Modern Mathematics Curriculum, Reaction Paper Example

This paper explores the geometric habits of mind and how it connects to the CCSS, and to the van Hiele levels.  Geometric habits of the mind will be the foundation upon which the role of using manipulatives in teaching will be built.  The Van Heile levels will also be discussed.

Geometric Habits of the Mind

The geometric habits of the mind allow for deductive reasoning.  It is the ability to develop ways of systematizing and abstraction by infusing courses with modern material with much older mathematical principles (Cuoco, Goldenberg, & Mark).  The habits of mind curriculum tries to “close the gap between what the users and makers of mathematics do and what they say” (Cuoco, Goldenberg, & Mark).

Habits of mind encourages false starts, calculations, and special cases (Cuoco, Goldenberg, & Mark). Students develop the habit of reducing things to dilemmas for which they have no proofs, suspending work on these dilemmas and on other details until they see if assuming the dilemmas will help. Habits of mind help students look for logical and heuristic connection between new ideas and old ones, and gives students a genuine research experience (Cuoco, Goldenberg, & Mark).

Curricula and syllabi should be constructed in a way that values how a particular piece of mathematics typifies an important research technique as much as it values the importance of the result itself.  This may mean studying “difference equations” rather than “differential equations” (Cuoco, Goldenberg, & Mark). It may mean less emphasis on calculus and more on linear algebra (Cuoco, Goldenberg, & Mark).  Habit of mind curriculum most definitely means the inclusion of elemental number theory and combinatorials (Cuoco, Goldenberg, & Mark).

The emphasis of geometric habits of mind curriculum is the thinking that is behind the mathematical equations, not the equations in and of themselves.  The algorithmic thinking behind the difference equations is what is most important (Cuoco, Goldenberg, & Mark).

The language of linear algebra may become obsolete, but the geometric language to describe algebraic phenomena will always be progressive and highly useful.  Habits of mind curriculum and general habits of mind extend across all disciplines, and are not exclusive to mathematics.

CCCS

The CCCS set out to address the problem of a deficient US mathematics curriculum that “is a mile wide and an inch deep” (CCCS).  Research studies over the last decade of higher performing countries in mathematics curriculum show that the United States must reform its curriculum.  The studies conclude that the curriculum must become more focused and coherent in order for the United States to compete with other, more mathematically advanced countries (CCCS).

CCCS seeks to implement a set of standards and curricula that are also coherent. William Schmidt and Richard Houang (2002) have said that content standards and curricula are coherent if they are: articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars. (CCCS)

Van Hiele Model

The Van Hiele model describes how a student learns geometry and is made up into five levels (Crowley).  The levels are: visualization (0), analysis (1), abstraction (2), deduction (3), and rigor (4).  Each level has five properties: fixed sequence, adjacency, distinction, separation, and attainment (Crowley).  Attainment has five different methods for helping a student master one level in order to move on to the next: information, guided instruction, explication, free orientation, and integration (Crowley).

The Van Hiele Model is the result of the work of Dutch educators Pierre Van Hiele, and Dina Van Hiele-Geldof.  They identified the five sequential levels by which children understand special concepts in regards to geometric thinking. The Van Hiele model suggests four levels of thought:  levels of geometric reasoning are sequential; levels are not age dependent, geometric experiences greatly determine advancement through the levels, and instruction at a level higher than the one the student is currently at may inhibit learning (Images).

Manipulatives in Teaching

Manipulative teaching is a technique by which instructors use regular everyday objects to engage in mathematical learning.  The objective is to introduce new mathematical concepts in an user-friendly way.  Examples of manipulatives that could be used in geometry are: place mats, blocks, geoboards, and geometric-solid models (Using Manipulatives).

Conclusion

Curricula and syllabi should be constructed in a way that values how a particular piece of mathematics typifies an important research technique as much as it values the importance of the result itself.  The emphasis of geometric habits of mind curriculum is the thinking that is behind the mathematical equations, not the equations in and of themselves.  Implementation of manipulatives in teaching geometry is highly recommended.

Works Cited

Cuoco, A., Goldenberg, P. E., & Mark, J. Habits of Mind: An Organizing Principle For Mathematics Curriculum.

Crowley, Mary, L “The Van Hiele Model of the Development of Geometric Thought,” in Learning and Teaching Geometry, K-12, ed. Mary M. Lindquist (Reston, Va.: National Council of Teachers of Mathematics, 1987), pp. 1-16.

IMAGES | Van Hiele Levels of Geometric Reasoning. Retrieved June 17, 2011, from Web site: http://images.rbs.org/cognitive/van_hiele.shtml

Mathematics, C. C. Common Core State Standards Initiative: Preparing America’s Students for Career and College. Common Core State Standards for Mathematics.

Using Manipulatives – TeacherVision.com. Retrieved June 17, 2011, from Web site: http://www.teachervision.fen.com/pro-dev/teaching-methods/48934.html