Academy schools are uniquely positioned for changing the landscape of education.

The first day of 9th grade. DaiCrede is very nervous. He’s heard about this high school, just rumors of course, that they let you choose what you want to learn about, there are no regular classes; you work with teachers in each area as you need the information to complete your project. There aren’t grades, just levels of achievement for learning targets. He’s not sure if he can make it in this environment. He is excited about the robotics lab, though. That would be cool…

Shelly can’t wait for the bus to get her to school today. She fingers her Advances in Analytical Chemistry magazine. Finally, a school that will let her pursue her dream of becoming a medical researcher! Just think, a whole day devoted to learning how to run a research program. No breaking the day into language, math, history! She can grab those things as she needs them for her project. She loved meeting with her teacher mentor during the summer to set up her project parameters. She’s nervous and ready.

Ned is nervous, too. He rides his bike, slowly circling the parking lot. High school. He’s not sure how having no classes is gonna work. How will he know what to do?

First Week: Orientation

 

This Academy High School has a Ninth Grade Academy for incoming freshmen. They will be going through a series of workshops this first week to learn how to navigate the process. By the end of week two, they’ll have chosen an initial Academy. During the third week, they’ll have chosen a project and have begun breaking their project into tasks – and identifying the various areas of knowledge they’ll need to access. They won’t go it alone, however. All students have been assigned two mentors: a teacher, and a student from a higher grade, (Mentoring is a learning target for all students, more on those later) who will help them navigate the project system and hold them accountable for task goals and deadlines. While this may not feel like a structured environment, there is a definite structure and protocols. The flexibility comes in as each student is allowed to follow his or her interests, change from one Academy to another as they explore, and begin to search out what they need to know as they need it. There is a framework for that!

That first week will see new students rotate through a series of workshops and learning sessions on the School’s Academies. These new-to-high schoolers get information on the various fields of study available to them. (Students may have chosen to come here- if their zoned high school is not an academy school, or if they were not doing well in a traditional school setting).  Returning students share their experiences with the process about to be undertaken; what worked, what they liked, what they found most challenging, what they would do differently. They testify to the powerful learning that takes place when they were able to choose topics for study that they wanted:

“Math becomes relevant when I had to figure out what was happening in my chemistry experiment. I went into the math room, laid out my work, and asked for help. Once we started, I realized I didn’t know the basics. I spent two weeks in the math lab, working through background knowledge, then learning how to apply it to my research. Whenever I got stuck, I would go back and get help.”

“I had to write up a report on my project. The requirement that it be a formal paper, with citations and everything was overwhelming. I didn’t know how to do that. I had to spend that last month in the English lab. I wish I’d started there first!  Everything I had done was in this big pile of notes. I had to learn how to write, how to make coherent paragraphs that made sense to others. I must have rewritten my summary statement a million times before I wrote my first draft. It was hard work, but I wanted to submit it to the robotics journal. I didn’t want to be embarrassed.”

“I have always wanted to go into nursing. I got to plan each of my 9 week projects around a different area of nursing. At first, it was hard to stay on task. I didn’t have to do anything but show up. I would log in to the electronic attendance monitor and consult my task plan. That plan kept me on track with assignments. If I didn’t know how to do something, I would go to a lab and get help from the teachers. It’s really different; me asking them what I need to know, instead of them telling me what I was going to learn that day.”

“The projects seem really hard. There is so much to do, it’s overwhelming. That first project is really where you learn how to do everything. The first few weeks of high school, you learn about the academies and you are encouraged to pick your project. There are lessons on how to break down your project by tasks. I took a quiz to find out what I was interested in for my first project. Some kids come in knowing what they want to do. I wish I had gone to the summer workshop. I could have started my project so much sooner!”

“My mentor teacher was a big help. He ran one of the math labs, but he always had time for my questions. When I felt like quitting, or when I would hit a wall in my project, he would let me talk it out. He listened, really listened, to me. As a sophomore, I’ll be completing two projects, instead of four. I’ll need a team of students to meet the advanced requirements. I’ve made some new friends on this last project that are in my Academy. I think we’ll work well together.”

“The best part of my project, water resources for our community, was presenting what I had found out to the local water board. I had some suggestions for making our water supply healthier, with the project costs and a timeline for implementation. My next project will be on finding a new way to deliver water to neighborhoods. Seems to me there has got to be a more resourceful way than digging and burying pipes in the ground!”

“I graduate tomorrow. I’ve already got a starting job in a large engineering firm. I was able to intern with them as a result of a project I chose during my freshman year. That led to another project during my sophomore year. My junior year, I got a small grant to research affordable housing – after I learned about writing grant proposals in the English lab. One of my projects was on making presentations to community developers, which I used to apply for the job I’ll be going to after high school. They were impressed with the amount of experience I’d gained in researching and development of my projects, they liked my presentation skills, and they will help me pay for my engineering degree while I work for them!”

Learning Targets are the New Standards for Career, College Readiness

 Teamwork, innovation, self-motivation, serious work ethic, honesty, integrity, and getting paid for something we love doing are some serious life/work goals. The Learning Targets replace standards by giving students real, usable skills for life. Critical thinking is absolutely necessary for making sound decisions. Being able to communicate through written, visual, or verbal methods is a must. These, and more Learning Targets, are imbedded in every project outline. Each year, the project should add appropriate learning targets, building on prior years. And who says a student must take four years? An Academy school has the ability to let a student move more quickly towards a goal, be it work or college. In addition to the projects, which can increase in complexity for those pursuing two, four, or six year collegiate or technical degrees, there are technical and apprenticeship projects and programs that would fit cleanly into this model, allowing our students to investigate different types of jobs through projects, and interact with the community during their research, giving them an authentic audience – the best way to incorporate honest evaluation, and spur our kids’ interests in their futures.

I work in an Academy school. I believe in this scenario, and I believe our students would be more than willing to embrace – and benefit – from this new way of looking at their education and preparation for life.

 I see a couple of things (and will probably think of more!) that would need to be done:

1. Start with the 9th grade, move up with them. Don’t try to change all four grades at once. Set reasonable learning targets for them, understanding that this replaces the 9th grade standards. Build in what they need to know in each project. These won’t be lightweight! Each project must take each child through the four core fields of history, science, language arts, and math, incorporate electives such as art, physical activity, technical (computer) or programming…. but it’s OKAY if some projects are heavier in one area than another. They will be completing four projects across the year. I think this mindshift on the leadership will be the hardest!

2. Train the teachers. Take two-three weeks to train your teachers in the Learning Targets and what mastery of these Targets should look like. Teach them how to facilitate the labs, because they won’t be teaching specific, planned lessons! Teach them the framework tasks of the project- so that no matter what the child chooses to learn about, all tasks from the framework are applied. For example, a child wants to learn about yo-yos: there would be tasks to learn the history, find the financial impact of the toy, write their findings in an essay with citations, consider the making of the toy, the science behind the motion of the toy, the aesthetics, or traditional design and decoration of the various models,perhaps the value in an exercise program, and how it could benefit the user. This is a do-able 9 week project for a 9th grader.

3. Inform the parents. Teach them what is changing. Tell them what we are looking for in terms of benefits. Give them the long view. Give them the questions we want them to be asking through the process, so they can feel, and see, their child growing and learning through the process. Create parent buy-in at the start.

4. Tell the kids, in middle school, what is happening, what we want for their future, what will be required of them. Give them a chance to go to a summer workshop, where they can investigate what their interests are, a short survey, some simple practice on picking topics, a brief look at the project tasks framework, what the labs will look and feel like, how to approach teachers and ask for help.

There is more, but I think this is a big enough chunk to chew on for today. Your constructive thoughts are welcome!

L-Q-E vocabulary (or how to drive your OCD students crazy!)

Algebra I, Unit 5, Compare and Contrast Linear, Quadratic, and Exponential functions…

This simple vocabulary lesson generated a surprising result: my kids thought the end result was too messy, so many of them didn’t want to follow the final step!

The activity uses a simple alphabet mind map. It is a 5×5 grid with a letter of the alphabet in each square. The last square contains X and Y. You can print a copy here, or have the students draw their own.

Here is a simple alphabet mind map.

The standards for this unit were that students should be able to identify linear, exponential, and quadratic functions from equations, graphs, tables, and contextual situations, and be able to compare each function with regard to rates of growth.

To achieve these skills, students needed to be able to identify the key characteristics and key vocabulary associated with each type of function. They also needed to be able to discern small differences in equations, the shape each functions takes when graphed, and the changes in a table that would indicate what type of graph the table would produce. Given contextual scenarios, they needed to identify which type of situation would produce a linear change, a parabolic track, or a classic J-curve from exponential growth or decay.

When we started, they could barely list the key characteristics, much less identify which function was associated with each characteristic, what those characteristics looked like, or how to tell them apart. They needed stronger, more fluent use of the vocabulary!

Step One, The challenge: using each letter of the alphabet, fill in the grid with the names of as many key characteristics of each type of graph as each student could think of. (We’d made lists over the previous several days, along with examples, so I knew they would be able to come up with several familiar words.) I encouraged them to start with any word they could think of that they associated with graphs or equations. As they wrote, I then passed out colored pencils for the second part of the task.

Step Two: After about 10 minutes of individual work, we came together in a large group, and I asked each student to share one item from his/her list. I encouraged the students to add new words they heard to their papers, and to use a different color  pencil than they used to write their initial lists. After going around the room about two times, we asked kids to popcorn choices that they had on their papers that hadn’t been covered. We had a few letters that remained without words, so we again asked for ideas from the whole group that would fit for those letters, reminding the students to stay within the linear, quadratic, exponential, and graphing parameters.

We found that we had to ask a few thought provoking questions to make sure some important terms weren’t left out.

(At this point, because we were talking about why these words were acceptable, what they meant, and how they were related to LQE, I had a pretty strong idea of where my kids needed additional help and lessons!)

As my students shared their words, I wrote them on a poster sized alphabet chart that I had prepared beforehand. I gave the students a few moments to make sure that they copied all the words from the collaborative chart onto their personal charts.

Step Three: each child labeled the outside of their chart with the words LINEAR, QUADRATIC, and EXPONENTIAL.

I explained that we were now going to match each term with the function to which it belonged by drawing a line from the word to the function. I warned them that some words might belong to more than one function!

They each picked a color to use for the line that would connect the appropriate words to LINEAR.  The first word under A, asymptote, was determined to be related to exponential, not linear. Not only did they have to decide which function, they had to say WHY and in what way the word connected with FUNCTION. The word Axis was next. Everyone could get behind that as a graph term that could belong to any of the function types, but we only connected words to one type of function at a time. We would come back to ‘axis’ two more times as we matched words with the other two functions! Colored lines were drawn from Axis to LINEAR. This happened with several more words, before the students began to realize this was going to get messy. I was drawing the same lines on my big poster, but I was totally surprised as students began color coding each word with dots, or making these neat lists on separate pieces of paper, sorting out each of the characteristics, because they didn’t like the tangled mess that was happening on my poster. (I explained that they were actually drawing the map for their brain, not their eyes. They were somewhat skeptical…). ‘I can’t read it,’ was the standard response!

Here was our resulting map!

 

I  encouraged the students to use a different color for each category, and we progressed in order through each function, so no one would end up confused. Throughout the matching, as students popcorned answers regarding which words to connect,  I continued to ask for agreement, disagreement, (thumbs up, thumbs down) and ‘why, how do you know,’ from the whole group. This was a very intense, fast paced portion of the activity, with even some of my most blasé students getting involved!

We followed this activity with a neat card sort, that was another intensive activity in and of itself, and was spread over two days. By the end of these activities, I could tell that more of my students were fine-tuning their selection processes, looking more closely at the details of each equation, graph, or table, and applying the key characteristics lists they’d made to their compare and contrast process!

Here were some lists they made of the key characteristics:

Close Reading in math; and the “after” math…

Literacy is everywhere, including math class. And I don’t mean those clever (and not so clever) word problems.

I used a simple story book, predictions, and paired discussions for my 9th grade Algebra I -ers. I passed out colored pencils for annotation, and paper for predictions.

We started with the brief prelude, a paragraph, about a ‘happy-go-lucky young man’ who meets an old man who tells the young man,’I have a gift for you.’

There were a few other clues in the paragraph, so after asking the kids to read and annotate the paragraph individually, a volunteer read the paragraph aloud. With no further discussion, I asked each student to write down their prediction of what the story was about. Then I asked them to share their predictions – and their reasons why they thought that – with a partner. Then they would listen to their partner’s prediction and decide if they agreed or disagreed.

I modeled having a conversation…

To remind them of how to have a meaningful conversation, I modeled listening, responding with questions about what was said, and commenting on the information. We talked about how conversations were about listening and responding, not a contest of spouting information. Everyone got to practice, with a little (okay, a LOT of,) prompting!

We turned to page 2. My students were now eager to read further (we had predictions to fulfill!), and since the book’s format was laid out in sections of two to four sentences, with lots of images, I was able to let all have a turn. We took the four sections on pages two and three rather slowly, and we pulled apart the actions of the old man (he gave the boy two seeds), and the young man (he cooked one seed and planted the other seed). We talked about the passage of time – winter, when they had met, and the boy planted the seed; spring as the ‘sturdy’ plant appeared from the ground; summer, when the plant produced two flowers; and fall, when two fruits became evident. Each one of my questions caused them to return to the text for details.

Before we turned to page four, I asked my students about their predictions, and whether they had changed their predictions based on this new information. The conversation among the students turned to the details they just reviewed: how long it took to grow the seeds, how one seed could feed someone for a whole year, and why couldn’t he get food where he lived. New predictions came from the ashes of the old, with several students predicting that the plant would grow huge, up to the sky, and the boy would climb it. This was an obvious scaffold onto a familiar story. They told me the key word ‘sturdy’ meant that it would be strong and big. We turned the page.

New predictions came from the ashes of the old…

As each student read a section, we learned that the plant the young man had grown produced two more seeds from the fruit. He cooked and ate one and planted the other. We noticed from the picture that he put a hammock in a tree. Apparently he was planning on staying put. I pointed out that this was something that usually happened when people planted, instead of hunting for food. Another scaffold, as they agreed with me that he would need a place to sleep while the plants grew.

Again, we turned the page. This time, the readers learned, Jack (that was the boy’s name) had managed to grow another plant from the seed, which yielded two fruits, which yielded two seeds, one which was dutifully cooked and eaten and one planted in the ground. No changes here, but we noticed in the drawings that Jack seemed to be getting fat. Also, the plant wasn’t getting any bigger, which some of the boys seemed disappointed with. I think they were still waiting for the plant to grow up to the sky. One boy persisted in asking why Jack just didn’t move to a place where he could buy some food. We made more predictions, adjusting our expectations based on what we’d read. I asked them if their mental picture of the story, and of Jack, was changing. One of the girls agreed, and then we turned the page.

Their mental picture of the story, and of Jack, was changing… 

The next two pages supported the story line. Jack continued for two years to cook and eat a seed and plant the other. The plant never got any bigger and never produced any more than two flowers, two fruit, and two seeds. It looked as though Jack would be living out his life, year by year, cooking, eating and planting seeds. I had the students make predictions anyway, and talk about what they thought Jack should do, given the circumstances. Some of the children thought Jack was stuck in a rut. On to the next page…

My readers were still eager to read. They continued, aloud, to devour the slim text (it was great to get a window into their abilities to read and decode the text – you know, for those word problems we will get to someday!) It was momentarily exciting to hear that Jack was as bored with his existence as some of my students! He said, and I am paraphrasing here, ‘if I always do what I have always done, I will never get anything different than what I’ve gotten.’ (Here I looked at my math students to see if they had taken anything of a personal nature from this comment. It appeared they hadn’t felt a connection.)

‘What,’ I asked, ‘Do you think he is going to do?’

‘He’s going to plant both seeds,’ spouted one girl. ‘But what will he eat while he waits for the plant to grow?’ I asked. They hadn’t thought about that, and in thinking about it, several students were seriously confused about how many seeds he could plant if he ate one, and he only had two to start with. (This may point to a reason that so many students struggle with adding negatives and positives.)

A serious discussion ensued as to what he was giving up if he didn’t eat the seed. Would it be worth it? We talked about how sacrifice is sometimes needed to affect change. One of the kids said he would be cranky because he was hungry – the voice of experience talking? The kids had gotten into the spirit of the lesson and were ready with their predictions. The Jack and the Beanstalk contingent were ever hopeful. There was still one boy who wanted to know why he didn’t just move to a place with food and forget about the seeds. We turned the page.

The story continued with Jack explaining that he decided to go hungry  so he could plant two seeds. To assist the verbal process, and give structure to the next round of predictions, I drew a table on the board and labeled the two columns ‘seeds planted’ and ‘seeds produced’. I then filled in the first line. Under ‘seeds planted’ I wrote 1, and under seeds produced, I put 2. I asked the kids to review the current situation: is this correct so far? After some discussion of my column titles, the students agreed. I asked them to fill in the next line. Now that Jack has decided to go hungry for a year, how many seeds will he plant? The students agreed he would plant both seeds.

I wrote a 2 on the second line under ‘seeds planted’.

‘So what goes under seeds produced?’ was my next lead in, ‘and why?’

Another look at the text produced the facts: two seeds produced two plants, each with two seeds. How many different ways could we count this, and still get answers varying from 2 to 6?!?

More fingers, more math. One girl supported her position loudly by mentioning multiplication; two plants times two seeds would be four. Several students had made the same conclusion by different routes, but getting the kids to share their explanations that they had discussed with each other was the hard part. The confidence that they can be right is so difficult to encourage! The student who mentioned multiplication was encouraged as another student agreed with her. A 4 was written in; satisfaction all around.

So, how many seeds will he plant next? And ‘why?’

‘Four’ was the immediate answer, until a single student voice reminded us that he had to eat. So began another discussion about how to calculate the number of seeds, and how many will he eat, and how many will he have to plant? Some students felt he was going to eat two of them. (We had doubled the amount of seeds grown – we must double the amount he could eat!)

We returned to the initial instructions on page 1. A student read the evidence and told everyone definitively that only one seed would feed Jack for a whole year. By this point, I was listening to these children teaching each other how to support their facts, correct their misconceptions, and expose and correct mathematical errors. Every child was involved in this discussion – every child!

One seed eaten, three seeds planted was the ultimate conclusion. (If I rendered their discussion here, you would be as bored as we were with three years of Jack eating a seed and planting a seed!) The chart was filled in, and we were ready for the next question: if he plants three seeds, how many seeds will he have to plant the next year?

While the whole process was a conversational struggle, before I left them to this final prediction, I pointed out the table and had them write it in with their predictions. We had just begun defining a function the day before, so I asked them if they thought this was a function? I was rewarded with a student identifying the input and output terms of seeds planted and seeds grown, as x and y. Several students then made the next connection to the fact that none of the inputs we had so far would repeat, so it must be a function.

Without giving away any more of the story, I will tell you that they were able to successfully calculate the answer. We stopped here as I had them make a final prediction about the rest of the story. We will revisit Jack and his adventures in the coming unit.

 

 

 

 

 

 

 

 

 

 

Here is the ‘after’ math:

Our school lit coach came by two days later and interviewed some of the students that had participated in the close reading. Here are some of their comments (I stayed across the room, out of their line of sight. I was curious to hear their honest responses.)

  • “I liked it because it was different from the way we usually do math.”
  • “Why can’t we do math like this all the time?”
  • “I felt like I really understood the story.”
  • “I remember more about how many fruits Jack planted.”
  • “The table helped me understand.”
  • “I think it will help me in my reading in other classes.”

Read that last bullet again.

‘I think it will help me in my reading in other classes.’

Wow. Now, that’s some kind of ‘after’ math!

Editor’s note: the standards for this lesson had to do with identifying functions; recognizing a function in multiple forms (such as tables), identifying functions from contextualized settings; the literary standards were identifying supporting information  and using contextual  clues to support mathematical arguments (a Mathematical Practice, as well). Additionally, the conversation supported mathematical reasoning, practicing vocabulary, and reasoning aloud. The constant predictions were embraced by the students, who stayed involved for what amounted to about 60 minutes of close reading. They had to know each outcome! No one read ahead, which surprised me a little. They really wanted to maintain the suspense! I can’t wait to revisit Jack and the rest of his story!

For those who notice the reading level, I wanted to use a text that would provide a low floor, and that would allow me a high ceiling – the actual math is writing and calculating an exponential equation. This was a great text for my ninth graders! The text didn’t cognitively get in the way of the activity.

To read Jack’s full story, visit Anno’s Magic Seeds, by Mitsumasa Anno. For more of her series of math books click here.

Knowledge Machines are here; How will you use them?

There was a time when school was about learning the three R’s: reading, writing and ‘rithmatic. Sounds like the beginning of a long ago time story, doesn’t it?

After reading this 1993 article from Wired, I realized that Papert’s ‘Knowledge Machines’ are, in fact, here.

Continue reading “Knowledge Machines are here; How will you use them?”

New Year’s will be in August, this year.

If you are a teacher, that is.

On August 8, hallways and rooms will fill with the wriggling eager bodies of their parents’ best! I can hardly wait! During the summer, I’ve been loading up on great ideas, reading about effective teachers, discovering new tech and new resources, and creating learning plans that will put them into practice!

I am teaching my students Algebra I this year; 9th graders, some returning 10th, and I want them to feel the excitement, the sizzle that I feel with math. This is a new year, a new crop of children, a new chance for me to share what I love- math – with children who never fail to delight me (and challenge me, worry me, turn my hair gray, and, well, you get the idea- but that’s another post!)

The year I have planned, this year, will be different. This will be the year that every student tests proficient on the EOC, aka Georgia Milestones. My lessons will start with Wonder/notice, there will be lots of student conversation, with roles for small group work, and conversation starter posters on the wall! My class will be fully engaged, will actually complete their assignments, will receive thoughtful feedback, and grades that really show how well they’ve mastered standards. I’ll make all the calls, on time, to the parents. 

My IEPs will have clear goals, my re-evals will be works of art! I’ll handle my discipline issues with skill and compassion. This year, I’ll have strong closure routines, include literacy in every lesson, hold awesome number talks, and have nimble responses to my formative instruction. 

This year, my room will be organized. I’ll have study centers, whiteboard walls, standing desks, and engaged, curious students! This year – well, this year will be everything I was hoping last year would be… 
So, you see, teachers really do celebrate New Year’s in August!

Why PBL? Part One

 

Why do our children have to complete four PBLs in a two month period, separate from their ‘regular’ schoolwork? Why can’t their ‘regular’ schoolwork be taught in such a way that they learn and can draw parallels to their world outside of school?

Not that the content should match their lives, but the way they learn that content; the way they organize and make it a part of who they are in school should have some relevance to how they organize and deal with the stuff outside of school.

These two parts of their lives should mesh, not be two such disparate worlds that they cannot be reconciled.

Here is one solution:

image

Making PBL Disappear: Why PBL? Part Two

Murder Mystery Solved with Trig!

Dateline: April 14, 2016

The murder of Maria, whose body was conveniently found at right angles to Leg Streets A and B, has been solved! Investigators found the weapon across the river, apparently thrown there by her assailant while he was running down Leg Street A in an attempt to escape. A quick thinking officer (who had majored in math at the police academy) was able to calculate an angle measure for the angle made by the throw from the perp and the street leading to the victim.  Another savvy investigator was able to determine the distance from the suspect to the location of the attack.

With the mathematical evidence in hand, investigators were able to triangulate a conviction. Math teachers everywhere weighed in, saying it has the proportionate ability to change the way investigators do business!

Dimensions of the prisoner’s defense will be released at a later date. Film at eleven.

Okay, so I don’t really have film (we forgot to assign the job of reporter!) What I do have are a room full of kids who can now set up the proper proportions for trig problems!

Here’s how the crime went down:

Scene 1: Before the murder, I handed six students a few props:


Each student had to use the prop to arrange themselves into a triangle. The other students watching were, um, helping. (that’s what they called it!)

A short q&a followed:

Me: Okay leg A, are you opposite or adjacent to angle b?

Hapless Student holding Leg A sign: “I’m opposite, um, no, I’m right next to him (indicating student holding the angle b sign)! What does adjacent mean, again?”

We were able to sort out the definitions, and the students holding the leg signs got pretty good at determining whether they were “opposite” legs or “adjacent”  legs. A big moment came as students noticed that they could be opposite OR adjacent. More importantly, they were able to articulate WHY the status would change.

 More importantly, they were able to articulate WHY the status would change.

Scene 2: The next six students were given the cards. This time, I stood back and let the first group help position the players. A little skirmish ensued as Leg A and Leg B were being positioned. After a brief discussion about whether or not leg locations could be interchangeable (did Leg A have to go in the same place as the first triangle?), it was decided that as long as a leg were placed on each side of the 90 degree angle, it didn’t matter what we called them.

The opposite and adjacent discussion began again. It was fun watching students correct these new players, or making them guess by giving them tantalizing clues!

(If you ever want to know what you look like teaching, give your students the reins. Mimicry is not dead!)

Scene 3: With everyone up to speed on definitions, the murder could now commence! Maria was positioned. Ryan was immediately suspect, as we put the crime scene tape in his hand and instructed him to escape a bit down the hall. The “weapon” was given another piece of crime scene tape and told to take off in the opposite direction. The “hypotenuse” was asked how far the “perp” had thrown the weapon. We stretched the crime scene tape from the suspect to the weapon location. It was at this moment that I heard several students say “Hey, we made a triangle.”

It was at this moment that I heard several students say “Hey, we made a triangle.”

(Scary, I know, right?)

After a bit more discussion, the students determined that we needed an angle and we needed the distance from the body to the suspect to set up a proportion to solve for the distance. Two students were dispatched with the piece of crime scene tape that had been held between the victim and the suspect (Leg A, for those of you following along). Twelve inch square floor tiles assisted in the crime scene measurement. I used my oversized protractor to come up with the angle measure, and we were ready to set up some proportions!

Back inside the room, our eager detectives checked their trig proportion info sheet (yes! They used their NOTES!) and settled on cosine, adjacent and hypotenuse. I stood back and watched them argue over who was going to set up the problem, exactly how to set it up, and how to enter the information into the calculator. Then I watched them convince one another which answer was correct.

Concrete to representational to modeling AND peer tutoring…I love it! I would say that a murder wasn’t the only thing that got solved today!

“It isn’t that I don’t like math. Learning takes time in math, and I don’t always get the time it takes to really understand it.”*

How many more of our students feel this way, but instead of telling us with words, they distract, joke, sleep, or skip class:

…Math is such an interesting subject that can be “explored” in so many different ways, however, in school here I don’t really get to learn it to a point where I say yeah this is what I know, I fully understand it. We move on from topic to topic so quickly that the process of me creating links is interrupted and I practice only for the test in order to get high grades.

Taking Time Learning Math:A Student’s Perspective by Evan Weinberg

Would I want to come to my class?
This question haunts me. What are my kids seeing, feeling, thinking? Why does this kid come, but stay totally uninvolved? Why does this child talk, constantly, but about anything but math? Where did curiosity go? Is my class a class I would look forward to?

My personal enjoyment of math comes from the struggle with ideas and the satisfaction I get from my connection of and understanding of the relationships among those ideas. It’s like a huge puzzle that will take the rest of my lifetime to fully understand. The student’s comments in Evan Weinberg’s post resonated with what I see happening with my students. They are not learning math so much as preparing for a test about math.

They are not learning math so much as preparing for a test about math. 

The current situation of ‘learn how to do this; learn how to do that’ mentality is slowwwwly changing over to ‘understand why this is so; why does this relationship work’ exploration. It will need a shift in how we teach, letting kids struggle and connect ideas (we must facilitate this exploration, but not down some tightly designed path), and changing our view of grades and mastery. I can’t say I don’t have the answer- I am working on an answer that works for me and for my students. And I’m sure I am not the only one teacher who has found the path that is taking them closer to the ideal.

This post grew out of my response to Evan’s column. His response,

“I completely agree that this is a shift, and it is ongoing. Clearly, despite the changes I’ve made to the way I teach, students still get the sense that the test is the important part, which means there is still a great deal of improvement yet to be made!”

*Taking Time Learning Math:A Student’s Perspective by Evan Weinberg

“…maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art…” 

This statement from STEAM: How art can help math students really struck a chord for me:

“The big point of it is that maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art,”Year 12 student Maxim Adams

I am currently teaching geometry (among other maths!) and we are in the sweet zone of shapes: triangles lead to parallelograms which lead from 2D to 3D to the idea of 4D (think twisted paper*). All of this incorporates the effects of angles on shape which affects (and effects) area. Some shapes are more ‘beautiful’ than others. Some shapes and forms lead us to wanting to know more…


I use paper folding to illustrate these shapes and connections. Patty paper is very light and a great beginning medium to illustrate the connections between shape and angle, how changing the angles changes the shape. Origami fascinates me, and I use it to fascinate my geometry students, and to lead them into exploring the relationships found there. The zig zag origami is complex and beautiful, and it takes a bit of time to teach, but as my students learn to fold, we are talking about math. Even their mistakes, when they have to go back and refold, teach them about the shapes and connections. And as STEAM director Melissa Silk says,

“I think it needs to be embedded from a very early age…When I’m teaching design, I’ll try to bring in a mathematical element. It’s worth students trying to make those connections without having to find it arbitrarily later in life.”

These ‘connections’ she speaks of are not innate. Students see without understanding. Teaching reveals and makes obvious; children begin to see math in everything in the world around them. It will allow them to make much deeper connections, more fluid connections, throughout their education and beyond.

 Art is not math, it’s not boring. Folding paper, for many math students, isn’t ‘doing math’. While they may struggle to fold straight lines, to make inverse folds, to make it look ‘right’, they are creating something beautiful, and if I can teach them nothing else, I can show them math can be beautiful!
 The idea, after introducing them to this complex looking, yet simple, folding technique, is to have them think about what it will take to replicate angles, curves and other shapes with their paper. Then they have to apply some design sense: how many folds? How long does the fold need to be? What direction do they need to fold? What angles will produce a curve? The folds are repetitions of a fold- what is the proportional relationship? In this image, there is a curve. Each fold is a point on that curve. It can be mapped on grid paper, it can be written as a formula. There is a connection!

Don’t be afraid to start with complex looking folds. It’s so powerful for students to struggle with something that seems beyond their abilities. Success is completely intrinsic and will lead to a student who is more confident. This feeling of success is habit forming. It is a primary building block in creating a student who perseveres!

(Check out 7 more art and math connections here.)

We can calculate the area of this beautiful 3D paper construction, we can create more of them by following the ‘formula’ or we can dramatically change or even destroy an image by changing the formula! This creative process stretches the imaginations of my students. The ‘what if’s are endless.

Many of my students will continue with this exercise beyond class, through drawings, foldings, and cut paper. There is a great simple foldable here that actually incorporates cuts to create a 3D effect. It’s a great way to jumpstart paper folding and cutting shapes to students that are not accustomed to paper-folding activities! I want my students to make graphic organizers that are beautiful to look at. Notes as art… What a concept!

Notes as Art… What a concept!

 

Pop Up Book graphic organizers

 

The possibilities for using this as a graphic organizer are endless!