How to help students understand high school geometry?
If you read the first part of this article, you can already see that the best measures to help students with high school geometry happen before high school. We need to 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. Some points to consider are:
- We need to include more justifications, informal proofs, and "why" questions in math teaching during elementary and middle school.
- In general, students need to think, reason, analyze, and use their brain in various school subjects (not just math).
This article will now concentrate only on the first point.
Understanding geometry concepts/Van Hiele levels
You can expect that children up through first grade are in the first van Hiele level – visual. This means children recognize geometric figures based on their appearance, not based on their properties. On this level, children are mainly learning the names of shapes, such as a square, triangle, rectangle, and circle.
During the elementary school (grades 2-5) children should investigate and play with 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 still a rectangle. Children should learn about parallel lines and understand that what makes a shape to be a parallelogram. Students should divide shapes into different shapes (such as dividing a square into two rectangles), combine shapes to form new ones, and of course name the new shapes.
Drawing also helps. Teach students to use a ruler, compass, and protactor, and give them lots of practice drawing squares, rectangles, parallelograms, and circles with the proper tools and as accurately as possible. For example, ask students to draw an isosceles triangle with a top angle of 40° or a rhombus with 4-inchs sides and one 66° angle. I use this a lot in my book Math Mammoth Geometry 1.
If all goes well, in middle school (grades 6-8) students will proceed to the third Van Hiele level (abstract/relational), where they can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. Thus, the students 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 teaching. The key is to emphasize the geometrical concepts and providing students lots of hands-on activities, such as drawing figures and working with manipulatives, instead of merely memorizing formulas and definitions and calculating areas, perimeters, etc. See below some example activities that will help children and young people to develop their geometric thinking.
How to help students to learn a single geometry concept
- Show students both correct AND incorrect examples of the geometric concept. Show the concept in different ways or representations (e.g. rotated, reflected, skewed). Ask the students to distinguish between correct and incorrect examples. This will help prevent misconceptions.
- Ask students to draw correct and incorrect examples themselves. For example, children in 4th grade can be asked to draw parallel lines and lines that are not parallel. In 5th grade, ask students to draw parallelograms and quadrilaterals that are not parallelograms.
- Tying in with the previous point, ask the students to provide a definition for a concept. This gets them to thinking about which properties in the definition are necessary and which are not.
- Allow the students to experiment, investigate, and play with geometrical ideas and figures. Use manipulatives, drawing, and apps or software (more on them below).
- Have students make their own geometry notebook, filled with examples, nonexamples, definitions, and other notes and drawings.
Computers and interactive geometry
A computer or a tablet is really helpful in the teaching of geometry, because it allows dynamic and interactive manipulations of figures. The student can move, rotate, reflect, or stretch the figure, and observe what properties stay the same.
For example, let's say you are teaching about isosceles triangles in 4th grade. You could simply use a word processor. For example, MS Word has a drawing toolbar which has an AutoShape for an isosceles triangle (it also has one for a right triangle and parallelogram). Ask children to draw some isosceles triangles and to drag them to make them bigger and smaller. Ask them also to rotate them. Ask, "What changes? What does not change? What stays the same? Can you draw this figure on paper?"
There also exist dynamic geometry programs and apps that are specifically designed to teach geometry in an interactive and investigational way. Such programs have been used in research experiments and in schools with good results. After you see what can be done with dynamic geometry software, it is very easy to fall in love with it – the idea is just great!
Here is a list of dynamic geometry software.
How can I help a student who is already in high school?
Perhaps your student is already studying geometry in high school and is having problems. Of course you cannot change how the student 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 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.
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.