Education ∪ Math ∪ Technology

Author: David Wees (page 44 of 97)

Three modalities of learning via computers

I think most learning opportunities through computers can be broken down into one of three basic modalities. These modalities may be mixed (as is the case when one uses adaptive assessment, for example) or they may be a stand-alone use.
 

  1. Computers as assessment
    Computer as assessment
    This is the lowest level of computer use, in my opinion, although it is probably the most attractive for many education policy makers. The ability to automate assessment, and to take assessment out of the hands of teachers is pretty powerful, for people who view students and teachers as inputs in an education machine.

    One issue is that even the best computer based assessments that exist are not able to examine student learning in any form useful enough for deep learning to occur. They also assume that tacit knowledge has no place in education, and that with enough information, we can peer inside the brains of our students to see what they are thinking.

    For obvious reasons, no educator should rely solely on computer based assessment for all judgements of students’ learning. They may be useful in some formative assessments.

     

  2. Computer as content delivery
    Computer as content delivery
    This use of a computer as a content delivery tool is seductive. After all, we consume a lot of content via computers as adults, and we certainly feel like we are learning effectively. Much of the learning we are consuming, however, is in the form of stories, which is a bit different than learning concepts via media. Derek Muller has a lot to say (and some research to support his perspective) about how effectively people learn science via videos.

    Content delivery is useful, no doubt, but the instructional model of "here is some information, now you know it" is problematic. It’s hard to tell if the person who has been exposed to knowledge has actually learned it, and in many cases, what they have is a shallow shell of knowledge. There are some attempts to address this issue, including implementing versions of learning which use computers as assessment and computers as content delivery together. So far, the research on these attempts is mixed.

     

  3. Computer as programmable tool
    Computer as programmable tool
    Computer as exploration tool
    In this model, the computer is a tool to be programmed, and the user is in charge of the output of the computer. This allow students to be in control of the computer, and therefore their own learning. They can turn the computer into a tool which serves their exploration, rather than the student being at the mercy of the designer of the software running on the computer in the first two modalities described.

    Programming should be thought of as in general, any process through which the student controls the output of the computer. Under this definition, any use of the computer where the use of the computer is not predetermined by an body external to the student could qualify as programming. Broadly defined, programming is the process of design and debugging, which is very much like the process that happens during science.

    Games fall into one these three categories. Some games allow for more exploration and programming of the environment of the game, and some games as merely vehicles for computerized assessment. Some games are intended to deliver content to the learner, and have merely disguised the learning environment from the user. To be clear, there is a certain amount of blurring between these three different modalities, and some activities involve all three.
     

I find that the third modality is the most powerful of the three, and the most likely to develop powerful learning opportunites for the student. Unfortunately, it is the most terrifying of the three, since the outcome of the process of using the computer seems to be the least certain. This may be why many educators shy away from this use of the computer. It can be frightening to let go of the learning process, and trust that, given some suitable boundaries and expectations, students will learn.

Objections to computer based math

At the conference I was at in London, we were discussing, what would a mathematics curriculum look like if the computational step of doing mathematics was something students did using a computer?

Update: The video from this session has been posted by the Computer Based Math organization. See below.

 

Here are some objections shared by Conrad Wolfram and Jon McLoone at the Computer Based Math summit that happened in London, England. I just thought I’d add my two cents, and offer some more possible objections not in this list.

  1. You’ve got to know the basics first.

    First, what are the basics, and why did we define them that way? Are they basic because the concepts are basic, or because they were discovered in a particular order? Are they considered basic because of computational complexity, or because of conceptual complexity?

    Seymour Papert, in his book Mindstorms, suggested that much of what we teach in mathematics classes is the result of historical accidents. He also suggested that we teach some concepts, not because they are the most valuable to teach, but because they can be solved with paper and pencil.

     

  2. Computers dumb math down.

    Jon McLoone has a terrific rebuttal to this argument. His basic premise: we’ve dumbed down mathematics education to limit us to what we are capable of doing with pencil and paper. I’d like to add that we have already turned mathematics education into sitcom-like instruction, where each topic can be taught in a single lesson (or a series of topics can be taught in a single unit), and where older topics are rarely, if ever, revisited. Having taught students how to use a particular topic, we then abandon it to learn new techniques.

     

  3. Hand calculating procedures teach understanding.

    While I think that it possible that hand-calculating can teach something, too often I see people learn recipes for doing math, rather than actually learning mathematical reasoning. I don’t see this procedure necessarily helped by computer based math, but I don’t see that it is hurt either. Whether students do a procedure by hand, or by their computer, if they don’t understand the underlining concepts, they will struggle to use the mathematics in any meaningful context.

     

  4. We’re already doing it.

    Really? There are some small pockets which are using computers as the tool for computation in mathematics, but not on any reasonable scale at the k to 12 level. Students do use calculators, but not consistently across the curriculum, and many potential applications of computers are not well represented by calculators.

     

  5. It isn’t math.

    Here’s a diagram I’ve created to help capture the process of doing math.
    The process of doing mathematics
    The big place in this process where computations happen is in the formulation (and sharing) step shown in the bottom right-hand corner. Note that actually doing the computation, according to this diagram, is only a tiny piece of doing mathematics. If you agree with my premise, that doing mathematics is more than the computations, you might be willing to accept that actually doing the computation step is a tiny piece of the mathematical process. Can we really say that students aren’t doing math if they hand-off that step to their computer?

    Do you think that children often get to do the entire process of math in schools, or are they often stuck at the computation step?

     

  6. You are making people over-reliant on computers.

    I’d like to have students doing more of the mathematical process. Not everything lends itself well to using a computer, and these types of things will still happen in classrooms. Some concepts and ideas are actually not often taught in schools (such as the applications of origami to mathematics) and should be. I want to see students doing more thinking in classes, not less. Mathematics is not entirely in the tool one uses to do computations; most of it happens in the head.

    So rather than seeing people be reliant on computers, I’d like to see some resources available (in the public domain) so that every computation students do on their computers has the "by hand" method carefully catalogged and available for students to use. I’d like to see the computations become part of a toolset, rather than what our students focus on learning.

     

  7. Traditional math is part of our culture.

    I’d love to see mathematical history taught as an option in schools, so as to preserve the culture of mathematical tradition. That being said, culture changes, and we grow and adopt new traditions. For example, almost no one uses quill pens anymore, and it’s certainly not a skill we teach anymore in schools.
     

  8. Good idea – but it can’t be done.

    People have already been teaching mathematics with computers as a tool for computation for a few decades now. I took a course myself in mathematical computation at UBC, and loved it. Rather than saying it cannot happen, since it has happened, we should look to see how we can expand and learn from the current iniatives.
     

Some further objections I can imagine people having:

  1. It isn’t fair; some students have access to technology, some do not.

    This is one objection that I think has some merit behind it. We need to ensure that if we do move toward a model where computers replace the by-hand methods, we need to ensure that everyone has equitable access. As Seymour Papert (and others) have noted, a computer is only a tiny fraction of the total amount of money we spend on a student’s education, and so objections based on money seem to assume that we need to keep all of our existing structures, and that we can’t shrink some of them to pay for computers. How much money do we spend encouraging disengaged learners to remain in schools?

     

  2. It isn’t healthy.

    It’s also not healthy to lack mathematical reasoning, literacy and analytical reasoning skills, but we let plenty of students graduate without these vital skills for life. We do need to balance screen-time versus other forms of more interactive and kinethestic learning, and this will be one of our challenges going forward in education.

     

What are some other objections you can imagine people having to this kind of change in mathematics? Can you extend my rebuttals to these objections?

New #BCed chat: What is the relationship between assessment and learning?

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.

Corporal punishment

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.

States which allow corporal punishment
(Source: Wikipedia)

Computer based math – hand or machine? DRAFT

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:

https://davidwees.com/content/mumbo-jumbo
https://davidwees.com/content/computers-should-transform-mathematics-education
https://davidwees.com/content/conrad-wolfram-teaching-kids-real-math-computers
https://davidwees.com/content/when-should-we-introduce-kids-programming
https://davidwees.com/content/maybe-we-should-be-aiming-computer-programming-instead-calculus-math


Summary:

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.
 

Bio:

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


Position:

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.

Error bars on grading

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.

Grades Sample Low High
Quizzes 5 4 6
  6 5 7
  7 6 8
  5 4 6
  6 5 7
  7 6 8
  5 4 6
  6 5 7
  5 4 6
  6 5 7
average 5.8 4.8 6.8
 
Homework 5 3 5
  5 3 5
  3 1 5
  3 1 5
  1 1 3
  1 1 3
  3 1 5
  5 3 5
  3 1 3
average 3.222222 1.666667 4.333333
 
Tests 40 35 45
  45 40 50
  35 30 40
  40 35 45
  30 25 35
average 38 33 43
 
Overall Grade 70.1 55.9 82.5

 

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.