Discusión de resultados

This research uses five kinds of communication types to describe the format of the written statements in the texts: definition, example, formula, investigation, and theorem. Table 5- 3 presents the examination result of the five types of the three textbooks.

Table 5-3: Types across Three Texts

USA Discovering textbook USA Pearson Textbook China People's Education textbook 1: Definition 26 (36.6% of Total) 19 (24.1% of Total) 36 (34.6% of Total) 2: Example 19 (26.8% of Total) 40 (50.6% of Total) 33 (31.7% of Total) 3: Formula 11 (15.5% of Total) 12 (15.2% of Total) 12 (11.5% of Total)

4: Investigation 15 (21.1% of Total) 3 (3.8% of Total) 6 (5.8% of Total) 5: Theorem 0 (0% of Total) 5 (6.3% of Total) 17 (16.3% of

Total)

Total 71 79 104

Units in total 72 89 104

*Each unit can have one count at most of each communication type.

In general, the Discovering Textbook has slightly lower content loads, as the total references are 72 counts. It has a unique character of using investigation (15 counts) as the primary way of conceptualizing geometry, compared with six counts and three counts in the other texts. All the “conjectures” are about symbolic expression of volume or surface-area; therefore, we categorized them as formulae. It does not cover or name any theorem at all. Although Cavalieri's principle appears in the text as an investigation activity, it does not specify it as a conjecture or theorem. In general, it shows a new trend of emphasizing students'

mathematical research ability and reasoning ability and emphasizing the content less. The total counts of communication types are less than the total units in this text because there is one unit which has sizable contextual information that belongs to none of these types. This unit cannot be clustered with the units nearby, as the topics are not related.

The Pearson textbook tends to use example (40 counts, 50.6% of total) as the dominant type of presenting content and half of the units are this kind. This text has few investigations and theorems, only three and five references respectively. It seems that the Pearson textbook

emphasizes drilling practices considering the relative proportion of the example counts to the total units. There are ten count differences between the totals of communication types and of of units. This is because, in the Pearson Textbook every section contains a block of statement called "Check skills you will need" at the very beginning. This is a block of content showing up at the very beginning of the section for students to check their prerequisite skills and most of the time it is not about 3-D geometry. There are ten sections, so there are ten counts of this sort in total. Because these ten blocks are officially stated in the ten sections, the researcher still included them as ten units. However, maybe it is better to exclude these ten units because they do not correlate with any of the communication types identified in this research.

The People’s Education textbook tends to have much more intense content loads compared with the US texts; it has 104 references in total. Definition, examples, and theorems are the main types used in the People’s Education textbook, the counts of which are 36, 33, and 17 respectively. This textbook seems to cover most types evenly, and each unit can be identified as one of the five types. It doesn't have extra unrelated information, and it is easy to identify a content cluster as a unit.

The definition is the prevailing type across the three texts, the percentage of which ranges from 24.1% to 36.6%. However, the definitions of a concept are sometimes different from one textbook to another. For example, the definitions of polyhedron across the three texts are quite similar, and they are shown in Table 3 of chapter three. In contrast, the definitions for cone vary among the three texts. The Discovering Textbook defines the cone using an analogy, which

states “Another type of solid with a curved surface is a Cone. Funnels and ice cream cones are shaped liked cones. Like a pyramid, a cone has a base and a vertex (Discovering Textbook, page 523).” The Pearson Textbook defines the cone quite similarly; it states “A cone is “pointed” like a pyramid, but its base is a circle (Pearson Textbook, page 619).” Howver, the People’s

Education Textbook uses rotation to define cone by imitating the definition of cylinder; it states "A solid generated by rotating a right triangle over one of its legs is called circular cone. The circle formed by the rotation of the other leg is the base of the cone. The curved area formed by the rotation of the hypotenuse is the lateral face of the cone (People’s Education Textbook, page 5).” It seems that the definitions in the People’s Education Textbook are relatively more

mathematically precise and rigorous than the US texts. In general, all texts emphasize definitions in geometry; however, the rigor of the definitions varies.

Example is the second prevailing type across the three texts, ranging from 26.8% to 50.6%. This shows that all texts place emphasis on showing students typical problems and modeling how to solve and write the process. Examples in the Pearson and the Discovering texts are very straightforward and mainly simple applications of the concepts. However, in the

People’s Education text, an example involves several small problems and several layers of thinking, and nothing is straightforward. This variation is noticeable in the examples in Table 4- 3.

The formula type seems almost evenly distributed across the three textbooks, with percentages of 15.5%, 15.2%, and 11.5% respectively. It is a typical and classical

communication format because of its precise and concise quality in capturing geometry ideas. However, the three texts differ in strategies and manners of communicating formulae to readers. The Pearson Textbook tends to give a formula immediately after showing one or two case

examples and presents the formula in a micro way, which means that every specific type of 3-D object has a formula for surface area or volume. The Discovering Textbook tends to not give the formula directly, but rather use investigation activities for students to explore and find the formula by themselves it leaves a blank for students to fill in the discovered formula themselves. In addition to that, in contrast to the Pearson Textbook, the Discovering Textbook tends to classify a formula in a macro way, such as for a generalized group rather than individual 3-D object. For example, prism and cylinder share one common volume formula, V=S*h, in the Discovering textbook, but they are treated separately as two different formulae in the Pearson Textbook. Most formulae in the US texts concern surface area and volume. The People's Education Textbook states formulas in an even more global manner. It uses four formulae to represent surface area and four formulae for volume of all 3-D objects, which are grouped into polyhedrons, cylinders, cones, and circular frustums. The other formulae involve arithmetic in spatial coordinates systems and space vectors.

The investigation type is a new communication and pedagogical approach, and it serves the function of inspiring students’ mathematical thinking and reasoning. It appears quite

differently in the three texts, the percentage of which are 21.1%, 3.8% and 5.8%. The

Discovering Textbook uses this approach the most. Discovering tends to offer a background that students can relate to or activities that students can have some concrete materials to manipulate and operate on as well as a relatively open question that can guide students to explore and think. The Pearson Textbook uses the investigation communication type least, having only three counts. Although Pearson inclines to use familiar or real-life contexts, the questions/activities in investigations are relatively closed and mechanical. Therefore, there is no space for students to explore and investigate by themselves. The Peoples’ Education Textbook tends to use real-world

problems to set a context for students to investigate, and the questions are relatively open-ended. However, the investigations are mainly through thinking and discussion, seldom through

manipulation and operation of concrete materials.

The theorem type is a conventional communication type and is a statement that concisely and abstractly captures a geometrical phenomenon. It is a reference when proving other

statements. However, it seems to almost disappear from the US textbooks. No theorems are mentioned in the Discovering textbook. Although five theorems appear in Pearson, they are mostly postulates, and no rigorous applications or proofs are given. In contrast, in the Chinese textbook, theorems are organized and connected with 17 counts in total. Four postulates are given first, and the rest of the theorems are deduced from them. The majority of theorems are proved by the traditional deduction method. For theorems that are not formally proved, some explanation is given to help make sense of the underlying logic. Theorems are applied to deduce and prove related theorems or examples. All theorems can be found in Appendix C.