Education ∪ Math ∪ Technology

Month: November 2011 (page 1 of 2)

Experiments in assessment

Here a few experiments in assessment I’m considering for next year.

  1. Compare the results between an oral assessment (as in, find out what they can tell me they know verbally) and a traditional test. . Question: How much of a difference does the mode of assessment make?
  2. Compare the results between a 10 minute quiz and a full length test. Question: Do I find out significantly more with a longer assessment?

  3. Give my students an assessment where I only give them written feedback and no numbers or check marks. Compare this with an assessment where I only give check marks, and another where I only give numeric feedback. Question: Are the numbers and check marks necessary?

What other experiments would you suggest that I try?

Mathematics in the real world: World Statistics

This is another post in my series on mathematics in the real world.



Thanks to a colleague of mine, I rediscovered the Google Public Data explorer. Within 10 minutes, I had constructed the above graph, which shows adolescent fertility rate for 15 to 19 year olds, versus life expectancy, measured against (look at the colors) average income for all of the countries in the world. If you click play, you can see a happy trend; life expectancy is increasing across the world for almost all countries, and the fertility rate is also decreasing.

This type of graph also lends itself well to questions from your students. For example, they may ask why so many teenagers have babies in some countries. They may also why there is a relationship (and from the above graph, it looks like the relationship is reasonably strong), between births from teenage moms, and life expectancy. They may also ask about trend itself, and why that is happening. Further, they may ask, how strong is this relationship? They may also confuse correlation with causation, which in itself can lead to an interesting conversation.

A natural extension of an activity related to this graph would be to have students construct their own graphs, perhaps even collecting their own data. What kind of social data do you think would interest your students?

What is math?

This image is an attempt to capture the important stages of doing mathematics. As pointed by other people, mathematics is not a linear process, which I am attempting to share via this image. I see analytical reasoning, flashes of insight, and exploratory calculations as the glue that holds these stages of mathematical thinking together.


The stages to doing math


How do you see the process of "doing math"? Is it possible that what sets mathematics apart from other disciplines is the formalism, and the calculations involved? How does this process compare to other things that we do in life?

Why math instruction is unnecessary

This TED talk by John Bennett raises an important question; why do we teach middle school and high school math?


I don’t know if using "puzzles" is a scalable solution for the problems in mathematics instruction in middle schools and high schools. It would probably work for many math teachers, but wouldn’t necessarily work for all math teachers. Puzzles and games are good for teaching analytical skills, provided you have someone around who models the use of analytical skills during the game. I’ve noticed, over many, many years of playing games, that many of my friends do not use much deductive reasoning during games. What I would support is much more use of puzzles and games during mathematics class than what is currently considered acceptable practice.

John’s argument that middle school and high school mathematics is unnecessary should actually be restated: our current middle school and high school mathematics curriculum is unnecessary. John is essentially arguing for a different curriculum, rather than discarding the practice of developing mathematical reasoning in students.

I think we need a variety of approaches. What we are doing right now works when students have a strong mathematics instructor, but isn’t working for every student. Instead of assuming that there is one solution to the mathematics education "problem", we should recognize that there are a variety of solutions. What works for John Bennett may not work for every mathematics teacher. I’d like to see these different solutions compete more with each other, and be able to do more research on the effectiveness of each of these approaches. We definitely need more flexibility in mathematics instruction, especially with regard to the curriculum outcomes.

I think we should be focusing less on curriculum outcomes, and more on the holistic goals of a mathematics education. I don’t think it matters if every student learns about the quadratic formula (for example), but all students would benefit from learning deductive and inductive reasoning, pattern finding, modelling of data, and problem formulation. Curriculum should be a vehicle for these goals, rather than the goal itself.

Computer based math presentation

Here are the slides from my presentation for the Global ed conference on Computer Based Math. I will share the presentation recording when I have it, as I had some great questions from the audience. If you are interested in discussing these ideas further, join the Linked In group. If you want to challenge my thinking on this, please feel free to do so. I don’t want to end up moving mathematics education in the wrong direction, but I am becoming more and more convinced that the full use of computers in education is the right way to go. See this page for some responses to common objections to the use of computers in mathematics.

Update: You can view a recording of the presentation here (requires Java).



Prepare students for life, not jobs

Schools as factories

Should the goal of schools be to prepare students for jobs when they complete school? The answer to this question seems to be obviously yes to many people, but this naive answer is problematic for a few reasons.

At the end of the 19th century, many educators agreed, the role of schools was to prepare students for a life of monotony in the factories.

Every manufacturing establishment that turns out a standard product or series of products of any kind maintains a force of efficiency experts to study methods of procedure and to measure and test the output of its works. Such men ultimately bring the manufacturing establishment large returns, by introducing improvements in processes and procedure, and in training the workmen to produce a larger and a better output. Our schools are, in a sense, factories in which the raw products (children) are to be shaped and fashioned into products to meet the various demands of life. The specifications for manufacturing come from the demands of twentieth-century civilization, and it is the business of the school to build its pupils according to the specifications laid down. This demands good tools, specialized machinery, continuous measurement of production to see if it is according to specifications, the elimination of waste in manufacture, and a large variety in the output.Elwood Cubberly, Stanford’s Dean of Education, 1905, Public School Administration, p338

Ninety-nine [students] out of a hundred are automata, careful to walk in prescribed paths, careful to follow the prescribed custom. This is not an accident but the result of substantial education, which, scientifically defined, is the subsumption of the individual…

…The great purpose of school can be realized better in dark, airless, ugly places…It is to master the physical self… to transcend the beauty of nature. School should develop the power to withdraw from the external world.” – William Torrey Harris, US Commissioner of Education, 1889 – 1906

John Dewey had a cautionary note for teachers that they should recognize their role in this situation.

Every teacher should realize he is a social servant set apart for the maintenance of the proper social order and the securing of the right social growth.” John Dewey, My Pedagogical Creed, 1897.


Consider now this quote from the current Secretary of Education for the United States.

We need more highly trained, highly skilled workers; we need to keep raising standards, raising the bar...” – Arne Duncan, US Secretary of Education, 2009 – Present

We still don’t have enough engineers.President Obama, June, 2011

When the goals of industry match the traits we think good people should have, then no one notices that our focus is wrong. What if the goals don’t match? What if the US economy needed 100,000 more janitors instead (note: Being a janitor can be a fulfilling and interesting career, my point is that the aims of a mathematics course would be quite different)? Are the aims of an education focused on the needs of industry the same as an education focused on the needs of our students?

We should be helping develop good people, which our world desperately needs. While it can be argued that a value judgement like “good people” is difficult (impossible?) to define, the key here is not that we carefully define the judgement itself, but that we recognize the change in focus. 

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?