Mathematical tools – Part 2

Were it not for convoluted language, plenty of lawyers would be out of work.  Educators, though, shouldn't be subjected to such torture.

William McCallum, one of CCSSI’s authors, wrote in the comments section of an article appearing on The Atlantic Magazine website, written by Barry Garelick, ``I agree with you that there is a lot of misreading of the standards out there in the field, and this is a problem.’’  Such arrogance.  The real problem is that CCSSI is poorly written, not only substantively, but also in its lack of clarity.

7.G.2’s ``Focus on constructing triangles from three measures of angles or sides...‘’ is at best, ambiguous.  Writing intelligible English is not the same as constructing logic gates, where the definition of ``or’’ invariably includes the possibility of both.  The parallelism in the sentence implies you are given either 3 angles or 3 sides, but we suspect it’s supposed to mean the following: ``Focus on constructing triangles given various combinations of three angles and/or sides.’’

Even our version overcomplicates a simple idea; there are only six combinations to analyze: AAA, ASA, AAS, SAS, SSA, and SSS.  The question then becomes: are there sufficiently common issues that justify lumping all six into a single standard, or was it simply done for convenience (gee, they look related)?

Beginning with SSS, we can say with confidence: this doesn’t belong in Grade 7; it’s far too simple.  Given three lengths, you can either construct a triangle or you cannot.  If you can, the triangle inequality theorem puts into mathematical terms what every first grader knows: cutting across the grass is faster than walking around the perimeter.  We wrote in Mathematical Tools – Part 1 that students should learn how to use a compass in Grade 4 and they should use the compass to demonstrate why the theorem is true.

Not only does CCSSI cover SSS too late, it doesn’t go beyond a simple binary analysis: either the triangle exists or it doesn’t.  CCSSI’s choice of the verb ``noticing’’ as the ultimate goal of the analysis lacks any element of challenge.

How will this standard be tested?  The National Assessment of Educational Progress (``The Nation’s Report Card’’) didn’t cover this topic, but we found a practice problem on the Tennessee Comprehensive Assessment Program (TCAP) Achievement Test–Grade 6 (produced by Pearson) which illustrates the type of question that we would expect:

With a binary analysis, the options for giving our students something to really chew on are limited, but that’s where CCSSI is taking us.  A student is supposed to ``notice’’ that, yes, a 4cm, 10cm, 13cm triangle can exist; the others are impossible.  It’s a one-step application of a simple concept, repeated.

As usual, we at ccssimath.blogspot.com prefer ramped up problems where a student needs to take a simple concept and thoughtfully apply it.

Example 1
You’re given sticks of length 7cm, 8cm, 9cm, 15cm and 16cm.  If you use 3 sticks at a time, how many different triangles can you create?  And a similar problem.

Example 2
A triangle has sides with whole number (integral) lengths.  If two of the sides have lengths 8cm and 12cm, what are the possible lengths of the third side?

It's a challenging problem based on a simple concept.  The O. Henry story ending to Example 2 is what sets it apart from the mundane; it’s difficult to put into words the parallel thinking that must occur in someone’s brain to discover the twist, but we’ve witnessed that moment of epiphany (when students realize that the 12cm and unknown sides switch roles).

Bonus: this problem can be re-posed using non-integral sides when students later study double inequalities.

Now let’s look at AAA: given three angles.  AAA is the odd duck of the six combinations because there are not zero, one, two or three, but infinitely many triangles (assuming, obviously, that the angles correctly add to 180º).


When would CCSSI have students be formally introduced to the concepts of infinite and infinity, as well as the mathematical meaning of ``undefined’’?

Answer: In mathematical terms, after an infinite period of time has passed; or more simply put, NEVER!

We previously berated CCSSI for glossing over the concept of zero, but this omission is preposterous.

Oh, sure, the word ``infinitely’’ does appear, initially in the sixth grade.  6.EE.8 states, in part: ``Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.’’

Is the number of points on a half-line the situation in which to introduce the concept of ``infinitely many’’?  We think not.  (The solution is not even countably infinite, which should come first, anyway.)

Infinite and infinity need their own standard: a proper introduction, and they should be distinguished.  Certainly ``an infinite number of’’ should precede ``infinity’’ and ``undefined''.  Whether it be how many whole numbers there are, how many times you can divide 0 into a number using repeated subtraction or long division, or how far an ant can walk (or how many steps it will take) around a circle, there are plenty of ways and nice activities to introduce these concepts, and infinite and infinity can and should appear before Grade 6.  Infinite and infinity are strange concepts to wrap one’s mind around, and they invariably induce odd feelings.  Getting students to think about infinite and infinity and to grapple with their implications shouldn't come with offhanded remarks; lessons should be explicit.

Oddly enough, in CCSSI the word ``finite’’ appears in Grade 5 (without a definition) with respect to decimals, and ``repeating decimal’’ makes its appearance in Grade 7, without a reference to ``infinite’’.  Where’s the cohesion?

Once again, CCSSI drops the ball on critically important concepts, and the ball bounces forever for an infinite period of time.

Okay, back to AAA.  

By CCSSI’s own reckoning, analyzing a combination of three angles really doesn’t belong in Grade 7 at all.  According to CCSSI, students don’t begin to analyze triangle angle measure until the 8th grade, so how can 7.G.2 come before 8.G.5 ``Use informal arguments to establish facts about the angle sum...of triangles...and the angle-angle criterion for similarity of triangles.’’?

The useful discovery from analyzing AAA is not ``noticing’’ that there are an infinite number of triangles that meet the criteria (CCSSI’s pointless goal), but that those triangles are similar.  Accordingly, we’ll touch briefly on similarity and its related concepts.  The sequencing and pacing of ratios, proportions and similarity needs to be correct, or else students and teachers will be left with a convoluted mess and little or no understanding will result.

We don’t understand why the term ``similar’’ takes so long to emerge in CCSSI.  Even young children recognize the mathematical concept of similarity and can choose two objects from a large group that have the same shape, albeit with completely different sizes.  As a general concept, similarity should come long before any formal treatment, and long before fractions and ratios.  Students should also learn from a young age that the mathematical definition of ``similar’’ is different than the colloquial meaning.

The sequence should be: similarity as a concept comes first.  Students can measure the lengths of sides of similar objects in Grade 3 or 4 and discover that if you divide the sides, the quotient is the same.  These problems can be revisited after students learn to divide decimals.  Both of these can be done long before they formally learn about ratios and proportions.

With the idea of similarity firmly in mind, examples with similar shapes can be woven throughout the curriculum: first as part of learning fractions and ratios, and ultimately in proportions.  We know proportion problems are difficult.  They certainly shouldn’t be stuffed into the same year that students first learn about ratios.

Formal introductions to ratios and proportions do not belong in the same year because it’s too much, too fast.  Like fractions and algebra, ratios and proportions historically have caused innumerable problems for students.  Great care must be given to teaching ratios correctly: concept, notation and problems.  Students need a lot of practice with ratios before they can solve proportion problems.  Moreover, both ratios and proportions should always be connected to the concept of similarity because the visual nature of similarity will help students understand and solve proportion word problems.

The following question was given on the 2005 NAEP:

Even with a calculator, only 21% of 12th graders could solve this proportion correctly.  Tellingly, the sample student responses that NAEP provides show no student making a drawing.

CCSSI gets the whole sequence wrong, and is not going to lead to better understanding of ratios, proportions and similarity.  CCSSI introduces both ratios and proportions in Grade 6, similarity in Grade 7, and the triangle angle sum theorem in Grade 8.  What a mess.  You’re reading it here first: CCSSI as written is not going to lead to improved results on problems like the one above.

To conclude, AAA also doesn’t belong in Grade 7; analyzing AAA and making a formal connection to similarity belongs soon after the triangle angle sum theorem, which CCSSI puts in Grade 8, but we think belongs in Grade 5.

Students shouldn’t use ``informal arguments to establish facts about the angle sum...of triangles’’, whatever those arguments might be; they should use protractors and structured activities to achieve concrete results.  Assembling all of the pieces for a complete understanding of the attributes of angles within triangles can be made into an investigation long before Grade 7: (1) the three angles of a triangle add to 180º, (2) knowing three angles, nay two, is sufficient to determine a triangle’s shape, and (3) similar triangles have the same ``shape’’.

Instead of CCSSI's ordering, the correct sequence should be: Ratios come after fractions; proportions come last.  Then proportions can be used to solve problems with similar figures.

We’ll discuss ratios, proportions and similarity in greater depth in another blog post.


Lastly, we take a look at SAS, ASA, AAS, and SSA.  To begin, we can relate from experience that the notions of ``included’’ and ``non-included’’ sides and angles challenge even high school students (e.g., tenth graders learning geometry proofs in New York State Regents courses).  Do these notions, and the congruence theorems, belong in the seventh grade, and what is the purpose?

Our contention is: discovering that various combinations of three sides and angles ``determine a unique triangle, more than one triangle, or no triangle’’ is not a valuable lesson at this age and does not lead to a useful deepening of learning by extending these ideas further.  The significance of such discoveries will be lost unless they lead to real applications, such as the ability to prove a geometric statement or a recognition that a geometric statement is false.  Leave the SAS, ASA, AAS (and SSS) congruence theorems until high school, if proofs are going to be taught.  Let high school students do the geometric construction and recognize the subtlety that SSA is a non-theorem.

Why do we think congruence theorems belong in high school, if they are to remain at all?  Consider the following ASA congruence proof, which appeared on the Grade 12 NAEP in 2009:

Only 2% of 12th graders who took this exam could do the proof.  Why would we want to push congruence theorems into Grade 7?

Congruence as a concept that can be applied to problem solving is quite limited (and, in fact, kind of boring.)  Similarity is far more interesting and creates all kinds of complications that can lead to challenging problems.  The following questions show the limitations of congruence in poseable problems:

69% of eighth graders got this NAEP question right in 2009, but we are not impressed with their ability on a one-step recognition task.  We wouldn’t even deign to call it a math problem.

New Jersey’s Department of Education posted this sample question for its HSPA high school leaving exam:

This is rote memorization.  Where’s the thinking?

Lessons teaching proofs of congruence using SAS, ASA, and AAS (and SSS revisited) are drawn out over a period of weeks in even a moderately rigorous geometry course.  It’s not a one or two day lesson.  As for SSA (the letters of which the cheeky teacher will reverse to drive home the point), the fact that this combination doesn’t determine a unique triangle is most effectively shown by...using a compass!  Yes, the tool that CCSSI doesn’t want to introduce until high school anyway.

We should clarify that leaving congruence theorems until high school does not mean that students shouldn’t be able to recognize situations when sides and angles are equal.  Students should be learning geometry constructions such as angle bisectors, perpendicular bisectors, lines through a point perpendicular to another line, etc., long before high school, but these can and should be understood through recognition of symmetry and other means, not through congruence theorems and formal geometry proofs.

In the high school geometry introduction, CCSSI states ``During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent.’’  That’s a complete distortion, 7.G.2 being the only relevant standard in Grades 6–8.

How do we know that some schools might use 7.G.2 to push congruence theorems into Grade 7?  Because it’s already being done.  Here is a guide for teachers we found on one Nevada county’s school district (we cut out the middle section for clarity):

With CCSSI’s age-inappropriate and scrambled progressions and its convoluted and contradictory language, students are going to be exposed to the wrong math at the wrong time.  We can understand why educators are confused, and no, Mr. McCallum, it’s not because they’re misreading.

7.G.2 should be dropped altogether, and the various combinations of angles and/or sides separated into age-appropriate activities.  Students should learn about and use protractors, compasses and other tools as necessary.  The point is that things shouldn’t be lumped together into one standard just because they can.  That will result in missed opportunities for real understanding and effective problem solving opportunities.