The instructions seemed adequate: plot your data, draw a best-fit line, and extrapolate the line to find the y-intercept. Graph paper and rulers were available on a table at the front of the classroom. To me, the implication was crystal clear that the graph should be done accurately on graph paper, and the line placed carefully and drawn using a straightedge. Evidently, this is only clear to students who learned math during the pencil-and-paper era.
Education reforms of the 1990s required that math students be taught use of calculators. Educators thought that because students would always have access to a calculator, time spent drilling skills like memorization of multiplication tables and practicing pencil-and-paper arithmetic could be better spent educating students about the concepts of how and why the arithmetic works.
Now that these students are in high school and college, we are seeing the disastrous results: the ability to see the answers from simple arithmetic was what enabled students to easily think forward algebraically. Without this basic tool, students experience great difficulty in tackling problems that require them to choose between alternative pathways, because they have to follow each pathway step-by-step to its end, and then go back compare the answers. This cripples students’ ability to take a global approach, which is a crucial skill to develop for solving complex engineering problems, project management, and effective leadership. It also leaves students unable to make good estimates, which renders them unable to catch errors that result in ludicrous answers. This problem is beginning to receive some attention—at the speed of our educational system it will probably be addressed within ten or twenty years.
However, a related problem that appears not to be on the radar at all is the way students are taught to use graphs. In the pencil-and-paper era, graph paper was the way to get a good-enough answer to problems that could not easily be solved exactly with calculations. Graphs also provided a way to show what the data points and equations represented. The valuable skills of interpolation and extrapolation were the bread and butter of a lot of science and engineering calculations.
In the calculator-and-computer era, it is easy for students to get their calculators to exactly calculate the equation of a best-fit line, and use the equation to interpolate or extrapolate precisely, without their having any real concept of what is actually happening. To get around the pedagogical issues, math teachers have their students sketch graphs to show their understanding of what the data points and equations represent, but the “correct answer” comes out of the calculator.
The casualty of the answer-from-calculator, graph-as-illustration approach is that students simply don’t see graphs as a useful tool for performing calculations. To them, a graph is simply a picture of what the calculator is doing as a way to check their understanding. A sketch of the graph, with the axes, divisions, points and lines all drawn freehand, has always been sufficient for them to show their understanding. What I wasn’t prepared for is their assumption that if I ask them to produce a numerical result from their graph, estimating the result from the sketch is still good enough.
To be fair, I did allow students who had enough background in statistics to use their calculators to calculate the best-fit line and y-intercept, provided that they also gave the slope and correlation coëfficient that their calculator provided. I felt that this set the bar in an appropriate place—students who haven’t learned the statistics concepts are still reasonably likely to comprehend the graphical approach and why it works, but the statistical equations would be beyond what I could teach them in a few minutes out of a physics class. Unfortunately, the ability to point them toward a graphical solution also seems to be beyond what I can teach them in a few minutes out of a physics class.
If you look through the Standards/Frameworks/Core at what skills are introduced when, you’ll probably see that kids are expected to learn more, sooner, than you did when you were a kid. For example, reading and writing are now subjects in kindergarten.
And yet, the problem you describe is a real one. If a child learns division in fourth grade, why can’t she plot points and fit a line to them in your class?
There are many facets to this problem. Who is introducing math to our children and how are they introducing it? (Can your child’s second grade teacher plot points?) What training, salary, and professional development are your childrens’ teachers getting? If the teachers are teaching to a high stakes test, what sorts of questions are on the test?
I have looked at the standards and at vertical articulation of concepts and skills–my job requires me to. Quite a few of the significant changes happened around the time I started teaching (2003), and more have continued since then.
I have a problem with kids being forced to learn concepts sooner and sooner. Decades ago, kids were taught concepts at an age when most of them were developmentally ready. Now, the trend is to teach the concepts at an age when the average kid is developmentally ready, which means at any given time, approximately half of the kids are not yet ready for what they’re learning. This leads to situations in which kids learn defensive strategies for getting the right answers without understanding what they’re doing. It also leads to kids feeling more frustrated and inadequate, which teaches them to give up without trying. We’re trying to address the latter problem by artificially bolstering kids’ self-esteem (e.g, by eliminating honors classes so the kids in the non-honors classes won’t feel inferior) instead of addressing the root cause of the problem.
I’m not convinced that the problem is teachers’ levels of training, salary and professional development. A generation ago , teachers received quite a bit less training and professional development than we do today. Teacher salaries haven’t quite kept pace with inflation, but adjusted for inflation, I’m only about ten percent worse off than I would have been in the early 1970s. Adding a few thousand dollars a year to my salary would be nice, but it’s a drop in the bucket, not a panacea.
And yet in the 1970s, kids learned to read, write, and do arithmetic more reliably than they do today. They may have been exposed to fewer concepts in their math classes, but more of them actually learned these concepts, because they were developmentally ready, which means they retained the concepts much better.
One big problem is that as long as the tests remain high stakes (for the students, teachers and schools), teachers will continue to teach to the tests. To do anything else is political suicide. As a result, children are doing a better and better job of developing the skills that the tests assess, at the expense of all of the things the tests are unable to assess.
Piaget, the great development psychologist, used to lament about the “American Question”: “How can we speed up development?” That attitude is still alive and kicking today, as evidenced by the standards. Instead of thinking of children as developing beings, who need to build the foundations of their understanding one step at a time–and the stronger the foundations, the stronger the eventual learning– we seem to think of children like they are widgets passing through a factory, and the teacher’s job is to man the assembly line and stuff their heads with the bits assigned to them and pass them on. Is it any wonder that most kids just aren’t interested in (or even feel alienated by) school?
[can I do this without writing a book?]
Start with the one-liners:
“You gotta know the territory” — the traveling salesman in “The Music Man.”
“Nobody ever went broke underestimating the intelligence of the American Public” — dunno where I got this.
The first one points to the importance of geometrical reasoning in creating problem-solving strategies. As a young professional, I was able to run circles around my arithmetic-oriented colleagues in debugging inertial-guidance software because my problem solving was rooted in a geometrical, yea haptic _grasp_ of the fact that there were only three degrees of freedom in the space of possible attitudes of the stable platform.
And of course, in this I had the benefit of a lagging educational system which would teach me mechanical drawing in high school and descriptive geometry in college. On top of a concept-rich family setting.
The second aphorism points to the fact that this shock has not been unprecedented, and will not be un-repeated as other skills come up short. The proportion of the population that can’t read graphs, or for that matter, tables, is scary. And the home is still a huge influence in kids’ integration of what they experience.
As for the educational methods, the dream scheme is things like the activity in the lab school at Wisconsin Center for Ed. Research where the kids were growing seedlings and plotting their growth on a roll of newsprint spread out across the floor. The abscissa didn’t have to pass through representation as numbers, it was the verbatim height of the seedling. But we haven’t found the economic/political means to put all kids in Montessori schools.
As for the political reforms to lance the boil of knowledge atomization to facilitate reproducible testing — I don’t think I have a coping topology for that problem space yet…
Keep it up!
“If I put anyone else in that classroom, they will encounter the same problems. You are doing a great job; go back and do more of it.” –Agnes Lo, my headmistress in Hong Kong.