This is the second part of a multi-part blog post on the concept of area.
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.
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.
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?
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.)
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’’.
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.
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’’.
has more efficient tools than counting, and why not use efficient
methods to solve a problem, and have students recognize and reject an
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.
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)
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.
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.
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
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.
any standards concerning area belong before multiplication, if not
2.G.2? Of course, there’s plenty enough to do with area before you
Here are some suggested activities we would use to introduce the concept of area:
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
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.
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
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.
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.
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.
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.
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)
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
Let’s dive headlong into CCSSI’s treatment of area measurement.
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.
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.
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.
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
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
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
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
How should previous discussions of area culminate with area measurement in third grade? Here’s how we see it:
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
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
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.’’
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.
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
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.
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