In
2007, the National Assessment of Educational Progress (also known as
``The Nation’s Report Card’’) gave the following task to 4th graders:

This
question demonstrates that even at the elementary school level, it is
possible to usefully integrate several aspects of problem posing: that
the problem be lengthy, that it bring together multiple math skills, and
that the path to the solution not be readily apparent. (We have
previously coined the phrase ``length, connectivity and dimensionality’’
to describe this triumvirate of features.)

As
our regular readers may surmise, we wouldn’t lead off with this
question if there weren’t something more to it. The specific task the
NAEP required students to perform barely scratches the surface of the
issues and learning possibilities contained in this fact pattern.

It
is an ongoing theme in this blog: as elementary school students learn
the basic operations, some abstraction should be taking place as well.
A perfect example of an abstraction lies embedded in the NAEP question:
at its most basic level, the graph quantifies concepts, but looking
deeper, certain patterns come to the forefront leaving the numbers to
retreat into the background. For instance, students could study the
graph to recognize and then describe a trend: middle-length names are
more common than very short or very long names. It is a quantum leap in
mathematical understanding when students make the abstraction that a
pattern or trend matters more than the exact quantities.

These
kinds of abstractions show how even at an early age, students can and
should be introduced to the notion that mathematics is more than about
finding the correct answer. Math is too often thought of by students as
being a dichotomy of right and wrong answers (a belief either
engendered by or perpetuated by weak-skilled educators), and it is this
singular perception that ought to be dispelled (to paraphrase the late
Mayor Richard Daley) early and often.

One
way to introduce students to the notion of subjectivity in mathematics
is through graphing: a series of math lessons leads students to learn
that for conveying certain kinds of information, choosing the right type
of graph matters; by reading graphs and making their own, students can
also learn that graphs can unwittingly, or even deliberately, distort
perceptions.

Example
lesson sequence: in the NAEP question, the vertical axis skips odd
numbers and the horizontal axis doesn’t start at zero. In what
circumstances might such omissions matter?

A classic example:

Students
can and should also learn to judge when a graph is a particularly bad
way to present certain information, such as: in the NAEP graph, you can
determine the total number of students in this class, but do you really
need a bar graph to do that, i.e., can’t you just add up the numbers of
students in the original data? Or even more to the point, in looking
for a pattern or trend, is knowing the total number of students even
necessary?

In
a well-conceived series of lessons on analysis, students can move
beyond the basic arithmetic to learn critical thinking about useful
applications of mathematics.

The
NAEP question works because it requires students to understand the
underlying data, i.e., the information that went into the graph and how
it’s presented, but the question doesn’t assess students’ understanding
of the point of the graph.

Which brings us to CCSSI, and what it offers and what is lacking in the way of graphs.

Until
Grade 5, when it introduces the Cartesian coordinate system (a topic we
will cover separately), there are only 2 relevant standards:

2.MD.10 states ``Draw
a picture graph and a bar graph (with single-unit scale) to represent a
data set with up to four categories. Solve simple put-together,
take-apart, and compare problems using information presented in a bar
graph.’’

3.MD.3 states, ``Draw
a scaled picture graph and a scaled bar graph to represent a data set
with several categories. Solve one- and two-step “how many more” and
“how many less” problems using information presented in scaled bar
graphs. For example, draw a bar graph in which each square in the bar
graph might represent 5 pets.’’

What learning will result from these two standards?

Julia at mathworksheetsland.com
has developed (and is making available at no cost) representative
lesson plans for many of the Common Core standards, and has given us
permission to use snippets here.

In
other examples, Julia’s worksheets asked questions like ``How many more
votes were cast for sausages than burgers?’’ and ``How many friends
voted for pastries and milk?’’

These
lesson plans and tasks certainly seem to align with 2.MD.10. 3.MD.3 is
only incrementally more complex, with its ``two-step’’ problems and
non-unit scales.

What
exactly is the value of CCSSI’s ``take-apart’’, ``how many more/less’’,
and ``compare’’ questions when analyzing a graph? In our estimation:
not much. Cloaking elementary arithmetic problems in the guise of a
pictograph or bar graph is not what students should be doing with
graphs. For instance, in the NAEP question’s fact pattern, you could
ask students to calculate that there are two more students with 5 letter
names than students with 4 letter names, or recognize that there are
twice as many students with 4 letter names as 3 letter names, but why
would you want to? Those are contrived and irrelevant exercises, and
not what students should be taking away from lessons around graphing.

Rather
than setting the highest level of attainment, the language in CCSSI
standards 2.MD.10 and 3.MD.3 form an appropriate classroom objective for
a beginning
lesson on graphing. Combined into one, the standards offer no more
than an introduction to the idea that numerical data can be represented
graphically. To put it more succinctly, 2.MD.10 and 3.MD.3 should be a
class’s first graphing lesson’s objectives, not its last.

***

We
here at ccssimath.blogspot.com would like to suggest a more rigorous
and useful learning sequence for Grades 2 through 4. Following this
sequence, students would be better-prepared to transition to Cartesian
coordinates in Grade 5.

Before
proceeding, we note that CCSSI discusses ``line plots’’ four years
running, in Grades 2 through 5, but first as a device for ``showing
data’’ and then for reinforcing arithmetic, but never for learning data
analysis.

How do we propose that a unit on graphing begin? Walk into a first or second grade class and you are virtually certain to find a poster that shows students’ names and birthdays. We’ve been to enough classrooms to know they are common.

There
are two problems with this: (1) the information is never presented as a
graph and analyzed and (2) these posters are created by the teacher,
both educationally useless.

It
would be a perfectly lovely (and hands-on) second grade activity that
students first create charts with vertical stacks of student pictures
(photos or drawings) born in each month. Then the activity can evolve
to create a vertical (leave horizontal pictographs for another time)
pictograph—or line plot, if you will—with a smiley face or other symbol
used for each student. That way, students will already be learning to
interpret visual, not numerical information, such as: in which month(s)
were the most students born? (See 2.MD.9)

Lesson
2: Let’s say you now have a vertical pictograph as described. Then you
can ask students a question similar to the NAEP question: Vicky just
moved to our city and joined our class. Her birthday is in April.
Modify the pictograph to incorporate this new piece of information.

What second grader couldn’t do this?

(We don’t mean our numbering scheme to imply each lesson should last one day; it’s the progression that is important.)

Lesson
3: Pictographs evolve into bar graphs; students can draw a line around a
vertical stack of smiley faces and be able to see the ``bar’’ that
results.

Without
even stating such, students will internalize that symbols in the
pictograph are one abstraction away from individual student photos and a
bar graph is one abstraction away from a pictograph.

Once
students are comfortable with bar graphs, it’s time for some deeper
analysis. Using graphs like in the NAEP question as a starting point,
days could be devoted simply to reading and interpreting information.
Of course, students should be converting data into bar graphs as well,
but CCSSI completely fails to acknowledge (and remind math educators)
that it is a far more common task to be asked to read a graph than it is
to create one.

Why
are data analysis tasks missing from CCSSI in elementary grades? We
have no idea. Even the NAEP question extends analysis beyond the
learning requirements set forth by CCSSI.

Lesson
4: Transition from separate categories (such as months, days or
animals) on the horizontal axis to quantifying an independent variable,
such as the number of letters in a student’s first name. That’s another
level of abstraction. Limiting students to counting the items in
multiple categories rather than quantifying an independent variable is a
needless dumbing down, but read the Standards closely: that’s what
Common Core is doing.

In
CCSSI’s elementary standards, data analysis never begins. Using a
graph as a mere contrivance is no math at all. Just because you can
create questions around a graph doesn’t mean you should. All of the
poseable questions that the CCSSI standards envision in 2.MD.10 and
3.MD.3 can just as easily be answered without a graph at all. Students
may enjoy a diversion from dreary worksheets covered with nothing but
numbers, but they really haven’t learned the utility of graphs from
lessons like that.

***

In
our proposed Lesson 4 above, students would have reached the point
where they have sufficient understanding to solve the NAEP question.
How did students, in fact, perform on this question?

Not
well, it seems, and NAEP helpfully provided one representative wrong
answer, pictured below, which may capture the essence of a deeper
problem in math education.

We’d
venture to guess that NAEP chose this particular error because it (and
variants) were commonplace. That a student understood the first step,
the implicit task of counting the number of letters in Victor’s name
doesn’t mean they could ably proceed. Relating the number 6 to two
separate concepts without discernment shows a fundamental disconnect,
and an analytical breakdown. Without an abstract understanding of the
concepts, the student didn’t understand that in such a graph, only one
of those concepts is graphically depicted and can vary, but the other is
fixed and already appears in a list.

The error is akin to putting a thirteenth month into the birthday month pictograph we described earlier.

58%
of 4th graders got this question wrong, meaning they either couldn’t
read or understand the graph or the task, or worse, couldn’t distinguish
one 6 from another. They may have been well trained in counting and
arithmetic, but aren’t evolved past rote procedures.

Obviously,
students need more and better lessons on the basic concepts of graphing
and its resulting abstractions, not less, as CCSSI is requiring.
There’s nothing in CCSSI from K to Grade 4 that would lead us to
believe students will be better prepared for questions like that on
NAEP. That’s unfortunate for our students.