Counting and its applications

Counting, notwithstanding the, er, tale of Clever Hans, is so basic that it seems to be an innate ability in several animal species, according to an article published in Scientific American™ in September 2009.
[Clarification: the word ``basic'' above is being used in context as ``fundamental'', not ``simple''.  See tweet below linking to this blog post that missed the point.]

Parents teach their youngsters to count to 10, or children learn it on Sesame Street, and there are numerous studies showing babies can conceive of small numbers and preschoolers can judge greater or less than without knowing exact numbers, but by understanding approximate counts.

In mathematical parlance, the counting numbers are also called the ``natural'' numbers, as existing in nature.

So how is it possible to teach counting, the most natural of mathematical skills, unnaturally?  The worst way is to force memorization of numbers and number names before the concept is understood.  What kindergartner do you know that owns 90 of anything, except maybe for Beanie Babies?  Open a Scrabble set and look at the 100 tiles.  Is that a number of objects that a kindergartner needs to comprehend in their child's world?  Memorizing the names of numbers at the age of 5 before actually understanding the concept of the number is simply put, backwards.  Mathematics should not be taught that way.  A child should understand the number along with the name, before reading, and then writing the number.

If particularly anal parents want to push too-large numbers on their child at an early age for fear of falling behind, so be it, but the paranoia should not be systemic.

CCSSI standard K.CC.1 ``Count to 100...’’ carries into the next year's standard 1.NBT.1 ``Count to 120...’’ along with a few variations on the theme.  We won’t pause long to chide CCSSI for such arbitrary parameters because there is a much more compelling standard in the parallel Measurement and Data strand that pertains to counting, which creates the potential for some real advancement of thinking and analytical skills in early childhood.

K.MD.2 states, ``Directly compare two objects with a measurable attribute in common...directly compare the heights of two children and describe one child as taller/shorter.''  1.MD.2 states, ``Express the length of an object as a whole number of length units...; understand that the length measurement of an object is the number of same-size length units that span it...’’

It’s not clear that the following useful link between strands and years was made intentionally, because strands were developed separately, but CCSSI gets the order and grade assignments correct.  Here's why:

A progression from the counting of objects in Kindergarten to measuring length as the first critically important application of counting in Grade One broadens thinking about a mathematical concept and has the potential for students to learn important interconnections, rather than simply relearning, reviewing or linearly extending previously learned material (which counting to 120 does), a triad which have long been the undistinguished hallmarks of American mathematics education.

How should teachers develop the topics of length and measurement?  CCSSI, which asserts it is a set of standards, not a curriculum, blurs the line between standard and curriculum with two suggestions: kindergartners should ``directly compare the heights of two children and describe one child as taller/shorter'' and first graders should measure the length of an object by ``laying multiple copies of a shorter object (the length unit) end to end.’’  It’s not clear where CCSSI intends these examples to fit in the overall scheme or whether it deems these two skills as sufficient.  Teachers are looking to CCSSI for some guidance and they are not getting it.  We hope book writers and teachers don’t take CCSSI’s suggestions literally as the definitive exercises to teach length and measurement.  Both the comparison exercise and the measuring exercise are abstractions, and as readers of this blog have seen, we emphasize understanding the concept before making the abstraction.

The steps in the learning sequence need to be spelled out more clearly for the benefit of early childhood educators. 

Luckily, children’s first exposure to the concept of length comes early, most likely in thinking about height: adults are tall; children are often complimented for growing taller; giraffes are taller than other animals; in New York, the Empire State Building is tall; on the plains of Kansas, a grain elevator is tall; and in Colorado, mountains are tall.  Height is an excellent central topic for a kindergarten picture book or a first grade primer, just in case young children haven’t thought about height enough.  After various classroom activities on the topic of height, then you can talk about, um, snakes.  Snakes are long, dachshunds are kind of long, trains are long, parades are long, Smart Cars are short.  Just picture an entire class with their arms outstretched–what fun!  The connections between height and length will become obvious.

Thinking about tall and short, long and short needs to precede comparing.  K.MD.1 seems to be making a stab at the concept of length with ``Describe measurable attributes of objects, such as length or weight.  Describe several measurable attributes of a single object.''  But this approach to the concept of length is posed so abstractly (it already sounds like a test question) that some high school students couldn't achieve the standard.  We think our activities-based approach improves on CCSSI.

After talking about length at length (ha!), students will be able to abstract that both height and length are really the same concept.  Then they will be ready to compare lengths.  (Concept lessons can overlap, but the abstractions are sequential.)  A math lesson can use children to compare heights and drawings to compare lengths.  Who is taller, Mary or Sally?  Which snake is longer?
(We don't object to comparing height or weight in kindergarten (see K.MD.2), but it's not so much a mathematical concept yet as just basic awareness and language skills.  We're sure that kindergartners do not need to recognize that a Cadillac is longer than a Volkwagen.)

1.MD.1 states, ``Order three objects by length; compare the lengths of two objects indirectly by using a third object.’’  These standards wildly jump the gun: in CCSSI’s rush to get ahead, it ignores the fundamentals.  The first part, ordering three, is far more advanced than comparing two, and the second part is totally unnecessary.  Comparing the lengths of two objects directly (see K.MD.2) is sufficient.

Exploring the attribute of length, through examples such as taller and shorter, and then longer and shorter, completes the establishment of the concept of length, and leads naturally into the topic of measuring.  But the steps from concept to abstraction are also distinct and sequential.

You could have a picture of a giraffe standing next to a bunch of rabbits standing on each other’s heads and ask, ``How many rabbits are standing on each other’s heads to be the same height as the giraffe?’’  Students think they are counting, but then they may realize they are also measuring before they even know the word.  It can be a watershed moment.  For the abstraction of measuring, again height comes before length; children have seen that people are different heights long before they recognize that walking to school takes longer than walking to a friend's house, and a room may be longer than it is wide.  Children understand rabbits and giraffes.  Only after they learn from examples the concept of measuring from counting unit lengths will students understand the purpose of the CCSSI-suggested exercise where they lay measuring units end to end to measure the length of an object.

To summarize, children should be
1. Recognizing the concept of length
2. Abstracting that height and length are the same attribute
2. Comparing lengths (``Is he a little bit taller or a lot taller?'' could be a transitional question.)
3. Counting units to connect the concepts of length and measurement
4. Using unit lengths to measure
These steps are distinct and each is necessary to mastering length and measurement.

Additional benefits of teaching length and measurement coherently: The topic of measurement dovetails nicely with another Grade One topic, subtraction.  ``How much taller?'' connects measurement with subtraction after students learn to subtract, and has the further benefit of abstracting subtraction because students will recognize that subtraction applies to both numbers of objects and numerically comparing the lengths of objects.


But now we propose a single change to a strand which, by introducing some complexity, will result in a significant deviation from CCSSI’s linear thinking.  Why introduce only the concept of length in Grade One as an application of counting?  If you expose students to this one application, they will connect counting to length, and make the longitudinal connection; but if you introduce students to two applications of counting, in close succession, they will abstract and understand that counting has not just two, but myriad applications.

CCSSI waits until Grade Two to broach the topic of area: 2.G.2 states ``Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.’’  This presents three problems, the first being that the basic skill of counting is being rehashed yet again.  By waiting until Grade Two to make the next linear extension of counting, the conceptual connection to length is too remote and will be lost; and finally, as we have seen before, it fails to lay a proper foundation in establishing the concept of area - there is no clear objective to what students will gain by counting same-size squares.

(Incidentally, is the word ``partition’’ supposed to appear in a second grade workbook exercise?)

By counting boxes, CCSSI simply compels students to review counting, albeit with some vague notion that they are going to pick up on the concept of area.  We don’t see the concept of area being evoked by such an example.

Instead, here’s an age-appropriate example that introduces first graders to the concept of area more effectively, without introducing abstractions or formulae.

This puzzle-like problem can be done with manipulatives, and although challenging, is well within the capability of a first grader.  It introduces first graders to the concept of area, using the previously learned skill of counting, but by counting in a new and different way.  We only seek to put the concept of area into the first grade; the abstraction can come later.

In what amounts to a severe dumbing down of standards, CCSSI waits until Grade Six to go beyond the calculation of area of rectangles (and ``rectilinear figures’’, see 3.MD.7d) by counting (Grade Two) and measuring and multiplying (Grade Three).  As the above example shows, the concept of area of even complex shapes can be introduced far earlier.  Sixth grade (6.G.1) is well past the age when the concept and formula of the area of a triangle (or non-rectilinear shapes) should be learned.  (The area of a triangle will be the topic of a forthcoming blog post).

2.G.2 states, ``Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.’’  CCSSI Grade Three third ``critical area’’ states, ``Students recognize area as an attribute of two-dimensional regions.  They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area.’’  The concept of length is already in Grade One, but we submit that the concept of area should be moved squarely (ha!) into Grade One as well and using more complex shapes than squares.  (Note: Using a balance with unit weights can also be a lesson topic for a science class, a third application of counting.)

By introducing both concepts of length and area in Grade One, a foundation is laid not just for understanding that counting is not the end, but rather a means to an end, an abstraction the value of which cannot be overstated, but also for far more interesting topics in geometry in ensuing years and at an earlier age without being age-inappropriate.