Annie gives a very short talk that highlights some of the issues in math education, and which I can tie to work various people have done on learning.
Everyone who is trained to become an educator has some fairly strong intuitive sense of what it means to be an educator. They have seen educators work, and they know how to copy the behaviours of the teachers they have seen. Unfortunately, often we want to change teachers behaviours, and so we must address the misconceptions that teachers have about learning head-on.
If you do not address the misconceptions that people have, chances are very good that they will incorporate the new information you present (in almost anyway that you present it) into their existing misconceptions and as a result, not change their behaviours at all. This is a problem that numerous educators have discovered (it seems independently of each other) and one which definitely has implications for teacher education.
Annie’s observation that her teaching college in 1988 was already talking about inquiry based learning, and some pretty serious reforms in mathematics education, and then her description of her beginning practices which were so different, gets at the heart of this issue. She was "taught" that inquiry based mathematics is an effective pedagogy, but she didn’t hear it. She probably did hear it, but she thought that her notion of what inquiry based education meant was the same as what she was doing. She was unable as a beginning teacher to see how different her techniques were than what she was being taught to do.
So if we want to change teacher education, we definitely need to assume that the student teachers coming in have an understanding of what it is to teach, and that much of what they understand is misguided and just plain wrong, and we need to incorporate the wrongness of this approach into our instruction of teachers.
It is my experience that we compartmentalize knowledge entirely too often in schools, labelling some ways of learning mathematics, other ways of knowing science, and still other ways of knowing the humanities. We compartmentalize knowledge so much in schools that I believe it leads to what I like to call Sitcom teaching, in which each lesson is a stand alone that does not depend on any other subject areas (or often even the previous lessons in the same class) in order to be learned by students.
This is a dangerous practice because it leads students to believe that mathematical thinking is somehow incredibly distinct from other types of thinking and to the logical conclusion that they can get by in life without being able to reason mathematically, or that it is even possible to live life without using mathematical reasoning. This is clearly false – the similarity between what we think of as different modes of thinking is much more than the differences. Just thinking of similarities and differences as I have pointed out in the previous sentence, a common activity in the social sciences, is using the basic ideas inherent in mathematical set theory.
A potential cure for compartmentalization is multidisciplinary learning, wherein the skills, knowledge, and modes of thinking are rejoined together to form whole projects.
Here are some sample multidisciplinary projects:
Build a community garden
In this project, students could learn science as they observe how the plants grow, and be encouraged to experiment with different amounts of light, watering, fertilizer, and soil preparation techniques to see how these variables affect plant growth. Younger students can count out seeds, and work with older children who help them carefully arrange these seeds into rows and columns. Students could learn how to calculate the lengths of shadows during the course of a day, and thus work out where are the best places for Tomatoes in their garden. Students could write letters of invitation to various community organizations to invite them to use the community garden. They could write grant applications to seek funding for their garden and petition local businesses to offer financial support for the garden. They could research how farming practices have changed over time. They could learn about the environmental consequences of our global food supply.
At the end of the season, they could harvest and cook their own food. They could give away the food to a local food bank.
Run a store Students could create budgets, order supplies, and keep track of inventory. They could research the health benefits (and problems) associated with various kinds of food. They could write proposals to change inventory selection. They could read about the manufacturing process for the goods in their store, and write letters to the manufacturers either requesting more information or a change in harmful practices. They could use their experiences in the store as a background for a short story. They could sell their art work, or books of their poetry. They could donate the proceeds to charity, or use them to buy supplies for their school.
Create an (rock) opera
Writing musical scores would let students learn more about fractions, sequences, and counting. Students could research different musical styles, and learn more about their culture. Students could create the music, the stage settings, design the costumes, program the lighting sequence, and create the program booklet for the performance night. Students could create multiple storylines and then find ways to bring their ideas together (where possible).
Of course, there are a lot of other project ideas not listed here. To implement these in most current school settings, teachers would have to collaborate and work together fairly well, and we might have to set aside some of our standard school schedules. However, we should never let the school schedule have too much control over the kinds of learning activities we do with our students.
What other benefits do you see to this approach? What are the problems with it?
My grandmother, Frances Shelley Wees, was an author and as such, she would often receive letters from people, particularly young women, asking her how she got started. A kind stranger found this letter from my grandmother to her grandmother (Mrs Hanson) in her possession, found me online, and emailed me a scannedcopy of her letter. I’ve transcribed it below.
My dear Mrs. Hanson;
I hope you will forgive me for not having answered your letter long ago. We have been back in Toronto for two weeks, but I’ve been settling us in a new house and getting my youngster’s clothes ready for winter–you know exactly what it’s like, I’m sure. And the grapes and the crabapples are still in the fruit stalls here and I’ve got a very jelly-minded family so I felt I had to do my housewifely duty by them.
I wish I knew exactly how to advise you to go about writing. I do think that what might be the right procedure for one would be wrong for another. In my own case, I’m sure a course such as the Shaw Schools offer would have been the wrong thing to take at the beginning, although, I think, like you, that I might profit by it now. I don’t, of course, know how old you are; but your letter sounds so sensible and philosophic that I don’t think you can be told the things one tells the twenty-year olds when they come asking. I tell them to go home and write and write and WRITE and forget about themselves and how famous they might some day be, and scrape off all the fancy polishes and work for simplicity and humility. You’re so much beyond them in your outlook on life. Maybe a story course, which would emphasize technique and give you a few rules, would be exactly right. Before technique, I feel, comes an attitude of mind, hard to acquire but invaluable. Until one has that attitude all the courses in the world are pretty much wasted, I should think.
I don’t really know anything about the Shaw Schools. I’ve heard of Mr. McKishnie. Their price seems pretty high. There is a first class course offered by the Home Correspondence School of Springfield, Massachusetts, which is only thirty-odd dollars. I’ve seen some of the lessons and criticisms, and thought them first-rate. Perhaps you would like to write to them and get their literature. The man who is at the head of it has been operating his school for a great many years and has written several of the leading text-books on short-story technique.
You ask about how I began. It might be misleading if I say I just began, with no training–I’ve not been to University and I never took any courses of any kind–because I had a husband who was (and is) a psychologist and a writer, and a number of friends who were more than generous with help and criticism. So that I got a good deal of guidance, some of which I did not at the time particularly appreciate. I honestly think that my greatest helps were the books on criticism and technique which i got from the libraries. I suppose I have read and minutely synopsized fifty or sixty of them, and of course have read countless others. Not quite countless; there aren’t so many. Perhaps you could get them through the extension library of your University (I’m not sure Saskatchewan has an extension service, has it? If not, you might try Alberta. Miss Jessie Montgomery, Extension Library, U.ofA. would do whatever she could for you, I know. She would send you a list of such books and if the regulations permitted would send the books themselves if you would pay the postage.
You have a number of advantages as a beginning writer, and your place of residence being one of the greatest. I find it much more difficult to write in the city..to shut myself away for the long periods of time necessary to do a good piece of creative work. Of course I enjoy all the interesting things going on about me, and eventually find them stimulating; but while I’m writing, all the stirring about me is irritating. I wrote my first five books in a little town in Alberta where I had my friends all warned that if they dared ring the telephone between certain hours I would put the curse on them.
I feel that this is a very inadequate letter, Mrs. Hanson, and I wish I could do something that would really help you. I know so well those early helpless feelings, the blackness of not knowing where to turn and yet having to go forward. I wish I could ask you to send something you’ve written for me to criticize, but time is a thing I don’t possess. Criticism is the thing beginners need most, I suppose. Although I’m not even sure about that. Maybe all they need is the firm resolution to be honest with themselves, to find out what they truly feel and believe about life and people then to write as simply as possible. If I were you I should most certainly attempt to sell the poems. You might try SATURDAY NIGHT, here in Toronto. You might try the Saskatoon paper, or any of the other western papers which use poetry. Get something into print as soon as possible. Seeing it there will open a strange and rather terrifying door for you…but the sooner that door is opened, the better.
Do send me a little note some time to say how you’re getting on. Thank you so much for your good wishes…I need them. I don’t find the path any too smooth. I don’t think a women does, when she has to manage a house and take care of a family. And there’s never as much money in writing as people think there is, not enough for the first years anyway to pay for responsible assistance. You have to mend socks with one hand and type manuscripts with the other, and carry whooping cough along in the next compartment to The Great Canadian Novel..and you have to like it.
Here are the big messages I see in my grandmother’s advice to this young writer:
To improve as a writer, WRITE,
Simple writing is more effective,
Attitude is critical, more important than knowledge,
Knowing the mechanics of writing is important as well, and you can learn these on your own,
Guidance and criticism is valuable, but so is self-criticism and self-learning,
Solitude and the ability to work in peace and some isolation is important,
Having an audience matters, and the sooner one has an audience, the better,
Space and time to write is helpful, but clearly not essential. Desire is essential.
She said it would be wonderful if I could do it again, but she also thought parents may not be able to attend because they would be too busy.
I had the thought that creating a blog could help with this issue. One could start by presenting some common questions parents have, and/or a short video response/presentation by someone knowledgeable in the field. Parents could discuss the issue further in the comments, ask more questions, and debate whether the response by the technology "expert" is in fact reasonable. Some entries could be nothing more than an image or video and some questions to debate in the comments.
It seems to me that many parents would have questions about technology use, and that constructing a common platform for many of us to use would be helpful. We all have parents who ask questions about technology, and although we would like to pretend we have the answer, we don’t always do.
Here are some sample questions that I have heard parents ask:
Is wifi safe?
Should I allow my child to play computer games?
Is social media safe? What is social media, anyway?
I’ve started a blog, which I call "Questions About Technology" (link may not be active for you for up to another 48 hours) I’d like to invite interested parties to join me.
Having spend the last ten years teaching students mathematical notation (while simultaneously teaching the mathematical concepts described by these symbols), I have often reflected on how efficient and amazing it is, and how unfortunately broken it often is.
Some notation shows off some of the power of mathematical thinking (for example, algebra), but some notation has clearly not been designed for clarity. In fact, my suspicion is that much of mathematical notation has been invented to save space.
Of course, a reason why one might one want to save space with mathematical symbols is because paper used to be expensive but I suspect this is not the main reason mathematical symbols are so tightly packed with information. It is also time-consuming to use more clear mathematical notation, and mathematicians love to be concise. In fact, I have often noticed that mathematicians often equate the length of a mathematical proof with its elegance, which over time may have supplied pressure to reduce the notation used to describe these proofs. A few mathematicians have contributed heavily to mathematical notation, most notably Leonard Euler, and these few mathematician’s desire for brevity has defined the notation we use today to communication mathematics.
Look at sigma notation for example. What does the letter sigma from the Greek alphabet have to do with finding sums of things? Absolutely nothing as far as I can tell. According to Dave Radcliffe, Sigma (∑) is short for summa (probably because they start with the same sound), which is the Latin for sum. Euler invented the symbol to use for summation, and we’ve been using it ever since. Essentially, we are using ∑ to mean sum for historical reasons.
The portion of this equation to the left of the leftmost equals sign is summation notation, which I have taught for years. I usually have to spend a class, sometimes two, explaining this specific set of notation. The brevity of the summation notation contributes little to the comprehensibility of this statement. It is essentially equivalent to the following:
Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86
Unfortunately this notation requires us to memorize the order of the parameters in the summation function, but this is functionally the same as the previous notation, except one more piece of information is given to us; we know we will be doing a sum of some kind without having to memorize the meaning of sigma. With some work, we may be able to improve upon this notation more, and provide even more clarity.
This notation is somewhat more clear the second option I suggested, since the parameters are defined within the notation. It is significantly longer to write than the original notation (takes up twice as much space) but it has a huge benefit of being significantly clearer. Further, one could imagine that if I were entering this notation into a computer, that the autocomplete function (which is common to code editors) could suggest parameters for me, as well as show me the definition of the parameter as I enter it. Finally, this notation is similar to how we define functions in computer programming (in some languages), and so when we teach mathematical notation, we will also be giving our students some ability to read computer programming code.
This issue about notation is not a trivial concern. The notation used to explain mathematical ideas is often a barrier to some students learning how to communicate mathematical ideas. Quite often students (and sometimes teachers) confuse learning notation for learning mathematics.
Furthermore, notation which is excellent on paper may be somewhat less useful on a computer. I have spent many hours looking for solutions to make adding mathematical symbols to websites more convenient and have discovered that there is no easy way to do this. Every method has drawbacks, and no method is as convenient as adding the same symbols to paper. My conclusion in terms of using mathematical notation with computers is that one of two things (or both) will happen. Computers will develop more touch senstitive interfaces, and software developers will create software that recognize the current mathematical symbols, or we will start to change mathematical notation to be more easily inputted into a computer.
The one huge advantage of our current notation is that it is somewhat universal. Essentially the same notation is used around the world, and by choosing a more amateur friendly notation, we will be creating localized versions of the notation for each language which is obviously problematic. In a computer, this is easily resolved by making the names of mathematical objects translatable so that whomever is viewing a mathematical document can select their language of choice. In print, this is more of an issue, and so we should reluctantly continue to use our existing notation until we have more fully transitioned from our traditional print medium, but the more we use computers to communicate mathematics, the more likely it is that we should fix mathematical notation.
Derek Muller sent me this link to a very popular video animation that attempts to explain fundamental forces in nature. You can watch it for yourself below.
The video uses analogy and some cute animations to attempt to explain how forces in nature come from difference between measurements of those forces in different parts of the university. For example look at the screen-shot taken from the video shown below.
If you look at this picture, does it accurately represent the statement given by the narrator? It seems to me that if you are going to use a visual to explain a concept, it should be clear from the visual what you mean. Visuals should support your explanation, and if your analogy strays too far from the concept you are trying to explain, your visuals do more harm than good. What was the first thing you thought of when you looked at this visual? I bet it wasn’t "Measurement by itself is meaningless, but as surprising as it sounds, that meaningless is exactly what causes the fundamental forces in nature" which is what the narrator says at this moment.
Here’s another screen-shot.
This visual says two things. The first is not stated by the narrator, but is suggested by the equation shown, specifically that what we are going to look at next is very complicated. The second is suggested by the crossing out of the word Quantum. In this case, the visual definitely describes what the narrator is going to do in the rest of the video – ignore quantum effects on the four fundamental forces. The bad news here is that ignoring quantum effects means that whatever follows is going to be out of date by 100 years of science, and not necessarily a very good representation of the apparent strangeness of the universe. In other words, what follows is a bad model that one will probably not understand.
Now let’s see what happens next.
My question here is, what of the previous 1 minute and 30 seconds do you remember? I’m going to suggest that you probably do not remember much. This new model is so vastly different than the old model the narrator starts with (and that previous model was not well explained, as you may recall) that the transference of the introductory model to this new model is not likely to happen. If you happen to be an expert in the area of the fundamental forces of nature, you may not notice this effect, since the earlier model is (maybe) describing something you already understand, and have already internalized. If you are not an expert, I very much doubt that a 1 minute explanation is going to make you one.
Further, if you look at this section, you may notice the model for currency transaction (which looks a lot like a function machine, an analogy mathematics teachers often use to explain functions to students) in the middle of the currency exchange. The currency portion of this implicit analogy probably makes sense, but the symbol in the middle may be lost on a lot of people, particularly since the narrator doesn’t take the time to explain what this symbol means.
Now this is where the narrator makes a huge assumption. He assumes that people have been able to make a somewhat over-arching generalization from his single example. He says, "Hopefully now you can see why measuring things differently in different places inevitably gives rise to a long range interaction, mediated by a particle." I doubt that anyone would be able to make that generalization without a fair amount of expertise in long-range interactions themselves.
It is a form of cognitive bias to assume that because an analogy makes sense to you, that it will make to other people. Analogies are useful as a sense making activity when the analogy describes a shared experience between two people, and very few people have an experience of currency exchange (surprisingly, only a small percentage of any population travels to other countries). In other words, using an analogy that people lack experience with is unlikely to lead to further understanding of a more complicated phenomena.
This particular video, when I watched it, had over 156, 000 views, and over 5000 likes, which suggests to me that one cannot take the popularity of a video and use it to gauge the effectiveness of the learning from the video. I recommend reading the comments on the video. You will see more than a few people who are confused by the video, or who add messages which are essentially unrelated from the video itself. The most popular discussion point I saw, in the 100 or so comments I read, was that this "minutephysics" video was in fact longer than a minute.
My complaints while directed at this one video are generalizable. Analogies used in videos should be related to common experiences, where possible. Visuals matter – using visuals which are confusing, or even wrong, not only distracts from the intended objectives of the videos, it can introduce other possible misconceptions. Avoiding people’s misconceptions in the videos, and attempting to present clear explanations means that people will, in general, incorporate the new information into their existing schema, leaving their current misconceptions intact.
This comment on the video essentially summarizes my main point (notice how many people agree with it).
I created a simulation so we could test what parameters we may want to use in the classroom so that students are most likely to see that the spread of a disease can be modelled effectivelyh, and see the probability of the infection being spread from person doesn’t change the type of mathematical infection curve much. Try the simulation here.
Some assumptions I’ve made with this simulation:
Individuals once infected, stay infected.
Each individual has an equal probability of being infected by anyone else in the population.
The probability of anyone being infected remains constant over time.
Individuals can be re-infected.
I don’t know if we will end up using this simulation with students, but if we do, I’d like it to be fairly clear so they can get started using the simulation without much intervention from me.
I was looking for research on whether word processors are effective when students are learning to write. So far the research is supportive, but I can’t find any research done recently. I suspect there must be research that is current and supports students using word processors. Please let me know if you have any research more recent than what I have below.
Bangert-Drowns, R., (1993). The Word Processor as an Instructional Tool: A Meta-Analysis of Word Processing in Writing Instruction, Review of Educational Research, p69-93, doi:10.3102/00346543063001069
Abstract: Word processing in writing instruction may provide lasting educational benefits to users because it encourages a fluid conceptualization of text and frees the writer from mechanical concerns. This meta-analysis reviews 32 studies that compared two groups of students receiving identical writing instruction but allowed only one group to use word processing for writing assignments. Word processing groups, especially weaker writers, improved the quality of their writing. Word processing students wrote longer documents but did not have more positive attitudes toward writing. More effective uses of word processing as an instructional tool might include adapting instruction to software strengths and adding metacognitive prompts to the writing program.
Lewis, R., Ashton, T., Haapa, B., Kieley, C., Fielden, C., (1999). Improving the Writing Skills of Students with Learning Disabilities: Are Word Processors with Spelling and Grammar Checkers Useful?, Learning Disabilities: A Multidisciplinary Journal, retrieved from http://www.eric.ed.gov/ERICWebPortal/detail?accno=EJ594984 on May 22nd.
Abstract: A study involving 106 elementary and secondary students with learning disabilities and 97 typical peers found that students who used spelling and grammar checkers were more successful than transition group students in reducing mechanical errors, particularly non-real-word spelling errors, and in making positive changes from first to final drafts.
Owston, R., Murphy, S., Wideman, H., (1992). The Effects of Word Processing on Students’ Writing Quality and Revision Strategies, Research in the Teaching of English, Vol. 26, No. 3 (Oct., 1992), pp. 249-276
Abstract: This study examines the influence of word processing on the writing quality and revision strategies of eighth-grade students who were experienced computer users. Students were asked to compose two expository papers on similar topics, one paper using the computer and the other by and, in a counterbalanced repeated measures research design. When students were writing on the computer, "electronic videos” were taken of a subsample of students using an unobtrusive screen-recording software utility that provided running accounts of all actions taken on the com- puter. Papers written on computer were rated significantly higher by trained raters on all four dimensions of a holistic/analytic writing assessment scale. Analysis of the screen recording data revealed that students were more apt to make microstructural rather than macrostructural changes to their work and that they continuously revised at all stages of their writing (although most revision took place at the initial drafting stage). While the reason for the higher ratings of the computer-written papers was not entirely clear, student experience in writing with computers and the facilitative environment provided by the computer graphical interface were considered to be mediating factors.
In this video, shared with me by Philip Moscovitch, a student has brought a type-writer into class. Is this perhaps, as Philip suggested, a protest against the use of an old pedagogy by bringing in an old technology? Does the use of a typewriter to record notes seem a bit ridiculous? Is it even more ridiculous that the student, as he states at the end of the video, can download the notes for the course?
A well motivated, literate student can learn as much or more from a good set of notes (or a decent textbook) for a course. Why come to class at all if all that is going to happen is a repetition of the notes?
Derek Muller: "Can you teach a general thinking skill?"
John Sweller: "I don’t believe you can. It can be learned, it is learned, and it is biologically primary…If you are talking about a teachable thinking skill, one you have to specify it, you have to provide evidence that it has been taught and learned and that you get a different response from people who have learnt that skill and been taught that skill and people who haven’t been."
So here’s my challenge. Can anyone find evidence of a "general thinking skill" that has been taught and then learnt by students?