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.
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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.
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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.