# Geometry without proofs

These days, high school geometry is light on proofs, writes Barry Garelick on Education News. Students may know the sum of the measures of angles in a triangle equals 180 degrees, but most can’t prove the proposition.

If done right, the study of geometry offers students a first-rate and very accessible introduction to the nature and techniques of logical argument and proof which is central to the spirit of mathematics itself.  As such, a proof-based geometry course offers to students—for the first time—an idea of what mathematics means to mathematicians, and how it is used.  Also, unlike algebra and pre-calculus, since geometry deals with shapes, it is easier for students to visualize what it is that must be proven, as opposed to more abstract concepts in algebra.

Most geometry textbooks give students “one or two proofs that are not very challenging in a set of problems devoted to the application of theorems rather than the proving of propositions,” he writes. Many problems indicate missing angles or segments as algebraic expressions. It is, to quote Mr. Spock, “illogical.”

While Mr. Garelick’s words are correct, they don’t imply that High School Geometry should be a REQUIRED course. It should be offered with proofs as an elective. The main math sequence should be two year “School Algebra” as NMAP suggested, with Statistics or Math Analysis as a third year.

Please note that the Common Core wasn’t suppose to be innovative in high school topic selection. It had to be “conservative” to get as many states to join as possible. This is no criticism of the Common Core, but a note of respect. Perhaps the next iteration with a push towards Math 123, not A1, G, A2 would be wise.

From a different perspective, in engineering, the concept of “lag time” is important. Classical Geometry lost its primacy in 1854. Proofs in topology may be just as interesting and considering our networked life, more valuable. Frankly, proving properties of inscribed angles, etc isn’t all that valuable anyway. A different elective course including Geometry, Topology, and possibly induction and recursion would be of higher value than the Geometry courses of the 20th century.

2. Cal says:

As is always the case, Barry has no clue about math in low ability classrooms, or what it entails. Proofs are still taught in high schools with honors geometry, or smart kids. They aren’t covered in low ability classrooms because the kids need algebra much more than proofs.

No kid, high or low ability, is ever going to use proofs again unless they major in math in college. To waste time teaching low ability kids anything more than simple 3-step proofs is an obscene waste of their time, assuming they could grasp it, and they can’t.

BTW, I do show my kids the proof for 180 degrees in a triangle. I just don’t make them do it.

• Mark Roulo says:

Proofs are still taught in high schools with honors geometry, or smart kids.

Things may have gotten better recently. But in 1998, Barry Simon at Cal Tech didn’t think that this was the case. He writes:

One of the pleasures of being on the faculty at Caltech is interacting with our bright undergraduates. For the past two years, I’ve asked the incoming freshmen in my calculus/probability class whether they had proofs in their high school geometry course. About 40% have not, and more than half of the remainder had at best a cursory few weeks.

Now … he might get the losers at Cal Tech (note that he is teaching freshman calculus), but these kids are still good enough to get in to Cal Tech. And maybe these Cal Tech undergrads forgot that they had done a year of proofs in their high school geometry class, but I doubt that, too.

• Cal says:

You should maybe expect that the kids lie. And a few weeks is plenty of time to do proofs. What on earth does he want? It’s not a course in proofs, but in geometry.

• I used proofs in college, even though I wasn’t a math major. (My college math class used them though, as did my general chem class). The framework also helped me with writing papers (where we had to give evidence from the text) and in philosophy classes. If we totally eliminate proofs, we need to put logic somewhere else in the HS curriculum. It’s important for kids to learn how to reason in an organized, linear fashion.

3. Barry Garelick says:

I have taught in “low ability” classrooms, Cal. And proof-based geometry at one time was not limited to honors students. The argument against proofs that you raise –that no one will use it unless they major in math–could also be said about geometry in general. It has been outmoded by analytic geometry, trig and calculus. There is a value, however, in learning the nature of argument even if you or people like you can’t see it.

• Roger Sweeny says:

“The argument against proofs that you raise –that no one will use it unless they major in math–could also be said about geometry in general.”

That argument can actually be made about high school in general. I think we should give it a hell of a lot more respect than we do.

I realize high school students are pretty much kids but imagine that at age 40 YOU were drafted to do something that didn’t provide a good or service to anyone else, and which didn’t teach you much of anything that you used after your draft service was over. Would you say, “Oh, well. It was for my own good. I’m sure they meant well.”

4. Cal says:

“And proof-based geometry at one time was not limited to honors students.”

I know. Gosh, can we think about what’s different about “regular” geometry students of today, and those of 15-50 years ago?

” The argument against proofs that you raise –that no one will use it unless they major in math-”

That was not the argument against proofs that I raised. The argument I raised is that kids need algebra much more than proofs, so if there’s a tradeoff, you take the algebra. And there is a tradeoff, and they will use algebra again–the next year, and in their college placement tests.

• Broken homes and parents without the energy to force them to knuckle down and pay attention. Working class kids in the 1950s could excel in school, even with uneducated parents, because there was discipline, respect, and parental support. Now, in the Belmont v. Fishtown world, the poorer kids have no support at home and no order or discipline in their lives….. There might be a reason why Fr. Flannigan and Milton Hershey counted the sons of single mothers as ‘orphans’, but it’s politically incorrect to do that today. Instead, we have to pretend that all structures are equally conducive to raising sane, healthy kids…

5. Crimson Wife says:

The math program I am using with my oldest student, Singapore Discovering Mathematics, does not include proofs in its geometry sections. Yet Singaporean students score among the highest in international comparitive math tests. I have yet to hear a convincing argument as to why the 98% of students who will not major in math should be forced to study proofs. If the student can solve geometry problems found on the SAT or ACT, why isn’t that enough?

• SteveH says:

This isn’t just about proofs. This is about the educational meme that says that kids need to know the wonders and understandings of math, but when it comes down to the details, those educators bail out with low expectations. You can apply this same effect to algebra. K-6 math curricula cause great problems because they attempt to teach math using vague ideas of conceptual understanding and critical thinking which have little to do with the formal mathematical understanding and skills needed for a proper foundation in math.

Then, when these kids get to high school algebra and really begin to struggle, people are all too willing to blame it on IQ. If I didn’t help my son at home, I’m sure his teachers would be happy to think that he’s just not that smart. However, they will be more than willing to take credit for his success.

I knew this was how people would spin his article.

• J.D. Salinger says:

Maybe your kids can have a career taking the SAT or ACT full time since you seem to think these tests are the be-all end-all of education indicators.

• Crimson wife says:

My student who will be doing algebra 1 next year is not going to be a math major, period. The future math majors are the kids her age who love participating in math competitions like MathCounts and AMC8 and do well in those. She has shown little interest and no unusual aptitude for that.

However, she does aspire to a field that will require taking the SAT/ACT and then the general GRE. So for her, scoring well on those entrance exams is a high priority.

• J. D. Salinger says:

There are some students who do not participate in MathCounts and AMC8 who end up being math majors. Some people don’t know what major they will end up as the “Momtattempts” comment below indicates.

I am so grateful that my school did not reserve geometry proofs for the elite. I was a A/B student but never an honor student and I was given a chance to shine in geometry with challenging proofs. If I didn’t have that experience then I would have never have been a math major in college. How does a high school sophomore know that they aren’t going to major in math just because they aren’t an honor student? I hate to give up on the late bloomers so early.

• SteveH says:

Again, the issue is not just about proofs. Schools can’t push lower expectations into K-8 with the assumption that few kids will ever need those skills. This just guarantees that they won’t ever have a chance to get those skills. The CCSS/PARCC highest PLD level (5 – “distinguished”) only means that these kids will do well in college algebra. They specifically state that as the goal. Their PLD document never refers to STEM, and they talk about how their test will define curricula back into the earliest grades. On one hand, we hear eduators talk about the glories of behaving like little mathematicians, but on the other hand, they set very low expectations for all, because, well, how many kids really use algebra as adults?

7. Cal says:

“I am so grateful that my school did not reserve geometry proofs for the elite.”

Unfortunately, your school apparently reserved the finer points of reading comprehension to a group that didn’t include you.

“. K-6 math curricula cause great problems because they attempt to teach math using vague ideas of conceptual understanding and critical thinking which have little to do with the formal mathematical understanding and skills needed for a proper foundation in math.”

According to the NAEP and pretty much every state test, kids do much better in elementary school math, despite your bleats about their inadequate teachers.

• SteveH says:

“According to the NAEP and pretty much every state test, kids do much better in elementary school math, despite your bleats about their inadequate teachers.”

So there is no problem? It’s all IQ? You lost that argument before, but apparently, it’s all you got.

• Cal, either you are confusing me with someone else or you have a reading comprehension problem yourself, or you do not understand what it takes to do a geometry proof. my statement clearly refereed to a ‘sophomore’ in high school. Not K-6.

8. Florida resident says:

Actually, the proof of the statement

“Sum of angular measures of three angles
of a triangle equals 180 degrees”

is difficult and requires, at some previous stage,
the use of 5-the Euclidean postulate.

2-D Geometry.

Definition. Given the original straight line,
and another straight line, passing through a point,
not belonging to the original one.
These two lines are called “parallel”,
if they, while staying in the same 2-D manifold,
nowhere intersect.

5-th Euclidean postulate:
Given the original straight line,
and a point, not belonging to the original one.
There exist another line, parallel to the original,
and this parallel line is unique.

Possible refutations of 5-th postulate.

Gauss geometry (geometry of meridians on the Earth):

Parallel lines in the above sense
do_not_exist, meaning that
any two “straight lines” intersect somewhere
(like meridians at the Poles.).

Lobachevski geometry:

Given the point outside
the original “straight line”,
there is more than one “parallel” line
(actually, an “acceptance” angle,
which contains the directions of those “parallel lines”,
and larger distance of the “point”
from “original straight line”
corresponds to (nonlinearly) larger
acceptance angle of those non-intersecting “straights”.

• Roger Sweeny says:

High school geometry is plane geometry: the geometry of flat surfaces. Gaussian and Lobachevskian alternatives don’t apply on flat surfaces.

Most people find Euclid’s 5th postulate fairly intuitive for a flat surface. If they have already seen that alternative interior angles are equal, the proof that the angles of a triangle add up to 180 degrees (a straight line) only takes a few simple steps.

• Mark Roulo says:

Most people find Euclid’s 5th postulate fairly intuitive for a flat surface.

Quite an understatement 🙂

It was so “obvious” that it took a few thousand years before anyone took seriously the idea that it didn’t have to be true.
Euclid’s 5th postulate won’t be the cause of any serious trouble for high school students.

• Roger Sweeny says:

A one minute proof that the angles of a triangle add up to 180 degrees:

• Florida resident says:

Dear Roger Sweeny:
I watched the proof you have linked to, it is excellent.
As you could see yourself, it uses 100%
of 5-th postulate by Euclid:
1) exsistence of a parallel line, and
2) its uniquness.

9. Bill says:

Back in 1979, I took geometry in high school, and it covered both plane and proofs. I don’t know about today, but it’s hard to imagine a geometry course in high school not offering at least exposure to basic geometric proofs as a standard part of the course.

10. Cal says:

“It was so “obvious” that it took a few thousand years before anyone took seriously the idea that it didn’t have to be true.”

Not true. Euclid himself saw there was something wrong with it, and people tried to prove the fifth postulate for thousands of years.

“So there is no problem?”

There’s not a problem you can prove exists in K-6 math, which is what you keep repeating. And yes, there is a problem, but the problem is with expectations, not “IQ”.

11. bky says:

One of my kids is taking high school geometry with proofs — they have done a lot of proofs, and I am glad off it, especially since he just took the algebra course in which there was nothing that I would call problem-solving. Constructing proofs is the first time he has had to really solve a problem in school (the problem being, how to prove this statement). Most of his homework consists of a vast time-consuming list of one-line exercises. But there is one thing very wrong with they way they do proofs. They completely ignore the proofs of the main theorems — the ones given in the book. Those are just assumed as axioms, so they never really study the way theorems are proved and how they depend on other theorems and the axioms. For example, many geometry proofs require the construction of an auxiliary line and otherwise require some strategic thinking. They don’t get into that at all. Which means that for these students, proving something is mostly an exercise in justifying a step, rather than about devising a strategy.

12. Peace Corps says:

Sad to say it, but I am thankful when my students remember that the sum of the interior angles of triangles is 180 degrees. By the way this is something that my son in fifth grade has already learned.

So, I guess I’m saying my big issues (bigger issues than proofs) are: 1) knowledge retention, 2) application of knowledge in ways that they have already seen, and 3) application of knowledge to “new” circumstances. I think Barry is saying that proofs will help with #3, but if I can’t get #1 and #2 from my students first…. is there any chance of reaching #3.

• SteveH says:

There’s not a problem you can prove exists in K-6 math, which is what you keep repeating. And yes, there is a problem, but the problem is with expectations, not “IQ”.”

There is no problem, but there is a problem?

If you lower expectations enough, all problems go away and you can feel real good about how much you are helping these poor, dear kids.

And of course, nothing could possibly be wrong with schools or curricula in K-6, so it must be IQ … oops, expectations … oops everything is great in K-6 so those expectations must be just fine. It’s all of those nasty people who expect kids to take algebra, because, of course (never mind everything you’ve said in the past), it’s “expectations”. Wink, wink, nudge, nudge. And never mind all of the work parents do at home to help kids get to the top math level in high school. They must just be fooling themselves. Yeah, that must be it.

• Norm says:

I was thinking to myself, “Uh-oh, SteveH is taking on Cal; she’s gonna start flinging poo at him…”

13. Cal says:

Yes, aren’t the parents who yammer on endlessly about how much work they do for their kids just amazingly tedious in their pleas for congratulation?

I actually tutored my kid in high school math, and I never posted endlessly about doing it.

Steve, apart from the fact that you work with your kid and are desperate to be praised for it, your posts are incoherent.

Incidentally, you’re the only one who has brought up IQ.

• SteveH says:

“I actually tutored my kid in high school math, and I never posted endlessly about doing it.”

Yes, the rich get richer and the poor get nothing but low expectations. Affluent kids get individual help, but poor kids get statistics. It couldn’t be that I’m trying to point out a problem. No, it must be a need for praise. Good analysis. It’s not IQ, it’s too high expectations. Ditto.

14. Jim says:

What is the significance of the year 1854?

15. Jim says:

To Cal –
You say “Euclid himself saw that there was a problem with it”
What is the textual evidence for such a remarkable assertion?

• Cal says:

It’s in Wikipedia, so it can’t be all *that* remarkable. Of course, given that we don’t know for sure there’s a Euclid, I might have personified it a bit much.

http://en.wikipedia.org/wiki/Euclidean_geometry

“To the ancients, the parallel postulate seemed less obvious than the others. They were concerned with creating a system which was absolutely rigorous and to them it seemed as if the parallel line postulate should have been able to be proven rather than simply accepted as a fact. It is now known that such a proof is impossible. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it.”

• Mark Roulo says:

One of the other arguments (besides the delay in using it) that Euclid might have been unhappy with the 5th postulate was that it was clearly much more “tangled” than the other 4 postulates.

1) A straight line segment can be drawn joining any two points.
2) Any straight line segment can be extended indefinitely in a straight line.
3) Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4) All right angles are congruent.
5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
http://mathworld.wolfram.com/EuclidsPostulates.html

16. Jim says:

To SteveH – Obviously the IQ level of students is a very significant factor in their education.

17. Jim says:

To Cal – Trying to read Euclid’s mind on the basis of the order in which the propositions in some editions are arranged strikes me as extraordinarily flimsy. The earliest essentially complete text of Euclid we possess was written down about 950 AD. The various Greek and Arabic texts we possess differ quite a bit in the order and arrangement of the propositions. Almost everything said about the history of early Greek mathematics is unsubstantiated speculation.
The Eudemian summary is probably reasonably accurate and Aristotle’s remarks are probably reliable but neither say anything about Euclid.
Otherwise the absence of real evidence about the history of early Greek mathematics should induce caution in making assertions about it.

• SteveH says:

“Obviously the IQ level of students is a very significant factor in their education.”

Duh. Obviously. It’s the calibration that’s the issue, and too many people use IQ to ignore other variables. While many affluent parents don’t accept low expectations, urban kids get relative statistics and platitudes.

• In fact, you get high IQ urban kids who don’t reach the academic heights of similarly-situated suburban kids b/c they lack a curriculum that lets them grow to their potential.

If a kid’s whole math curriculum is geared to ‘try to finish Algebra by 12th grade’ he’ll never realize AP Calc is an option….

• SteveH says:

CCSS/PARC defines their top PLD “distinguished” level 5 as:

“They are academically well prepared to engage successfully in entry-level, credit-bearing courses in College Algebra, Introductory College Statistics, and technical courses requiring an equivalent level of mathematics.”

Level 4 is called “strong” and it’s the lowest level to meet their college and career readiness standard, defined as:

“Students who earn a PARCC College- and Career-Ready Determination by performing at level 4 in mathematics and enroll in College Algebra, Introductory College Statistics, and technical courses requiring an equivalent level of mathematics have approximately a 0.75 probability of earning college credit by attaining at least a grade of C or its equivalent in those courses.”

That’s “strong”. Talk about lowering expectations.

Their PLD level document says nothing about STEM preparation in a PARCC curriculum. It doesn’t exist.

Note that ACT decided to split from PARCC and provide its own products. Some states appear to be dropping PARCC and going with ACT. What do parents want to see; that little Suzie is getting a top level “distinguished” rating each year, or that she is on track to get at 25 on the ACT?

• Roger Sweeny says:

SteveH,

I don’t know if you can answer this but do you know where that 0.75 number came from? Did somebody actually follow a representative sample of people who scored 4?

• Sean says:

Roger: I don’t know where CCSS/PARCC gets the 0.75, but ACT derives their “college readiness” cuts empirically.

• SteveH says:

I don’t know, but they seem to be interested in a goal of no remediation. They talk about how this is difficult to gage since many colleges have different standards. Because of that, they can’t define something that guarantees no remediation. Maybe that’s where the .75 comes from.

At least the ACT is not starting from ground zero in terms of calibration. The ACT now has their Explore, Plan, and ACT programs to offer to states. I know that Michigan pays for all students to take the ACT, but I thought they were aligning with PARCC. Alabama has left the PARCC consortium for ACT.

It also appears that the College Board is developing their own products to compete for CCSS dollars. They have ReadiStep, PSAT/NMSQT, and SAT. (I think Coleman is ready to move the SAT towards the ACT.) They also have something called Pre-AP for grades 6-12. The College Board has the inside track for providing states with programs and feedback for higher-end students. They have the credibiity and data from AP tests. The ACT and College Board products offer so much more than PARCC, where the top level barely gets kids ready for college algebra, but they call it “distinguished”, and that low top level drives the curriculum back to the earliest grades.

• Roger Sweeny says:

“It also appears that the College Board is developing their own products to compete for CCSS dollars.”

Surely you can’t mean that. The College Board is a non-profit, and non-profits only care about the public good. They don’t care about money.

• Cal says:

I really don’t care much whether you think I should read Euclid’s mind or not, particularly in a context where we aren’t sure Euclid existed. The point is that my statement is generally accepted by many, not something I’ve invented, and I’m totally uninterested in whether or not you think it’s true.

18. Bill says:

“Students who earn a PARCC College- and Career-Ready Determination by performing at level 4 in mathematics and enroll in College Algebra, Introductory College Statistics, and technical courses requiring an equivalent level of mathematics have approximately a 0.75 probability of earning college credit by attaining at least a grade of C or its equivalent in those courses.

IMO, I don’t know why algebra is called a college level course, and basic stats should be a part of any decent algebra II/Trig or Precalculus course.

Perhaps colleges should quit admitting unprepared students or under-prepared students after they take a placement exam in english and math (I had to take these types of exams back in 1981 as a incoming freshman).

If you can’t score high enough to place into at least stats, finite math, pre-calc/calculus or into English 101 or 102, you really don’t have any business being admitted to college at all.

While I might seem harsh here, it’s a crime to admit students who by definition would need anywhere from 2-6 courses to get prepared to do college level work to be admitted to a given major, and probably a lot more work to be admitted to a STEM major.

Without trying to read between the Garelick’s lines, I took away 2 things:
1) the study of proof in high school Euclidean geometry courses has been severely diminished in recent years, and
2) this is ironic, since reform-based math ostensibly values conceptual understanding over repetitive exercises. Yet today’s hs geometry courses are basically a series of repetitive exercises based on theorems that are treated as axioms.

Debating whether or not geometry is a necessary college prep math course isn’t relevant. The fact is, that about 50% of the ACT math is geometry-based. As long as it’s on the college entrance tests, geometry will be part of the hs math curriculum.
I happen to fall in with the crowd that thinks this is a good thing. I think it’s even better when proof is part of the course. A solid geometry curriculum will have both applications and proofs. There is no reason why low-ability students can’t take a class involving proofs. So students who can’t do proofs get a B instead of an A. I don’t think that’s the end of the world.
Historically, the study of geometry was part of the medieval trivium so that students could learn applications of logic. If there’s one thing I hear teachers across content areas complain about, it is that students have no logical skills. Students struggle to write coherent paragraphs, much less coherent papers because they have no framework of how a logical argument is presented.
HS geometry is also supposed to be an opportunity to apply the algebra skills previously learned. Many of the proofs involve nothing more than writing some equation using segment/angle notation instead of a single variable, substitution, and there you are. So yes, students “need” algebra I suppose, and geometry is a means of applying it.

20. Richard Aubrey says:

iirc, when I was in HS, more than half a century ago, all students were required to take Alg 1, and plane geometry.
I enjoyed plane because we were solving puzzles, in a manner of speaking. It is possible we didn’t do proofs, although some of the discussion seems vaguely familiar, it is possible they were called something else, it is possible I’ve forgotten.
However, my classmates who were interested in STEM seemed to manage it in college, becoming engineers and scientists. So either we got it and I’ve forgotten, or we didn’t and it isn’t necessary.

21. Cal says:

“So students who can’t do proofs get a B instead of an A.”

hahahahahaha! That’s really funny. Like that’s the tradeoff.

Incidentally, the geometry that the ACT and SAT test is the geometry that is being taught. In the essay, Barry m makes this clear:

“instead, they contain many problems in which missing angles or segments are indicated as algebraic expressions. For example, opposite sides of a quadrilateral that is identified as a parallelogram may be labeled x + 2 and 2x – 6; the student is asked to find the length of the segments. ”

The SAT/ACT do not test proofs, although the ACT occasionally tosses in a contrapositive question about logic. I cover logic, but not as part of proofs.

22. Jim says:

Cal – The statement that Euclid perhaps did not exist is absurd. Of course he existed. However we know little about him or his views on the philosophy of mathematics other than what is clearly revealed in the surviving texts. Attempts to attribute to him this or that view on the fifth postulate or whatever based on supposed obscure hints in the many different texts in Greek and Arabic that have survived is an exercise in tea leaf reading.
The fact that many people may believe a proposition is not a substitute for solid evidence.