Learning Strategies, Continued
The Learning Progression for Geometrical Measurement
For students with disabilities severe enough to significantly impede learning, concepts and processes of measurement can often be quite vague if they are introduced without concrete reinforcement. Geometrical measurement involves many separate procedures and takes regular practice to master. Without such mastery, there can be little expectation for a student in a self-contained setting to make marked improvement in mathematics instruction or to overcome significant learning challenges in general. Geometrical measurement itself is defined by the National Council of Teachers of Mathematics as “the assignment of numerical value to an attribute of an object, such as the length of the pencil” through unit partition and unit iteration. As stated above, middle-school geometry serves as a bridge between visual concepts taught in elementary school and abstract, high-school concepts. Felix Klein provides a useful and elegant definition of geometry: geometry is the study of the properties of figures that are unchanged by rigid motions.15 If a student has yet to develop counting skills and has also not developed an understanding of the properties of shapes, measurement and manipulation of shapes would be hard to perform with proficiency. Kim et al. offer a framework, titled “Learning Progression for Geometrical Measurement in One, Two, and Three Dimensions,” which serves to address the need for effective geometrical instruction and which can be applied to instruction for students with special needs.16 The framework provides the underlying structure for this curriculum unit and includes the following processes:
- Unit partition and unit iteration
- Spatial ability
- Efficient-sized measurement units and spatial structuring
- Abstraction
In using unit squares and unit cubes, the activities in this curriculum unit help students understand measurement as partition into units. Further, through activities related to polyominoes, rectangles, and rectangular prisms, they will increase their spatial ability. In manipulating physical objects and in transferring them into visual representations, students will begin to understand that measuring is the decomposing of an object into “efficient-sized measurement units” or, as Kim et al. state, the “taking a part of an object as a unit and then placing the unit alongside the object in terms of unit subdivision and unit iteration, respectively.”17 Having students move between practice in two dimensions and practice in three dimensions will also be a strategy used in this curriculum unit. As Kim et al. show, this movement, from dimension two to dimension three, is valuable in forming spatial reasoning skills and forming what Kim et al. call “spatial ability,” or the ability to understand the relationships extending from a two-dimensional representation of an object to the object itself, which is understood according to “terms of their three-dimensional properties.”18 In other words, students with spatial ability know that parts of shapes not represented in a depiction are in reality still present. Open-ended questions that can increase spatial ability and that will be used in this curriculum unit include, “What do you notice about the face of your planter if I turn it around?” and “How would you draw this shape now on your paper?” These questions should make explicit the fact that there are some faces that, while not being visible, are still present, so that students begin to develop an ability to visualize spatial relationships while reading two-dimensional representations of three-dimensional objects.19 Finally, abstraction will occur when students transition to using the formulas for area and volume after practicing measurement with manipulative units.
The Use of Polyominoes as an Effective Manipulative in the Special-Education Classroom
My research indicates that, when coupled with direct-instruction and open-ended tasks, the proper use of manipulatives can serve as a powerful tool in special-education settings. Carbonneau and colleagues’ 2013 meta-analysis of studies on the efficacy of manipulatives explains that when manipulative use is accompanied by high levels of instructional guidance there is an increase in academic performance and student understanding.20 Additionally, Boaler and colleagues support a pedagogical move away from teaching math as mostly an algorithmic practice where students simply “memorize and calculate.” 21They argue for a movement toward mathematics pedagogy where visual processes are encouraged and where approaches such as the use of manipulatives are coupled to a deep and deliberate thinking process, initiated but not ended by instructors.22 Yet, while many educators in mathematics rely extensively on their use, data show that the use of manipulatives is not always productive. Again, in their review of research, Laski et al. found that many mathematics strategies involving hands-on activities impede learning.23 To address this issue, Laski et al. put forth the “Four Principles for Maximizing the Effectiveness of Manipulatives,” a list in which they enumerate the requirements for effective use of manipulatives in mathematics. The four principles follow:
- The consistent and coherent use of objects longitudinally where there are successive observations of a phenomenon and its relations to other phenomena.
- The use of “highly transparent concrete representations” followed then by the use of “more abstract representations.”
- The avoidance of objects that have what I would call connotative dissonance or, in other words, alternative meanings which may interfere with both the significance of the concept on which the lesson circles.
- The direct and explicit presentation by the instructor of the connection between the manipulative and the specific mathematical concept it embodies.
The phrase “highly transparent concrete representations” refers to an object or image in mathematical instruction that has significant verisimilitude and which details in an exact manner the concept being taught, so that both a visual and conceptual connection can form between the two. Concrete representations that are fully transparent present neither conceptual static nor distraction for students because their design is devoid of redundant or irrelevant characteristics. In the case of this curriculum unit, the rectangle and the rectangular prism are both objects that must be encountered in everyday life, and that without much effort can display transparent representational coherence or, more succinctly, a representational naturalism. Additionally, simple unit squares and unit cubes will be highly transparent manipulatives. Laski et al. define the common term “manipulative” as “concrete materials used to demonstrate a mathematics concept or to support the execution of a mathematical procedure.”24 As Laski et al. write, the manipulative as concrete object represents the concept as abstract knowledge.25 Thus, hands-on activities can be the use of what is often called a “foldable” worksheet, where the foldable has no representational value within the framework of the mathematical concept of instruction. On the other hand, manipulatives are objects such as one-hundred squares which represent numerical concepts such as additive properties and decimation and that, therefore, facilitate numeracy. In effect, manipulatives make explicit what in a concept may be implicit or abstract, while hands-on activities often simply impede learning.
Considering the above scholarship, I have distinguished between “hands-on” activities and “manipulative” activities, avoiding the former in designing this curriculum unit. Again, there is a substantial difference between the term “hands-on” and the term “manipulative,” which often goes ignored in special educational settings. To be clear, the term “hands-on” designates an activity involving materials that can be obliquely related to or entirely detached from the concept of the lesson. The term “manipulative”, in contrast, indicates the use of concrete materials within a lesson where those materials are deeply and inextricably connected to the content and concept of the lesson. In the case of this unit, manipulatives will be one-inch unit squares and one-inch unit cubes which have no other symbolic value for students. With the use of unit squares, which I perceive as highly transparent objects, students will configure polyominoes and decompose planar shapes to explore concepts of symmetry, rotation, and additive properties of area.27 In order to create a polyomino students will take the middle of a square’s side and join it to the middle of at least one other square’s side. Citing Solomon W. Golomb, Roger Howe presents the following definition for polyominoes:
“a figure made by joining squares, with: i) two squares touching either only at a corner, or along a whole edge of each; and ii) connected, in the sense that you can get from any square to any other square by passing through the middle of edges (and avoiding corners).”27
The use of polyominoes provides ample opportunity for students to visualize additive properties and to practice measurement using units. Yuan, Lee, and Wang show as much in their 2010 study revealing marked improvement in learning when middle-school children used polyominoes, whether these were digital or physical.28 And because there is such a large number of activities that can be performed by students using these manipulatives, there is ample opportunity to affect an open-ended instructional style.
Figure 2. The above illustration presents two examples of polyominoes, two non-examples of polyominoes, and all five versions tetrominoes (i.e. polyominoes made of four unit squares). It should be noted here that there are twelve possible versions of pentominoes and 35 different versions of hexominoes where rotations and reflections of these shapes are not considered distinct.
Project-Based Learning and Open-Ended Tasks in Teaching Area, Perimeter, and Volume
As delineated by Stephanie Bell, the definition of Project-Based Learning (PBL) is “a student-driven, teacher-facilitated approach to learning” where students “pursue knowledge by asking questions that have piqued their natural curiosity.”29 Bell explains further that PBL raises the academic performance of students and increases their self-efficacy. Both increases can be understood as resulting from learning centered on individual responsibility, group accountability, independence, self-monitoring and sharing out student work with significant community members. While peer-to-peer collaboration can be an important component of PBL, it is not an essential one. That is, through the actual construction of the planters the individual student can learn and engage this curriculum in meaningful ways by working independent of others. In the case of this curriculum unit, the process of guiding student’s planter-building process and the process of directing students to calculate the volume, area, and perimeter of the objects individually can be understood as both a collaborative and project-based process.
Along with developing an understanding of the relationships of shapes and their components through both student-generated representations and ordering tasks, this curriculum unit will rely heavily on open-ended tasks similar to those espoused by Dorothy Varygiannes in her recent article “The Impact of Open-Ended Tasks.”30 In this article, Varygiannes shows that there was an explicit push to include open-ended tasks in classroom instruction upon the adoption of Common Core standards. She writes that providing opportunities for “outside-the-box” thinking was a way of “inviting our students into the mathematics conversation…and providing a ‘safe haven’ for them to contribute and to increase confidence.”31 Student application of conceptual understandings spurred by higher level thinking within an open-ended framework was considered a prime indicator of student success in Common Core. Thus, open-ended tasks should not be undervalued in planning. In this unit, open-ended tasks will include asking students, “How many rectangles can you make with an area of 16 (and whole number side lengths)?” This open-ended activity will invoke prior knowledge of the properties and definitions of rectangles and expand student understandings of these properties before beginning processes of measurement in later activities. Furthermore, questions such as “Why did the perimeter of your shape change?” and “Why did the area of your shape stay the same?” will be posed to students during their activities. A noteworthy open-ended task from Siegel and Ortiz’s lesson mentioned above was having students draw a series of rectangles with different dimensions on graphing paper in order to assist their calculations and measurement. Students can decide for themselves the shape and size of the rectangles, and once done, find different ways to measure area and perimeter. This open-ended process of creating visual representations is so valuable that the authors express the following: “Pictorial representation is an incredibly important tool…that my students are encouraged to implement on a near-daily basis.”32
In conclusion, the review of taxonomy, student’s daily production of images, PBL, evidence-based manipulative use are all significant features in the “From Polyominoes to Planters” curriculum unit. Combining these strategies will demand of students a high level of cognitive engagement and will allow them to understand shapes, their interwoven relationships, their dimensions, and how to measure them. Further, the categorization and reproduction of shapes has importance for students with disabilities, because without this, the process of measurement and identification can be a nebulous set of procedures to them. Only after students master the representation and classification (e.g. students’ creating property lists of rectangles) will they learn the formulas needed to calculate values of perimeter, area, and volume. In using these strategies, the curriculum unit will move from concrete to abstract instruction in accordance with “The Learning Progression for Geometrical Measurement” discussed above.33
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