlays. I also noticed that three hexagons met at each vertex. Since a hexagon, with six congruent sides and angles, has a total angle value of 720 degrees, the sum of the three interior angles (120 degrees for each) was 360 degrees (Austin, 2009). So the other prerequisite for tessellations was met. 


My next regular tessellation was created with twenty green triangles (four rows of five) Again there were no gaps or overlays. These designs were formed with a slide, rotation, slide, flip sequence. Upon observing the finished product I noted the trapezoidal and hexagonal shapes that had been created out of combinations of three and six triangles respectively. Again, focusing on the vertex and the surrounding six interior angles, I affirmed that they totaled 360 degrees.
The orange square pattern blocks were used to cover the plane with neither gaps nor overlaps. Thus, the square can be said to tessellate in yet a regular design. I also discovered that to tessellate a series of trapezoids, I needed to slide, flip and rotate the figures.


In order to introduce a non example, I smuggled in a few five sided pentagons.
I attempted to tessellate with these figures but gaps appeared in the configuration. I checked the sum of the interior angles around the vertex, and voila, the reason for the flawed tiling was revealed. The interior angle of each pentagon is 108 degrees and by adding the three interior angles, I calculated the total to be 324 degrees (Austin, 2009). The gap (36 degrees worth), was measured by comparing the difference between that total (324) and the magic number of 360.


I was now prepared to try my skill at semi regular tessellations. Using a combination of the five geometric shapes contained in the pattern blocks (trapezoids, rhombi, squares, hexagons, and triangles, I was able to replicate these congruent figures by flipping, sliding, and rotating. Tessellations were the end result. Some of the inherent patterns contained symmetry. For instance, a combination of four hexagons and four trapezoids resulted in 180 degree rotational symmetry, as well as horizontal and vertical line symmetry. (In line symmetry, each point on one side of the line has a corresponding point on the opposite side of the line).


For my next undertaking I combined a hexagon with four triangles and attempted to name it. To name an arrangement of regular polygons around a vertex, one first must locate the polygon with the least number of sides (the triangle). Next, find the longest consecutive string of these polygons around the vertex (in this case, four triangles in succession). Lastly, continue in either direction, (here, it mattered not), and indicate the number of sides of that each polygon contains (Totally tessellated, 1998). My formation which was comprised of four triangles and a hexagon was consequently named three, three, three, three, six.


Reflecting on my tessellation efforts, I find that I have acquired a deeper understanding of spatial development, shapes, angles, and even colors. Transformations (flips, slides and rotations) provided me with opportunities to increase my spatial senses by enhancing my orientation with different perspectives. I have also discovered that the equilateral triangle, square, and regular hexagon each have the common ability to fill the space around a vertex perfectly. Accordingly, one can deduce that three, four and six sided polygons, with their interior angles being exact divisors of 360, will always tessellate. Parenthetically, by calculating the difference between the sum of interior angles of the polygons in place around a vertex, and 360, I can mentally designate the type of shape that will fit into the configuration. Furthermore, I  found that I can tessellate with all the pattern blocks, including the rhombus and trapezoid. The reason for this is that each of the interior angles of the included shapes is an exact divisor of 360 degrees and would therefore fit into a compatible sequence around the vertex. Consequently, since the sum of the interior angles of all quadrilaterals equal 360 degrees, it is logical to assume that all quadrilaterals will tessellate. Finally, through my endeavors, I was able to discern appropriate color combinations with the proper balancing of warm (red, orange, yellow) and cool (blue and green) hues.


Not the least of my outcomes is the ability to connect my new knowledge to the real world. Examples of tessellations in everyday life abound. Some of the more common examples are: tile floors, walls, brick houses, wallpaper, flags, honeycombs, quilts, and clothing patterns (Zaslavsky,1996). Tessellations have cross curricular applications as well.  These delineations play a pivotal role in many facets of science, engineering, architecture, geology, metallurgy, biology, cryptology, and the visual arts.


The National Council of Mathematics Teachers (NCTM), was quick to spot tessellations as a player and has been visionary in integrating these designs into the educational curricula. The NCTM’s geometry standards include four content areas: Shapes and Properties, Transformations (the shapes’ slides, flips and turns), Location (the shapes’ placement on a plane or space), and Visualization (seeing shapes from different perspectives) (Van de Walle, 2007). Tessellations seem entwined throughout all four strands of the standards. Knowledge of shapes, their properties, their movement, their location, and viewing them from different perspectives, all fall under the purview of the student as he/she identifies and manipulates shapes to create tessellations. Consequently, I am quick to concur that tessellations are indeed major players in the realm of geometry.


References


Austin, C. (2009). Geometry companion. Victoria, British Columbia: Trafford Publishing.
Hatfield, M., Edwards, N., & Bitter, G. (1997). Mathematics methods for elementary and middle
school teachers. Boston, MA: Allyn and Bacon.       
Laureate Education, Inc, (Producer). (2007). [Motion picture].Program Eight, “Symmetry and
Tessellation.” Baltimore: Warrick, P.
Totally tessellated. (1998). Retrieved June 12, 2009 from http://library.thinkquest.org/16661/
Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.
Zaslavsky, C. (1996). The multicultural math classroom. Portsmouth NH: Heinemann.

                         

Throughout this course I have been presented with many valuable concepts and strategies which have augmented my understanding of both measurement and geometrical concepts. First and foremost, I have acquired the insight that these two concepts are irretrievably interfaced. Empirical evidence indicates that measurement is one of the central tenets of geometry. Axioms, postulates, formulas, and procedures are all bonded to standardized units and the accurate use of measuring devices. Therefore, student familiarity with measuring units and devices are sine qua non for mastery of geometrically related material.  Both the entities also correlate with the transcendental teaching concept of learning by doing, rather than memorization. As such, students involved in these endeavors should be active rather than passive learners. They should initially be given the opportunity to explore and make discoveries about measurement and geometry using concrete manipulatives (Van De Walle, 2007). In this manner they learn the why as well as the what. Conversations centered on discoveries, involving students and teachers, are indispensible corollaries of this process. During these exchanges, students should be encouraged to predict, negotiate the meaning of definitions, clarify their thinking, and justify conjectures (Moyer, 2001). As a result of this process, they will develop a deep understanding of the concept. Further, they will be able to generalize and extend their knowledge to other areas of measurement and geometry. Importantly, conversations such as these also provide opportunities for misconceptions to surface which consequently can be addressed by the educator (Renne, 2004). Finally, this overall scheme will facilitate the students’ progress through the van Hiele levels of thinking - more about this later (Van De Walle, 2007).


The National Council of Teachers of Mathematics (NCTM) has found evidence that students today are encountering substantial difficulty understanding the concept of measurement. Data from international studies has indicated that students are weaker in the area of measurement than in any other mathematical topic (Van De Walle, 2007).  The NCTM has found that at least some of this failing is due to teachers placing too much reliance on worksheets and pictures, while not paying enough attention to hands-on experiences. It is in performing these concrete operations that the students develop an understanding of the measurement  concept (Van De Walle, 2007)


Measurement is, in essence, a comparison (Laureate Education, 2007a). Specifically, it involves the comparison of an attribute of an item (length, volume, weight, etc.) with a unit that has the same attribute. Length is the first attribute children learn to measure. Before any measuring activity can begin, however, the children must have a clear understanding of length as an attribute of the object being measured (Van De Walle, 2007) This ability can best be evaluated by observing the students doing math in informal, everyday play activities, with minimal teacher guidance (Kribs-Zaleta and Bradshaw, 2003). After understanding the concept of attribute, the students can utilize Dr. de Walle’s three step measuring procedure (Laureate Education, 2007a). The first step is to identify the attribute to be measured. The second step is to select a unit with the same attribute. The final step is to compare the measuring unit to dimensions of the object being measured (Laureate Education, 2007a).


In the primary grades informal units (footprints, measuring ropes, body parts, etc.) are initially used to measure objects. There are good reasons for this prioritizing. First, informal units makes it easier to focus on the attribute being measured (Van De Walle, 2007). Secondly, they help students focus on the lesson objective of understanding what measurement is, rather than being distracted by the concept of standardized units (Van De Walle, 2007). Finally, informal units inject a degree of fun into the lesson. This can be a highly valued motivational tool (Van De Walle, 2007). Once these initial measuring concepts are understood, the students can transition into using standard units, with their precise numerical values, with a high degree of confidence (Van De Walle, 2007).


Estimation plays a significant role in measurement, as well. Highly utilitarian in everyday life, estimations are educated guesses that conjure up relationships among units, encourage predictions and activate prior knowledge and familiarity in an almost a priori fashion. Accordingly, activities which enhance estimating skills are strongly encouraged in the classroom.


Discovery, rather than memorization, is one of the maxims of this course. In order to adapt this inquiry-based method to measurement, I choose to cite an exercise that highlights its constructionist approach. The activity is called Changing Units (Van De Walle, 2007). In it, the children first measure the length of an object with a specific unit (a Cuisenaire rod). They would then be presented with another rod that is either much longer, or shorter, than the original unit. Their task is to predict the measured length of the same object using the new unit. Would it be a larger or smaller number? Students would then write down their predictions and include their rationale. After discussing the predictions and explanations, the children would then be presented with the task of actually measuring the same object with different unit lengths. Afterwards, they would once again discuss the results. The somewhat different totals could then be used as a conduit to stress the fact that non standard units give us an “about” or approximate measure. Through their own direct experience, as well as communicating with and learning from their peers, the children will therefore discover the importance of using similar sized units to measure length.


I will now reflect on the subject of geometry. This reflection will accommodate not only content but the development of geometric thinking, as well (Laureate Education, 2007b). The NCTM’s geometry standards include four content areas: Shapes and Properties (geometric figures and their attributes), Transformations (the shapes’ slides {translations}, flips {reflections} and turns {rotations}), Location (the shapes’ placement on a plane or space), and Visualization (seeing shapes from different perspectives) (Van de Walle, 2007). The Van Hieles’ five-level hierarchy of geometric thought informs, as it renders eminently teachable, each facet of these content sub-strands (Laureate Education, 2007). In order to fully exploit the potential of their individual geometric learning curve, students and teachers need to combine the characterized content areas in the standards with the procedural strictures of the van Hiele levels of thinking (Van de Walle, 2007). In this tenor they can master the numerous geometric skills, and their attending activities, in a logical, sequential manner. Ancillary to this, multiple experiences with an imposing variety of activities are necessary to bring this “learning through understanding” to full fruition (Van de Walle, 2007).


The methodology espoused by the van Hieles promotes five sequential levels of geometrical reasoning (Van de Walle, 2007).  Progress from one level to the next is more dependent on the richness and quantity of mathematical experiences than on chronological age (Laureate Education, 2007b). Additionally, the use of physical materials and drawings are de rigueur at the first few levels.


Central to their theory is the observation and discernment of the sophistication of students’ thinking, as they engage in geometric tasks (Van de Walle, 2007). As a result of these assessments, the instructor can implement teacher guided, inquiry-based activities, in which the pedagogy matches the child’s level of thought. Accordingly, during these activities, it is imperative that teachers use geometric terminology that is in synchrony with the students’ levels of thought. Without this proper correlation, the students and teacher will be in two different geometric places, figuratively speaking, and consequently, the children’s learning curve will be seriously compromised (Malloy, 1999). The overarching, dual objectives of this five stage approach are as follows: mastery over the present level’s concept, as well as preparing the child to advance to the next hierarchical stage.


At the Visualization level, students generally recognize figures by appearance alone, oftentimes comparing them to an example. At this juncture, the manipulative in front of them represents the concept. As such, they cannot relate it to the universal idea of the shape (Laureate Education, 2007b). In addition, transformations such as rotating and flipping will cause the shape to lose its identity (Malloy, 1999). Consequently, students may think that a rotated square is a diamond and not a square because it looks different from their visual image of square. The neophytes’ vocabulary is also at the minimalist stage. Standard nomenclature is rare. Hence, a parallelogram may be a “slanty rectangle” or an angle, “the arms of a clock.” (Van de Walle, 2007).


As one can readily perceive, the objects of thought at this level are individual, concrete shapes. In order to reach the products of thought, or goal, for Level 0, activities at this level should involve lots of sorting and classifying (Van de Walle, 2007). Seeing how shapes are alike and different, and placing those in the appropriate groups should be a primary focus. Moreover, the students should decide on how to sort rather than the teacher (Van de Walle, 2007). This allows the child to perform the exercise with ideas they understand and lends to them a certain amount of ownership. During the course of level 0 activities, some standard terminology will become known, such as rectangles, squares, triangles, cylinders. Properties of these shapes, such as the number of sides, parallel sides, symmetry, right angles, and congruence, are included in the exercise, but only in an informal, observational manner (Van de Walle, 2007)
Students working at the Analysis stage, or Level 1, are capable of considering all shapes within a class, rather than an individual shape (Van de Walle, 2007). In order for a student to reach this level, he/she needs to classify shapes according to properties, such as the number of sides, congruence, perpendicular lines, and symmetry. The learner should also be able to classify many shapes with these properties, not just the ones he is manipulating (Laureate Education, 2007b).


The next level, or Informal Deduction, injects a greater degree of logical reasoning into the learner’s problem-solving schema. At this point, students are capable of recognizing relationships among properties of shapes, as well as classes of shapes, and are able to follow logical arguments using such properties. They use informal, deductive reasoning and arguments and engage in “if…then” and “what if” investigations (Van de Walle, 2007). 
The last two levels, Formal Deduction and Rigor, generally exceed the parameters established for this course. In Formal Deduction, students go beyond just identifying characteristics of shapes and are able to construct proofs, using axioms and definitions.  The highest level, or Rigor, finds the learner comparing different axiomatic or geometric systems (Van de Walle, 2007). 


My own facility with solving both measurement and geometrical problems has been sharpened as a result of this course. Much of my progress is a result of being exposed to Dr. de Walle’s concrete, discovery approaches. 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 (Laureate Education, 2007a). 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 (Laureate Education, 2007a). We can more easily adapt them to the unique proportions of these polygons. As we explore the threads 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 (Van de Walle, 2007).


Once we have tweaked this nomenclature of the dimensions of the rectangle, we can attain a deeper understanding of the concept of area. No formula is necessary at this juncture. (Van de Walle, 2007). 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 to obtain the squared area. By counting the square units contained within, we can thereby affirm the formula for the area of a rectangle to be base times height. We can then extrapolate this concept to embrace three dimensional objects as well, by substituting “area of the base” times the height.” (Van de Walle, 2007).
The non-constant relationship between area and perimeter has always been a source of consternation for students (Ferrier, et al, 2001) Part of this problem lies with their lack of conceptual understanding of both perimeter and area. By recalling a first- hand experience with an inquiry based exercise involving perimeter and area, I feel not only more confident in my own conceptual understanding, but also in my ability to teach the subject.


We were given a rectangle with certain length and width dimensions. We were asked to find both the area and perimeter of this shape. We were further asked to double and triple the initial dimensions while noting the concomitant and possibly correlative growths of area and perimeter. Looking back, and reflecting on the numbers, I determined that as I doubled the length and width of a rectangle, the perimeter doubled. As I tripled these dimensions the perimeter tripled. The reason for this is that when I double or triple the width and length, I am doubling or tripling the outside lengths of the units. Since perimeter involves adding up these outside dimensions, the increase will match the additions to the length and width. The area, which involves multiplication and square units, is a different story. When I calculate the area, I have to account for all four sides of each unit. When I add, say, one unit to the width, I must then multiply this increased number by all the numbers and sides in the length. Looking at it another way, if I add one unit to the width and one unit to the length, then though the perimeter increases by a mere four, the area has increased by an entire column of width units squared plus an entire row of length units squared. This helps to explain the larger increases in area when compared to the growth of the perimeter. As a result of this “dissecting the concept experience” I now have a thorough understanding of how area and perimeter relate to each other in rectangles.


My ability to teach concepts relating to measurement and geometry has also improved. I have learned that intermingling small-group work, whole-class discussions, individual writing and hands-on experiences creates multiple opportunities for students to explore, hypothesize, demonstrate, clarify and extend their thinking, as regards measurement and geometry (Renne, 2004). This methodology, in turn, leads to higher order thinking. It internalizes the understanding of concepts, and leads the learner to the “zone of proximal development” which is the area between current understanding and possible future acumen (Laureate Education, 2007b). This is exactly where we wish the student to be and it is our responsibility to see that the child emerges from this zone and proceeds in the appropriate direction.

 

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