“They had invented a set of rules, which if you followed them without thinking you could produce the answer…” ~ Richard Feynman
On this coming Monday night, from 7pm to 8pm PST, Mr. Wejr and myself are planning a Twitter discussion about the relationship between assessment and learning. The topic is fairly broad, and should allow for anyone interested in assessment to participate.
Make sure to use the #BCed hashtags in your tweets if you are participating, and watch out for daylight savings time. We’ll both make sure to announce the chat during the day.
We hope you will join us! If it is your first time joining a Twitter chat, see our BCed wiki page with links to some resources to get you started.
When I think of the conversations we have on Twitter about the use of technology in education, I wonder why we don’t have this conversation more often; it is not okay to hit children, especially when those children are not your own (I don’t like this qualifier, but I recognize that parents have more rights than we do as teachers, whether or not I agree with what they do with those rights).
We need to get the places in red below to codify this in law ’cause for some reason, they haven’t yet.
I’m to be on a panel for the Computer Based Math summit happening in a couple of weeks, and I have to construct a 5 to 10 minute presentation on the following question:
Where do we draw the line between what should be done "by hand" and what calculations can be done on a computer in mathematics education?
If you could help me with some feedback (and potential challenges) on my position (see below) that would be helpful. I’d like to thank the various people who have influenced what I’ve written so far as well.
My existing posts on this topic:
Conceptual knowledge is necessary to be successful at mathematics, but I believe that for many of the algorithms we teach students, there is little difference between using a computer to do the algorithm and using pencil and paper. Some of the algorithms themselves have embedded conceptual knowledge, and are of course important to learn, but should be learned for understanding how the algorithm itself works, rather than necessarily memorizing the algorithm.
David is a mathematics teacher and learning specialist for technology at Stratford Hall, a small independent school in Vancouver, BC. He is an experienced international educator, having worked in the USA, England, Thailand, and Canada. He has his Masters of Educational Technology from UBC, and Bachelor degrees in Mathematics, and Secondary School Education. He has written numerous articles for magazines, and blogs regularly at https://davidwees.com
I want to challenge the broad assumption that seems to exist, at least in k to 12 education, that there is a best set of content for learning mathematics. Aside from some numeracy skills, and arithmetic, the vast majority of the mathematics we learn tends to focus on algebraic (and eventually calculus) thinking. I suggest that what would be better would be to focus on mathematical thinking, and to allow much more room for many different kinds of math to creep into our schools. Learning algebra, for a dedicated individual interested in using it in a science, math, or engineering career, is not that difficult and would only take a year. Instead of the issue being hand versus computer, we could focus on ensuring that students learn how to think mathematically, in a variety of different ways.
Specifically related to calculating using a by-hand method or a computer, both are mechanical operations; without understand the algorithm, one cannot really be considered to be doing math.
Paper, pencil, and language itself, are all forms of technology. If the technology changes, the way the algorithm is done changes. When we use a computer to do a calculation rather than doing it by hand, we are merely trading one algorithm which students could potentially understand or not understand for a different one.
Critically, pushing around symbols on paper is just a symbolic representation of the real math taking place within one’s head. When one does a calculation, whether it is by hand, or by machine, an important feature of whether or not one can be said to be doing the calculation is whether or not one can predict the potential output from the algorithm, or if one understands the process they are using. By prediction, I mean, have the ability to recognize nonsensical answers, and to have a feel as to the approximate size of your answer at least, if not always the exact value.
It is important to recognize that this is not a new perspective. Consider this statement from the Agenda for Action produced by the NCTM in the 1980s.
"It is recognized that a significant portion of instruction in the early grades must be devoted to the direct acquisition of number concepts and skills without the use of calculators. However, when the burden of lengthy computations outweighs the educational contribution of the process, the calculator should become readily available."
Obviously we can easily substitute calculator for computer. So the NCTM draws the line between that which is educationally useful versus a “burdensome” calculation. Clearly this is a fuzzy line and needs clarification, which is part of the purpose of this discussion.
Control over what one does is a key aspect of “doing something” and is often the chief complaint against using a computer to do mathematics. “If you just enter it into the machine, you aren’t doing mathematics, the machine is doing it for you.” A story might be useful here, so you can understand my perspective on this.
One of my friends is an oceanographer, and at the end of the summer, he and I had a conversation at a party about what he does for a living. I asked him if he does any math as part of his job, since I am, of course, naturally interested in where mathematics is used outside of school. He replied, “No. My computer does all of the math for me.”
He explained to me that he spends about half of his time creating mathematical models to describe ocean currents and climate on a small scale, and then uses the computer to crunch data and compare it to his model. For example, he recently proved that of three data collecting stations a company he is working for deploys, one of them is unnecessary since the other two can predict the conditions at the 3rd station with 88% accuracy.
So here is this person who is creating complex models involving differential equations, writing Matlab scripts to crunch data, and comparing the output of the scripts to his models, and then communicating his analysis to his employer, and he doesn’t consider himself to be doing mathematics because the calculation step is done by his computer.
I think we probably agree that my friend has done a great deal of mathematics, and that what he does for a living models some of the mathematics we’d like our students to be able to do. His creation of a model, programming of that model into his computer, analysis and organization of the resulting data afterward is all highly mathematical, and is the kind of stuff that we could consider to be done “by hand.”
What I also see from this story is that my friend is most definitely “in control” of what he is doing. He has both control over the process he is following, and over the machine which is helping with calculations he could not possibly do “by hand.”
Further, when you program the machine, you are in control of what it does. If you make a mistake in your program, the computer complains.
So we require then an ability to predict and understand an algorithm, an ability to use it to model contextual situations, and an ability to use the output of an algorithm to reason and communicate mathematics. We also require, as a system, much more flexibility in the mathematics taught at the k to 12 level.
Educators make mistakes when grading. It happens. Sometimes we mark a student’s work lower than we should, compared to their peers, and sometimes we mark it higher than we should. The question is, what effect does this have on a student’s overall mark?
Here are some sample grades. The sample column is the original grade, the low column is a mark 1 lower than the sample, the high column is a mark 1 higher than the sample.
The overall grade was calculated here by finding the averages of the three categories (quizzes, homework, and tests – standard categories in many classes) with quizzes worth 20%, homework worth 20%, and tests worth 60% of the overall grade. These aren’t particularly unusual grades. Note, however, how wide the possible error is in the final grade, which could potentially actually range from 55.9% to 82.5%, which is a 26.6%, or a HUGE amount in any grading system.
Of course, teachers aren’t likely to mark everything low, or everything high. One could make an assumption that both of these cases are equally likely, and then instead of using the likely minimum mark, and the likely maximum mark, we could try and aim for 2 standard deviations from the mean of the possible grading outcomes. In other words, what’s a likely range?
I created a script (warning: takes a while to run in some browsers) which randomly generates a sample of 10,000 overall grades, starting with the baseline above, and randomly adding errors in grading for each assignment, assuming that teachers were equally likely to assign a lower grade as a higher grade, and as getting the grade exactly correct (this assumption is probably false, but I had to start somewhere). For one sample of 10,000 grades, the minimum grade is 60.2, and the maximum grade is 77.5, suggesting that the distribution of grades isn’t symmetrical (teachers are more likely to assign a grade which is too low to students who are at above 50% overall, and too high for students who are at below 50% overall). The standard deviation of these scores is 2.32, which means that 95% of the time, the grade will fall between 64.6% and 73.9% (the mean of the data set was 69.2). This is a range of likely values of over 9%!
Note that this script doesn’t account for a host of other reasons that the grades for this individual student could be in error. It doesn’t account for lost assignments, misread names, addition errors, etc…
How many teachers know that there are error bars on the percentages they are expected to give to students? Maybe if we reported this student’s grades as 70.1% ± 4.6%, students and parents might recognize that grading is more subjective than they realize? Maybe we could stop the practice of assigning letter grades to students work based on strict boundaries?
I remember than in grade 12, I was assigned a grade of 84% overall in English 12, with an A being an 86% in my school. This meant that I missed out on a major award at university (it was my only B in grade 12) and that I had to write an entrance exam to get into my first year English course (I passed). I’ve obviously done fine despite this grade, but I remember it often, and it is a reminder to me of the often arbitrary nature of teacher assigned grades.