Practical and Proven Professional Development
C O N N E C T   W I T H   U S :
  • Home
  • Professional Development
    • Upcoming Events
    • Staff Development DVD's
    • Workshops & Conferences
  • Blogs
  • Store
    • Products
    • Order Form
    • Sole Source Document
  • Resources
    • Handouts for Purchase
    • Free Videos
    • Math Videos
    • Student Gallery
    • Velveteen Teacher
    • Free Resources
  • Contact Brad
  • Brad's Other Books

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

8/31/2013

18 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.

18 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
Bill in Brooklyn
9/19/2016 11:14:44 am

I wish there were some examples in the article.

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
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



Leave a Reply.

    Author

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

    Categories

    All
    Curriculum
    Pedagogy
    Resources
    School Climate
    STEM
    Videos

    Archives

    December 2017
    July 2017
    April 2017
    January 2017
    November 2016
    October 2016
    September 2016
    August 2016
    July 2016
    June 2016
    May 2016
    April 2016
    February 2016
    January 2016
    November 2015
    September 2015
    July 2015
    June 2015
    May 2015
    April 2015
    March 2015
    February 2015
    January 2015
    December 2014
    November 2014
    October 2014
    September 2014
    July 2014
    June 2014
    May 2014
    March 2014
    February 2014
    January 2014
    December 2013
    November 2013
    September 2013
    August 2013
    June 2013
    April 2013
    March 2013
    February 2013
    January 2013
    December 2012
    November 2012
    October 2012
    September 2012
    August 2012
    February 2012
    December 2009

    RSS Feed

TTT Press

Home
Blog
Professional Development
Store
Resources
Other Books by Brad
Contact Brad

What our customers are saying:

Thank you for a well-organized, meaningful, and engaging presentation that gets to the heart of the matter: connecting context and concepts in instruction.  Terrific!
Linda Buck – Principal
I love when I can go back to the classroom and use what I just learned.  Thank you very much for your expertise and enthusiasm.
Kim Clay – middle school teacher
© 2013 by Brad Fulton and TTT Press