The concept of area – Part 1

This is Part 1 of a multi-part blog post on the concept of area.

In 2005, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

This question was given to fourth graders, 47% of whom answered correctly.  This result would have been mediocre if tiling had been taught by fourth grade and good if tiling had not been taught (i.e., students were able to figure it out on their own)—but we don’t know which.

We do know that fourth graders, under Common Core, will not yet have been introduced to the concept of the area of triangles, either in the context of a tiling question or as a general formula, which remains years off.  The delay is an unnecessary dumbing down of the curriculum; under either scenario above, 47% correct shows the question is age-appropriate (We will talk more about the formula for the area of a triangle in another part of this multi-post.)

Tiling before fourth grade, according to CCSSI, is only going to be done with squares (3.MD.7a).  Otherwise, only vague references to composition of triangles are made, in the bizarre suggested task for kindergartners (!) to ``join...two triangles with full sides touching to make a rectangle’’ (K.G.6) and the entirely age-inappropriate 1.G.2 standard, which includes, ``[c]ompose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles)...to create a composite shape, and compose new shapes from the composite shape.’’

(We suggest readers look at the complete standard 1.G.2, which includes three-dimensional right rectangular cones.  We’ve even found one school district that is extracting the words from each math standard and making them vocabulary words for that grade.  Now first graders will have to know the following words: addend, analog, array and data.  We kid you not.  Do you yet see what a monster CCSSI is creating?)

That’s all the background in composition (and decomposition) of non-rectilinear shapes that students are going to get before Grade Six.  Even the simple notion of putting two triangles (as halves of a square) together to equal one square appears nowhere before then.


In this blog post, we take advantage of NAEP’s occasional latitudinal comparisons: the identical question above was also presented to eighth graders (77% answered correctly).

We were not surprised by these results; this is a fairly mechanically solvable problem, with little conceptual or abstract understanding of area needed.  Students merely need to count the shaded squares, and combine two pairs of triangles into squares.  We would hope students have learned to do this before entering high school.

Other than mechanical tiling problems, relying on nothing more than rudimentary addition and that two halves make a whole, what other area problems can students do reasonably well?  Consider the following:

NAEP 1992
This question typifies another really low level area skill, calculating the area of a rectangle, which also can be solved mechanically, and in a single step.  That only 20% of fourth graders answered this correctly, but 65% of eighth graders in the same testing year got it right is at least a positive trend, if not encouraging.

However, to paraphrase Aristotle, the ability to tile (add) and calculate LxW (multiply) does not a conqueror of area make.

Let’s now look at some less-than-simplistic concept of area questions (and by simplistic, we mean simplistic by American education standards).

Consider the following:

NAEP 1992
What we found interesting in the report was not the results, but the trend: while 72.4% of fourth graders got this fairly easy question right (not surprising), in eighth grade (same testing year) it was a barely besting 74.2%.  (In both grades, answer D was far and above the most common wrong answer.)  Here’s a problem that requires no calculation, but only a rudimentary understanding of the concept of area and an ability to visualize.  We are calling this less than simplistic, if you recall.  We can only wonder: have we wrung common sense out of students by the time they reach high school?  Do students think ``area’’ means ``convolutions’’?  Did they think this was a trick question?  Or have students simply never really understood the concept of area in the first place?

Certainly this problem makes evident that a basic technique for area questions, the fundamental ability to visualize regions as the sum of smaller, simpler regions is either not properly taught, not properly learned, or both.  In these types of area problems, the question is going to require more than a simple calculation; there needs to be least some additional thinking.

Area problems nicely illustrate the conundrum because they are visual, but in general, students are not learning and employing the basic problem solving technique of breaking things down in order to make more complex problems easier (in some cases, possible) to solve.  In the broad realm of ``breaking things down’’ is an essential skill necessary for these area questions: we have seen time and again the inability to visualize or draw what is commonly called an ``auxiliary line’’ to help solve geometry problems—a shortcoming that is pervasive.

Let’s look at an example of what we mean:

2009 NAEP - Grade 8
With the added complication (compared to the previous carpet problem), the results of this question were dismal (22% correct).  The surprise, though, was the most popular choice of wrong answer.  While we might have thought the difficulty was caused by the need to understand and correctly apply the Pythagorean Theorem or be familiar with the ubiquitous 3-4-5 right triangle (or simply that a potentially distracting diagonal length was given, the number 5 being necessary to use, but not in the final calculation), that the most students (32%) completely ignored the 5 and chose answer B (which would have been the area of a rectangle of length 8 and width 4), shows a complete lack of understanding of concept, not simply confusion about a triangle’s base, height and hypotenuse.  Students who chose answer B demonstrated no understanding of area at all.  (We think 42 sq. cm would have been a better ``distractor’’ answer, but it seems so many were drawn to 32 that our trickery would have been for naught.)

The results of this question tell us that students are not learning or practicing visualization skills (where the path to the answer is not immediately clear) and are complete unversed on age-appropriate, useful problem solving techniques.

Of course, students must be exposed to repeated examples before they become brave enough to venture out on their own and try drawing lines, because the correct decomposition may not always be immediately apparent, but this practice is almost completely lacking in American education, both by students and teachers as well, as the above problem clearly illustrates.

To recapitulate: if students had simply drawn the obvious vertical auxiliary line that the problem screams for, they would know the answer could never be 32.

Drawing auxiliary lines is also part of our notion at ccssimath.blogspot.com that problems should have ``length, interconnectivity and dimensionality.’’  (Please read our post on Wholesale whole-number murder and redemption.) The way to solve a problem should not be immediately apparent, but American students are accustomed to seeking the quick way out, or being allowed to, which is even worse.  For a problem to have any significant educational value, there need to be some intermediate steps, requiring some tension which leads to insight, before a way to proceed becomes clearer.

Let’s not fault the students.  It’s our responsibility to lead the way.  We the bloggers have noted the absence of not only ``auxiliary line drawing’’ technique ability in students, but also lack of creativity in writing such problems that call for such technique, evident throughout our study of curricula, lesson plans, test questions (above type of question excepted) and textbooks.  See our problem (also used in the Wholesale posting) for an example ``requiring’’ an auxiliary line.  NAEP is exposing the issue; we have to find the solution.

It seems silly to state the obvious, but students can be trained to solve complicated area questions, and by extension, learn in general to persevere (to use the CCSSI term) when the solution to a problem is not immediately evident.

Let’s see how bad the lack of concept and visualization skills really is.  Consider the following:

2007 NAEP - Grade 8

In this short constructed response question, only 13% of students completed the task correctly.  It shows that while eighth graders can calculate the area of already tiled triangles, they cannot tile triangles on their own, and they cannot visualize the correct arrangement.  (We also know they can calculate the area of a rectangle, if they ever got that far.)

There is a certain tension in this problem because it has dimensionality (our word meaning the path to the answer is not obvious) and that gives the problem value.  Students may try something first and get it wrong, such as tiling the triangles into a large square.  They will (hopefully) see that the square violates the conditions of the problem and will have to start over.  But having to start a problem over from the beginning because the path taken didn't work out is frustrating; when students can't see the correct approach right away, they just give up and ask for the answer.  The capitulation and coddling is systemic; American students are not often asked to struggle with a problem, and not often allowed to struggle.  We don't know which is worse.  Either way, they are not practiced in struggling, so when such a question appears on an exam, this NAEP example shows they are ill-prepared to handle the tension.  It is because the correct approach was not immediately apparent that made this question so difficult for students that lacked basic problem solving skills.

As our final example, in 1996, 4th, 8th, and 12th graders were all (!) given variations of the following question:

NAEP 1996

From 4th to 8th to 12th grades, the percentage answering this short constructed response question increased from 16% to 35% to 42%.  We think, arguably, that with respect to their ages and expected abilities, the 4th graders performed the best.  This question was still considered of ``medium'' difficulty for graduating seniors.

That there needs to be some notion of auxiliary lines and cutting and repositioning of shapes, all visualization skills, this and the previous questions we presented clearly demonstrate a pervasive problem, and the solution needs to be approached emphatically, starting early, with questions increasing in complexity every year.  Area is one topic that doesn’t reach closure; it allows applications of ever-evolving complexity.  Let’s not forget that calculating the area of complex shapes is one of the fundamental aspects of integral calculus.  We should create a strong foundation from the get-go.

C1. To conclude this installment in our blog, we are skipping past the presentation of some obvious examples of shapes that require decomposition to determine the total area, etc., to an example of an area problem that combines our preferred elements of length, interconnectivity and dimensionality:

We won’t say here at what grade level this problem should be introduced, but it’s probably a lot earlier than the typical math teacher will think it belongs.  We at ccssimath.blogspot.com have confidence in our students to rise to the challenge; it’s CCSSI that’s giving them short shrift.