Why Students Don't Understand Geometry And How We Can Fix That

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Why Students Don't Understand Geometry and How We Can Fix That

8/31/2013

31 Comments

“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. 31 Comments my name suvagiya rinkal link 2/8/2015 11:51:52 am

i don't understand my student life

Reply rayyna link 11/16/2017 08:53:04 am

honey me too iss okay

Reply Terri Husted link 7/16/2015 11:57:37 am

Please check out my book Understanding Geometry by Critical Thinking Co. Thanks!

Reply Parasu Anantharam 2/10/2016 09:41:38 am

Brad, Thank you for your article on Geometry. Would you please offer examples of intermediate activities that will help students to bridge the gaps in their geometric ladders? Thank you!

Reply Sanai C 7/2/2024 03:40:27 pm

This would be very helpful i feel like examples activities would help.

Reply Alisha Quintanilla 7/3/2024 11:51:06 am

Hey! Adding examples of intermediate activities will be very helpful, I really like how you used the statement bridge the gaps in their geometric ladders I feel that when I was younger there was a gap in my education that was never fufiled.

Reply Bill in Brooklyn 9/19/2016 11:14:44 am

I wish there were some examples in the article.

Reply Sanai C 7/7/2024 07:57:52 am

I said the same thing that way i can have a visual on the article the picture could help way more.

Reply Bitch Ass 6/5/2017 10:33:32 am

I love this Shit! Thanks for the excellent as fuck explanation!

Reply interested reader 7/23/2017 04:54:48 am

you douche

Reply rayyna link 11/16/2017 08:53:59 am

word

hiasef link 1/24/2018 10:54:43 am

bhoiasef

Reply fuck you 1/15/2018 08:38:01 am

its easy

Reply suq madiq 9/20/2018 03:05:07 pm

suq madiq

Reply suqmadiq 10/2/2025 02:24:14 pm

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Reply hasfe link 1/24/2018 10:54:13 am

uoefwa

Reply Kiira 4/9/2018 06:53:19 pm

as one that is studying for the coming regents while also doing take home tests and inschool tests at the same freaking time, i can honestly say that i want to break down and, well: cry. Teachers always tell me that "practice makes perfect" but i honestly don't think that this lil' lost boy here is going to get any more 'perfect' with any more practice.. as helpful as this article is, or 'was,' seeing that it is 2018, i don't see any probability of me passing math in the near future

Reply ohgodhelpme 9/18/2018 10:51:01 am

geom is ruining my life... i have a 68 in it and i need a 75 by the end of the week to keep playing soccer. any advice?

Reply Gr1ndNever$top$ link 11/29/2018 04:10:43 pm

Manage your time and the SZN will be breeze. Study, ask for help and complete everything to the best of your ability. Hard Work Pays off if you wanna stay on the team!

Reply $lime Stress link 11/29/2018 03:59:20 pm

Stress fuckin sucks am I right? Juss don't procrastinate and always perservere

Reply This Isn't It Chief 11/29/2018 04:12:29 pm

My fellow chief, it sure does. You have spoken words of truth. Thanks coach.

Reply Eatmylongpeepee link 2/7/2019 02:42:23 pm

I wanna cum in this person, jk I'm dumb at geometry, help.

Reply frsgdgdr link 9/22/2019 03:44:39 pm

gsrgsr

Reply BRUH 9/29/2020 10:12:26 am

my geomtry teacher sucks D she goes way to fast and when I finaly understand 1/10 of it she moves on, gives back to back quizzes and tests and just sucks

Reply Leocadia 11/10/2020 10:02:06 pm

This is a bit old, but I feel it's still relevant. I need to bring my 65 in geometry up to an 80 by december, but for some reason, I never understand it. I do online school, and the teacher just rushes through and never pays attention to us in class. Although, I admit, I do have a hard time paying attention, and I get distracted fairly easily. I don't know what to do to focus better because my mind wanders. I'm passionate about theatre, and in order to stay in my elective classes and exempt my geometry final, I have to get an 80 or above. Unfortunately, because I'll spend so much time focusing on math, I'll forget to do work for other classes, and my grade will drop in those classes. I'm stuck in a paradox with an inevitable outcome. I'm getting stressed and I want to quit school entirely because of this. Nobody told me high school would be this hard, and I'm not prepared at all. I need help in geometry but I don't know what exactly I'm not understanding.

Reply Shirley M link 1/13/2021 05:45:40 am

Nicee share

Reply Doesnt matter 3/26/2021 12:10:11 pm

Im in the same boat my teacher absolutely sucks and I have to pass with an 85 or I fail the year I have 0 hope left and its taking a missive mental toll I honestly dont care anymore its just a 5 minute slide presentation then 1 daily work assignment then 3 homework assignments and a test every Friday All in all my district sucks and geometry or really any maths courses in highschool should not be mandatory

Reply yes 9/10/2021 03:20:03 am

geometry makes me want 2 kms

Reply Sanai Coleman 7/2/2024 03:35:06 pm

I feel like this article is very great and gives great examples on how some students need more help with the subjects they are learning. And how this article can help some future teachers

Reply Alisha Quintanilla 7/3/2024 11:44:53 am

Hey! I agree, This article pretty much breaks down examples to the importance of learning geometry and the steps. I feel this article is very beneficial for future teachers because of all of the details.

Reply Alisha Quintanilla link 7/3/2024 11:42:53 am

After reading this article I feel that everything mentioned was very relevant in terms of student's not understanding geometry. I feel that I am one of those students, and because of my lack of geometry I now hate math as an adult and struggle. Me personally, I struggle with the the arithmetical way of approaching math. I feel that I also have a gap somewhere in my studies because math just does not stick. I am constantly having to learn, and re learn math. Overall, I do not like geometry and math in general. But I am willing to learn and change my mindset because as a teacher it is crucial that I understand material. If there was anything in geometry that I feel is the most important when teaching chuildren is helping students learn the properties.

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Brad Fulton is an award winning teacher and nationally recognized provider of professional development with over three decades of experience in education.

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Why Students Don't Understand Geometry And How We Can Fix That