“I always did fine at math, until I got to geometry in high school.” The vast majority of people can identify with that statement. However there is a correspondingly vocal minority who responds, “Math never made sense until I got to high school geometry, then I could see it!” For this reason,
many students hit the geometry wall in high school and their mathematical journey ends. This wall often prevents them from continuing in mathematics courses and having a successful transition to college.
There are a number of reasons why this wall is so hard to conquer. First of all, there seems to be evidence that people tend to have a natural proclivity to either an arithmetical way of approaching math or a visual and geometric one. For the former group, the ability to reason spatially is not as easy as it is for the others. As teachers, we can’t control a person’s natural abilities. Some students learn foreign languages more quickly than others; some are more naturally coordinated.
The good news for educators is that these deficits don’t mean that these skills cannot be developed and learned. For many students, their lack of geometry understanding is due in part from a lack of opportunities to experience spatial curricula.
Many textbooks and many district pacing guides emphasize numeracy, arithmetic, and algebraic reasoning. Geometry (along with data and statistics) is often tucked into the final chapters of the book and the final weeks of the year after state testing. Because we educators are pressed for time and need to reteach and review concepts that weren’t fully understood, we often fail to get to those chapters. Thus students enter high school geometry with the following skill set: “I know the names of shapes, and I had to memorize the area formulas, but I don’t remember them.”
It turns out that the seminal work on geometric thinking was done by a Dutch couple, the van Hiele’s. They made two significant discoveries about how we learn geometry. First, there are five sequential levels of geometric thinking. (More about them in a moment.) Secondly, and this is the real good news, moving from one level to the next higher one, is not so much a matter of cognitive development dependent upon age but rather hinges upon exposure to these geometric experiences.
Here are the five levels of acquisition of geometric thinking. Students at one level cannot leapfrog to another but must move sequentially through the layers. (The van Hiele’s numbered their five levels 0–4 while American researchers have reclassified them as 1–5 to allow for a level zero in which a child has no geometric knowledge.)
1. Visualization – Children can identify shapes based on appearance not on properties. Students at this level may not see a square as a type of rectangle nor even see it as a square if it is rotated slightly.
2. Analysis – At this level, students begin to associate properties with their shapes. The student who struggled to identify a rotated square will now see that it has four congruent sides and four right angles and is therefore a square. Similarly the level one student would struggle to recognize a triangle with a vertex pointed down and a base at the top whereas a level two student sees that the three sides make it a triangle.
3. Abstraction – Now students can begin to think about the properties and apply them to arguments that involve inductive reasoning. The student who sees that four different triangles all have an interior angle sum of 180° would use that pattern to reason that all triangles must have the same interior angle sum.
4. Deduction – At this level, students use deductive logic to prove their conjectures from the previous level.
5. Rigor – This goes beyond the former level to explore proofs by negation and non-Euclidean geometry.
As you can see, most students in elementary grades are operating at level one; they recognize shapes. However they don’t always do this with fluency and accuracy. I once displayed a square to some 4th and 5th grade students and asked them to name the shape. They had no problem telling me it was a square. However when I rotated it to look like a baseball diamond, about nine in ten said it was now a diamond. The rest assured me that it was a rhombus. Only one student out of over 100 was able to tell me correctly that it was still a square, and I had only rotated it.
By contrast, high school algebra is taught at levels four and five. Because students must move through these levels sequentially, it is as if we have asked them to climb a ladder that only contains the first and the final two rungs with a great gap in the middle.
This illustrates the struggles we face in teaching geometry. Identifying a square is on most state’s kindergarten standards. However only 1% of the students I asked knew what a square was five years later. Obviously this is because when their text or teachers showed squares, they typically had a baseline parallel to the bottom of the page. Students had not attended to the properties of the squares.
However, when I posed the same problem to 8th graders, nearly all were able to identify the shape as a square even when rotated. This shows us that their acquisition of this knowledge did not occur as a result of our kindergarten instruction but rather was due to experiences they encountered in later grades.
Again this is good news, for it tells us that we can accelerate this growth by offering students these crucial experiences in geometry. However, since most textbooks do not provide these opportunities, it falls to us to create these lessons. Fortunately they are out there. In a future blog, I will offer examples of intermediate activities that will help students to bridge the gaps in their geometric ladders.
There are a number of reasons why this wall is so hard to conquer. First of all, there seems to be evidence that people tend to have a natural proclivity to either an arithmetical way of approaching math or a visual and geometric one. For the former group, the ability to reason spatially is not as easy as it is for the others. As teachers, we can’t control a person’s natural abilities. Some students learn foreign languages more quickly than others; some are more naturally coordinated.
The good news for educators is that these deficits don’t mean that these skills cannot be developed and learned. For many students, their lack of geometry understanding is due in part from a lack of opportunities to experience spatial curricula.
Many textbooks and many district pacing guides emphasize numeracy, arithmetic, and algebraic reasoning. Geometry (along with data and statistics) is often tucked into the final chapters of the book and the final weeks of the year after state testing. Because we educators are pressed for time and need to reteach and review concepts that weren’t fully understood, we often fail to get to those chapters. Thus students enter high school geometry with the following skill set: “I know the names of shapes, and I had to memorize the area formulas, but I don’t remember them.”
It turns out that the seminal work on geometric thinking was done by a Dutch couple, the van Hiele’s. They made two significant discoveries about how we learn geometry. First, there are five sequential levels of geometric thinking. (More about them in a moment.) Secondly, and this is the real good news, moving from one level to the next higher one, is not so much a matter of cognitive development dependent upon age but rather hinges upon exposure to these geometric experiences.
Here are the five levels of acquisition of geometric thinking. Students at one level cannot leapfrog to another but must move sequentially through the layers. (The van Hiele’s numbered their five levels 0–4 while American researchers have reclassified them as 1–5 to allow for a level zero in which a child has no geometric knowledge.)
1. Visualization – Children can identify shapes based on appearance not on properties. Students at this level may not see a square as a type of rectangle nor even see it as a square if it is rotated slightly.
2. Analysis – At this level, students begin to associate properties with their shapes. The student who struggled to identify a rotated square will now see that it has four congruent sides and four right angles and is therefore a square. Similarly the level one student would struggle to recognize a triangle with a vertex pointed down and a base at the top whereas a level two student sees that the three sides make it a triangle.
3. Abstraction – Now students can begin to think about the properties and apply them to arguments that involve inductive reasoning. The student who sees that four different triangles all have an interior angle sum of 180° would use that pattern to reason that all triangles must have the same interior angle sum.
4. Deduction – At this level, students use deductive logic to prove their conjectures from the previous level.
5. Rigor – This goes beyond the former level to explore proofs by negation and non-Euclidean geometry.
As you can see, most students in elementary grades are operating at level one; they recognize shapes. However they don’t always do this with fluency and accuracy. I once displayed a square to some 4th and 5th grade students and asked them to name the shape. They had no problem telling me it was a square. However when I rotated it to look like a baseball diamond, about nine in ten said it was now a diamond. The rest assured me that it was a rhombus. Only one student out of over 100 was able to tell me correctly that it was still a square, and I had only rotated it.
By contrast, high school algebra is taught at levels four and five. Because students must move through these levels sequentially, it is as if we have asked them to climb a ladder that only contains the first and the final two rungs with a great gap in the middle.
This illustrates the struggles we face in teaching geometry. Identifying a square is on most state’s kindergarten standards. However only 1% of the students I asked knew what a square was five years later. Obviously this is because when their text or teachers showed squares, they typically had a baseline parallel to the bottom of the page. Students had not attended to the properties of the squares.
However, when I posed the same problem to 8th graders, nearly all were able to identify the shape as a square even when rotated. This shows us that their acquisition of this knowledge did not occur as a result of our kindergarten instruction but rather was due to experiences they encountered in later grades.
Again this is good news, for it tells us that we can accelerate this growth by offering students these crucial experiences in geometry. However, since most textbooks do not provide these opportunities, it falls to us to create these lessons. Fortunately they are out there. In a future blog, I will offer examples of intermediate activities that will help students to bridge the gaps in their geometric ladders.
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