The Reflective Educator

Education ∪ Math ∪ Technology

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Month: July 2011 (page 1 of 3)

Sharing ideas about math in the real world

I’m very interesting in finding ways mathematics is present in the world outside of the classroom, which I’ll call the "real world." Obviously what students do in the classroom is part of the real world, but too often in math instruction school math is completely separate from the contexts kids experience in their day to day lives. I’d like to build a collection of resources for math teachers so that we aren’t all scrambling in the dark looking for ways to incorporate more contextual learning in our teaching. Note that I’m not at all opposed to teaching mathematics which is highly interesting but has no real world context. I’m just opposed to teaching math which neither has context or any pizazz.

Here are some of the ways you can share your thoughts and suggestions on real world math.

  • Join the Flickr group here. Make sure that each of the photos you include has a description of why it represents a mathematical idea. Your students could also join this group and share their own ideas!
     
  • Blog about real world mathematics. Use the tag ‘realmath’ in your blog post (all one word) so that other people can find it by searching. Post a link to your blog post on Twitter (or any other social network) with the hashtag #realmath.
     
  • Post ideas under #realmath directly on Twitter (no blog post required).
     
  • Contact me through the form above and (assuming it’s appropriate!) I’ll post your idea about real world math here.

Please suggest other ways we can share ideas for context based mathematics and I’d be happy to include them here.

Math in the real world: Randomness in nature

This is another post in a series I’m doing on math in the real world.

 

Nature has lots of things which look random, but which are influenced by environmental factors. For example, when pine cones fall from a pine tree, they will tend to fall in a ring around the tree, but will bounce when they hit the ground, so the terrain impacts the distribution of the pine cones on the ground. Barnacles are essentially randomly placed on rock, but on shore-lines they are almost always found on the side of the rock facing the ocean.

Generally we tend not to understand randomness very well. When we create something that we want to be random, we tend to over-emphasize filling of empty space and spreading out the information, and under emphasize the number of pseudopatterns and clumps in the data. 

Here’s a classroom idea. Have students go around and take pictures (or record in a journal, perhaps with a pencil drawing) of things which look random. See if they can find examples of things which are actually random, and things which are evidence of human activity. In the slideshow above, you should spot one example of a rock collection which most decidedly not random.

Math in the real world: Architecture and Construction

This is another post in a series I’m doing on math in the real world.

 

The amount of mathematics required to design, plan, and construct one of these houses is amazing.

The plans have to be done in scale, and the building is often shown in both perspective and orthogonal views. The budget for a housing project is often estimated using square footage formulas, but can be quite complicated when more carefully calculated. Ensuring that the house is up to building code, or is LEED certified can involve more sophisticated calculations and analysis. For custom designs, the customer needs to be involved in the process, which means that good communication skills are crucial.

When creating the plan for actually creating the real version of the house, each aspect of the project has to be carefully scheduled. This is so that one isn’t trying to wire the house for electricity after one has laid the drywall. Creating this schedule often involves collaborating (with multiple contractors), problem solving, and logical reasoning skills. The design itself has to be checked for structural flaws. For example, it is also critical to check at this stage that load bearing walls which actually be able to bear the weight of the floors above them. 

The construction itself involves measurement, and understanding tolerances in measurement. Some measurements also involve using trigonometry, or the Pythagorean theorem. Workers have to plan carefully, and when they run into issues, creative solve problems.

In your school there are a few ways to use construction problems. Students could use a 3d design program like Google Sketchup to create a model of their house. From this model students could calculate the measurements of the house, check for structural integrity, and even create a careful budget of the cost of the house. Students could even create a design, and then implement their design (either in miniature, or possibly at full scale if you have a useful project that needs to be completed).

 

Books Every Math Teacher Should Read

This was originally posted at the Elevated Math blog, and I’m reposting it here.

I’ve been doing a lot of reading this summer, and I’ve come across a few books in the past few months that I think every math educator should read.

A Mathematician’s Lament by Paul Lockhart

I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.” ~ Paul Lockhart, p33

The original essay that inspired this book is still available online here, and if you can’t find the time to read all of Paul’s book, I recommend at least reading the essay. Paul talks passionately about some serious problems in mathematics education today, most notably that much of what is taught in schools is not actually mathematics itself, but a caricature of mathematics.

Mathematics Miseducation by Derek Stolp

[M]athematics, as it is taught, does not give children any view of reality, let alone a rational one.” ~ Derek Stolp, p33

Derek argues first that mathematics, as it is taught today, does not warrant inclusion in our curriculum, but then demonstrates some clear ways that mathematics education could be changed to make it viable again. He has the best argument for a constructivist approach to mathematics education I’ve read so far.

The Math Instinct by Keith Devlin

Overall, the shoppers’ performance was rated at an average 98 percent in the supermarket compared to a mere 59 percent average on the test.” ~ Keith Devlin, p187

Keith demonstrates how animals and nature have developed some amazing strategies for using mathematics in highly creative and important ways. He then suggests how this relates to our own ability to do mathematics, and shares some very compelling research on poorly we transfer math skills learned in school to our own lives, but use mathematical strategies nonetheless.

How Children Fail by John Holt

I asked Monica the other day how many thirds were in a whole. She said, ‘It depends on how big the whole is.’ If we could look into the minds of our students, in how many would we find that thought?” ~ John Holt

While this book is not specifically about mathematics education per say, John does have some insights into the difficulties learners experience. He often has useful anecdotes taken from his own teaching of mathematics about the frustrations and problems learners have in this area. John also talks about the disconnection between what kids learn and the world, the problems with reward and punishment systems, homework, and a host of other issues in education talked about in depth by other educators.

What’s Math Got To Do With It? by Jo Baoler

I just read “What’s Math Got To Do With It?” by Jo Baoler, and I would actually move it to the top of the list. It’s a must read for math educators, not because it has the most eloquent argument against our current form of mathematics, but because it has some actual solutions that educators can implement right now in their existing practice that will make a difference. Jo also does a fabulous job of avoiding pointing fingers, reflecting on research that has been done, and grounding her observations in improving practice in easy to read anecdotes that everyone can understand. I’d also strongly recommend her book for parents who are concerned about math education as well.

Some other books which I have not yet read, but which are on my to read list are:

  • From Reading to Math: How best practices in literacy can make you a better math teacher by Maggie Siena
  • Young Children Reinvent Arithmetic: Implications of Piaget’s Theory by Constance Kamii Young
  • Knowing and Teaching Elementary Mathematics by Liping Ma
  • Innumeracy by John Allen Paulos

What books are on your “must read” list for every math educator?

Update: On the original post Grace ( @msokeeffe ) added Seymour Papert (Mindstorms, The Children’s Machine) and Robert P. Moses’ Radical Equations: Civil Rights from Mississippi to the Algebra Project.

A practical way to reduce teacher burn-out: Say thank you

Dr. Adam Grant shares an easy way to reduce teacher burn-out. We need to help teachers realize the work they do makes a difference in the lives of their students. He shares a story about how a university call centre saw much reduced stress rates simply because they heard the five minute story of a university student who benefited (with a scholarship) from their work. I recommend watching his Ted talk from TEDxPhiladelphiaEd.

The quiet revolution in education

(Clay Shirky: How social media can make history)

While education reformers like Michelle Rhee, Joel Klein, Bill Gates, and others will tell you that education is stuck in the status quo, right underneath their noses there is a quiet revolution occurring in education.

The revolution is happening through social media. Every day thousands of hours are spent by educators, even during the summer, to improve their personal practice through discussion and sharing of resources. Every day more and more educators are joining the fray, choosing to sign up for social media sites (like Twitter) so that they can become part of the conversation on education reform. While the number of educators not yet sharing their ideas dwarfs the number sharing, those that are sharing are vocal about the benefits that they are getting and inviting their colleagues daily to join them.

There are probably 50,000 educators using Twitter alone, and if each of these teachers posts just 1 average length tweet a day, that’s about 500,000 words written each day on education by people in the trenches. If each teacher on Twitter reads just 10 tweets a day, that would mean that more than 5,000,000 words about education are read each day via Twitter (The actual numbers are likely to be much higher than these conservative estimates).

Outside of Twitter, educators are connecting through Classroom 2.0, Future of Education, and literally thousands of other Nings and professional development sites. There are almost certainly thousands upon thousands of conversations between educators, about education, happening on Facebook every day as well.

Educators are doing much in the non-digital world to connect as well. Edcamps and Teachmeets, which are free professional development conferences, have sprung up all over the world. Educators are organizing TEDx conferences, like TEDxUBC, TEDxDenverEd, and TEDxPhillyEd, to name just a few (I attended all three of these).

This is all done outside of the more traditional professional development avenues, and it is having an impact on education. Teachers are flipping their classrooms, engaging in education hackjams, discussing educational practices in massive weekly Edchats, presenting their innovative educational practices with thousands of other educators via online webinars, and much, much more.

Much of this quiet revolution is happening during times when educators would have traditionally been off work, during their summers, their breaks, and at their homes, challenging the idea that educators aren’t willing either to change, or to spend their own time doing it. Educators are not being paid to participate in the opportunities discussed above, nor are they being given much support.

Instead of blaming all of the current problems of education on educators, maybe it’s time to support the thousands of innovative educators out there taking matters into their own hands? None of the accountability systems in place, or being developed, accounts for the incredible professional sharing occurring globally in education today.

The grassroots efforts are a much more effective way to introduce systemic change than top-down efforts ever will be since peer pressure is always stronger than authorative pressure.

Math in the real world: Trees

This is another post in a series I’m doing on math in the real world.

 

The growth of trees is actually a fairly mathematical process that at least involves fractal theory, graph theory, and topology. You can actually generate very realistic looking trees using a computer. See the video below for an example of simulated tree growth.

Here’s an idea. Take your kids outside and find some trees (even bushes or ferns will do in a pinch). Explore (and catalog) what rules different trees seem to follow as they branch. See if you can follow those same rules with pencil and paper to produce tree-like drawings. For bonus points, take some pictures of some younger trees, and use your rules to predict where the next branches will start, then follow up in a year to see if you were right.

Kevin Slavin: How algorithms shape our world

Kevin makes some great points in this TED talk, and his talk certainly speaks to the need to teach an understanding of how algorithms work. In my opinion it is important to teach the study of algorithms explicitly, rather than implicitly through just memorizing them. We should focus more on how algorithms are used and why they work. What if kids learned how to create their own algorithms? 

Math in the real world: Which piece of cake?

This is the second in a series of posts on math in the real world.

 

Which piece of cake should you choose?

This is a problem that often happens at birthday parties. Should you pick a corner piece, a side piece, or one of the middle pieces?

The answer depends on whether you want to optimize for volume of cake, volume of icing, or a balance between enough icing, and enough cake. You can either view this as a volume and surface area problem (or use calculus to determine the optimal piece of cake).

Side piece of cakeMiddle piece of cake

It’s not hard to see that a corner piece has the most available area for icing, but it is less clear whether a middle piece or a side piece has more icing. How much of a difference does the angle in the cake make?

One obvious way to bring this problem (literally) into your classroom is to make a cake and share it with the students, which is more useful if you talk about the shapes of the pieces in advance, and get kids thinking about which piece of cake they should take, depending on their preferences. At the very least it is likely that students will never look at a birthday cake quite the same way again.

Math in the real world: Carnival probability

This is the first in a series of posts on mathematics in the "real world."

 

Should you pay to play these games?

Aside from the obvious answer, that these games are fun, and so whether or not you win, the games have an intrinsic "fun" value, one wonders how fair these games are to play. It’s easy to deduce that the games are unfair (why would an organization trying to make money use them otherwise?) but how unfair are they?

This is essentially an expectation problem in the wild, and attempts to analyze games of chance like these by mathematicians led to the formation of probability theory itself. 

One way to answer this question is to go to a carnival where there are a bunch of these games, and record some experimental data. You can either use your own money to play all of the games, or stand around surreptitously with a notebook and let other people do the experiment for you, while you record results.

Another way would be to set up your own versions of these games, then host a carnival at your school (or in your classroom), and have participants play the games (perhaps using monopoly money?). If they don’t record results, the activity is fun, but not very mathematical, so I strongly recommend some record keeping takes place during a school sponsored carnival event. You will also need some time at the end for people to analyze, then discuss their results. I recommend, if you do this as a school activity, checking in with your students about common misconceptions about probability that they have, and make sure their analysis of the results spends at least a bit of time debunking some of those misconceptions.