Category: teaching

Pedagogical Aikido

“Where did the 850 cans come from?” I was in the middle of sharing the iReady enrichment lesson (14) with my fourth graders when one of them asked me this question.

Have you ever had a student ask a question in order to postpone learning? If you’re a teacher, then that’s a silly question. Of course! 

This is one of the few things that I remember from my elementary and middle school days. It was a thrilling challenge to try to come up with just the right topic or question that could throw the teacher off track. 

We would hope and pray for a story. Then, we would artfully flatter and ask questions that would lead our pedagogue down the rabbit hole of memories, further and further… away from the lesson at hand. 

Fast forward forty years. Today’s students still play the same tricks on their teachers! This past week I was engaging some fourth graders in math enrichment, when one of them tried steering me off task. Little did they know, that I practice Pedagogical Aikido. 

Redirecting Energy

Aikido is a form of martial arts that is known for using an opponent’s energy (ki) against them. Masters of this study practice redirection. 

Although I have not formally studied Aikido, I love its principles and attempt to use the philosophy of redirecting thought and energy within the walls of my classroom as much as possible. 

For example, the other day when my student asked about the origin of the 850 cans in our math problem, I allowed the student to think that he had derailed the lesson. I told him that this was an excellent question. “850 cans is a lot of cans. Where would a school get that many cans for a fundraiser?”

The martial art Aikido uses a triangle to teach the redirection of energy. There are three components that work together to use an opponent’s attack against them, saving your energy and neutralizing the situation. It all starts with Balance, known as tachi waza (Aloia, 2020).

“How many students does our school have?” I asked the class. 

I could have squashed the student’s inquiry, telling him something like, “I don’t know where the number of cans came from. It’s hypothetical. Let’s just move on!” Or, “It came from Curriculum Associates, the authors of our math program. Don’t ask silly questions.”

If I had done that, I would have disrespected the student. A dismissive teacher or one who blocks the question head on is too hard, too strong; the lesson too one-sided. By allowing for the question in the first place, and then entertaining it, I had my center of gravity low to the ground. My metaphorical feet were spread wide apart and knees bent. The question didn’t topple my lesson. I was balanced.

In answering my question, the students were surprisingly accurate. Our school has around 700 students. “How many cans would we have if each student brought in one can?” I prompted. That was easy. “But, not every student will bring in a can… And, some will bring in more than one.” The easy back and forth of these simple concepts established a flexible, down to earth ease of thinking. It also revealed the problem. We don’t know where the 850 cans came from.

Next, it was time to Break Balance. This is the second part of the redirecting-energy triangle. “The opposite of balance is imbalance, or kuzushi. To break an opponent’s balance, one must first redirect their energy to one’s own advantage” (Aloia, 2020)

I shouldn’t be surprised, but I was very impressed, nonetheless, at how quickly my students figured out how many classrooms our school had. It was the advanced fourth grade math students receiving enrichment, after all!

I had begun the imbalance kuzushi by getting the class to come up with the total number of classes in the building. After figuring out that our school has five classrooms per grade and our school teaches six grades, if you include kindergarten, we discovered that there are 30 classes represented.

“Let’s say that our school collected 850 cans. How many cans would each class bring in?” The students had no clue where to start.

Antonio Aloia (2020) explains that kuzushi has two arms. The physical off-balancing of an attacker, parrying the opponent’s strike and redirecting the momentum of the assault, coupled with a strike of their own is what one normally thinks of when imagining Aikido. Um, of course there isn’t any literal physical contact with students, let alone “attacks,” but presenting this new problem of dividing up the number of cans by the number of classrooms was a cogitational assault of sorts.

The other arm of kuzushi is a psychological off-balancing. This is where a martial artist would “Distract a would-be opponent by bringing their attention to something else, be it an object on a building or something farther away and behind the opponent” (Aloia, 2020). Pedagogically, this happened when I changed the student’s original question from “where” to “how”: “Where did the cans come from?” turned into “How could a school come up with so many cans?”

While the martial art of Judo involves throws, Aikido keeps your opponent tight and controlled. Perhaps counter-intuitively, it is concerned with the well-being of the attacker. So, rather than toss my students aside to flounder with the problem of dividing 850 by 30 on their own, I guided them through the process of figuring out the answer.

I asked them how many cans there would be if every class brought in 10 each; 300. “Okay, maybe that was the first week of the fundraiser. If each class brought in another ten cans during the second week, how many cans would the school have collected?” We were up to 600 cans. They were starting to catch on. 

One of the students used Google to divide 850 by 30. Rather than scold him, I asked him if it were possible for any of the classrooms to bring in .333333 of cans. This was a silly question. “What happens with the remainder from the division answer?” I asked. They didn’t know. “For our purposes, we will assume that the students from every classroom brought in 28 cans. The teachers brought in the rest.” My students were okay with this explanation. 

The third side of Aikido’s redirecting energy triangle permeates everything. It is ki or energy. Don’t think of it as power or force, though. Ki is more like momentum.

“How big are our classrooms? How many students are there in a classroom?” I got several answers on this. We decided to use the number 20. “Let’s say that a quarter of the students don’t bring in any cans. If the rest are responsible for bringing in 28 cans, how many brought in two and how many brought in one?” My students just looked at me. I told them to try and figure it out on their own, and then I’d show them. 

One student crushed it, and I had her show the class what she did. Then I modeled drawing a picture to solve the problem. 

After all of this, I told my students, “Now that we have collected all of these cans, we need to put them in something to bring them to the food pantry that we are donating them to.”

“If Dylan went out and bought a bunch of boxes… Thank you Dylan! (Dylan is all smiles at this point; He may or may not have been the person to ask the question that started all of this;) And, if Dylan’s boxes are all the same size, holding six cans each, how many boxes would Dylan have to get?” I let them wrestle with that a little while. 

When I was prepared to let them demonstrate their math on the board, I turned to the slide that had the original question on it. They reread the word problem as I decided on who would come forward to share their work first. A few students groaned and some others called out. “That’s the problem we just did!” 

“Yeah?” I feigned ignorance. 

I used someone else’s name when I told the story about getting bigger boxes; Ones that held 8, instead of 6 cans. “How many of those boxes were purchased?” 

As it turns out, we never got to fully explore the last question, but a couple of students tried solving it in their heads. I had completely Aikido-ed them! Lol.

Redirecting energy can be an even more effective motivator than a cool lesson. Take their energy, spin it around, and use it against them. Students will feel like they’re in charge of their own learning, and in a way, they are!

Source

Aloia, A. (2020, June 19). Reflecting on Jujitsu Pioneer George Kirby’s Advanced Techniques for Redirecting an Opponent’s Energy. Martial Arts of Yesterday, Today and Tomorrow. https://maytt.home.blog/2020/06/19/reflection-on-jujitsu-pioneer-george-kirbys-advanced-techniques-for-redirecting-an-opponents-energy/comment-page-1/?unapproved=2695&moderation-hash=f6966939a4ca212a2123a94cabda8d13#respond

Buttery Batch of Math Cookies

This is the introduction that I used on my fifth graders.

In preparation for teaching a math enrichment lesson to my fifth graders, I looked at the iReady “Extension” activities in the Ready Math “Teacher Toolbox,” and I found a problem that I liked a lot. (iReady and Ready Math are products of Curriculum Associates. My district has been using it for several years, and I like it a lot.) This lesson (14) is all about using fractions to solve word problems.

Here’s an image of the worksheet that a teacher could photocopy or share via Google classroom. Because I have the luxury of actually teaching enrichment lessons, I decided to do some explaining before handing over the problem. Also, I opted to make a few tweaks, too. In my experience recipes usually call for specific measurements of butter, not a number of “sticks.” Therefore, I covered up the word sticks in the problem and wrote in “cups.”

This changed the outcome of the answer quite a lot. Now, students would not have enough butter to complete the recipe. They could access new sticks of butter, but if they did that, then solving the problem wouldn’t require wrestling with all of the fractions presented in the partial sticks. That’s when I imagined the real-life experience of baking cookies after having worked all day at making a big meal, like Thanksgiving.

Needing soft butter for recipes is a real thing. Also, who doesn’t love consolidating? We can clean up all of those partial sticks of butter and make cookies at the same time!

I shared what a typical day of cooking in preparation for a large Thanksgiving meal looks like at my home with my fifth grade students, setting the stage for having several fractions of sticks hanging around. With the instruction to use up the warm butter first, and then dip into the cold butter from the new package, I set my students loose to calculate how much butter would be left.

Many students jumped on adding up all of the fractions. They began figuring out compatible denominators, so that they could combine every partial stick and find out what they had in all. “But, do you have to do that?” I asked them. No one wanted to venture a guess.

“What are you asked to find?” I pressed.

“Two and half cups of butter,” someone accurately answered. Without saying anything, I drew two of the worst cups ever drawn on an interactive board, followed by half of a third. I made fun of my drawings, which everyone helped with, pitching in their own digs. Once that settled down, I pointed out the lines I’d drawn through the middle of each measuring cup.

“Why’d I do that?” Earlier, we had discussed that fact that one entire stick of butter was the equivalent of half a cup. The students understood better than they could put it into words, so I articulated the concept for them, “Each half of a cup was one stick of butter.”

Then, we looked back at the fractions. It was easy to see that 1 2/3 + 1/3 would be able to fill one whole measuring cup. That leaves us with three fractions with differing denominators. “Before working out a common denominator to add up all three, think about what you are trying to do,” I instructed. “What is your aim?”

I showed the students that 1/2 a stick of butter + two of the 3/4 would equal one whole. “That would take care of half of a measuring cup,” I told them. Also, I should mention that I crossed out halves of measuring cups, as we discovered combinations of partial sticks of butter that would fill them.

“If we used up two of the quarters to combine with the 1/2 a stick and create a whole stick, how many quarters are left?” One quarter. “And then, we have 5/8 of different stick left.”

They instantly got it. We were 1/8 short of a whole stick of butter. In the end we needed one whole cold stick of butter, plus 1/8 of an additional stick to add to all of our warm butter fragmented sticks to fill our two and a half measuring cups.

The Ready Math extension lesson (14) has a second question that I left as is. The catch is that my students used our additional left over cold butter (2 7/8 sticks) from my adapted first problem to solve it. I let them struggle with this one for a few minutes before I showed them the short cut of drawing pictures.

“You might think it childish to draw pictures,” I began. Fifth grade is the oldest grade in my school, so these were the seniors of the place. “…But, I find it easier to manage some problems when I sketch what is happening.” I had been watching them crunching numbers, making common denominators again, and subtracting fractions. Now, within a handful of seconds, I showed them how many quarters could be made from two sticks of butter! I pointed out the idea of labeling the quarters in order to keep track of my thinking. I wrote a B above each “batch” of cookies. Sure, I could just count the quarters, but when it came to the last stick, it will be important to identify what portions of butter will complete a batch.

As I divided the last rectangle into eighths, I asked, “What am I doing to this last stick of butter?”

Rather than answering my question, they were chomping at the bit to be the first to spew the solution to the problem. “Eleven and 1/8!” more than one fifth grader shouted at the same time.

“No, that’s incorrect,” I casually, but cautiously counseled. Rewording what they had yelled in order to make plain the problem with their answer, I said, “You cannot make 11 AND 1/8 batches.” The emphasis on the word “and” did the trick.

“You can make eleven batches, and you’ll have 1/8 of a stick left over,” a student corrected.

“Perfect,” I affirmed. “Drawing pictures might seem silly, but look at how simple it is to see the answer. We didn’t do any denominator work past doubling up the number of sections in the last stick. I hardly did any math, beyond simply counting!

“When you are taking standardized tests, you get scrap paper. Use it. Draw pictures. Illustrate word problems. Take the time to label parts of your illustrations. Make sure that you understand what you are being asked. What is your goal? What are you supposed to find? It’s not just a number. It is the solution to a problem. In real life, it is a key that will unlock a problem. Be a problem-solver; Not a human-calculator,” I told them.

Photo by Elliot Fais on Pexels.com

In conclusion, my aim is to turn these advanced math performers into problem-solvers. With this goal in mind, I try to make lessons that force students to use what they have learned in their regular math class in a way that is not only compatible with what they would find in the “real world,” but forces them to understand how to use the skills. I often allow my students to use calculators because the problems I prepare for them require more knowing what to do with the numbers than practicing running through algorithms. AI can learn how to crunch numbers, but will it be able to successfully manage a kitchen full of amateur chefs laughing, telling stories, and making meaningful memories, all the while measuring butter for cookies after already cooking and eating a Thanksgiving dinner?

To combat the threat of AI, don’t try to make humans better than machines. That just makes them more like machines. I say, grow the human-ness of students. This is getting pretty deep, so I’m going to go eat a buttery cookie while I chew on these ideas for a future blog;)

Nemesis Brings Balance: A Story

In preparation for teaching my 4th and 5th graders about conflict in literature, I did some research that uncovered something novel. I had always thought that someone’s nemesis was the antagonist of the story, the enemy, the ultra-bad guy, the anti-hero. This isn’t exactly the case. Before I share what I learned, I have a story for you.

Photo by Steshka Willems on Pexels.com

There once was a baseball player born with a natural affinity for the game. It was more than that. He had a gift. His parents knew he would be great, and they gave him everything he would need to fulfill his destiny. 

Everyone who saw this boy play commented on his skills. When he threw the ball it seemed to know where it was supposed to go. His glove was a magnet. No toss escaped it. The bat in his hands may as well be Thor’s hammer, Mjolner. It punished pitchers year in and year out. 

A funny thing happened early in the boy’s career. He was still in school when a coach who understood the greatness before him told the ballplayer’s parents, this one will go far as long as he never assumes his talents, but continues to practice and better himself. This coach had glimpsed raw talent before, and knew that without tuning, the song of the marvelous instrument turns into a grating noise. 

The young baseball player had no problem making it to the big leagues. The people who played ball with him were graced to know his genius for the game. Even his opponents felt privileged to have been able to rival him. 

Numerous articles were written about the rising star. Every journalist craved an interview with the boy, but he eluded them all. One journalist in particular resolved to take photographs of the athlete in action. If she wasn’t granted the privilege of speaking with the ball player, she would communicate his greatness to the world through picture. 

The journalist, not new to baseball, had never seen the human body move the way this star player moved. Her photographs captured the power of the throw like none other. Looking at the glove that never missed a toss portrayed in portrait amazed every viewer. 

Even though no one secured an audience with the baseball player, this photographer wished to put words to her photos and sought an interview. Rejection after rejection caused the journalist to turn to other means of query. She would write letters to the baseball player. She tried to get his manager to intervene. The journalist befriended other players in hopes that they would put in a good word for her. Eventually, the journalist even bought presents to give to the baseball player. Perhaps she could bribe him to give her some sound bites, at least. 

Other journalists saw this photographer pining for the privilege to interview the star as pathetic at first. They feared the photographer would fall apart, focusing so much attention on one person. They warned her. It was too late. And then, she was gone. 

Photo by Tim Eiden on Pexels.com

The owner of the magazine that the photographer worked for heard the speeches of brother and sister journalists at the photographer’s funeral and decided to do something about this tragedy. Why wouldn’t the baseball player just talk to his employee?

The magazine owner compiled a book of the best photographs and had it delivered to the baseball player. There were no words. Not even a title was printed on the cover! Everyone knew who was featured in the photos. It didn’t take much research to find out who had captured the pics. 

What no one foresaw, though, was the way in which the photos would imprison the baseball star. He had never bothered to read the newspaper articles written about him. He hadn’t ever seen the famous photos. When he viewed the pictures of his playing, he was mesmerized. Is that how he looked to others? 

Photo by Lisa Fotios on Pexels.com

He removed photos from the book. Framed his favorites. They were placed all over his house. He found it difficult to pass one without stopping to look. Eventually, the baseball player couldn’t leave his home without missing the photos. After a while, he stopped leaving all together. 

The once great star didn’t eat. He didn’t drink. He wasted away to nothing. When his friends went to check in on him, they didn’t find a former baseball player. All that was left was a flower; A daffodil. 

Did you recognize my modern myth for what it was, a retelling of Narcissus? I wrote this version for my students. I wanted to emphasize the idea of undeserved good fortune, a theme I had overlooked in the past. 

More than rivaling a hero, Nemesis is a “balancer of the scales.” I’ve thought that Nemesis was synonymous with antagonist, but she isn’t. I didn’t remember her role in the Narcissus myth, but she was the one who led Narcissus to the pool where he first beheld his reflection. She heard the prayers of the nymph-friends of Echo, who died of unrequited love. 

I always focused on Narcissus getting what he deserved, but there is much more to the original story. First of all, his parents had been warned that Narcissus would live long if he never knew himself. I wove that into my tale by having the baseball star’s coach warn against relying too much on good luck and gifts. Talents need to be honed, stretched, developed. 

Next, I never recognized the role of Nemesis in the story! She deliberately brings Narcissus to the pool where his downfall takes place. She knows the prophecy. She understands the importance of Narcissus remaining ignorant of his beauty.

It isn’t pay-back, though, that drives her to initiate Narcissus’s learning about his looks. It is a need for balance. Nature was too kind to Narcissus, just as my baseball player was too talented. In my tale, the magazine editor brought balance by making the athlete see the photos that the gifted photographer captured. I hope you recognized the photos for what they were; “Echoes” of the action.

Give Your Teaching Life by Using Real-Life Situations to Teach

It’s math enrichment time again. The pacing guide has the fifth graders learning to add and subtract fractions. Our curriculum has an enrichment lesson that students can work on independently. It’s a grid that has empty spaces that need to be filled in order to help every row and column add up to the same sum. I like these types of exercises because they are puzzles and make the practice feel like a game. 

This is the enrichment activity provided by iReady. It’s a good tool for independent practice, but I wanted to provide a “Math Experience.”

I view my job as a math enrichment educator as deepening the understanding of already mastered math skills. Puzzles and games are a great way to make the skills easier to access, faster to use, and more accurate. They are excellent tools for regular ed teachers to provide for those students who have demonstrated competency in a math concept. This style of enrichment activity is good for keeping the advanced students busy while the teacher catches everyone else up to speed. For my part, I aim at helping students see the math from a different perspective, though. 

I cut all of the tiles for the ceiling before attaching them.

It makes a grid of mixed numbers!

Diamonds are beautiful, rare, and expensive gems used to make jewelry, but did you know that they are also the hardest substance on Earth? In fact, the fragments that are cut away from a diamond when it is shaped for a piece of jewelry were used to make saws and drill-bits that can cut through rock (2023). I’ve had the experience recently of using a diamond-studded saw blade to cut through ceramic tile and marble to make a new shower in my house. Just like adding and subtracting fractions can be used to complete a pretty grid, they are also very useful in everyday life. I hoped to make my students’ understanding of fractions more rich (enriched) by having them use their skills to help me measure tile precisely. 

This was one of the most difficult home remodeling projects I’ve tackled. I should use it to teach my students, I thought to myself. So I did.

Welcome to math-enrichment, real-world experience-mode. This project was full of fractions–really! I was excited to share it with my students. 

The first thing I did was think about the different ways I had to work with fractions in order to make the ceramic tiles fit my shower walls. I came up with a math problem that involved quite a lot of fraction work. This will be good, I thought, but perhaps there’s a simpler one to begin with. Even though it’s not the first part of tiling, I decided to have my students measure the window sill, a piece of white marble that had to be exact because the sides of the frame were already tiled. 

I put a photo of my window, minus the sill, into a Google jamboard. The photo shows my tape measure. “I took this picture, so that I would remember the exact measurement when I was at the granite store ordering my marble. I don’t want the gap to be too big on either side of the marble, or it will look weird,” I tell my students. “What I want is for the gap to be the same size as all of the other gaps between tile in the shower, ⅛ of an inch.” After explaining the parameters of the problem, my students have a goal: Make the marble look nice. It will look nice if the measurement is just right. “The space we have to fill is exactly 28 ½ inches wide,” I informed my students. “What width will the marble be cut in order to leave ⅛ of an inch on either side? Go!”

Anyone with experience measuring with rulers and tape measures for cutting materials will understand that you can simply back up your finger or the material on the tool to find the answer. Fifth graders have not had this experience, yet. Also, we were sitting in a classroom, instead of holding a tape measure up to the empty window frame.

After letting my students wrestle with the fractions for a few minutes with a couple of them finding success, I showed the class how to look at the problem, by holding up a yardstick. I placed my finger at the 28 ½ inch mark. I slid my finger over a fraction (⅛) of an inch and asked them how much I just “cut” off of the marble. They saw that it shrank by ⅛ of an inch. “Now, we need to take ⅛ of an inch off of the other side, but do we move our finger ⅛ of an inch from the beginning of the yardstick?” I paused to let them imagine what would happen if we did that. “No, we include the subtraction of that ⅛ of an inch right here,” and I slid my finger over another fraction (⅛) of an inch. “Now, how wide is our marble?” 

Some of my students had written 28 and 2/8 on their papers. When we looked at the yardstick together, we discussed that it is better to say 28 and ¼ of an inch… “Not just because you are supposed to reduce your fractions! But, because one is easier and faster to count than two; plain and simple.” This set us up for our next problem, the one I had come up with originally, but put off until after having practiced some measuring, first. 

Time for some tiling!

“Now, we need to cut some ceramic tile to fit my shower wall,” I pitched. “Remember, we want to keep gaps of ⅛ of an inch between everything; between the tile, between the corner of the wall and the tile, between the tile and the metal edging; everything.” (This isn’t just easy for students to forget. It is easy for someone measuring for actually tiling a shower to forget. I should know! I can’t tell you how many times I had to re-measure or recalculate because I forgot to account for the grout!) 

“Okay, so our wall is exactly 30 and ¾ inches wide, from the corner of the shower to the metal edging. I’ve already tiled the window wall,” I explained. “This 30 and ¾ inches is the space that we need to fill with tile and grout.” I showed them a tile and told them that it is two feet by one foot, rectangular. There was a photo of my shower wall in the Jamboard, too. I wanted them to visualize the final product. While I had written the entire problem into a Google jamboard, I shared the problem with them orally, also. It was easy for me to communicate, because I just did this! It was fresh on my brain. 

“We will be alternating the tile,” I told them. “We begin at the bottom of the shower wall and work our way up. The first row will begin with a whole piece of tile placed in the corner. The next row will begin with half of a tile starting at the corner, and so on.”

Surprise! The tile isn’t really 2 feet by 1 foot; not exactly.

“Guess what,” I exclaimed. “I have a treat for you: The tile that you thought (I thought;) was two feet wide isn’t! Like everything else in life, it is a little short; a quarter of an inch short to be precise. It’s your job to figure out exactly how wide to cut the tile that will fill the gap that is left within the 30 ¾ inches space. Don’t forget about the ⅛ inch grout between everything! Go.” 

Student mouths hung slack-jawed. I used the Jamboard to demonstrate the math. I drew lines that represented the sides of the shower wall. “This is the corner,” I told them, pointing to the line on the right. “And, this is the metal edging that is the end of our tiling.” I pointed at the line on the left when I said this. “Now, how wide is the space between the two lines?” I prompted. 

It took some waiting, but finally one of the students ventured, “Thirty inches?” 

“Thirty and what?” I reminded them.

“Three quarters of an inch,” they completed. I told my students that this was the width of my tub (30 ¾ inches).

“Okay,” and I wrote the complete distance in the space between the two lines. “Now, let’s begin by allocating ⅛ of an inch over here.” I drew a little circle next to the line that represented the corner. “What comes after the ⅛ inch of grout?” Pause. Wait. Patiently persist in waiting. “Look back at the problem.”

“A tile,” someone says out loud.

“Yes,” I affirm the correct answer. “We put a whole tile on the wall. How much distance does that take up?” I went ahead and wrote on the Jamboard while they were thinking: 24” – ¼ of an inch.

“23 and ¾ of an inch,” a student answered before I was done writing. Pleased, I put the length into the designated space in our row of tile. 

“Now what?” I asked.

“We figure out how wide the remaining space is,” a student said in a half statement/half question.

“Yes, but…” I wanted to provide positivity, but needed to help them toward accuracy. “Don’t forget about the ⅛ of an inch on either side of the remaining tile. You have to leave space (⅛”) between the two pieces of tile and also a space (⅛”) over here.” I drew tiny circles and labeled them on the Jamboard. “Use all of these measurements to figure out where I will cut my tile to place on the wall,” I told them. “Be careful!” I warned. “I don’t want to waste any tile. I have just enough.”

There were a few ways to figure out the answer. You could add up all of the fractions. And then, subtract the mixed number from 30 ¾ inches. You could also subtract each individual “space” away from 30 ¾ inches to see what is left. This method is like sliding your finger down the yardstick. After letting the students try the math on their own, we discussed. 

If you thought that the previous problems were difficult, buckle up. “Now, I have a real treat for you,” I told my students. “That first whole tile; The almost two feet wide tile; We have to cut a hole in it. The spout for my tub has to go through it.” Groans. “That’s right! I want the pipe, which is exactly one inch in diameter, to be in the center of my wall. What does that mean?”

“Halfway,” someone says. 

“Yes.” I let that sink in. “How far from the corner is the pipe?”

One of my fraction experts quickly tells me that it ought to be placed 15 ⅜ inches from the corner. I was pretty impressed, and had the student explain how he had converted ¾ into 6/8 in order to divide the fraction in half. 

“That’s right, but that is the center of the pipe,” I tell them. After explaining the term diameter, I point out that half of an inch will be on one side of 15 and ⅜ and half an inch on the other. “Where do we cut the hole for the pipe? …Don’t mess up! We don’t want to waste a great big whole tile!!”

After they fight with fractions for a minute, I remind them of the ⅛ inch of grout that precedes the beginning of the whole tile. Many groans follow this reminder. 

The whole lesson ends with me warning the students that, “After all of that measuring and math, you better be sure to cut the tile on the correct side of the line you draw, because your diamond blade will shave 1/16 of an inch off of whatever you’re cutting. In other words, you could figure out that the hole for your pipe has to be cut 14 ⅞ from the edge. Mark that measurement on your tile. And then, when you go to make the cut, slice the tile on the wrong side of the mark, which would make the distance from the hole to the edge of the tile 1/16 of an inch short. What would that be?” I posed the question, but didn’t actually expect them to figure it out. They knew I was messing with them, and they all gathered their things to leave. 

“Wait! What about our next row of tile?” I laughed as they left shaking their heads. 

Tony Wagner, the author of “Creating Innovators: The making of young people who will change the world” (2012), describes the practices of some highly effective schools. One of them is Olin College, a small engineering school that is pioneering hands-on learning.

“In classes at Olin, the primary goal is not the acquisition of knowledge. The goal is to develop a set of skills–or, in Jon Stolk’s terms, competencies–by solving a problem, creating a product, or generating a new understanding. Knowledge is important, but it is acquired on an “as needed” basis. It is a means to an end. Traditional academics often criticize this approach for being too utilitarian and lacking an appreciation of learning for its own sake, but the evidence is that Olin students are very well prepared for graduate school and better prepared for work, with managers who have been surveyed by the college reporting that Olin students who’ve just graduated act as if they’ve had three to five years of experience. Learning research shows that students understand and retain much more of what they learn when they have studied and used the knowledge in an applied context.” (Wagner, p. 175)

“When will we ever use this in real life?” is a question many math students will utter when trudging through seemingly pointless pedagogy. Infuse life into your teaching by showing how the lessons are used in real-life.  

Sources:

Development History of the Notched Rim Lapidary Diamond Blade. Barranca Diamond. (2023). https://www.barrancadiamond.com/home/history.html#:~:text=Richard%20Felker%2C%20a%20pharmacist%20develops,natural%20and%20manmade%20stone%20products.  

Wagner T. & Compton R. A. (2012). Creating innovators : the making of young people who will change the world (1st Scribner hardcover). Scribner. 

Math Games: Dessert for Dinner?

What if you could produce a dessert packed with protein and healthy nutrients; I’m talking even more beneficial than a typical meal. Would you serve this delectable dish for dinner every day? My conclusion may surprise you.

Last week was Parent-Teacher-conference-week at my school. Students had half-days, and families either visited the building or used virtual conferencing tools to converse face to face with educators. This was the very first time that I bounced around from teacher to teacher, visiting the conferences of my gifted students’ parents. While there are many ideas that I could comment on, the one that stands out most was from the parent of one of my math enrichment students. 

The family has a third grader who is gifted, and that is why I was attending the conference. But, his little sister, who is in first grade, attends my math enrichment lessons, and it was something that she said that got me thinking. Her parents told me that they asked their daughter what she did in math enrichment class, and she told them, “We play games.” 

“Is that all?” I imagine them pressing, being the good communicative parents they are. Yup, is the first-grade answer:) 

This is a recent pic of 2nd grade learning to play Dominos.

I laughed when I heard their tale. I explained to the parents that I was teaching the first graders Dominos. After defending the fact that there is a lot of mental math and problem-solving, there was some light banter between parents and the regular ed teacher about only knowing the stacking and tumbling side of Dominos. 

Because their gifted third grader had already experienced lessons showing the critical thinking development of Dominos, it wasn’t necessary to get too defensive. They were “on board” with my use of games for strengthening math skills. But, the idea of my lessons being categorized definitively as nothing more than games gave me pause. Initially, I was perfectly okay with tricking students into learning through having fun. What teacher would turn down that strategy? “Can we have some more math enrichment, please!” the students whine. “Um… Yes!” every math teacher in the world would utter. 

Then I thought about the idea of turning everything into a game. Wouldn’t that be wonderful for the students? But, would it be healthy? Hmm… 

This is where the thought experiment at the top of this blog originated. I was musing over my math lessons being perceived as games, and I dreamed up the analogy of only eating dessert. Prepare to enter a rabbit hole of research. I’ll try to keep it palatable😉

History of Dessert

Asking “Why does dessert even exist?” feels a little like questioning the purpose of gold or jewels. Isn’t it obvious? It’s awesome! 

Believe it or not, dessert did not always exist, however. Similarly to gold and jewelry, it was discovered, and has evolved over time. The French are responsible for turning entremets into dessert (Gerson, 2019). Before there were sweets to end a meal, entremets were served as “interval” dishes, literally “between-foods” courses (Teppen, 2015). They were meant to cleanse the palate. They may be sweet, but not necessarily. 

Eventually, a final course of fruit, called le fruit, was formalized (Gerson, 2019). Only, before serving it, the table must be completely cleared. This cleaning of the table was called desservir, the French verb for “to clear.” More than tasting wonderful, the original final course of fruit developed into something lovely to gaze upon. Some desserts even consisted of “Elegant metal and glass structures holding whole apples or plums. Other times, meticulously crafted sugar figures became the center of dessert displays, and might not be eaten at all. Dessert specialists in the eighteenth century were supposed to understand architectural design and be capable of replicating it in sugar paste” (Gerson, 2019). 

These creators of dessert, as it came to be known around the time of the French Revolution, when the Bourgeois assimilated the term, were originally more like artists than chefs. Maryann Teppen (2015) writes of an entire battle scene, complete with tiny sugary soldiers with guns and canons, that told the story of Louis XV’s demise crafted out of sugar. It is hard to imagine your dinner table being cleared; plates, napkins, silverware, and foods being “dessert-ed” away; only to be replaced by an elaborate, sugary scene of violence that you feast your eyes upon but don’t touch!

Modern dessert serves a different purpose. BreezeMaxWeb (2022) suggests it psychologically signals the end of eating. Consuming a small, sweet treat at the conclusion of a meal might communicate to the body and brain that we are all done, and there is no need to nibble superfluous snacks. The End. 

A practice that I began a couple of years ago has helped me lose some weight and become more healthy; I will eat an apple at the end of every lunch. Many years ago I heard that apples help clean your teeth, and apparently there is some truth to that (Apples: Dental Hygiene Facts, 2017). Once I’ve eaten my apple, I cannot/will not eat anything else. I don’t want to undo my teeth cleansing. This has helped me de-snack my afternoons.

Let me reintroduce the concept of math games, here. Could a game be used to transition from one course of subject matter to another? Would playing a game cleanse the cognitive palate, and prepare students for something completely different? Of course! Would this be an appropriate way to signify we are done with the subject? I think so.

Delayed Gratification (Deferred Satisfaction)

How many parents use dessert as a reward for finishing a well-balanced meal? That treat is the ribbon at the end of a race. Some contests require more work and take longer, but when getting to the game of a lesson is the goal, students may trudge longer, work harder, and persist through all kinds of problems. Those students who finish first might learn patience through having to wait for their peers to catch up with them before the whole group can consume the dessert of a lesson together. 

Self control. Training. Conditioning. “If I let you eat this piece of cake, do you promise to gobble up all of your peas and carrots without complaining?” doesn’t just sound silly. I probably don’t have to tell you that this is an ineffective reward model;) 

These are this year’s 3rd graders (math enrichment), learning the game, Cribbage.

But, what if the dessert is carrot cake? What if the dessert is healthier than the dinner? Then what? “Eat all of your cake, or you won’t be given any peas…” Wait, what?!

Is there something to be said for learning to crunch through cardboard in order to earn cake? According to a longitudinal study spanning 40 years (Casey et al., 2011), learning and practicing self control early on in life can lead to better academic performance, less behavior problems, and even higher SAT scores. Casey and company (2011) describe in their paper, “Behavioral and neural correlates of delay of gratification 40 years later,” some ways kids can curb the pull of stimuli by learning cognitive control. There are mental strategies and tricks that people can use to provide buffers, dampeners, and walls to contain and maintain self sovereignty. Students may never learn or develop these important skills if they are never asked to wait for anything.

Through reading this research I wondered if teachers, themselves, are bypassing the delay of gratification when they jump right into games to teach. What educator looks forward to grumblings from their students? I propose that most  would prefer praise of pupils happy with pedagogical practices over the squabbling of scholars required to earn a fun activity. Are we educators partaking in dessert before dinner when we teach with games?     

Dessert Before Dinner

Before we beat ourselves up too much, let’s bring our metaphor along with us as we explore a couple of Jamarillo’s fun list of 11 Reasons to Eat Dessert First (2023). This may initially seem like a self-serving exercise, justification, or defensive maneuver, but hold on. Jamarillo raises the point that food can sometimes be a serious psychological hangup. “When we have disordered eating, we can often develop food or meal fixation.  Dessert is one of the most common food items restricted. This can lead to binge restrict cycles and disruption of hunger cues” (2023). 

Is it possible for students to develop “learning disorders” by experiencing “binge-playing” with learning games after enduring unnecessarily long restrictions? Just as Jamarillo (2023) suggests that dessert-first-eating can help overcome eating disorders through stimulating hunger, tapping into nostalgic memories, practicing navigation of bodily needs versus wants, and learning to respect cravings, beginning a lesson with a learning game can help students who struggle academically to open up to pedagogy.  

One thing more, and this might be a great way to end this blog, Jamarillo (2023) ends her short article with the fact that dessert is an ambiguous course. It can be a sweet, but doesn’t have to be. Fresh, raw fruit could serve as dessert. Pies, pastries, a tiny chocolate or candy, sweetened veggies, and yes, of course cake can all constitute desserts, whether eaten at the beginning of a meal, middle, or end. 

In conclusion, my first grade student may imagine all she does is play games during math enrichment time, but this learning dessert is rich with problem-solving proteins, mental math nutrition, and healthy higher-order thinking! With the short amount of time I have with my students, I have to make my challenges tasty. And, I’m okay with that;)

Sources

BreezeMaxWeb. (2022). Why Is Dessert Important After Eating Food?. Casa Romana Sweets. https://casaromanasweets.com/why-is-dessert-important-after-eating-food/#:~:text=When%20you%20eat%20dessert%20after%20your%20meal%2C%20it%20signals%20to,moving%20after%20you%20eat%20it.  

Casey, B. J., Somerville, L. H., Gotlib, I. H., Ayduk, O., Franklin, N. T., Askren, M. K., Jonides, J., Berman, M. G., Wilson, N. L., Teslovich, T., Glover, G., Zayas, V., Mischel, W., & Shoda, Y. (2011). Behavioral and neural correlates of delay of gratification 40 years later. Proceedings of the National Academy of Sciences, 108(36), 14998–15003. https://www.pnas.org/doi/full/10.1073/pnas.1108561108 

Cherry, K. (2023, November 5). The Meaning of Delayed Gratification: Deferred Satisfaction and Its Rewards. Very Well MInd. https://www.verywellmind.com/delayed-gratification-why-wait-for-what-you-want-2795429 

Apples: Dental Hygiene Facts. Summit Dental Health. (2017). https://summitdentalhealth.net/apples-dental-hygiene-facts/  

Gershon, Li. (2019, August 21). The Invention of Dessert. JSTOR Daily. https://daily.jstor.org/the-invention-of-dessert/  

Jaramillo, S. (2023). 11 Reasons to Eat Dessert First. Peace and Nutrition. https://peaceandnutrition.com/11-reasons-to-eat-dessert-first/  

Miller, K. (2019, December 30). What Is Delayed Gratification? 5 Examples & Definition. Positive Psychology. https://positivepsychology.com/delayed-gratification/  Tebben, M. (2015). Seeing and Tasting: The Evolution of Dessert in French Gastronomy. Gastronomica, 15(2), 10–25. https://doi.org/10.1525/gfc.2015.15.2.10

Creating Word Problems: Math Enrichment

It’s October, and the second graders in my school are learning the basics of solving word problems. The arithmetic is super simple single-digit algorithms. Likewise, the stories couching the numbers are unadorned with character development, setting, or plot. Time for some #MathEnrichment!

My idea was for the 2nd grade gifted students to write their own word problems. Before creating our own, I thought I’d model some. I wanted to provide some math that was challenging, but doable, albeit with my help. As it turns out, my math riddles had the second graders perplexed to the point of paralysis.

It wasn’t just the numbers. Somehow, I’d forgotten what Ready Math had taught me! There is a specific way to read word problems. Even the smartest of the smart; kids who can wrestle with and make sense of the math; won’t be able to decipher what is being asked of them if they aren’t taught how to comprehend what is going on in a math story. It is like an Olympic runner being dropped into the middle of a forest and expected to sprint to the finish line that she can’t even see.

I constantly tell my students that challenges are fun. These kiddos weren’t buying it. Challenges are only fun when there is some hope.

Like an idiot, I dragged my students through my word problems, doggedly showing them what the numbers were doing. They were good sports. When I let them write on the Google Jamboard, they perked up. In the end, they left my room with number hurricanes storming their cerebrals. I was left to pick up the mess of math misconceptions strewn about the streets of seeming failure.

That was last Friday. Over the weekend I remembered; There is more to solving a word problem than crunching numbers! The Ready Math curriculum instructs teachers to have the students use a 4 step approach to solving word problems. When you break the process down this way, it is much more manageable.

  1. First, read through the word problem and decipher what the story is about. Don’t worry about the numbers. What is the topic? Are we talking about reading books or alien monsters that can control your actions with their minds?
  2. Next, identify the important information. What are the tools you’ll need to fix this problem? Don’t be fooled into thinking that unnecessary numbers or information will be needed. Sometimes there are superfluous facts stirred into tricky math stories to trip you up!
  3. Before doing any math, you must figure out what you are asked to find. (This step might come second. I can picture needing to know the end goal prior to identifying the important info.)
  4. Finally, we begin doing some math. Show all of your work/thinking.

There’s a fifth or bonus step that I told my 2nd grade gifted students about, that has to do with communication. Just like we include publishing in the “Writing Process” and the “Scientific Method,” we are not done solving our word problem until we share the answer. Make sure to label the numbers with whatever unit of measurement or name of thing you are talking about!

The answer of this word problem doesn’t even have any numbers in it! But, you better show and be prepared to explain your work for full credit.

Tuesday morning, the day after Indigenous People/Columbus Day, we were back at it with more challenging word problems. When I first showed my students this Jamboard, I had the “sticky notes” layered on top of the word problem. As we read and discussed what each one meant, I moved them to the sides and shrunk them, so that they all fit on the right for reference. Only then, did we read this word problem.

My 2nd graders kept yelling out numbers, like there was a contest for who could solve the problem first. It was humorous to tell them that every single number that they would say, no matter what it was, would be wrong. This was perfect for drawing their attention to step 3. “What are you asked to find?”

“Are you supposed to provide a number as an answer?” I asked my students. This got them thinking. And, even after solving how many candies each kid had, they still needed to compare the numbers in order to really finish the problem. It wasn’t enough to just know how many each had.

This is only the first of many word problems to come for our 2nd grade gifted students.

Once we had successfully solved our Trick or Treat word problem, it was time to make up our own. Before getting creative, we decided on the numbers and operation. We would have the mathematicians subtract seven from twenty.

I wanted to include everyone’s ideas. That is why the math story has flowers named after a student’s pet bunny, a main character named “Kid Pineapple, robots, laser blasts, and lots of the word “stinky” in it. Ha ha. We had a figurative blast coming up with our story.

We look forward to making up and writing down many more for our other 2nd grade friends to solve… The Ready Math way.

Persephone Play

When you mention Greek Mythology in a 3rd grade classroom, students get all excited. They have heard some of the famous names, like Zeus and Poseidon. They might be familiar with Mount Olympus and the idea of characters (gods) having powers or being responsible for natural phenomenon. Very few have ever heard any of the specific stories.

I recently shared a play with the Polite Pirates (my students) that I’d written a few years ago. Originally, I wrote it to be performed during a winter holidays celebration. It is the story of Persephone being kidnapped by Hades, causing her mother Demeter (goddess of the harvest) to get so depressed that she allowed the whole Earth to wither and die (Winter).

After reading it through once, students switch parts. It is okay for a girl to play Hades, the richest of the gods (he is governor of all the Earth’s minerals… gems) and a boy to read Demeter (goddess of the harvest), etc.

Before creating teams of four to read the play, I explained a little about Greek Drama. Rather than have a narrator, there is a “Chorus” that sings the narration. As you can well imagine, this was a silly experience. We read the entire play through once as a class. I had to count to 3 each time the Chorus was “singing” so that we were somewhat together.

One other thing that I incorporated into this drama that was very different was having two groups reading/performing at the same time. This is called dual dialogue. It can be difficult to pull off, but the text lends to rich teaching opportunities. Why would Hades and Zeus be talking at the same time as Persephone and Demeter? Where might they stand, if this was being performed on a stage?

I had to point out the comedy of some of the dialogue and stage directions. Even though you read the text within parenthesis to yourself, and not out loud, it is important to actually read it and imagine what is happening in order to get the point of the story. And then, I made the chorus use hyperbole in its declarations, along with some fourth wall breaking when Hades interacts with it. With these explanations, a little coaching and modeling, we had a lot of fun reading this play. I welcome you to use it (Google Doc of play) in your classroom. It’s a great end of the year activity.

The subject of being kidnapped and taken to the underworld, albeit very dark is made light with loads of comedic relief. Also, I incorporate a game of Monopoly, so that 9 year olds have more to connect with.

Cast of Characters:

CHORUS – the voices that narrate the play 

PERSEPHONE – Goddess of Spring, daughter of Demeter

DEMETER – Goddess of the Harvest, sister of Zeus & Hades

HADES – Ruler of the Underworld, brother of Zeus & Demeter

ZEUS – Leader of the Greek gods & ruler of Mount Olympus (also, Demeter and Hades’s brother)

CHORUS: (As if making an important declaration; introducing a dignitary) Who is more powerful than death? He comes to take us all. NO THING can escape him.

HADES: (Pretending to be humble) Well, you know… I am pretty powerful…

CHORUS: There isn’t a living thing that can withstand your power, oh mighty Hades!

HADES: (seems a little uncomfortable) Yeah, well, that is actually part of the problem.

CHORUS: Hades, ruler of the Underworld, keeper of secrets, all powerful bringer of…

HADES: (cutting CHORUS off, mid praise) Yes, yes, yes… that is all great, and everything, but keeping secrets is not all it’s cracked up to be. 

ZEUS: Are you complaining, brother, Hades? I thought you liked being all-powerful ruler of the Underworld.

HADES: Oh, hi Zeus. It’s not that I am complaining or anything. I’m just a little lonely down there, all by myself… me and my secrets… You know.

CHORUS: Hades, lonely, desperate, seeker of companion…

HADES: (interrupting what seems to be an increasingly negative appraisal of himself) No, no, no… It’s not like that. I’m not desperate or anything… 

ZEUS: (in a soothing voice) Do you need a friend, brother? I’ve been noticing you leave your Underworld unattended to visit Mount Olympus more often recently.

HADES: (kicks the ground sheepishly) I just see Demeter and her daughter having such a nice time together. All I ever hear are tree roots growing. The only beings I can talk with are the worms of the ground.

ZEUS: You signed up for that job. You wanted to be (exaggerate importance) “All Powerful”. I let you rule down there, even though I didn’t think it was a good idea. 

HADES: I know, I know. Okay, you’re going to make me say it: I wish I had a friend.

CHORUS: (jeering tone) Hades, wants a friend! Hades wants a friend! Hades wants a…

PERSEPHONE: (to Persephone) Did you hear that?

DEMETER: No, what?

PERSEPHONE: It was probably just the wind.

CHORUS: (very quietly, as if from far away) Hades, wants a friend! Hades wants a friend! Hades wants a…

(Here, two separate speaking parts happen at the same time. Take turns speaking, but make it obvious who you are talking to.)

HADES: Oh brother… 
ZEUS: Do you need a shoulder to cry on?
HADES: No, I need someone to share secrets with. They are no good all bottled up inside!
ZEUS: Are you asking me to send someone down to the Underworld to keep you company?
HADES: That’d be nice.
ZEUS: Who do you have in mind?		PERSEPHONE: There it is again.
DEMETER: I didn’t hear anything.
PERSEPHONE: I think I want to go check it out, mother.
DEMETER: I don’t know, Persephone. I have quite a bit of harvesting left to do here. I will be sad without you.
PERSEPHONE: I won’t be long. I just feel like someone is in need.
DEMETER: Don’t be gone long.

PERSEPHONE: (not seeing Hades who is walking away, backwards) Hello? Anybody there? …Boy it’s dark down here…

(Hades moves away from Persephone as she moves closer to him, until she bumps into him.)

HADES: Oh, hi there.

PERSEPHONE: Were you the one calling on the wind for a friend?

HADES: No, that was the Greek Chorus. They narrate Greek plays.

PERSEPHONE: So, you weren’t the one in need of a friend?

HADES: Well… (a little embarrassed) I’m not going to turn you away.

PERSEPHONE: What’s your name?

HADES: Oh, excuse me, where are my manners? My name is Hades.

PERSEPHONE: Aren’t you the ruler of the Underworld?
HADES: (a little shy) That isssss  technical my title…
PERSEPHONE: I bet you get lonely down here.
HADES: You could say that.
PERSEPHONE: Do you play board games?HADES: I’m bored all of the time!
PERSEPHONE: No, board games, like chess, checkers, Monopoly….
HADES: (excited) Let’s play Monopoly!
PERSEPHONE: That sounds fun. I’ll be the top hat. 
HADES: That’s classy…		DEMETER: (looking for her daughter) Persephone? Where are you?Where did you go?
Zeus, where is my daughter? 
ZEUS: What’s wrong, Demeter?
DEMETER: I’m looking for Persephone, and can’t find her.
ZEUS: (thinking to himself) Hmmm, I wonder if Hades brought Persephones to the Underworld to keep himself company.
DEMETER: The Underworld! WHAT?!
ZEUS: (sheepish) Well… Hades was feeling lonely.
DEMETER: (to everyone/threateningly) I am not going to let anything grow again, until Persephone returns to me.
ZEUS: (soothingly) I’ll talk to Hades.

CHORUS: And, Demeter, true to her word, neglected all plants and animals while she searched for her daughter; Field and forest dried up.

ZEUS: (observantly) This is not good.

CHORUS: Demeter looked everywhere for an entrance to the Underworld, by which she might fetch her daughter. 

ZEUS: (speaking to Hades) Um, Hades… excuse me.

HADES: (speaking to Persephone) That’s your third roll, and no doubles. 

PERSEPHONE: Ah, man! Now, I have to pay the $50 to get out of jail.

I’m hungry.

HADES: (speaking to Persephone) I’ll go get you something to eat. 

ZEUS: (speaking to Hades) Hades, we have a problem.

HADES: (notices Zeus) Oh, hi Zeus! Persephone and I are enjoying a game of Monopoly. Do you want to join us?

ZEUS: I’d love to, but I can’t. Wait, how long have you been playing this?

HADES: Oh, I don’t know. Maybe three months or so. 

ZEUS: That’s a long game! No, I can’t join you; Demeter is so upset at Persephone’s disappearance that she has refused to allow the earth to rain or grow or live this whole time.

HADES: That doesn’t sound good. I thought the Underworld was getting more action than usual. 

ZEUS: I’m sorry, but Persephone needs to go back to her mom.

CHORUS: Hades agrees to send Persephone back to her mom, but first gives her 3 pomegranate seeds.

PERSEPHONE: Thanks for the nachos, Hades. They were a little crunchy, but yummy. 

CHORUS: The pomegranate seeds caused Persephone to remember her Monopoly game with Hades.

HADES: Come play Monopoly with me again, Persephone.

CHORUS: Demeter was overjoyed when she saw Persephone… So happy, in fact, that not only did all of the plants grow back, but they grew beautiful flowers.

DEMETER: I am so happy to see you!

CHORUS: Now, every year, Persephone visits Hade in the Underworld during the winter months. Her absence causes Demeter to get depressed…

DEMETER: I get so depressed.

ZEUS: This causes everything to die. 

PERSEPHONE: The world goes cold.

HADES: We play Monopoly for three months every year.

DEMETER: When my daughter returns to me it is like she springs out of the earth.

CHORUS: This is how the ancient Greeks explained the seasons.

May the 4th be with You: Symmetrical Starships

Today was May 4th… No, it was May THE Fourth! As in, “May the fourth be with you…” This meant the Polite Pirates (my students) simply had to do some Starwars-themed activities.

In preparation for a fun puzzle project, I taught the Polite Pirates about symmetry. I told them that they would be drawing spaceships. They were very excited.

“There is a reason the wings of an airplane are symmetrical,” I began. “Have you ever seen this trick?” I asked my class, taking a piece of paper and placing it just below my mouth. 

As I blew hard right over the top of the paper, the entire paper flew up and fluttered under my breath. “Oooohs” hummed across the room. 

I drew a picture of the paper on the board and explained, “When air travels quickly across the top of the paper, you create a vacuum (Physics Force, 2023). The fast moving air pulls the air that is resting in front of the paper up to join it. With no air in front of the paper, it gets sucked up into that empty space. 

Without any good pics of my teaching from earlier, I reintroduced the lesson right before dismissal to snap this photo of amazement.

“Your household vacuum is a machine that creates a vacuum in order to suck up dirt and dust,” I tell my students. “A motor spins a fan that pushes air out of the machine. When the machine is empty of air, that is when it is truly a vacuum.” Mouths make Os of understanding. 

I drew a crude picture of an airplane wing. “When air flows under the wing, it is going straight. That part is flat.” I draw arrows of air moving straight. “The front of the wing curves up and back. Air has to bend and is angled up when it travels over the wing. This causes the air that is resting above the wing to travel up also, creating a vacuum, or low pressure. As the air goes up, the wing goes up (Shaw, 2021).”

I picked the piece of paper back up, and once again demonstrated air pressure pulling the paper up. Of course, they all wanted to give this a try! After a minute or two of varying successes, I drew an entire airplane on the board.

This Polite Pirate planned out his space craft on paper before arranging polygons on paper.

“If one wing were smaller than the other, the plane would raise unevenly,” I explained. “The aircraft would start spinning. It is important for the wings to be symmetrical, so that the airplane remains balanced.” 

[All of this explaining took longer than I had planned, but it was worth it. The students were really into the science of “lift,” and it lended extra importance to the idea of symmetry.]

After attracting the Polite Pirates attention once again, I explained, “You are going to engineer your own starship space cruiser,” I whispered in a serious, hushed voice. “The main quality that your creation must have is symmetry. Whatever you design on one side must exist on the opposite side of your ship.”

Tracing the shapes with pencil proved tricky.

Each student received one large piece of paper. Students could work in pairs if they liked. Bowls full of plastic polygons were placed in the middle of groups of desks. The Polite Pirates took several polygons each and began assembling space ships. 

I had told them and wrote on the board; At least 8 polygons per ship, and at least 3 different shapes must be used. 

Students were to hold down the plastic shapes with one hand, and carefully trace an outline of the perimeter with pencil. Then they removed all of the polygons and went over the pencil with a marker. 

“You will trade your picture with a partner and try to recreate each others’ space ships, arranging the colorful plastic polygons on the papers,” I explained. 

Students took photos of their creations, so that they had answers to their puzzles.

Creativity swept the classroom! Colorful space cruisers covered desks and flowed across the floor. 

This Polite “Padawan” Pirate even drew an insert to show the movement of her spaceship.

A pair of students asked me if their shape was symmetrical when they had to add a blue triangle to a row of parallelograms in order to make a long thin trapezoid. I told them to ignore the color. “Can you cut it in half and fold the one side on top of the other hiding it completely?” When they saw that they could, they realized the symmetry of the shape. 

The one thing that I’d do differently is limit the number of shapes. While my students had a blast using tons of polygons to make gigantic space ships, the answers to puzzles were too open-ended. A student could use a hundred combinations of shapes to fill a large empty space. The better puzzles were the ones that had a set answer.

It would have been better for space ships to have only a few right answers. Then the refilling of the perimeter with colorful polygons would require more concrete problem-solving.

All in all, this ended up being a memorable, fun learning experience. The Force was very strong with all of the Polite Pirates today.

Sources:

Physics Force. University of Minnesota . (2023). Retrieved May 4, 2023, from https://physicsforce.umn.edu/content/paper-lift-0

Shaw, R. J. (Ed.). (2021). Dynamics of Flight: Kids Page. NASA. Retrieved May 4, 2023, from https://www.grc.nasa.gov/www/k-12/UEET/StudentSite/dynamicsofflight.html#:~:text=Airplane%20wings%20are%20shaped%20to,wing%20up%20into%20the%20air.

Deep Teaching with Story Telling

“All of that, just to teach this little lesson?”

How deep does your teaching go? One way to help students understand a concept and remember the lesson is to share a story. On Thursday I was teaching an idea about fractions that was difficult to grasp. We had been working on understanding this concept all week. I had drawn models on the board and number lines on students’ papers. A few simply were not getting it. I was at a loss. 

The lesson had to do with the fact that whenever the numerator is the same as the denominator, a fraction is equal to one whole. It doesn’t matter if it is 365 over 365 or 5/5, they both equal one whole. How? While I could stop at providing the rule, I like to explain the “why” of math. The following story ensued. 

This is my dad, well past his book-binding days.

When I was in high school, I had the worst job! (This got everyone’s attention.) At least it was my least favorite job. I worked in a factory. What we did was kind of cool. This factory bound books. It was a book bindery. My dad worked there. He was a manager, so he was in charge of a team of people who operated different machines. He got the jobs, planned out how to complete them, gave people orders, made sure things ran smoothly, fixed machines, and was responsible for shipping out completed jobs to happy customers. 

The book bindery had two parts. My dad’s part worked on orders of new books. The other part of the business would repair or re-bind old books. This part would get a shipment of books from university and school libraries in the summer. Workers would use a huge cutting machine to slice the spine of a book away. Then the front and back cover would be removed. A new cover would be made and glued onto the old pages that had been either glued or sometimes sewn together. The new cover would get stamped with the name of the book, author, and publisher. There was a different gigantic machine for each part of this process. This is where I worked for a summer right after graduating from high school. 

846-06111912 © ClassicStock / Masterfile Model Release: Yes Property Release: No 1930s ELDERLY FACTORY WORKER IN OVERALLS SMOKING PIPE PUNCHING TIME CARD

What was so bad about it? Sounds kind of neat, right? There was NO freedom. You had to “clock in” at 8 AM, sharp. It was best to do it a couple of minutes before eight. If you were late, you’d get a “talking to.” A manager would come by and tell you that you had clocked in late too many times. One more and you were gone; You’d be fired. The manager might allow you to explain yourself, but there was no empathy. The clock was unforgiving, and you need to be on time. “Clocking in” meant getting your stiff paper card from a metal sleeve holding many cards, pushing it through a slot on the top of a metal machine displaying the time on the front. You’d push your card down until a cha-gump was heard. At the same time the sound happened, you would feel something grab your card. When you pulled it out, a time would be stamped on it. 

One more thing. It wasn’t a good idea to stamp your card more than a couple of minutes early, unless you had permission. The owners of the bindery did not want to pay anyone more than they had to, and if your card had any extra time on it, they would be responsible to give you money for that time. 

A book with the cover removed, you can see the stitching.

After clocking in, I would get to my workstation to pick up where I left off the day before. I was lucky. Whether it was because my dad worked in the other part of the bindery or I was good at it, I don’t know, but I got to operate “The Blade.” This was the gigantic cutting machine that sliced the spines off of old books. I also used it to trim the edges of pages, so that they were clean and straight for rebinding.

In order to work the machine, you would place the book onto a metal surface and push it against the back wall with the spine facing you. I would adjust the depth of the cut by turning a knob to move the book closer or farther away. When I had it just right, I would press a petal down with my foot. This lowered a metal wall that clamped the book down and held it in place. At this point I could see whether I had lined up the book just right or not. I might need to fine-tune the settings before making my cut. [I had to be careful, because if I cut off too much, the book will not have enough space on the inside of the pages for anyone to read it. If that happened, I’d just wasted an old book. You would get into big trouble if that were to happen. There are no do overs! If you didn’t cut enough off, then you could do it again, but you are wasting valuable time.] With the book held tight by the big clamp, I would push two buttons on either side of the front of the machine with the thumb of each hand, and a giant guillotine of a blade would swoosh down right in front of the clamp, slicing the spine away from the book.

Why the TWO thumb buttons? Let’s say you wanted to push a book against the back with one hand while slicing the binding away. You could accidentally cut every finger off of your hand with one fatal swish of that blade! Forcing you to use both hands at the same time ensures zero accidents. 

Unfortunately, other machines had work-arounds; ways to bypass the safety procedures; and even the cutting machine could be fooled. You could tape a piece of cardboard over one button, tricking the machine into thinking that one of your thumbs was pushing it in. Basically, there were opportunities to become seriously injured on the job. While management might give someone a hard time for doing something dangerous, they would also heap tons of pressure on everyone to achieve inhuman amounts of productivity. You constantly felt like you weren’t getting enough done fast enough. If only there was a way to quicken what you were doing. What if you eliminated one of the timely safety precautions…?

At 10AM a bell would ring, and we could take a fifteen-minute break. The workers would pile into a break room in the middle of the factory where picnic (ironic name) tables were set up. We did this even if it was a gorgeous summer day. It was probably better to NOT know how beautiful it was outside! At the tables we ate snacks, drank coffee, and chatted. There wasn’t any talking on the factory floor; chatting would slow down production, so this was a time for finding out what coworkers had done the night before. We read newspapers to find out what was happening in the world outside of the book bindery. 

10:15AM did not find us exiting the break room. We had better already be out of there and at our stations when the next bell rang. Anyone found lingering would get a talking to. 

Another bell would ring at noon. [While I was sharing all of this information with my students, I was drawing a timeline of my prison-like day on the dry-erase board.] You would go back to the break room. You better have a packed lunch, because there’s no running to McD’s, even though it was only 2 miles away. I tried it once and the stress of making it back to work on time equaled more indigestion than it was worth!

The factory floor was cement, which caused your feet to hurt. I piled up cardboard boxes to stand on, and I wore sneakers with thick soles, but once your feet hurt, there was no fixing them. Additionally, even though there wasn’t much of a dress code, we did have to wear pants. It was summer time when I worked at the book bindery. Even though the place had air conditioning, the owners were constantly adjusting the temperature to use as little power as possible. It always seemed a little too hot. 

It was difficult to avoid counting the hours between breaks and leading up to quitting time. When would the final bell ring?

Now, we get to the math of the story. I look to my students for an answer. Who can help me get out of here? What time did the final bell ring? I began my eight-hour day at 8AM. One student guessed, “6PM.”

“Was that a guess, or did you work that out?” I query. “Come on. Think. Eight to Noon would be how many hours?”

Another student half-guessed, “5PM.”

“You’re getting closer,” I hint.

You can probably imagine that someone figured out that eight hours would take me from 8AM to 4PM, but we have to account for the half hour of lunch. The owners aren’t going to pay me to eat. That was my time. The 15 minute coffee break in the morning was included in my work day. (Probably, this was meant to fuel productivity with a caffeine jolt, not to mention relieve the tension of not being able to talk all morning.)

Let’s say a guy has a medical condition that requires him to drink some medicine on the hour, every hour. If he takes a drink each hour of an eight-hour work day, his day is split into 8 parts; 8 hours = 8 parts. He doesn’t miss a dose, so he has had 8/8 drinks per day. The 8/8 is one day. 

I didn’t need to stop and take a drink. I only stopped during the allocated break times. With only two breaks, my day was split up into three parts. I work all three parts, so I work three out of my three parts. My 3/3 day is only one day, also. My friend and I both work a full eight-hour day, but mine is simply divided up differently. 

8/8 = 1

3/3 = 1

8/8 = 3/3

I’ve drawn timelines (number lines) on the board. It is easy to see that the same amount of time is broken up differently. 

If my friend keeps his medicinal drink in a 32 ounce water bottle, and his dosage is one ounce per hour, how many days will he be able to use the bottle before he has to refill it? 

How many doses will one day be? With each day broken up into eight equal parts (hours), the denominator will be 8. The total number of ounces (32) will be the numerator. The improper fraction will look like this: 32/8. Is there enough medicine for more than one day? A lot more. How much more?

How many eights go into 32? Or, how many eight-hour days can the 32 ounces be stretched over? You could do repeated subtraction. The water bottle will last four days. 

This Polite Pirate would be celebrating her birthday over the weekend. She got all dressed up on Friday to celebrate. I told her that she looked like a “boss.”

Before leaving this story and transitioning to a different lesson, my students had to know why on earth ANYONE would work at such a horrible place. I told them that there are pros and cons to nearly everything. First of all, this could very well be the only job that some of the workers could get. The book bindery employed many people who did not speak English. Once they were trained on how to operate a machine, they could do their job efficiently, and it didn’t matter that they could not communicate via the same language as the owners. Occupations that require more communicating might require people to know English. 

But, I knew English, so why did I work there? It was a summer job, and although the hourly wage was not very high, it was the only place that offered a full 40-hour work week to a temporary employee. Other businesses weren’t interested in investing training in a worker who would only be there for a few weeks. Also, if I worked over 40 hours, which the boss wasn’t too keen about, I’d get “time and a half.” Ooooh, I feel more math coming on… Groans. 

Many businesses only offer benefits, which include health insurance, to “full-time” employees. This title belongs to people who contract or agree to work a 40-hour week. Sometimes it is worth working a less attractive job, so that you can keep your family safe by having health insurance. This is the American way. 

I’ve told the Polite Pirates (my students) all about running my own painting business before becoming a teacher; There’s lots of math in those conversations! At this point in my explanation I point out that while running your own business means you are the boss, and you have freedom, it is a lot of work! Had the stress of making sure that I had future painting projects to do, because if my work dried up… Then what? There’s no money coming in! So, I had to do a lot of marketing, and that costs money. Then I had the pressure of finishing projects on time. Sometimes I had to work more than 40 hours in a week. Because I set prices with customers before beginning the projects, I didn’t make any extra money if I worked longer hours! And, what if I priced it badly? What if I thought that a job would be lucrative if I charged 300 dollars, only to find out that the product needed to complete the job would cost me $250? Don’t even get me started on spilled paint…! How much of that profit would be left if I had to buy a customer a new carpet? 

Working at a factory is, believe it or not, liberating from the stress of all of that responsibility. You punch in your time clock, put in your hours, punch out, and leave all of the thoughts of work at work. What you didn’t complete will be waiting for you to finish tomorrow. Let the manager stress out about how a job is going to get done. 

Finally, although this all sounds kind of awful, I am glad that I had the experience of working in the book bindery. I learned all about how books are put together, which was interesting. But, more importantly, I got to see first-hand a type of life that I may not have known had I not worked there. 

As I rose from my chair to erase my notes from the board, my students understood that all of that was to teach a simple lesson on fractions. “All of that, just to teach us about whole numbers…?” a few students said in surprise. Yup.

Building Bridges: A Hands-On Math Lesson

This blog serves two purposes: First, I just shared a lesson with the Polite Pirates (my class) that went so well that I want to share it with everyone. And, second, due to its complexity, several students need additional clarification. I am hoping that by writing this down, I can make clear how the business of bridge-building works.

Yesterday morning I dug out the colorful, connectable, plastic blocks I’d stored away in a bin under the counter. I told my students that they would be building bridges. Cheers rang out. They were to work as a team to construct a way for a car to travel from one pile of dictionaries to another. The Polite Pirates cheered with joy.

“There’s a catch, however. You have to buy the blocks!” I exclaimed. The class groaned. “Each one costs ¼ of a dollar. (I’ve been teaching fractions and mixed numbers.)”

“How much is that?!” a few students grumbled.

“You know how much a quarter of a dollar is,” I accused. 

“Twenty-five cents?” a student clarified.

“Yes. You will work as a team of engineers. Use your Spiral Notebooks to keep track of the number of blocks. You’ll need to figure out the total cost of your bridge.

“Oh, one more thing: Your bridge is going to make you money. That’s right! Sure, it will cost you to build it, but once it is done, you can charge a toll for cars to use it. For every foot of bridge you are allowed to charge one dollar. 

Making money got their attention;)

This is the slightly confusing part (one of them, anyway): If about twenty vehicles travel across your bridge per hour, how long will it take to make (dramatic pause) one thousand dollars? 

Before setting them loose, I showed them that this problem was doable. (The looks on their faces were incredulous.) “Let’s say you build a 10 foot bridge. How much money can you charge to cross it?”

“Ten dollars.”

“Right, but that is $10 per car. If twenty cars travel across your bridge in one hour, how much money do you earn?”

Thinking… “Two hundred dollars,” a student offers. 

“Good; You are correct. Where did that number come from?” I prompt. I want the class to know how to do these calculations.

“I multiplied ten by twenty.”

“Right. If your bridge makes $200 every hour, how long will it take to get to a thousand dollars?” Their minds were working, now! 

“Five!” several students shouted in unison. 

“Don’t forget that you need to use some of that money to pay for the building blocks that you used to construct the bridge,” I remind them. And, they’re off! 

The Polite Pirates had a blast working together. I was impressed that they almost instantly formed the idea of making supports to hold up longer sections of bridge. In this way they could earn more toll money. Of course, they had to count the blocks that they used to hold up their bridge when tallying up the cost of building materials. I didn’t let them use anything other than the building blocks for construction. 

Once the bridges got to be several feet long, every single student on the team was needed to hold the bridge in place as more supports and lengths were added. I overheard one third grader explaining to his partners that even though it would cost more to use extra blocks, they would make more money from tolls, because they could make their bridge longer. I reinforced this idea by sharing, “Sometimes you have to spend money to make money, folks!”

After a timer I had set went off, I had everyone stop building and count up the number of blocks that they had used. One team split up the task by divvying up the sections to be counted. They then added all the numbers together. Next, it was time to calculate the cost of all of those blocks. One team had used 355 blocks! How could they figure out 25￠ per block? 

I reminded them that they already knew what a quarter of 100 was. “Separate the 300 from the 55,” I told them. “Each 100 would be how much?”

“Twenty-five dollars,” someone answered. 

“That’s right. Now, how many twenty-fives do you have? We’re talking about 300 blocks.” I wrote 100÷4=25 on the board. When someone suggested that they needed 3 twenty-fives, I put X3 under the 25. “These might seem like really big numbers, but you already know what ‘three-quarters of a dollar is,” I prompt.

“Seventy-five cents!” a few blurt out. 

“Not cents, though…” I can see the gears turning behind my students’ eyes. It feels like I can hear the steam coming from their ears. They even gasp with understanding. 

“Seventy-five dollars.”

“Now, for the 55 other blocks. Is there a number close to 55 that is divisible by four,” sounded like Greek to them. I reworded my question in a more leading way. “Can 48 be evenly divided by 4?” This connected with their math facts. A student raised his hand.

After deciding that 48 blocks would cost $12, we tackled the leftovers. “What’s left?” I asked the Polite Pirates who were sitting so patiently on the carpet in the front of my classroom. Counting up from 48 to 55, we discovered there were seven blocks left. “How can we figure out the cost of these?”  

When this question was met with blank stares, I quickly drew seven (very ugly) squares on the dry erase board. I drew a hasty circle around the first four. “Each of these cost 25￠. How much money is four quarters?” Lights blinked on in every students’ eyes. 

“One dollar…!”

 “And…” I prompted for the cost of the remaining three unaccounted for blocks. 

“A dollar and 75￠,” a student finished the thought.  

“So, these seven cost $1.75, the 48 blocks cost $12, and the 300 blocks cost $25 X 3,” I summed up our calculations thus far. “What will this team need to do next? Tell the person next to you.” Then I sent everyone back to their seats to work out the costs of their bridges. I told them to figure out the prices independently. Then compare your work with your teammates. In this way you can double-check your math accuracy. 

I walked around the room, helping students with their division facts. There was a wonderful hum of productive struggle. Some individuals figured out that they would have to add more than one additional hour on to their original answer in order to account for the cost of the bridge. Most found that only one hour would do the trick. 

Now that all of the math was done, it was time to write about it. I had posted a question in the Polite Pirates’ Google classroom: “How long will it take for your engineering firm to make one thousand dollars?”

They could work with partners and discuss their writing with their team, but each student was responsible for producing their own explanation of what they did. I left the math that I’d shown them on the board, so they could copy it into their notes, or just write about what we figured out together. 

One of the things I like most about using Google classroom on iPads is how easy it is to use the “Speak to Text” feature. It’s true, you have to teach and practice rereading and editing your text, in order for students to use this effectively, but it speeds up typing entire paragraphs. It also helps facilitate a more “Discourse” -style text. When my students purely type, they are less likely to include opening statements, and they will leave out key details. Through the process of “Telling” their iPads what they did, you get a more structured description. And, because it is easy and fast to do, students don’t have a problem including more details. 

Here are just a few samples of the amazing answers that the Polite Pirates typed into their Google classroom assignment.

I hope you enjoyed hearing/reading about this lesson that my students experienced this week. If you have ideas on ways to improve or modify it, let me know in the comments. Thanks, and take care.