Year: 2013 (page 8 of 15)

(Image credit: drewleavy)

I talked to someone recently about first aid training, and they expressed their frustration at how ineffective first aid training usually is.

Unfortunately, according to my friend, many people who teach first aid actually have very little practical experience using first aid. As a result, the agencies that are responsible for first aid certification give their instructors "idiot-proof, deadly boring, text-filled presentations" to use in their training so that every first aid course is at least minimally useful. According to my friend, instructors are expressly prohibited from using their real life experience and telling stories of themselves actually using first aid, and also from explaining the reasons behind the protocols used in first aid. Further, most people taking first aid training have little to no interest in first aid themselves, they are almost certainly required to take a first aid course as part of retaining certification in their line of employment.

It occurred to me that this situation is remarkably similar to the position that we find ourselves in mathematics education, at least in the k to 12 level. Mathematics teachers often lack experience either making mathematical discoveries, or even applying mathematics as might an engineer or physicist. Consequently, curriculum is packaged in such a way so that nearly anyone can follow the text and make sure the students get at least a minimally effective mathematics background. We aren’t prohibited from using stories to relay our experience in mathematics, but we often have pretty frustrating limitations on what mathematics we can teach, and at the end of their k-12 program, how our students will be assessed. Finally, almost none of our students is really interested in mathematics; most of them are in our courses because they are required to be there.

Fortunately, first aid training is occasionally successful. My friend suggested that about 80% of the time, people who have first aid training are more useful in an emergency situation than people without the training. At my school, a 6th grade student, learned the Heimlich maneuver during a recent first aid training, and then used it to save the life of his mother two weeks later (An aside: according to my friend, the correct response when someone is choking, after encouraging them to cough, is 5 sharp blows to their back with the heel of your hand, right in the middle of their shoulder blades, followed by abdominal thrusts, then CPR if they fall unconscious). Not teaching first aid is obviously not an option, even when the programs are usually limited in effectiveness, but I wonder, what would happen if we let people who were highly experienced in using first aid have a bit more flexibility in how it was taught?

In the same way, our mathematics education is not a complete disaster. Many people go on from their subpar mathematics education to be able to use mathematics in a meaningful way, and some of those people even make new mathematical discoveries. However, the surest proof I have that our system is less than adequate is the enormous number of people I have met who will happily admit that they were terrible at mathematics, hated it, and now never use it.

The question that I think defines my career as a mathematics educator is, what can we do about this issue? It is of limited use to complain about a problem, especially one as well discussed as mathematics education, without proposing some sort of solution.

What if mathematics educators who had actually used mathematics to solve problems, or had developed new areas of mathematics, had more freedom in how they were able to help their students learn mathematics? If someone has significant experience in topology, number theory, mathematical modelling or any other area of mathematics, why not let them teach this area of mathematics to their students, perhaps leaving out some other area of our curriculum so that they would have time to do so. What if we taught mathematics in such a way that our primary goal was to inspire our students into further study of mathematics?

Whenever John (not his real name) entered my class, we had to open the windows. He smelled really bad, most of the time. He also wasn’t in class very often, and I never knew when he would be school, and when he wouldn’t be in school. There seemed to be no rhyme or reason to his attendance.

When John did attend class, he had to be given a new pencil and book in order to write down any notes or work any math problems. Eventually, I just kept a pencil and a notebook in class for him since he never did any homework.

He was an attentive and bright student, but he missed so many classes that he struggled to keep up. In fact, "keeping up" became impossible for John. When he wrote tests, he would do the questions nearly flawlessly that he had seen in class, but there were rarely enough of these kinds of problems for John to demonstrate mastery of the curriculum.

In about February of the year, John stopped coming to class, and I never found out what happened to him. I was talking to one of my colleagues about John, and she mentioned he was homeless, and I felt completely embarrassed that I never realized.

John is not alone. There were 1.6 million children in the United States in the year 2011 who experienced homelessness for at least part of the year. Many of these children struggle in school because of issues similar to what John experienced, and then continue to suffer throughout their lives in the vicious circle that is poverty.

I suggested to someone recently that we should build schools, especially in areas likely to be high risk, to be more like community centres, with access to health-care, food services, emergency shelters, social services, and after school care programs all under the same roof. Services for children could then be managed centrally by the school and the odds that a child could slip through the cracks and not get what they needed would be diminished. It would also be a lot less expensive than the current emergency-room health care system in the United States. The response of the person was basically along the lines of "It’s the parent’s fault. They need to be more involved."

To this I respond, we should not punish children for the periodic inability of their parents to make their lives work. If we are to be a humane society, then all children in our society should be cared for, whether or not their parents are always able to provide this care.

When you rely on a service, particularly a "free" (ad-supported services aren’t really free) service, you always run the risk that whomever is maintaining and controlling the service will shut it down, or strip it of functionality so as to make it less useful (or even useless) for you. This happened to Google Reader, it almost happened to Delicious, it happened to Posterous; this list will continue to grow over time.

The lesson for me here is not to rely on these third party servives, but take the time to maintain and control my own services. Right now, I maintain my own blog, my own online bookmarking site, and my own RSS reader. Running these services myself has been very easy as I have had minimal interruptions in the service of my website, and the code-base for these services is also under my control (so I can modify them as I need) since they are all open-source projects.

When OX Text is released, I am considering increasing the investment I put into my website (currently about $150 a year) so that I can consolodate all of the various projects I am maintaining onto one site, and migrate away from Google Docs. I just found Roundcube, which looks like it has the functionality I need to replace my current Gmail webmail. 

I would like to migrate completely away from all services that are maintained by third party organizations because these organizations have a history of misusing the data I provide to them, and suspending support for products that do not meet the organization’s goals, regardless of the popularity of the product with their customers.

What other alternatives do you know about for popular services?

I’m working on a couple of short videos comparing the standard algorithm for a multiplication and addition, and considering some ways of using other algorithms which are more likely to make sense for students.

To be clear, these presentations do not enough justice to student-created algorithms, which I would strongly recommend as a starting place for exploration of algorithmic reasoning. These two videos are merely an attempt to compare two different algorithms against each other, and to provide some support for teachers interested in learning more about why we might want to use something other than the standard algorithms in our classrooms.

I also want to make it clear that how I am describing these distinctions in the videos is not how I would introduce them to children (or adults for that matter, if I have the time). Instead, I would start with setting up realistic situations, whole group, and small group discussions around investigating these operations, using both manipulatives and symbols to represent numbers in the algorithms. A really useful activity, for example, is for children to be making these comparisons themselves, so that they can look for patterns in different operations, and abstract these patterns into general rules they can follow to make their use of any algorithm easier.

 I see these algorithms as a useful way to get started with abstract reasoning, provided they are framed in a certain way, as described below.

Multiplication

Addition:

Dave Cormier, in an excellent, excellent presentation, made the point that learning multiplication tables belongs to the "simple" domain of knowledge (which in his defense was probably an example he chose to help his audience understand the Cynefin framework). I think I need to understand his definition of simple better, because I do not see multiplication, especially when it is typically learned, as particularly simple.

I’m not trying to pick on Dave in particular, I have heard "multiplication is simple" from a lot of different sources, Dave’s presentation is just the most recent place. Look at the Cynefin framework below to get a sense of how Dave is classifying areas of knowledge.

(Image credit: Dave Cormier)

I’m going to argue that multiplication as an area of knowledge more appropriately belongs to the complicated domain (especially for children).

First, the symbols used for multiplication (numbers) are a complex idea, and it is not entirely clear to me that every child really understands the relationship between numerals and numbers by the time they are introduced to multiplication. Numerals are a more complex topic than I think most people realize.

Here is some data to support this claim.

I went around my school and found 12 people (6 grade 11 students and 6 adults) to volunteer to write down the numbers from 1 to 10 in a row on a blank sticky note. I then took photographs of the sticky notes, and reorganized the numerals so that they were grouped by numeral type, rather than by person. It should hopefully be clear from these samples that when we teach students how to recognize numerals, we are really teaching them to recognize a class of objects which have similarities, but that there is no single way to write any numeral.

The idea of numbers themselves is complex. Different cultures may have different internal models for numbers, with some cultures being reported not to have an internal linear sense of numbers. It may even be that our sense of numbers doesn’t just vary at a cultural level, but at finer grain structures than entire cultures. Your sense of what numbers are is likely influenced by your parents, your local cultural, and our global cultural representations of numbers.

Multiplication itself exists in a culture, with different cultures viewing multiplication differently. There is currently no wide-spread agreement on whether multiplication is best represented with arrays, as repeated sums, as an area model, or as an algorithm used to calculate in these contexts. Given that mathematics educators do not all agree on something as fundamental to multiplication as how we define it, it hardly seems reasonable to call it simple.

Further, there are a large number of different algorithms for multiplying numbers, and many different possible representations. Some of these algorithms focus on mental arithmetic, others on visual representations, and others on numerical calculations. There are benefits and drawbacks to each of these methods, leading me to conclude that there is no, one best way to multiply numbers.

To top it all off, students, not all of whom even know how to tie their shoes, are expected to start mastering this skill and idea at age 8. I think that the idea of "simple" is a relative term. What might be simple for an adult might well be incredibly complicated for a child.

At our school we currently have open-access to the printer by our students. Unfortunately our students waste a lot of paper, usually reprinting something 10 times without checking to see if the printer is actually working and/or has enough paper. We work with students to try and improve their use of the printer, but in the mean-time, we have a lot of paper which has already been printed on one side, and which would otherwise go to waste.

Here are some ideas on what schools could potentially do with all of that paper. 

1. Use it for paper folding.

   

   Paper folding has strong links to mathematics and art. It could provide a context for a mathematical activity so that students see the mathematics they are learning applies to parts of the world, rather than being just symbols on paper. Students can also use paper folding as a way of expressing themselves through art.

    

2. Students and teachers can obviously use the extra paper as rough paper for writing, drawing math diagrams, or a whole bunch of other things.

    

3. Send the paper to a school in need. Our partner school in Kipevu, Mombassa, Kenya typically runs out of paper half-way through a semester if they are lucky.

    

4. Some of the paper may have information on it which is sensitive in nature. You can first send the paper through a paper-shredder, and then use it for weaving, which can be handy when you need baskets around the school. What school wouldn’t benefit from having more storage containers available? 

   
   (Image credit: josey4268)
   
    

5. Flip it over and print on it again. The only problem here is that printers are typically more likely to jam with paper that has already been used, so this may be a problem, depending on your model of printer.

    

6. Use it to line the inside of animal cages in your science department.

    

7. Cut it into smaller rectangles and use them for exit slips.

    

8. Turn the paper into envelopes and have students send "Get well soon" and "I miss you" messages to their classmates that are sick, or to their penpals in another school.

    

9. Turn the paper into paper airplanes and have an iterative design contest. The objective of this type of contest is to collectively build the best paper airplane you can by cycling between creating paper airplanes, observing how well they fly, and then redesigning the airplanes. From this activity students will learn that design is messy, collaborative, and fun.

Any other suggestions?

I’ve started a presentation on grading, which anyone with the link can edit and add their perspective. The presentation is split into four areas:
 

• What are the goals of grades?
   
• What evidence supports the use of grades in schools?
   
• What evidence does not support the use of grades in schools?
   
• What are some alternatives to traditional grading systems?

Please feel free to add your perspective to this summary of the debate on grading, ideally citing evidence for any strong claims that you make if possible.

https://docs.google.com/presentation/d/1tu3y6DU5IrT-Y68tva_mZHM7HYeGvpZMfRBMdxY3x38/edit?usp=sharing

Remember that there are a lot of emotions behind the different perspectives on this issue, so to refrain from personal attacks, etc… My objective is to gather evidence and perspectives, rather than start any flame wars.