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# Week 6: Humanistic mathematics

Part 1

Find a source, such as an article or a video, that discusses humanistic mathematics or some of its aspects. Link your source and write a brief summary.

Part 2

Find a math-rich craft you like. Sketch a project you would do if you had to teach math through that craft, to a group of students who also like that craft. What topics lend themselves well to that craft.

Inspirational example 1: hyperbolic crochet coral reef movie.

Inspirational example 2: Huzita-Hatori axioms of origami geometry (it's more powerful than Euclid's axioms - you can even trisect angles!)

Official trailer of origami math movie "Between the folds":

Part 3

Some people define mathematics as problem-posing and problem-solving only. Others consider math-rich art, poetry, and crafts to be mathematics as well. As educators, we face this argument (from parents, administration and students) every time we attempt to teach with projects. Name several steps we can take to make sure we keep ourselves and our students intellectually honest (doing real math) while conducting humanistic mathematics projects.

Part 4

Name several ways technology can help students "mathematize" the craft you chose in Part 2.

• Part 3 & 4: I found it best to use an example. Math is near impossible for me to do without having visual examples...whether its paper and pencil or looking at concepts in a book or online, etc. I just can't do any math thats beyond basic in my head. So, for part 3 & 4, I couldn't get origami out of my mind. So here is how I would approach the question of how to assure you relate your math concepts for example...& lets just say to keep it simple, measuring angles' degrees.

here is a really good link that shows how to start to creat a simple origami on a square piece of paper:
http://anyone-can-origami.com/basic-origami/

If you were teaching angles, the students could be required to measure all the different angles while creating the origami. Or you could do the opposite, which would be fold the square so that you have a 45 degree angle in one place & a 90 degree fold somewhere else.

Origami has so many levels of math, that you could truly use it straight through to college level learning. Here is another link to demonstrating higher levels of math through origami:

http://plus.maths.org/content/power-origami
(scroll towards the bottom fo the page to see the math equations.
An origami has so many human crafty ways of keeping a students interest yet all the while, there are math lessons throughout it. Its a never ending human experience with math, science & humanity all wrapped in one.

With technology, I can show a million examples of creations stemming from math. Being online, sharing lessons, brainstorming with other teachers, allowing students to experience with many different forms of learning and being able to tell use as their teachers what keeps their interest is such an incredible gift we now have available to us.

I literally NEVER did an origami. If it wasnt for the world wide web, and this class, I wouldnt know how to create an origami...much less teach it to my students.

• Lisa, I found the "million examples" you list focused on social aspects fo technology. Modern tools enable VISUAL sharing in particular! When I first started to learn origami, I had to decipher the symbolic diagrams, for example, - - - - meaning "valley fold." While I mostly enjoyed these diagrams as puzzles, I would also get frustrated with more complex folds. "But what do you mean push it through and turn it inside-out and upon itself?!" - I would yell at the text instructions. When the number of origami videos exploded on YouTube, my life became so much better. I still like to decipher symbolic diagrams as puzzles, but if I get stuck, I can see the movement live (someone folding it in the video).

Which brings me back to math. So many parts of math are PROCESSES. Animations and videos allow us to share our processes. We made some examples in this course, in Prezi and Scratch...

Modeling software is usually very social, because we can send files around! But it's not just about sharing, it's about making. Robert Lang, one of big origami artists, talks about the role of modeling software in his craft and his math: http://www.langorigami.com/science/computational/treemaker/treemaker.php

The fold diagrams are so beautiful to me, with the mathematical beauty of structure and symmetry:

• I'm sorry if I tortured you with too much!! I truly appreciate all of your replies! Your passion for this course is FANTASTIC & CONTAGIOUS!! :) -Lisa Lisa Rittler email: LisaRittler@gmail.com cel ph# (215) 740-6036 Our OceanCity, NJ condos are available all year!
• Heh, no torture - I liked your phrase "million examples"! It's a hyperbole, sure, but hyperboles are very appropriate for the topic of humanistic math, aren't they?! For my part, I always appreciate MANY examples as a learner. As a teacher, it's a good practice to use gazillions of them. The very minimal possible number is three, for everything we do. You'd be shocked and amazed how many textbooks break "the rule of three" and provide only one or two examples for new techniques and ideas they introduce.

• I love origami! IMy husband and two year old fold paper airplanes all the time. We actually got daddy a book for Christmas with all different examples. I never really made the connection between the folding and math concepts though. I am driving my husband crazy now when I use math terms like forths when we make the planes! But my son is two and understands what I mean!

• Part 1: I found an article called Interdisciplinary Teaching Strategies in the World of Humanistic Mathematics. This article talks about humanistic mathematics as an interdisciplinary undertaking and states that a major goal of humanistic mathematics is “for students to gain an appreciation for mathematics as a creative, collaborative, and exciting endeavor and to move away from thinking of mathematics as a monotonous set of rules used to solve tedious exercises which seem to have no useful purpose.” I thought this was a nice way of summarizing what humanistic mathematics is all about, and I especially like the idea of math educators finding ways to present the subject as creative, collaborative, and exciting, since I think many students would not use those words to describe math.

The author outlines three examples of humanistic mathematics teaching techniques. The examples are from university level math classes, but I think the approaches can be generalized for many levels of learning. The first example uses polyhedral models to study group theory. I thought this was an interesting project since it shows concrete examples of concepts from group theory, which can be a very abstract subject. The second example uses art and images to aid in understanding 4-dimensional space and beyond. I love the idea of this concept presented as a progression of images. The third example examines math problems in a historical context, which is an idea that I really love. I think that math history is fascinating and can really add a layer of depth to the subject, but it is so often neglected in math classes. This example tied in neatly with this class because it discussed solving historical math problems using modern technology.

Part 2: One topic I’m really interested in exploring from a teaching perspective is spatial skills and spatial awareness. I’m very personally invested in gender equity in mathematics education and the underrepresentation of women in STEM fields. I’ve done a lot of research about this over the past year, and while I think there are a lot of complicated issues surrounding it, I am especially interested in practical techniques educators can use in the classroom to make a difference. The Department of Education published a practice guide called Encouraging Girls in Math and Science that outlines five research-based pratical suggestions for teachers, and one of the suggestions is related to spatial skills. From the guide: “…the mathematical test items that show the greatest difference favoring boys are spatial in nature. Spatial skills can be improved with training.”I think it's a topic worth exploring in the classroom, and one that is beneficial for both boys and girls. A few of the specifics mentioned in the guide are mental rotation, spatial perspective, and embedded figures.

I thought that one interesting way to approach spatial skills from a humanistic perspective is through photography. Mental rotation is asking students to look at a two dimensional representation of a three dimensional figure and then determine if that figure can be rotated in such a way that it matches another two dimensional representation. Photographs are two dimensional representations of three dimensional objects, so I think there are many connections that can be explored. Having students take photographs gives them the opportunity to explore the ideas of scale and perspective, skills which have applications in both math and photography. It would be interesting to take some 3-d objects of various shapes and sizes and let students see how many ways they can take a photo that shows one object as a completely different 2-d shape.

Part 3: One way we can be sure our students are learning real math is to create clear math objectives for a lesson or project and assess based on those objectives. A clear measurable objective that is specifically about math would give us as educators a way to keep the project focused. If we could show parents or administrators that a project starts with a standards based objective and ends with assessment of that objective I think it would help them understand that there is real math learning happening.

I also think it would be helpful to share research that shows the benefits of a particular humanistic method for teaching math. From the outside, projects that are craft-oriented might seem like a waste of precious time in math class, but if we can point to research that shows these strategies have been shown to increase students’ understanding of math concepts, those projects might seem like a more valuable part of a math curriculum.

Part 4: Digital cameras make it very cheap to take lots and lots of photos, so technology makes this kind of project much more practical than with film photography. Students could upload the photos and use editing software to crop photos and further explore spatial perspective. To mathematize this kind of project, students could use 3-models of geometric shapes to take photos that match 2-d pictures. For example, students could be given several different drawings of a triangular prism and be asked to take a photograph that matches the image. I think that exploring the relationship between the shape of the object and the two dimensional representation can strengthen spatial skills.

I think this project would fit nicely with 3-d geometry software like discussed in this article (link goes to pdf file). This software is very math focused and allows students to manipulate objects in space, which goes back to the mental rotation mentioned in the practice guide. I think that by using this in conjunction with photography, students can focus on math and also think deeper about the concepts they explored with photography.

• You can design a lot of mathy group or network (online) games centered on photography. Building on your example, a group can go on scavenger hunts for objects with a given projection (for example, cubes can project as squares, parallelograms or hexagons, depending on where you stand). It looks like combining photography with simple editing tools is the way to go, which is easy to do with the touch technology. For example, students can trace outlines of their shapes on the screen.

Here is one photography game I ran, a scavenger hunt: http://mathtrek.naturalmath.com/mathinthemachine/

For spatial skills, girls may be at disadvantage because spatial skills develop, early on, through movement. And little girls are discouraged from moving "too much" and tend to be more stationary. "Math playgrounds" with smart climbing structures appealing to girls may help! Also math dance...

It's a lot of work to assess projects, compared to exercises. This is a reason teachers may use projects less. I would use rubrics to assess projects.

I like the way the software package you linked assembled a collection of didactic 3D tools. Each tool does different things! Actually, most of these tools - not in one package though - are already available online in some form. For example, virtual Legos are similar to their cube builder (though more powerful): http://ldd.lego.com/

• I love the idea of photography and thanks for the site with the scavenger hunt! I love Legos so thanks for that too.

• Part 1:

http://www.comicmath.com/calvin-and-hobbes-math-comics.html

Some math by Calvin and Hobbes :) I find this so funny because it truly is something so many parents can relate to. We want so desparately to have an answer to every question our child or as a teacher, our students may ask. What's so nice about this comic is that I immediately wanted to come up with a mathmatical equation to solve the boy's question! I thought this would be so fun to work on in a classroom!

• Comics can make math very personal - and this is definitely a humanistic aspect!

This IS a good question. How do they know? At the end of the day, dad isn't all that wrong, because part of the material science is experimental. They don't use trucks and bridges for models, though.

• Thanks for this comic! We all want to have the answer when a kid asks.

• Part 2:

http://www.learner.org/interactives/renaissance/fibonacci/gold2.php3?scale=8&direction=above&square=6

I just love the fibonacci pattern & how we can find it all over in nature. I think students of many ages can enjoy creating these kinds of spirals. I like the idea of kids looking at different images online, like flowers and then just computer images of using this pattern to create beautiful art!

Lesson:

1-I'd start with assuring that students truly understand the concept of the fibonacci sequence (fib. seq.).
I'd give out a worksheet with fib. seq. missing #'s on them and have the students fill in the numbers.

2-Next i'd show images of fib. seq. and fib. patterns found online.I'd ask the students todothis as well. I'd ask questions about how they think it relates to the sequence. Gathering information. Asking them if they think they could use geometric shapes to create som kind of fib. patt.

3-I'd demonstrate similar to the link above using squares, rectangles, etcto create a spriral.

4- I'd havethe students work in groups to do the same.

5-finally as outside of school assignment, I'd ask them to take pictures on their phone or look online or even around their house for things that look like they have a distinct pattern or even possibly a fib. pattern.

• Lisa,

As a child I remember taking standardized tests that had some variation of "Guess the pattern and fill in the missing number".  These questions drove me nuts; I don't recall being told that an equation could describe them, which would have helped me big time.  I never even heard of the Fibonacci sequence until I began college, though I've always noticed the pattern in nature (pinecones, sunflowers, I wanted to be a botanist at one point enough said).  When I was growing up, public school math education wasn't what it is today (especially for "girls").  It's great that you want to introduce students to this concept; as you said, it's everywhere.  Since there are so many examples, it's a great starting point for the topic of numerical sequences in general.

• Sue, the way you approach tasks in this course is very analytical and deductive. Pattern tasks are often presented as guessing, or inductive reasoning where people "randomly" try different things and see if anything emerges. It does not surprise me you would dislike such approaches, and that knowing there are systems behind patterns (equations, functions) helps you.

There was a big discussion, even a sort of a war among math ed researchers, spanning several years, articles and conferences, about the simple claim that pattern tasks are useless in math ed. People described scenarios similar to your childhood story. Pattern tasks do not contain enough tools, these people claimed, to help kids see the formulas, math ideas and other STRUCTURES behind the pattern. Other educators said that you could support students in understanding this math behind patterns, and provided examples of such support, like those Fibonacci tasks.

• Part 1

I found this video that really taps into an appreciation of the subject of math. These are some of the quotes that jumped out at me that relates to the humanistic idea of math. http://www.youtube.com/watch?v=kWaabsx11B4

“If you can ever talk to God, math would be the language.”
“Mathematics is the language of the physical world.”
A lot of the math discovered throughout history is still true today. “There is a permanence to math that is very beautiful.”
“People who are math people do math because it has a beauty to it.”

This video’s primary purpose is talking about the Art of Problem Solving's Summer Math Intensive at Bard College. This is where students can develop their love of math and natural talent in mathematics. The teachers in the video talk about how important math is in human development, which I think relates directly to the humanistic curriculum. This is a 3 week experience that builds the students love of math and develops problem solving skills on a much deeper level. I want to go!

Part 2

A math-rich craft that I think would be really interesting for students to create is scale models or drawings. This can help students see the ratios between the actual objects and the scaled objects as well as how important the proportions are to create the model or drawing. Based on the interests of the class, I have two options for this activity.

Option 1: Space and NASA

The first thing I would do in my lesson is use this video from NASA Connect, a channel that investigates the connections between math and science and space! This video includes questions to ask the class before starting the activity as well to get the students to think critically about the topic before they start creating. The video also provides an explanation of how scale maps are used and how to set up a proportion. http://www.youtube.com/watch?v=Cv7_CVD6_Yk The topics that relate to this video are ratios, setting up and solving proportions, space and NASA, scale drawings and maps, and scale models.  After the students have watched the video, they will then be given the task to create a scale model of an object in space or a NASA-related object, like a space shuttle.

Option 2: Ancient History

To begin this activity, students will investigate the pyramids of Egypt using the following site. http://www.pbs.org/wgbh/nova/pyramid/geometry/index.html Students can click on the pyramids and read about the different parts of the pyramid in the picture. It is important to know the actual figure that they will be scaling down. Students will write down three interesting facts from the website. Students will then click on "Build a scale model of the Great Pyramid" and the "outline provided" link. The students will then print, cut out, and assemble the Great pyramid, with the measurements facing out. Students will then complete the table on the "Build a scale model of the Great Pyramid" page of the website to compare the actual heights and scaled heights. Then students will create their own scale models of the Khafre and Menkaure pyramids with the scaled dimensions calculated in the table. To tap into the students’ creative side, students will then be instructed to paste their pyramids on an appropriate-sized poster board to be displayed. They may decorate this board in order to properly showcase the location of the pyramids. The topics that relate to this activity are Ancient Egypt, calculating scaled dimensions, and creating scale models.

Part 3

I think that projects are an important way for students to extend their understanding of a concept beyond the homework and test and also incorporate other important skills for their development. Some of my top scoring students on tests and quizzes struggle with the projects I assign because it requires other skills that might not be as strong. For example, the pyramid project that I created for Part 2 requires students to measure, draw, cut, fold, and assemble. These fine motor skills are important to incorporate into the curriculum to give students practice, especially depending on what age they are. I have students that take an excessive amount of time to cut, which may be because they were not given enough opportunities to practice and refine these skills. Other skills that can be incorporated into projects is problem solving and real-life experience. I had my Pre-Algebra students creating small businesses and writing up business plans with an expenses list and explanation of the price of their item in order to get a profit. This is the type of experience that students need to see in the classroom in order to be able to see where it can apply in real life.

What I do to make sure that parents and students see the math skills involved in the project is create a rubric. The rubric also includes the point values of the project so the students and parents can see what parts of the project have a higher point value, which makes it the most important ideas in the projects. I make the creativity and evidence of effort shown about 5-8 points and the mathematical skills usually make up the majority of the points. For example, if I would create a rubric for the pyramid project I did for Part 2, this is what it would look like:

_____/3 points for Interesting facts noted
_____/15 points for Excel chart completed and correct based on the information given
_____/15 points for accurate scale model based on the calculations recorded
_____/7 Creativity and evidence of effort shown on the display poster
_____/40 Total Points

Part 4

A technology that could be used in conjunction with the pyramids or space project would be to have students use GeoGebra to create the points and lines that make up their scale models. This could be the step before the students create the actual models so the students can have a blueprint of what they are creating.

Another piece of technology that the students can use is creating a Prezi presentation of what they learned about space or Ancient Egyptian pyramids and include the steps for creating scale models of the pyramids.

Another way that students can show off their display boards is creating a video or a virtual tour of their display board and the pyramid scale models they created. I can already see some of my students using Lego people to “walk” through their pyramids and share some interesting facts about them. I have seen some of the videos they can create and they are amazing! A high school student from my school won a Lego video contest and his video is premiering in a Lego film this summer. So cool! We really do have to amp up our projects so these tech savvy students can really show off their talents!

• Hi Gina,

Love your idea about building scale models!  Being able to physically verify the results from their calculations (in the form of the completed model) emphasizes the connection between 'math thinking' and 'math doing'.   Using one's hands to fulfill a creative idea (their own or anyone else's) is very rewarding for some.  Others feel that creating the plans that  craftspeople/ tradespeople will bring to life is more rewarding.  Building scale models offers enjoyment for both types, as well as providing an opportunity to socialize a bit while they're working.

The pyramids also offer a great opportunity to explore the relationship between culture and math (week 7 topics), as the pyramids are examples of the Egyptian society's access to both scholarly engineers and slave labor.  Again, great idea - there are so many learning opportunities!

• Thank you, Sue! I love connecting culture and the history of math into my lessons because I have so many history buffs in my classes.

• Scaling is a very rich math topic, because you deal with proportions - the cornerstone of algebra, according to several math ed research groups. As you describe, mapping and physical models are two projects capturing different aspects of scale. Mapping is more symbolic, whereas physical models are more concrete.

The gifted population you work with tends to have a sizeable number of motor issues. It may be because kids are attempting more ambitious projects (so their issues are more in the way). Or it may be because gifted kids and their parents are more aware of their own learning (and issues with learning). It may also be because gifts come with risks - making kids "twice exceptional." Whatever the reasons, you have to think of accessibility...

Toward this end, I would like to suggest more tech to you, namely, 3D printing and the modeling programs that go with it. Here is one of my favorite educators who work in this area, called "digital fabrication" - Paulo Blikstein from Stanford. He works with disadvantaged kids a lot. His lab is fantastic. http://www.youtube.com/watch?v=ylhfpDAniqM

• Wow! I was amazed while watching the video about the different inventions the students were doing and how they were inspired to create them. I did not think this far into the tech world because I was so focused on students creating the models, but you are absolutely right that gifted students do have challenges with fine motor skills especially. I have some students that can solve an equation with variables on both sides but struggle when cutting out a net for a cube. I do think we need to make sure to keep those activities in class so we, as teachers, can help those students gain the practice with their fine motor skills and help them develop. But I also think that students can dig deeper when they are not focused on the cutting and pasting. I was truly amazed by this video. Thank you for sharing!

• Part 1

(Disclaimer) So I have a slight obsession with the golden ratio and the Fibonacci sequence. I think it is insane that this number and pattern just appear in nature seemingly randomly. Also it shows up in architecture, the human face and body, and many other aspects of the world around us. The world in which we live can be broken down into shapes, and equations, NUMBERS!!!!!!! They are all connected in ways that at a glance you would never see, but with a deeper look it can all be explained with math. Here is a short video I found that show this idea.

I also found this article by Roger Haglund a professor at Concordia College in Moorhead, MN. www2.hmc.edu/www_common/hmnj/haglund.doc .He talks about some of the reasons why current math teaching methods might not be working. When he began teaching at this college, he offered to start a math class, Explorations in Mathematics, for students who did not meet entry level requirements. He used a humanistic approach to try to get the students involved and motivated. The article includes surveys which showed these students, at the end of the course, had a better appreciation for math and seemed to enjoy the class. If using humanistic mathematics can make our students more involved and LIKE math, I am in!

Part 2

I like to draw. I am not that good at it but it is fun and relaxing. Drawings can be broken down into parts. By this I mean, you can take a sheet of paper and section it off into smaller parts. One idea I had was to have students bring in any cartoon, or picture of their choosing. Then they must redraw it using math! So they could make a scale in which they could segment the picture so they could redraw it in easier to manage segments. This can lead to many math concepts. I could tell the students they must shrink it by 50%. They would have to figure out how much smaller that would be and how to make the segments. I could also have them double the size of the picture (I would have to make sure they had pictures small enough, or I had paper big enough.) For a little more of a challenge, I could ask them to draw the image flipped or rotated in some way. This would be more of a challenge because it is not just simply redrawing the image; they would have to consider what would move where. As mentioned before, I could use my love of the golden ratio and have them bring in picture involving the ratio. We could then talk about how the proportions work and how to use them to help in redrawing the image.

I love the video of the crochet coral reef. It is really neat how the project started as a simple thing and took on a life of its own. The use of technology allowed for many people to be involved that may not have known about the project at all. I knit myself, never could get crochet, but I wish now I could do it as to add to the project! Origami is also something I find fascinating. We could challenge students to make different objects using certain number of folds.

Part 3

I think if we base the project in math topics then we could justify the humanistic side of it. As with my project for part 2, it is based off learning how to transform an image. Instead of simply using a triangle to show the stretching and shrinking, flipping and rotating we are using art. As with anything, I would establish some form of assessment. There would be a rubric for the project to ensure the students are getting the math behind the idea. Some students may simply be good drawers and feel they do not have to use the grid system, but if that is part of the project they need to do it.

Part 4

I was thinking that after they redraw the image by hand, we could scan the originals and recreations into the computer. We could then use some type of software to see how accurate our recreations are. I think this would be fun for them to do. I am sure I will get the “why couldn’t we just do it all on the computer?” from students. But I will simply tell them I needed artwork for my walls. J No I would explain that doing the actual image manipulations makes it more clear as to what is happening. We could also work with proportions and they could do more complicated resizing. For example shrink the image by 3/8 or enlarge by 43/7. These would be hard and ridiculous to do by hand!

• Had a chuckle about how you knit and don't get crochet - I crochet but am still befuddled by knitting, though many patient souls have tried to teach me to no avail!

• My mom tried her best to teach me crochet but no go. I feel like you can either do one or other but not many people can do both.

• You raise a very serious question - "Why still do things by hand when we have computers?"

• Handmade things have artistic and sentimental value. By hand, you can add certain variation, details and aspects that computer software does not support. Each medium has its restrictions, computer art included (for example, you can do fine details with pencil drawings, but you can do color with oil pastels). But little errors, freehand variations and such of handmade objects add them human charm.
• Doing things by hand is instructive. It's a good way to learn

I would like to add a reason that may seem obvious:

• The way your body acts on physical objects is not yet reproducible via computer interfaces.

There are some experimental virtual reality interfaces that attempt to "go there." Meanwhile, consider an example from your "drawing by part" task. Let us say you want to draw a hand (which is a hard task). In the physical world, you can touch one finger with the other hand and feel its three bones, that signify the three parts. You can bend the finger to feel how the parts are connected. You can take a marker and trace its tip around the finger's joins to feel that line making a roughly circular shape. These EMBODIED actions aren't yet available through computer interfaces, though people are actively working on making it happen.

One aspect of your task I appreciate is that it can be scaled to all ages and stages of development. You can do it with toddlers, say, making rough characters out of simple geometric magnets. But you can do the same task at graduate level, with sophisticated functions used to make 3D mesh, like this dragon model:

• I never gave thought to the different mediums of drawing, pencil versus paint and such. I wonder if it would effect the proportional outcomes. I do like to do things by hand. Don't get me wrong I love technology and my computer software that helps me do things more efficiently but when I learn a new concept in math I like to learn the long way, no short cuts or help.

I do try to think of different age levels and abilities when I think of lessons. Since I am not yet teaching I am not sure what age I will have to plan for. Thanks for the dragon visual, it is cool and would be hard to draw by hand.

•

PART 1

I found a brief article (actually the transcript of an introduction to a faculty discussion about humanistic mathematics) by Gizem Karaali, an Assistant Professor of Mathematics at Pomona College.  This particular faculty discussion defines humanistic mathematics as "the human face of mathematics".  Karaali makes many interesting points during his introduction, such as: those in different areas of mathematics "really do not talk to one another", the terms that groups of mathematicians use to discuss their respective specialties often creates a language barrier, and "our goal is nothing short of breaking the barriers between these allied disciplines".  The goal of the faculty discussion is to create ways to "encourage scholarly work that transcends disciplinary bounds" and to brainstorm about "how can we integrate the arts and the humanities into a capstone mathematical experience for our students?  Conversely, how can we integrate mathematical experiences into a humanistic classroom?"

PART 2

I love music and welcome any opportunity to integrate it into mathematics instruction.  I already discussed this a bit in this week's gathering, so I apologize if I'm repeating myself (and also apologize for being rather talkative to begin with).  Also, this is only a rough idea of something I hope to do someday; any suggestions about my 'fantasy lesson' would be appreciated.

For younger students, I would explore fractions by integrating music into a music lesson called "Is it possible to hear a fraction?"  Students who are already studying an instrument might find this initially redundant, so I would encourage them to help me lead the discussion and/or bring their instruments in to provide examples.  To begin, the lesson would demonstrate the relationship of the 'beat' (time signature) to fractions (hopefully there would be a percussionist in the class to demonstrate as rhythm isn't my strong point).  We might write out beats of songs in fractional form (express music as a function of time).  As students become familiar with this concept, we will discuss how musicians who play a song together intuitively use math to make their fractions (notes) 'match up' though each musician is working with a different 'set' of fractions (percussion vs. keyboard vs. vocals, etc., this is much easier to hear than write about; listen to your favorite music and you'll know what I mean).  Again, students would be encouraged to either perform or provide their favorite music as examples.

The lesson will take an interdisciplinary approach when we explore how fractions (in the form of time signatures) are often associated with particular dances (3/4 for waltzes, 4/4 for the nightclub dance scene, 6/8 for an Irish jig); dance students (or instructors) would be encouraged to comment.  Following that, we will discuss common mathematical 'fraction relationships' that characterize particular genres of music.  At this point, I would love to co-teach with language and/or history teachers to further explore how culture, the spoken words of language, mathematics (via fractions), socioeconomic position (similarities/differences between those who composed assisted by instruments vs. those who didn't have that choice) and the desire for aesthetic enjoyment combine to form what we call 'music'.

For older students, I would use this framework (and encourage student contributions), but also include a discussion about how wavelength influences the sounds that humans perceive and explore the basic mathematical principles of sound, which would be great co-taught with with their science teachers.  The discussion of fractions would be expanded to include an introduction to harmonics and how these fractions influence the sounds we perceive (guitarists and bassists would probably be happy to give a demonstration).  Musical intervals have a mathematical basis; this and their effect on humans' aesthetic enjoyment will be explored and combined with all of the topics of my 'younger student' interdisciplinary unit.

I know this is WAY wordy, thanks for bearing with me and reading through.

PART 3

I'm going to answer this question as a teacher whose curriculum is largely geared toward producing 'acceptable' standardized testing results (such as those required by NCLB or state agencies).

We need to teach students not only how mathematics is integrated with other disciplines, but also how 'society' might separate it, as this is what often happens with standardized tests.  Therefore, students need to be able to apply - and demonstrate - their math learning in both 'forward' and 'reverse'.

Parents are often quite anxious about standardized testing - it begins in elementary school and continues right up to the SAT's (or ACT's).  Teachers need to take the time to teach (and reassure) parents about the benefits of a nontraditional, interdisciplinary approach.  However, this reassurance can only be validated by the student's ability to demonstrate their learning in both traditional- and non-traditional settings.

Therefore, I think it's essential to ensure that students can demonstrate their learning in either 'forward' or 'reverse'.  At this time, I don't know of any way to accomplish that other than having students practice mathematics in both settings, and assess them in both settings so that they will get an idea of what the 'rest of the world' is looking for as far as future employees.

Students who are able to experiment with both approaches may find they enjoy one approach more than the other; this realization may be very valuable when the student must choose a career path.

PART 4

My 'fantasy lesson' could use tech help in many ways.  A metronome is essential.  Rhythm tracks can be computer-generated; such software is readily available.  Software that compares audio input to a mathematical ratio (time signature) is also readily available, though this might create a high-pressure situation if students are involved, so I probably wouldn't ask students to demonstrate this (they'd probably get a kick out of seeing how their music teacher measures up, though).  Software that translates audio input into graphic depictions is also available; students can actually visualize sound, which reinforces the connection between what we hear and mathematical functions that we can see).  The user can also alter sound by manipulating these graphics.  Sounds from different audio inputs (saxophone, drums, anything) can also be synced (either if the musicians don't/can't or if the user (such as a dj) wants to create something new).  Students can explore different outcomes without actually having to perform the piece (or ask someone else to perform it).  Students would also be taught about software such as Auto-Tune, which combines math and programming to produce a desired sound.

In closing, I want to thank my 8th grade General Music teacher, Ms. Engler, whose classes were the inspiration for my lesson.  She motivated us to practice good behavior by offering the reward of devoting one 45-minute class period at the end of each month to discussing whatever music students brought in (or wished to play on their instruments).  I'll never forget how surprised I was when she demonstrated that some songs by Pink Floyd and Teena Marie had the same time signature (I was a classically-taught violin player who was taught that 'popular' music had nothing in common with classical (wrong!) and that thinking outside the box would certainly result in musical destruction).  Despite those 'words of wisdom', I'm still here, learning ragtime, bluegrass, and Irish, and I'm not ashamed to admit that making fractions add up to a whole on paper is always easier than making your two hands cooperate and 'add up' when playing against a metronome. I would never ask a student to perform in class unless they were completely comfortable with the possibility of failure; as teachers, we need to reassure all students that failure (or possibility of) is an essential part of learning.  As an education student, I look back on how Ms. Engler integrated her unit on jazz history with Black History Month (and what we were learning about in English and American History) and I am in awe of her organizational skills and commitment to making sure that her students learned about an essential part of music history and its relationship to the socio-economic-political climate of the time.  She gave us 100% and then some, and related it all to math by means of fractions (time and intervals).

• Glad you are finding interesting people! Gizem has great ideas specifically for the scholars, but also for educators. I interviewed her and Mark Hubert when they re-opened the Journal of Humanistic Mathematics in 2011: http://mathfuture.wikispaces.com/HumanisticMathematics More recently, this January, the two of them and Sue VanHattum organized a poetry reading at the Joint Mathematics Meetings, the largest math and math ed gathering of the year: http://mathmamawrites.blogspot.com/2013/01/san-diego-event-math-poetry-reading.html

Please do make your responses as long as you need and want. For my part, I am always happy when people dig deep into a topic. I have been thinking for these two weeks, since I first read your response, about tech resources that would help you. Scratch actually has some capabilities for making sounds, but I am not sure they fit your design, centered on fractions. However, if you go into ratios and proportions, Scratch would work. There are several open and free composer programs, where sounds are represented visually. Kiddie software is especially visual (and appropriate for links with math) but all composer programs I have seen will support investigations of the sort you started to sketch.

I think I already linked this video, but I love its depth. It is inspired by the book, which is not very accessible (not kid-friendly at all), but beautiful, called "Godel, Escher, Bach": http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

The video visualizes the structure of Bach's Crab Canon: http://www.youtube.com/watch?v=xUHQ2ybTejU

What I would love to see is a video showing how little, more technical details such as the signatures (e.g. 3/4) go into grander patterns, such as the symmetry of a fugue. I am not a musician, so I can't make that bridge myself.

There is a "software toy" (not to confuse with games) that is old, but very well-designed, called SimTunes. It lets you paint the board, and then little musical instruments run around it and play by color. You can probably see how to implement some of your tasks with that interactive, for example, paint one line twice as long as the other and see their ratios!

• Thanks for the resources.