1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook
Your guidance will be extremely appreciated!
During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.
The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.
I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.
According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:
| Grade | Explanations (from Curriculum Focal Points by NCTM) |
| Grade 1 Geometry: Composing and decomposing geometric shapes. | Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry. |
| Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes. | Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry. |
| Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes | Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle. |
| Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area | Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems. |
| Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity | Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. |
| Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes | By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders. |
| Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle | Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra. |
Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.
(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)
In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:
3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%
These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.
Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.
So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.
No comments:
Post a Comment