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I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a Van Hiele Level 0 to 1 thought process. For instance, when talking about parallel line angle relationships, once they realize that all pairs of angles (when you have 2 parallel lines and a transversal) are either congruent or supplementary, asking them to justify why the angles are congruent is "They're parallel and both look acute."

I continually remind them that explaining their ideas based off of earlier ideas is BETTER than using visual information, and provide not-to-scale diagrams to emphasize that fact, but I'd like to create a poster to point to to say something like "It's good that you noticed they look the same, but we're looking for at least a level 2 explanation here." I'd like to make explicit to my students my goal of moving them "up the scale" to a higher level of thinking.

For ease, the Van Hiele levels as presented to educators can be phrased as follows:

Van Hiele Levels (Source)

How can these be rephrased concisely and accurately in student-friendly language? For instance, I'm not sure how I feel about the rephrasingseen in even the table I provided. I have some ideas of my own, but would prefer to collaborate.

Is communicating this goal with students a good idea, in your opinion?

At what level do higher-level educators expect their students to be by the time they reach their college level courses, so I could possibly add this to the poster?

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    $\begingroup$ One flaw in this summary -- what is the difference between the "appearance" of a shape and "properties" of a shape? In other words, what are "properties" of a shape? How would you translate the word "properties" into "student-friendly language"? $\endgroup$ – Jasper Jan 7 '16 at 23:11
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    $\begingroup$ I agree the existing summary is flawed! By "appearance," that means the students might judge a shape with rounded corners to be a rectangle, or base their understanding of "rectangle" on the idea of "something that looks like a door." Or, in my case, "congruent" are "angles that look like they are the same size." At the next level, the student might say "Rectangles have 4 right angles" but might not categorize squares as a type of rectangle. Students might say "Alternate interior angles are congruent" but would consider it a memorized fact different from "coresponding angles are congruent." $\endgroup$ – Opal E Jan 8 '16 at 0:11
  • $\begingroup$ Or, another level 2 thinking: they might understand that two triangles are congruent, but then say they aren't similar. $\endgroup$ – Opal E Jan 8 '16 at 0:13
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    $\begingroup$ Anyone with a math degree has reached level 4 (in the image) at bare minimum, or you wouldn't be able to pass an intro proof class. Did you read the wikipedia article? It more clearly explains the levels; the image is merely a summary. $\endgroup$ – Opal E Jan 8 '16 at 18:33
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The argument has been made that this is sort of a misappropriation of the terms, because the levels are meant to define levels of understanding rather than levels of detail. I'm assuming what you're aiming for is that you can say to your student(s) "I'm looking for a level 2 explanation here", and they would provide you with an answer that would prove that they have achieved a level 2 understanding of that concept.

Thus, your table is going to look slightly different because it describes the type of answer you must give in order to prove a level of comprehension, rather than describing the level of comprehension itself. Note I may have misunderstood the levels above to some extent as I am familiar with the concept of levels of comprehension but haven't seen it formally listed in this form before (irony!). I think, even if you disagree slightly with my interpretation, the overall exercise will still be useful (I hope!):

Level 1 - It looks like it's true (The Duck Test)

"If it looks like a duck, and quacks like a duck, it's probably a duck."

It's a good phrase because it's memorable and it well-defines the concept. It also reinforces the lack of concrete data associated with this level of comprehension: "I can't prove it's true, but my understanding is that it's unlikely to not be true". This proves level 1 comprehension because... well, they have made a correct basic assumption based on the appearance of the problem.

Level 2 - I can deduce that it's true

"I have gathered some data on the objects in question, and can deduce that this is the case"

Note: Deduce, not prove.

To show Level 2 comprehension, the student must be able to provide detail to support their Duck Test. In the original example given, it would be to say "I have calculated the angles, I know they're both acute so my duck test holds". They don't have to then prove that the corollary (angles are congruent) holds to show this level of comprehension. This proves level 2 because they have used actual data to make a correct assumption (well, technically an assumption).

Level 3 - I understand the Importance of these properties

"Not only can I deduce that this is true, I can extend my deduction to related abstract concepts"

To show Level 3 comprehension (IMO, there may be disagreement here), the student must show that they understand the relevance of what they are seeing. What are the consequences of the these two angles being congruent? Can the student make the jump of comprehension to the other properties that must therefore be true (related pairs of supplementary angles, the types of geometrical shapes that would be formed from sets of congruent angles, which lines must be parallel if these two angles are congruent). Rather than just seeing it's true, how can I USE the fact that it's true?

Again, note that the student isn't required to prove any of this yet, just understand the significance of the data. I might be able to deduce from the sound on the roof that it's raining pretty hard, but will it occur to me to take an umbrella when I go outside?

This proves level 3 because they can volunteer consequences of the properties they are seeing, showing an understanding of the abstraction - they understand what it means, not just what it says.

Level 4 - I can Prove that it's true

"I know that this is true, and I can prove it"

The student shows comprehension of both the consequences of the properties and the base principles behind the properties. They can use these base principles to prove their duck test rather than just deducing from "stuff I know is true". F.Ex., I can prove that these angles are congruent using my knowledge of angles and parallel lines, rather than simply deducing it's true because I've been told that two parallel lines bisected by another line at an angle forms a set of congruent angles.

This shows level 4 comprehension because they can prove their deductions rather than relying on "facts" that they have been told (Ability to prove the theorem from axioms, definitions, and previously proven theorems, as opposed to merely knowing how to use the theorem).

Level 5 - I don't know that this is true (yet), but I can prove it

"I know that this is true (and I can prove it), but can I prove whether this is true?"

Student shows absolute comprehension of the subject, to the point that they can apply what they have been taught to prove things that they haven't been taught. Using their knowledge, they can make good assumptions about other things that might be true, and go on to prove and disprove them.

This shows level 5 comprehension because the student understands enough about the system to try and extend their knowledge beyond its current level.

"I know that this holds in this case, but can I be sure that it always holds?"

"This configuration looks interesting, can I use my knowledge of geometry to prove anything about it?".


It may not be exactly what you're looking for, but it's what I've understood your question to be asking for, and I hope it helps regardless! The unfortunate truth is that (in my experience) most students won't ever reach level 5, particularly in a compulsory subject, because it requires a significant level of enthusiasm to reach it. Maybe by showing the decent but relatively uninterested students that there is something to aim for beyond level 4 (which is sufficient to get you through pretty much any exam), it will encourage more of them to take that leap. If that's the case, then this will all have been worthwhile!

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  • $\begingroup$ I'd say it's definitely helpful, and I don't expect my students to reach level 5 by any means! I'd say level 5 is more of an ability to handle axiomatic systems that are less intuitive than plane geometry, such as spherical or neutral geometry, and understand that axioms are arbitrary. This is a level of understanding reached at the undergraduate level, if at all. $\endgroup$ – Opal E Jan 8 '16 at 18:20
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I think one problem with this whole approach is that one of the fundamental tenets of the van Hiele model is what the van Hieles called "the principle of separation":

a teacher who is reasoning at one level speaks a different "language" from a student at a lower level, preventing understanding. When a teacher speaks of a "square" she or he means a special type of rectangle. A student at Level 0 or 1 will not have the same understanding of this term. The student does not understand the teacher, and the teacher does not understand how the student is reasoning, frequently concluding that the student's answers are simply "wrong". The van Hieles believed this property was one of the main reasons for failure in geometry. Teachers believe they are expressing themselves clearly and logically, but their Level 3 or 4 reasoning is not understandable to students at lower levels, nor do the teachers understand their students’ thought processes. Ideally, the teacher and students need shared experiences behind their language.

(Source.)

Assuming this is correct, it would seem to follow that a student currently reasoning at level $n$ cannot possibly understand what you are asking for if you tell them you want them to be at a level higher than $n+1$. They would be literally incapable of understanding what you mean, because individuals must pass through the levels in sequence. The levels only make sense retrospectively, to a person who has already passed through them.

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  • $\begingroup$ I think another poster understood what I was asking for well -- I want to be able to concretely describe what sorts of actions or statements they can make that show me what level of thinking they are at. If I have a student at level n, I want to push them towards a level n+1 explanation with a concrete example of the type of statements I am looking for, and don't plan on pushing them to a level n+2 explanation until the show they are at level n+1. $\endgroup$ – Opal E Jan 8 '16 at 18:24
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    $\begingroup$ An additional complication is that many people believe students are at different levels for different aspects of geometry -- I'd say my students are at a much higher level when it comes to simple plane geometry than coordinate geometry because of their level of experience. They can reason more abstractly about lines and angles than they can about quadrilaterals because of where we are in our sequence. $\endgroup$ – Opal E Jan 8 '16 at 18:26
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This model is really describing three different concepts.

1-2: What does it look like? This is "basic" shape recognition.

2-3: Practical geometry. How many sides does a shape have? What kinds of angles does the shape have? Can shapes be put next to each other without any gaps?

4-5: Proofs, for the sake of proofs. The examples might have to do with geometric shapes. But the exercise of doing the proofs could be done just as well with algebra, or with matrices, or with sets.

The model is missing a step between 3 and 4: Constructive geometry, like the methods used by origami masters, old-fashioned hand draftsmen, and ancient Greek compass-and-straightedge geometers. These are practical exercises, which someone at level 2 can see the use of, and someone at level 4 can check the accuracy of.

A translation

  1. Visual. Name that shape. What does it kinda look like?
  2. Descriptive. Tell me about that shape. How many sides does it have? Are the sides straight? What kinds of angles does it have?
  3. Categorizing. What do these shapes have in common? Can they be joined together? What gaps would they have?

3.5 Construction. Can you modify a shape to have particular properties? Can you make drawings of a 3-D object carefully enough that someone can make another object based on your drawings? What are the tolerances?

  1. Proofs. Can you prove things about the shape? If you say something is true about kind(s) of shapes, can you be sure that nobody can find an exception?
  2. Creative proofs. What rules can you change about the proofs? How does this change what you can prove?

Real-world applications

  • Most human beings need to be at least at level 1. There is a reason that babies can recognize shapes before they can talk.
  • Most hands-on workers need to be at least at level 3.
  • Designers of physical things (including tailors, seamstresses, drafters, engineers, and architects) need to be at least at level 3.5. (The missing "constructive geometry" level.)
  • Students studying mathematics, science, computer programming, or engineering in a college-preparatory or college program need to be at least at level 4.
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  • $\begingroup$ What level of reasoning would you say a doctor or other medical workers should be at? My students are at a specialty school for the medical sciences. $\endgroup$ – Opal E Jan 8 '16 at 17:47
  • $\begingroup$ My guess is that medical workers are "hands-on workers", and need to be at least level 3. Doctors and pharmacists need level 5 to understand the potential consequences of various interventions -- but not as a geometrical issue. Furthermore, doctors and pharmacists need to understand statistics, because the issues they deal with do not have the 100% certainty of proofs. Doctors, nurses, and pharmacists need to take lots of science classes as undergraduates, so they need level 4 to pass those classes -- but again, not for geometric applications. $\endgroup$ – Jasper Jan 8 '16 at 17:58
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Unable to resist continuing my comment-rant: it is perverse to disbelieve a thing that appears to be true. Sure, appearances can be deceiving, but... mostly not so much. It is also perverse for mathematics teachers to create situations in which appearances are deceiving... since, in my opinion, the goal is not to teach paranoia, but to teach the amazing things that mathematical methodologies can uncover. Often (notoriously in physics) an idea is "only" a heuristic, but a very powerful one (as Dirac's quantum theory...)

It is also perverse to claim that obviously-true things "must be proven before we can move on". Yes, it is incidentally a test of our "proving-power" to prove "obvious things", ... but, here is the point, this is certainly not the first or primary point of mathematics. That is, mathematics really took off after Newton's epiphanies that it explained nature. Prior to that, the bourgeois games of the ancient Greeks (preserved for 2,000 years) were mostly a game played by those with sufficient leisure. Didn't explain anything about the world, and that didn't matter. Newton's and Leibniz' calculus "had issues", that were not fully resolved until mid-20th century, when A. Robinson and E. Nelson showed that infinitesimals are legitimate. The epsilon-delta debauche of the late 19th century was a slightly awkward first attempt...

That is, I worry that this purported summary of students' competence is based on a significantly misguided notion of the nature and history of mathematics.

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    $\begingroup$ My concern is: many of my students, when drawing diagrams to model real situations, draw an inaccurate or not-to-scale diagram. ENGINEERS draw inaccurate and not-to-scale diagrams. They need to know that just because when they drew their diagram the angle looked like a right angle, it doesn't mean they can assume the angle is a right angle. Just because THEIR picture doesn't have a 35 degree angle, doesn't mean we can't draw an angle, label it 35 degrees, and figure out something about the triangle it's inside. They need to be able to handle not-to-scale drawings of their OWN creation. $\endgroup$ – Opal E Jan 8 '16 at 17:46
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    $\begingroup$ I agree, people need to understand the vagaries of "pictures". However, it is manifestly unclear to me that "mathematics" in the school-subject sense is at all an optimal response. Maybe courses in "drafting" (which used to be part of "shop" class), making orthographic projections of real physical objects, would be more useful. I am skeptical that layers of formality address the issues, since it appears that most people just never quite get the knack of "deduction" much less "proof". The popular senses of these words hijack any formal use... $\endgroup$ – paul garrett Jan 8 '16 at 18:34
  • $\begingroup$ I have the opposite opinion. I think what you call "bourgeois games" is precisely the point of mathematics, and (with some exaggeration) Newton shouldn't be considered a mathematician at all. The most important purpose of education isn't to improve our mastery of nature or technology; it's to build a democratic society that won't execute Socrates or fight the Peloponnesian War - and mathematics is one of the continuations of Aristotle's work towards this goal. $\endgroup$ – Alexander Woo Jan 13 '16 at 7:11

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