Unauthentic, Imaginary World Math

Photo by Cowboytoast

"Real world" and "authentic" are two of many educational buzzwords overused right now. What if instead of making sure that everything has a truly "real" context we give students a creative opportunity to explore the "unreal."

The inspiration for this post comes from a new blog by Randall Munroe, author of xkcd, called What If?. In this blog he answers hypothetical questions by doing the actual math to answer them. So far he has shown things such as how much force does Yoda have? and what would happen if you gathered a mole (unit of measurement) of moles (the small furry creature) in one place? These questions are not real or authentic but the math and science is.

But these questions are fun and interesting! Students love to talk about fantasy and science fiction such as zombies and vampires.

So why not expose your students to a few of these kind of questions and have them try to "prove" their answer. Afterwards show them what Randall Munroe came up with. Then have students come up with their own questions and write out their reasoning and solutions. This activity would tap into their creativity but also demonstrate their mathematical computations and more importantly their mathematical reasoning. It also would be a literacy task in math. Finally and most important in my opinion it may also be an avenue to engage a student's passions in math class that Jeff de Varona has been asking about.

Lego Robot Erase

My students, Cullen, Jordan, and Jordan designed, built the robot, and implemented this program where the robot follows a black line and then erases it.

Math from Canada

My student teacher, Kyle Webb, in Canada sent us a video problem to solve. Check it out on his blog.

We solved and sent a screencast back to him. This was my favorite thing that we have done this year so far.

Wood chips

I should have wrote this before my last post to better describe how my math class has been going. We are currently working on a basic geometry unit of area, perimeter, volume, and surface area. I was excited because this would be "easy" to teach with out a textbook.


I started off by explaining a problem I had of needing to know how much wood chips I needed to cover some landscaping that I did with a class last year. We discussed what we needed to know to figure this out and then went outside and measured the circle. They told me we needed to know the length and width of the circle so I had them measure where they told me those were. I did not try to correct their improper terms. We came up with 7 meters and 6.9 meters. One student noticed that they were approximately the same. We ran out of time for the day.

The next day we started in class and discussed the wrong use of terms. I had them search and find the area of a circle and we talked about what the formula means. Then I asked which of our measurements was right. They argued that 7 meters was correct because it was a whole number. We finally concluded that we did not know which one was right and that we had to go outside and take more measurements and then average them. We also talked about the fact that the circle was eye-balled when created and not perfect.

We ended up solving the problem and then solved two more wood chip problems for some rectangle gardens. Through out these lessons I asked lots of questions and guided their learning but did not give out any information. The students either came up with the answers themselves or surfed the web for them.

My evaluation of this teaching method was that I did not see the high engagement that I had hoped for by the class. My top students were with me and the bottom students seemed to be daydreaming or not really participating. I don't have a great story of the student who always fails getting excited and being successful.

Next I needed to cover parallelograms and triangles which are harder shapes to find in the real world. So we did some visual proofs together in Geometer's Sketchpad so they could play around and see why the area formulas work.

Again I have to give the district unit test so I gave them some practice problems with area and perimeter of parallelograms and triangles. They were totally lost. They could not remember the formulas or even use them when I gave the formulas to them. I ended up going around the room and individually teaching how to use the formulas.

We measured a bunch of food boxes and found their surface area and volume. I demonstrated how to use the formulas on the board and the majority of the class still needed me to re-teach individually.

So in response to the comment from Matt Townsley on last post about teaching at a deeper level. I have tried (I am not giving up!) but in the end I have to prepare the students for the district test. That is why I found ThatQuiz to be a useful tool for students to check their work on the basic problems that they need to know. I know it is not technology integration but doing the same old thing just on the computers. I do think the immediate feedback to students of whether or not they found the right answer is helpful. And unfortunately these are exactly the kinds of problems on the required tests.

All right push me back some more readers :)

Standardization Kills Real Learning

I have not written about my math class much yet because I have been frustrated. My goal was to use the textbook as little as possible and to use authentic learning sources. The reality is that my scope, sequence, standards, and assessments are all mapped out for me with little wiggle room.

My first unit was on factors, multiples, and prime factorization. The more I think about these topics I find them to be quite abstract and separated from the "real world." The best real world example I could come up with was matching up hot dogs, ten in a package, with buns, 8 in a package, for multiples (Thanks to Becky Goerend for that tip on Twitter) to which another teacher responded, "I just let the extra buns rot in the frig."

This kind of example and others like it in the textbook just feel like the contrived story problems that drive students nuts. No one actually takes the time to figure out the right number of hot dogs and buns because nobody wants to buy 40 of them unless they are having a pretty big party!

I could have used multiples today when we bought candy for my son to bring as birthday treats for his class to make sure each student got the same amount. Instead we bought enough for each kid to have one package and we will eat the leftovers :) This is where math becomes too abstract and irrelevant to students because the questions that are asked in the book would never be worried about in the real world.

Although I do not have to use the textbook, each of our ten unit assessments (read tests) are already created for me by the district. I am required to use these tests. So on top of preparing (read teach to the test) students to take the MEAP next week (Michigan's assessment for NCLB) I feel that I have to teach to the test for every unit. I can not make an alternative assessment such as creating a mathcast or some other portfolio type project.

The push in this country to standardize everything in education to guarantee that each student receive an identical education is a fallacy and just plain ridiculous. It is time for the pendulum to swing back to professional teaching that is creative and individualized. We need to trust teachers to teach the right content at the right time for each student instead of trying to teach everybody as if they are in the same place at the same time. We need to start treating students as humans who are naturally curious, intelligent, and motivated by authentic learning experiences instead of as lab rats.

I am tired of hearing about how we are behind all of these other nations based on some test. The United States is still the creative center of the world. Last time I check the elite of the world stil come to our universities. This will eventually change if we continue down this overkill of standards and cookie cutter assessments that kill curiosity and creativity in our kids.

Teaching math backwards

As I have mentioned I will be teaching my first math class this year-6th grade (not counting student and substitute teaching). I want to drop the textbook as much as possible, but have to admit that I am a bit intimidated to commit to not using it at all. I see textbooks as a crutch for teachers because I think that they are terrible. But textbooks do make it easier to plan and teach and I am not sure I can prep without it every day.

Today I was challenged by two blog posts to give it a realistic try. The first is by Nancy Stewart about 8 principals that she will use in teaching her special education math class. She mentions ditching the textbook which is encouraging just to hear that others are taking the same step. She also gives some good ideas on how to at the same time make math more real world and personally relevant to students.

She also mentions one of my favorite bloggers Ira Socol and links to this post. Math teachers you must read this post as Ira paints alternative ways to teach math and turns much of my thinking upside-down. He shows examples of using sports, construction, cooking, and money to make math authentic and meaningful. I am sure he would argue that my times table idea is a waste of time.

The most challenging thought to me was a comment (you must read the comments) by Homer the Brave:

"Start with philosophy. Teach kids about logical systems. Teach them how to understand a provable statement and how to spot a fallacy. Then say, 'We're going to now apply this same set of rules about philosophy to math.' Then teach algebra. The details of arithmetic will then follow, imbued with purpose and meaning.

So basically we teach math backwards arithmetic, algebra, and then philosophy. I personally was an excellent math student in school because I was very good at memorizing and working algorithms. It was not until taking classes about teaching math in college that I understood the philosophy and reasoning behind the math. Homer argues that we should start with the philosophy (logic) and then move to algebra with arithmetic last. This is very new to me but does make sense.

Of course this change would have to happen at the district curriculum level. The easy cop out for me is that I must teach to the standards assigned to me. But I can at the same time as teaching the standards, teach the logic and philosophy behind the math. I can teach authentically without the textbook as much as possible. I am up for the challenge. How about you?

Students prove the premise of UP

My 6th grade technology class has just finished making hot air balloons out of tissue paper and launching them. We also wrote letters to Leon Gambetta helping him escape Paris in a hot air balloon after reading Horses with Wings by Dennis Haseley.

I then used this article from Wired for a lesson last week.I posed the question of whether the premise of the movie UP was possible? The class voted and then I told them that we were going to prove it one way or the other.

I then made them figure what questions to ask and what we needed to know to "prove" it. They had the Internet as their resource. It took most of the hour, but they figured it out. Learning to ask the right questions was an important part of this lesson that I emphasized to them. They had to figure how much the house weighed, how big the balloons are, how much lift helium has, and the volume in a balloon.

We then talked about the errors in our hypothesis such as neglecting the mass of the string. We also talked about other practical concerns mentioned in the article and in the comment feed such as the fact that houses are not designed to be lifted. Lastly we looked at the clusterballoon website of people who launch themselves. The students were fascinated by this site.

Overall it was a very good lesson and I plan to repeat it again next year with one change: bring in some real helium balloons and have students perform an experiment to discover the lift of helium by measuring the balloon's size and tie it to a weight on a triple beam balance.

Reflections on our Landscaping Class

Our art teacher, Kim, and I combined our art and technology classes for the final quarter of this year and team-taught a landscaping class. This post is a reflection on the class, good and bad, looking towards another version of it next year. It was challenging, frustrating, and rewarding at the same time. We had a general idea of what we wanted to do and accomplish, but we really wanted the students to have as much a voice as possible in the class. Therefore it was difficult to "plan" because we really wanted to leave that to the students.

We started the class knowing that we would use my background in concrete construction to have the students build some kind of decorative pavers in a garden space that we would create outside of our rooms. We tried to spend time in research with books and the Internet. Kim taught them about Georgia O'Keeffe and Frank Lloyd Wright. The students drew the space from a two-point perspective and as a blueprint on graph paper. The students really struggled with the abstractness and ambiguity of the project.

We realized the students could not focus on the big picture of the project, so we switched approaches and had the students draw life size versions of their personal tiles. We made some guidelines for them that the tiles had to be based on nature. The students really took off with great ideas.

While they worked on drawings in Kim's room, I brought small groups to my room and the students built their forms. They choose the shapes, measured, cut, drilled, and screwed the forms together. For many of them it was a first time using any of these basic tools. It was great to see some of the students who had some skills in this area jump in and take charge.

Next we began pouring the pavers. Concrete is a challenging material because of the time constraints. You can not just pack up at the end of the hour and finish the next day. So we were able to pour 2-3 each day taking over ten pour days to finish. The students mixed the concrete with a drill, poured it in the forms and finished the concrete. A few days later we stripped the forms.

The process of figuring out the layout design of the tiles was one of my favorite parts of the project. Now that they had the tiles they could "see" the problem of layout. We cut out life-size cardboard shapes of each tile and they wrote their names on them. Then we went out side and told them to lay them out. Then we talked about how it looked and they edited by moving them until they "found" their design. This concrete, no pun intended, process of design was much easier for middle school students than trying to create something on a blueprint before they even really understood what they were making. We took pictures of the cardboard placement so we could remember when we bought out the real pavers.

The absolute best part of this project was seeing the skills of some of our hands-on kids validated at school. They may struggle in traditional classes, but they were the leaders of our class and you could see their self-esteem rise. They worked hard and were proud of their results. Another great part of the project was our blog at http://vwoodgarden.blogspot.com/ Students wrote about their individual tiles and we posted it next to a picture of their tile. The students were so impressed by all the dots on the clustrmap on the first day (thanks to my PLN on twitter!)

As I move to a paperless classroom, I am reminded the most important part is students. I have always emphasized project-based learning that is based on collaboration and problem-solving. Next year we plan to have the students design and build benches and some sculpture stands. We are also going to have students collect food scraps from the cafeteria and start a compost pile.Of course as has been said many times before, learning is messy, but so worth it. The students are proud to leave behind this contribution to their school.

Ok readers, how will you be creative and challenge students in real world ways next year??? Kim and I asked ourselves what are we doing sometimes and this was by far the class that took the most energy and planning. It was hard work, but oh so worth it!