Category Archives: authentic learning

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.

Unions

I read this post at the Cooperative Catalyst by Educational Revolutionist and left the following comment:
I think you are confusing two separate issues: unions and learning. The purpose of unions is to protect the rights of workers in regards to wages, benefits, working conditions, etc. Union dues are paid by teachers to protect their rights and wages. Unions are not directly designed to promote learning or guarantee student rights. Unions are part of the democratic “checks and balances” against big business. Review the history of Carnegie and other robber barons. Sure they donated lots of money to libraries to and created education institutions. Sounds alot like Gates, Broad, and crew to me.
But as Mary Beth has pointed out, positive working conditions and benefits leads (but does not guarantee I admit) to better learning situations for students. These benefits lead to more qualified teachers entering and staying in the field. Other rights such as planning time lead to better learning opportunities. As my dad always quoted a friend who owned a used construction equipment business said, “I can’t afford a $7/hour mechanic. I need a $20/hour mechanic.”
Unions do fight for more funds for education all of the time. Are they self-interested? Sure, but the money goes to many things in schools also.
Lastly, I agree it would be incredible if teachers would rally, protest, and boycott against NCLB, RTTT, and standardized testing too.

BTW “Get up, Stand up” by Marley came up on my Itunes shuffle as I wrote this. #karma

Real Reform Goes Backwards

I for one am tired about hearing about broken public schools. Are they perfect, of course not. Neither is any other human institution. But what really disturbs me is that the people who want to “fix” public education have the worst ideas ever. They want to make schools more like business and hire head-hunters (superintendents) to fire the bad employees (teachers). Then the head-hunters move on to work for Fox news or some think tank. These people do not truly care about children but about money. Therefore their goals are to get rid of the evil unions that protect teachers’ benefits so they can hire cheaper teachers (outsource them just like the manufacturing industry in this country). Standardization measured by tests is the magic cure!

The problem is that schools are not factories and should not be treated like them. And most importantly students are not widgets that can be taught by anyone using the same script. These reforms disrespect both teachers and students by treating them as all the same. Just put the student in the proper machine (program) and out they pop at the end- educated (able to pass standardized tests). But students need relationships to grow and learn. Schools should be more like families than factories.

I think real reform starts by going backwards to a “classic” liberal arts education. In Ancient Greece students were discipled by the master as part of a community. Relationships were an important part of education. They learned by asking questions. They studied logic, poetry, geometry, and exercised all as one experience. They were not divided by age and subjects were not compartmentalized. There was not classwork and homework. All of life was learning, one integrated experience.

By caribb

Students today need to be given time to think, deeply. They need to play, ask questions, discover, solve real problems, and discuss. They do not need to memorize facts for tests. They need to think critically from multiple points of view. Students need to be given choice in what to learn and how to do it. Schools need more individualization and less standardization.

What can we learn from the Ancients? That living is learning and relationships matter. We need less federal bureaucracy and more local control. We need schools that look different because the communities they serve are unique. If we truly teach students to think and learn on their own and in community with each other they will do amazing things. We need science, math, and arts all mixed together. I believe that we should give students freedom to be creative and to engage in ideas with each other. We should scrap textbooks (this form of standardization has dominated even longer than the state and national tests). We should teach from real world problems and current events (we have plenty of them to choose from). Students should engage in real questions and work for real solutions. We should use hammers, nails, wood, computers, dirt, flowers, paper, cell phones, microscopes, cameras, and animals. Students should perform labs where the teacher does not know the answer. Students should study current events and then research the history to understand why things are as they are now. Students should use math to calculate solutions to world poverty, lack of clean water, and adequate food.

So what is the role of the teacher in all of this? To guide the learner and challenge them with new ideas and experiences. Most students are not self-motivate learners by themselves (because schools have bored this out of them); they will need mentors and guides to show them how to learn and to challenge them with new ideas. Master learners (teachers) should create fascinating learning opportunities as starting points and then encourage students as they pursue deeper concepts. Too many teachers use curriculum, standards, and textbooks as a crutch and rarely present students with authentic learning. Real reform happens when we abandon the pre-packaged education being sold by textbook companies and start the adventure of giving students great learning experiences.

So what’s your excuse? Reject standardization now. You may not be able to change the structure of your school schedule or the the architecture of the building, but you can change the way your classroom works. Don’t know how to start? Start by talking to your students. Find out where their interests are and build from there. Start by talking about the news and the issues in the world. Ask students what they can do about it. Put away the textbooks and engage with your learners. You might be surprised by where you end up.

The Alchemist and Louis L’Amour

By Alexandre Baron

Last weekend I read The Alchemist by Paulo Coelho for the first time. Yes, I know it is not new by any means and I would guess that many of you have read it long ago. There are many layers that one can read into the story but I would like to point out two that struck me.

I think most people would say that the overall theme of the story is pursuing one’s passions and dreams. The obvious question to me is “do we allow space in schools for pursuing dreams?” It seems to me that we are too busy covering curriculum and meeting standards. I am sure that I have said it before, but I hate standardization. The reason is that standardization drowns out passion and crowds out dreams.

The thing that struck me most about the book was about the channels of learning in the book. Santiago learns in many ways: reading, from experts (the king Melchizedek, the gypsy, the crystal shop owner, the Englishman, the leader of their desert caravan, and finally the Alchemist), from experiences, from following his heart, but most of all from nature. He learns from watching his sheep and camels. He learns from the desert. Contrast Santiago to the Englishman who primarily learns from books. I think the Englishman represents Western text-based learning whereas Santiago is a more ancient, Eastern, holistic learning from nature. Santiago learns by living life and observing life and nature everywhere. Western schools need to be more like Santiago.

Santiago also reminds me of my favorite childhood author, Louis L’Amour. He was a western writer and I loved his stories. In his memoir, Education of a Wandering Man, L’Amour details his “education.” Much like Santiago he left home when he was 15 and explored the world. He had jobs as a boxer, sailor, lumberjack, elephant handler in a circus, skinner of dead cattle, assessment miner, a tourist guide in Egypt, and a tank officer during World War II. He sailed the world and was shipwrecked in the West Indies. The other thing that L’Amour did was read-all of the time. Let me share some quotes from his memoir:

This is a story of an adventure in education, pursued not under the best of conditions. The idea of education has been so tied to schools, universities, and professors that many assume there is no other way, but education is available to anyone within reach of a library, a post office. or even a newsstand…


Somewhere along the line I had fallen in love with learning, and it became a lifelong romance…


this book is about education, but not education in the accepted sense. No man or woman had a greater appreciation for schools than I, although few have spent less time in them. No matter how much I admire our schools, I know that no university exists that can provide an education; what a university can provide is an outline, to give the learner a direction and guidance. The rest one has to do for oneself.
If I asked what education should give, I would say it should offer a breadth of view, ease of understanding, tolerance for others, and a background from which the mind can explore in any direction.
Education should provide the tools for widening and deepening of life, for increased appreciation of all one sees or experiences. It should equip a person to live life well, to understand what is happening about him, for to live life well one must live with awareness…We can only hope they come upon an issue they wish to pursue.

By Sandy Redding



To me this is another example of how all of our ideas to “change” education are really not new. Leading thinkers understood this years ago before computers even existed. I can’t help but think that L’Amour would find even less use for schools as they still exist today with the easy access to knowledge through the internet. Lous L’Amour was a self-made man in many ways, but he understood that learning is available for anyone who passionately pursues it.

My favorite quote is his reasoning for dropping out of school and leaving home: I left for two reasons, economic necessity being the first of them. More important was that school was interfering with my education.


He goes on to explain how the factory model of school would not let him skip basic classes and take higher classes that he was more interested in. So he dropped out and pursued his own learning. He later says, that dropping out is a good option only for those who are willing to read hundreds of books on their own. Certainly does not seem to be watering down learning, does he?

How can we create a climate that encourages students to dream and pursue passions rather than “interfere with their education?”

PS: I was very surprised to hear my pastor tell this story to close his message today. It sounds to me like “The Alchemist” used this as the basis of its plot.

How I laid out the square

Here goes my attempt at explaining how we laid out a square parallel to our school around a circle with diameter of 21′. I must confess that I did the math and started by having some students help me in class. I ended up having two students stay after school with me for about an hour to finish the layout.
First,I may have been unclear in the previous post but the circle already existed that the square needed to go around. So we measured 12′ 6″ off the building in two spots outside of the circle to establish the east line of the square parallel to the building. We put stakes in and ran a string line. Next we measured over 21′ from that line to establish the west side of the square also parallel to the building. That was the easy part. The tricky part was finding the corners.

(The building is on the east side and west is up on this sketch)

Our reference points were the furthermost northern and southern points of the circle. These points are the midpoints of the north and south sides of the square. But how did we establish  another point to create a line that it is perpendicular to the east line? If you are thinking Pythagorean Theorem that is part of it, but how do you find the corner? You guess, of course. Mathematicians, in the real world sometimes you have to estimate.

We multiplied a 3-4-5 triangle by 3 to get the dimensions 9′-12′-15′. We measured 12′ off from the east string in line with the midpoint. Then we held a string from the east string to the west string so that it barely touched the stake and we “eyeballed” it square with those lines. We marked the intersection between this string and the east string with a sharpie on the east string. We measured over from this mark 9′ and put another mark on the east string. Next we measured from this mark to the stake (the hypotenuse) and it should have been 15′. Of course we guessed on our corner point so we were off one inch and it measured 15′ 1″.  We corrected this by shifting the corner and this point north 1″. Now the hypotenuse measured exactly 15′ and we put a stake in at the northeast corner.

Now it gets easier. We pulled a string from the established northeast corner to the west line so that it barely touched the stake at 12′. This allowed us a very good estimate of the northwest corner. Next we measured over 21′ from both northern corners on our string lines to establish the two southern corners.

Our last step was to measure the diagonals which should be congruent. We ended up about an inch off so we double checked our overall dimensions and found one stake was leaning in so it was not quite 21′. We straightened it and our diagonals were within 1/2″.

The next day we measured over 2′ 6″ from all four sides of our square and pulled lines to find their intersections which gave us the four corners of our outer square. I really do not think that this problem was solvable by my 8th grade class but I wish I would have taken the time to walk them all through it.I think it was a great opportunity to show my students that math does apply to real problems in blue-collar jobs not just at universities or in a lab somewhere.

PS If I was doing it over I would have just “picked” center point on the east line pulled off the building where it touched the circle. Then I could have measured 10′ 6″ both ways and established both of the east corners and used Pythagorean theorem to find perpendicular lines off from both corners. It would have been much easier 🙂