How to help students understand high school geometry?
If you read the first part of this article, you can already see that the measures to take should happen before high school. The best approach involves changing how math and especially geometry is taught BEFORE high school. Some points to consider are:
- Improve geometry teaching in the elementary and middle school so that students' van Hiele levels are brought up to at least to the level of abstract/relational.
- Include more justifications, informal proofs, and "why" questions in geometry teaching during elementary and middle school.
- In general, make students think, reason, and use their brain in different educational tasks (not just math).
This article will now concentrate only on the first point.
Understanding geometry concepts/Van Hiele levels
You can expect kids up through first grade to be in the first van Hiele level - visual. This means children recognize geometric figures based on their appearance, and not based on their properties. On this level, children are mainly learning the names of some shapes, such as square, triangle, rectangle, and circle.
During the elementary school (grade 2, 3, 4, and on) children should investigate geometric shapes so that they will reach the second van Hiele level (descriptive/analytic). That is when they can identify properties of figures and recognize them by their properties, instead of relying on appearance.
For example, students should come to understand that a rectangle has four right angles, and even if it is rotated on its "corner", it is a rectangle. Children should learn about parallel lines and understand what is a parallelogram. Students should divide shapes into different shapes (such as dividing a square into two rectangles), and combine shapes to form new ones, and of course name the new shapes. The shapes to be recognized should be rotated so they appear in different positions.
Drawing also helps. Students can learn to use a ruler, compass, and a protactor, so they can then draw squares, rectangles, parallelograms, and circles.
If all goes well, in middle school (grades 6th-8th) the student would proceed to the third Van Hiele level (abstract/relational), where he/she can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. And thus, the student would be prepared for the formal proofs and deductive reasoning in high school geometry.
Experiments have shown that this is indeed possible with the right kind of geometry teaching. The key is to emphasize the geometrical concepts and providing the kids lots of hands-on activities like drawing the figures and working with manipulatives, instead of only memorizing formulas and definitions and calculating areas, perimeters, etc. See below some examples of activities that will help children and young people to develop their geometric thinking.
How to help students to develop understanding of a single geometry concept
- When studying a concept, show correct AND incorrect examples, and in different ways or representations (rotate the pictures upside down etc!). Students are asked to distinguish between correct and incorrect examples. This will help prevent misconceptions.
- Aks students to draw correct and incorrect examples of a geometry concept. For example, ask them to draw parallel lines and lines that are not parallel. Tying in with this, ask them to draw a parallelogram and a quadrilateral that is not a parallelogram.
- Tying in with the previous point, you can ask the students to provide a definition for a concept. This can get them to thinking about which properties in the definition are really necessary and which are not. For example, ask them to define an "equilateral triangle".
- Allow the students to experiment, investigate, and play with geometrical ideas and figures. For this you could use manipulatives, lots of drawing, and computer programs (more on them below).
- Have each student make his/her own geometry concepts notebook, with examples, nonexamples, definitions and other notes or pictures.
Computers and interactive geometry
A computer can really help in geometry teaching, since it allows a dynamic, interactive manipulation of a figure. A child can move, rotate, or stretch the figure, and observe what properties stay the same.
For example, let's say you are teaching the concept of an isosceles triangle in 4th grade. You could simply use the Drawing Toolbar in Microsof Word, which has the AutoShape for isosceles triangle (as well as for a right triangle and parallelogram). Let children draw one or two and then tell them to drag it from the white handles to make it bigger/smaller, and also to rotate the figure. Ask, "What changes? What does not change? What stays the same? Can you draw this kind of thing on paper?"
There also exist dynamic geometry software that is specifically designed to teach geometry in an interactive investigational way. Such programs have been used in research experiments and in schools with good results. After you see what can be done, it is very easy to fall in love with such a program - the idea is just great!
How can I help the student already studying high school geometry?
Perhaps your student is already studying geometry in high school and is having problems. Of course you cannot change how he/she was taught in the past. Since this is such a common problem, many publishers have come out with textbooks that emphasize "informal" geometry and geometry concepts, instead of proofs. You could use one of those books, and simply forget about the proving.
Yet other books include proofs, but not in the same quantity or same emphasis as in previous years. These include for example Harold Jacobs Geometry: Seeing, Doing, Understanding. The link goes to my review of this book.
And even with good preparation, high school geometry and the proofs can still be difficult. All in all, there is no quick and easy answer to the difficulties in this course. Remember that even math teachers in schools struggle with this problem of getting students to understand and construct proofs. Maybe the explanations on Ask Dr. Math: FAQ About Proofs can be of some help.
I have reviewed several geometry books:
Geometry: Seeing, Doing, Understanding by Harold Jacobs (high school)Geometry: A Guided Inquiry by Chakerian, Crabill, and Stein, and its supplement "Home Study Companion - Geometry" by David Chandler (high school).
Dr. Math geometry books - these are inexpensive companions to middle and high school geometry courses.
RightStart Geometry is a hands-on geometry course for middle school where much of the work is done with a drawing board, T-square, and triangles. It is more pricey, but of good quality.
These two books are my creations:
Math Mammoth Geometry 1 for grades 3-5 emphasizes hands-on drawing exercises and covers basic plane geometry topics for those grades. Price: $7.50 (download), $12.70 (softcover printed book)
Math Mammoth Geometry 2 for grades 6-7 continues the study of geometry after Math Mammoth Geometry 1, continually emphasizing conceptual understanding, besides calculation-type exercises. Price: $5.95 download, $10.40 (softcover printed book).
Here is one high school geometry book that is "traditional" in its emphasis on proofs:
Geometry by Ray C. Jurgensen
Why is high school geometry difficult? - the first part of this article, explaining the Van Hiele levels.
Actually, I loved Geometry, but I was in the Honors course with a brilliant teacher. As a tutor and now teaching homeschooled children, I teach the same way he taught us. Mr. Kasper taught us to flowchart the proof from either end, citing the theorem in initials beneath each step. We could flowchart some of the more difficult proofs in half the time that the two-column proofs take, simply because we had a visual layout that easily led to the next step. We did learn to do the formal two-column proofs, but we always did them from the flowcharts, accomplishing them more easily. I've tutored kids that do not understand the two-column proofs but catch the concept with the flowcharts quickly. On the other hand, I have had two students who needed the formal columns.