The concept of area – Part 2

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

NAEP’s stark revelation of the extent to which students fail to understand the concept of area and lack basic problem solving skills warrants not a simple tweaking of current methodology, but a complete rethinking of how the topic of area is presented.

Existing mathematics curricula compartmentalize math skills into discrete chunks, the main benefits accruing to educators: it’s easier to teach and easier to write tests.  The math bite approach to imparting skills is worse than the much-maligned notion of teaching to the test; it institutionalizes a testing regime, at the expense of developing students’ thinking skills.  Even if they learn every standard, the parts remain disjointed without students gaining the real math wisdom that comes from synthesizing parts into a whole.

In earlier posts, we have tried to convey this notion of synthesis.  We discussed our general philosophy: how applying concepts in useful ways abstracts the concept.  We continue to advocate the concept → formal introduction → applications → abstraction sequence.  Once concepts are synthesized, i.e., reach the abstract stage of understanding, actual thinking emerges and those concepts are ready to be applied in new, previously unencountered situations—and isn’t that the real point of education?

Because area is a longitudinal geometry topic that winds through CCSSI from elementary through high school, it can be analyzed for this distinction: taught piecemeal or in an integrated fashion?  We have examined CCSSI’s approach to the topic of area and we find little that holds promise.It starts (we think) in 2.G.2, which states, ``Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.’’  We discussed in our previous blog post Counting and Its Applications how applying counting to measuring length and solving shape puzzles abstracts the idea of counting, while the latter also stirs an inchoate concept of area.  But finding the ``area’’ of a rectangle in the second grade by counting squares is NOT an application of counting that we envisioned, and does nothing to impart any real concept of area, one because there’s no logical path out of this exercise, but especially because it does not say that’s what you’re doing.  2.G.2, as presented, is no more than a menial, dead-end counting exercise, a rehash of a concept that already should have reached closure.

2.G.3, which suggests dividing a shape into ``equal shares’’, heads towards fractions, but its last sentence ``[r]ecognize that equal shares of identical wholes need not have the same shape’’ lurches awkwardly into an entirely different direction, i.e., that students should understand two differently shaped regions can have the same area.  Is 2.G.3 ultimately about fractions or about area?  We have no idea, but if it’s about area, why doesn’t it mention the word, and besides, how can a second grader visually recognize (other than being told) that two differently shaped regions are ``equal shares’’, i.e, they have equal area, without some related exercises to prove the point?  (NAEP already showed that eighth and twelfth graders have difficulty with this.)

2.G.2 seems to ``measure area’’ and 2.G.3 seems to ``compare area’’; we cannot decide whether these are CCSSI’s unwitting or intentional attempts to discreetly introduce area as an attribute, albeit without mentioning the word ``area’’.

If nothing else, the order is backwards, because in the correct approach to learning measurement, one should always compare two things informally before formally measuring them.  We discussed this order of learning in our previous discussion of the concept of length.  But whichever order CCSSI followed, it overlooks the essential introduction to the concept of area without the constraints of mathematical operations.  It is the initial introduction to the concept that builds the foundation of understanding from which all other aspects of area emanate, and NAEP has shown that concept is one of students’ weakest skills.

We don’t think the way it’s presented, either second graders or their teachers will get the point of 2.G.2 and .3’s exercises.


Other than being a directionless introduction to area, we vehemently object to 2.G.2 as a counting exercise on another ground.

Sometimes using counting to solve a math problem is just plain wrong, and when counting is applied in the wrong way, this engenders or reinforces bad habits.  When we see an institutionalized teaching of bad habits in mathematics, we have come to think of it as ``bad math.’’ (original title, right?)  We’re going to say it: CCSSI is fostering ``bad math’’.

Mathematics has more efficient tools than counting, and why not use efficient methods to solve a problem, and have students recognize and reject an inferior method?

In the 2012 Super Bowl, there was a hilarious commercial about a high school graduate who thinks he’s been given a car by his parents.  He stops a man who is out exercising and asks the man, ``Why jog when you can drive?’’

The irony is that while the commercial’s absurdity is deliberate, CCSSI’s is unwitting.

We previously discussed how using ``proto-algebra’’ to find an unknown value introduces the mechanics underlying the higher concept of algebra, and why that is valuable.  Using counting when (a) counting is a concept that has long reached closure, (b) it is not leading to a higher concept, and (c) a higher concept is readily available—is bad math.  It’s bad math on all three levels. (Footnote 1)

In a rejoinder to 2.G.2, we ask, ``Why count when you can multiply?’’

We don’t want children using counting, or reverting to counting when higher order operations are available.  Students should never, ever count when multiplication is called for. (Footnote 2)

In fact, the first time students see a rectangle partitioned into unit squares and are asked the number of squares, they should be presented with the opportunity to categorically reject counting as the means with which to solve this problem.

The question could be posed as ``Should we count up the squares?’’  And when this question is posed soon after learning multiplication, a majority of students (hopefully) will recognize the better method.

Teachers, try this with your class and then have your students sign a petition to remove 2.G.2 from CCSSI.  Students should never be told to, or get into the terrible habit of counting squares in a rectangle to find its area.  However, critical thinking, specifically, learning to distinguish good and bad methods for solving a problem, is a beneficial aspect of an effective education.

The upshot of our pointed remarks is that the formal introduction to area as a measurable attribute and how to measure should wait until after multiplication is learned.


Do any standards concerning area belong before multiplication, if not 2.G.2?  Of course, there’s plenty enough to do with area before you measure it.

Here are some suggested activities we would use to introduce the concept of area:

1. Take a small glass of water and spill it on the floor and have students look at the puddle.  Then hold a bigger glass of water and ask what will happen if you spill it.  Ask what the difference is in the two puddles.

2. Two farmers want to spread some sunflower seeds.  Who will need more seeds?  If we use two plots where the shape is different, but students can determine that the size is the same, then they will know that the farmers need the same amount of seed.

From these examples, the discussion leads to the word ``area’’.  Students can learn that the word ``size'' has many meanings, but one of those meanings is ``area.''  Then looking at various shapes, students can estimate which has a bigger area.  That will lead to a problem where it’s not clear which shape has the bigger area, such as a roundish shape and a long and skinny shape.  Or that two shapes are vastly different, but could have the same area.  (Insert 2.G.3 here; now seems pretty limited in this context, yes?)  Then that leads to a discussion of how one might determine which has the bigger area.  Only after understanding the concept of area and visually comparing areas are students ready to learn about measuring area.

Before they formally measure area, in class activities with paper and scissors, students could have two shapes and cut them into pieces, rearrange the pieces, and do matching exercises to see which shape has pieces left over.  The number of activities is limitless.

Students should understand the concept of area, be able to compare areas that are obviously different, and finally realize that there are better ways of comparing areas other than visually, just as they did previously with the concept of length.

Second graders that learn about area this way would be better equipped, even before formally learning to measure area, to solve the NAEP rectilinear shapes question we discussed in Part I than the fourth and eighth graders who actually took the test.

Once students are practiced with paper and scissors exercises, they will even be prepared for the problem below, which is a magnitude of difficulty above the NAEP area problems we presented, but readily solvable by students with a clear understanding of the concept of area.


Not until Grade Three does CCSSI put area firmly on the agenda.

We know we are being redundant, but it bears repeating: area as a concept and area measurement are not the same, but CCSSI is skipping the concept and plunging into measuring.  ``Recognize area as an attribute...’’ is not enough.

3.MD.5 states, ``Recognize area as an attribute of plane figures and understand concepts of area measurement.’’  (Footnote 3)  Even though the bold-face heading ambiguously states ``...understand concepts of area...’’, seemingly broaching the general concept of area, it in fact has leapfrogged to measurement of area as the header begins ``Geometric measurement...’’  (Footnote 4)

We hope educators will recognize this deficit and insert the concept in there somewhere, perhaps as we suggested above.  We just hope the first, second and third grade teachers standing in left, center and right fields don’t look at each other as the ball drops in for an inside-the-park homer.

Let’s dive headlong into CCSSI’s treatment of area measurement.

It’s obvious to us, but maybe not to the authors of CCSSI: one does not always measure area with squares.  CCSSI’s emphasis on this approach in 3.MD.5, .6 and .7 is narrow and misguided, a fact to which your nearest farmer or candy confectioner will attest.  Teaching area measurement as a tiling of squares is an example of a concept being introduced with testing in mind, rather than life-long learning.  Students will form the notion that area is composed of little squares, and be conditioned to mechanically solve area questions without sufficiently understanding the underlying concepts.  We imagine they will have difficulty with area units like acres and hectares (see 3.MD.5a), the mathematical concept of ``packing’’, and finding the surface area of a sphere, which has no squares at all.

3.MD.6 is the same silly counting exercise as 2.G.2, but simply adds units and calls it measuring.  Reciting unit names does not impart meaning.  This reminds us of our previous critique of CCSSI asking kindergartners to count to 100 without knowing what 100 means.

3.MD.5b is age-inappropriate and has little to do with what third graders are learning.  Covering or ``tiling’’ using squares with no gaps or overlaps, is a more elementary version of the triangle puzzles problem we previously suggested for first graders.

3.MD.7d ``Find areas of rectilinear figures by decomposing them into non-overlapping rectangles’’  is certainly a useful skill for students to know, but this is really two separate skills, the latter in broader form (decompose complicated shapes into simpler shapes) a skill that students should have learned long before the third grade.

``Draw a line that divides this picture of a house into a triangle and square’’ would be a nice exercise for first graders.  We previously talked about the lack of skill at drawing auxiliary lines; why not start students early?

3.MD.7d also states, ``Recognize area as additive.’’  The word ``additive’’ can be thought of in a numerical sense, but also in a conceptual sense and the distinction ought to be made.  Students should understand that area is additive as a concept long before adding the numbers obtained from measured areas.  In first grade, if a student draws a square, she can be asked to ``add a triangle to the top of the square’’  No student is going to draw a triangle that overlaps with the square or isn’t connected.  A student, given a picture of a rectangle (representing a box) containing 6 chocolates in a 2 x 3 array, can be asked, ``If one box holds 6 chocolates, and you add (by drawing) another box alongside the first, how many chocolates can the boxes hold?’’  Students will be developing notions of the additive nature of rectangles long before learning to measure area.

CCSSI completely overlooks the value of emergent learning through a sequence of problems of ever-increasing complexity.  By plopping ``area is additive’’ into the third grade, a lot of time for effectively learning such an obvious but important concept has been wasted.

Limiting area questions to counting and tiling forces students to think about area in a contrived way, teaches bad habits, and does nothing to teach visualization skills, which is critically important to the concept of area.


How should previous discussions of area culminate with area measurement in third grade?  Here’s how we see it:

Real thinking skills develop from learning about area through a logical sequence and the integration of several concepts.  Before multiplication, students (should have) learned about area as a concept.  They are presumed to already have learned about length and measuring length; they will need to apply those skills (measuring area will be a useful application of length and will help to abstract the concept of length).  The last concept students will need is multiplication, which is the most recent concept they will have learned.

Take concept of area + measuring length + multiplication skills + pinch of angel dust and stir

Students are now ready to merge these separate concepts into one idea, area measurement.  (Footnote 5).  Let’s consolidate area measurement of rectilinear figures into two steps:

1. Add the bolded words to 3.MD.7a to read ``Students [f]ind the area of a rectangle with whole-number side lengths by tiling it and multiplying the number of squares in a row by the number of squares in a column, and show that the area is the same as would be found by multiplying the side lengths.’’

2. Replace 3.MD.7d with ``Students realize that finding the area of more complex regions requires visualization skills and a combination of mathematical operations, and are able to solve such problems.’’

Voilá!  These standards as modified will bring closure to rectilinear area measurement (and will help complete the sequence for multiplication from concept to abstraction)...but the topic of area will as yet be in its infancy.

In our next blog installment, we propose (and justify) adding a major area topic to Grade Three.


Footnote 1.  Even if 2.G.2’s intent were to use counting as a bridge to multiplication, it’s wrong.  Addition is the bridge to multiplying, not counting.

Footnote 2.  We don’t want children, adolescents and young adults counting on their fingers to add either, but we see it all the time in classes up to the college level.

Footnote 3.  In the Grade Three introduction, ¶3, CCSSI states, ``recognize area as an attribute of two-dimensional regions’’, but 3.MD.5 states, ``[r]ecognize area as an attribute of plane figures and understand concepts of area measurement.’’  On one hand, we wish CCSSI would make up its mind, two-dimensional regions or plane figures, but on the other hand, we think both descriptions are misleading and limited.  The inconsistency of CCSSI’s terminology is just sloppy, and sloppiness pervades the standards.

Footnote 4. CCSSI repeatedly jumps the gun without laying the proper foundation.  K.MD attempts to introduce the notion of measureable attribute for length and weight, and 2.G.1 talks sides and angles (we already took CCSSI to task for discussing angles before students know what an angle is), but none of those introductions is so misguided as the way area enters the picture.

Footnote 5. We look askance at CCSSI’s Grade Three introduction, ¶3, which states, ``By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.’’  The area of rectangles is NOT ``connected’’ to multiplication, area measurement is much broader than that; we think CCSSI’s phraseology is terrible.