A blog by a veteran math teacher who tries to learn new things. Most posts are about teaching math and problem-solving, but sometimes, it's just about life.
I recently discovered that Popular Mechanics, a magazine I remember hearing about from when I was a kid, has a riddle of the week, given by Jay Bennett. I used to love Click and Clack's riddle of the week from Car Talk (anyone remember Dewey Cheetham and Howe? I am dating myself again!), so I was pretty excited when I found Jay's puzzles.
I decided that we would dedicate a day to going through the problems in my Honors Problem Solving Seminar. It was pretty cool to see that we had already solved a majority of them, and even cooler that there were a plethora of ones that we haven't solved yet.
When we got to "The Airplane Problem," I saw it was deemed "hard."
I remembered when this came out as a Ted-Ed Riddle (see video at the top), I heard it was one of the hardest riddles. So I never watched it.
It's interesting, considering that on day 1 of Problem-Solving, we talk so much about grit and struggling to work through a problem, and yet, I really did not want to tackle the airplane problem.
There are definitely times when we can't solve a problem in class, and we have to look it up. I think it's good for students to see that the teacher can't solve every problem, and sometimes there is a twist that we just never would have thought of.
Well, my students dug into the problem. We talked it through for a while and after about 15 minutes, I said, are we ready for the answer?
K, a great thinker, said, "I need more time." She was on to something. OK, I said and started shuffling my papers, cleaning my desk, graded a question or two...I was done. I couldn't do it, so I stopped. A few times, I stopped the students, feeling bad that the problem was deemed so hard and maybe they were 'wasting' time, thinking maybe it was one of those impossible ones that you just don't totally understand even after watching the video, i.e., I'm still lost on the three gods riddle.
I said, "Should we watch the answer?" K said, "Wait, I need more time." OK, I said and really began to look at her work. Within a few minutes, she stepped back, admired her work, said, "I think I finished", and proceeded to explain step-by-step, how to solve the problem that I gave up on.
SPOILER ALERT! Here is her work:
As we watched the answer on the video, K nodded at each step, as it matched her solution. I thought, wow, K just taught me a thing or two about grit today. K put the "possible" in "impossible" for me.
I am so proud of K's accomplishments in this class. It's great to learn things from your students, both knowledge, and perseverance!
Tomorrow is the first day of school! I am teaching Honors Problem-Solving Seminar (see this for the first unit of the textbook I created, problem set 1, problem set 2, and the blog post by my students from last year.), Pre-Calculus AB (our newly named accelerated Pre-Calc, for kids moving on to AP Calculus AB next year), and Algebra 2 Honors. I have four classes, and we are lucky enough to have a Math Lab staffed one period a day by a math teacher. So that's my 5th period in a 7-period block schedule.
I was going to blog AFTER I got my thoughts together about what I am doing this week (we see all students Wednesday for 30 minutes, and then we see our first set of blocks on Thursday and our second set on Friday), but I realize I need this blog to gather my thoughts together RIGHT NOW. :)
Wednesday:
I have am using Apple TV in lieu of a SmartBoard, so I am using an iPad with Notability as a teaching tool. It's a little confusing using a laptop for writing lessons and an iPad for presenting, but I am getting the hang of it. We are a 1-1 school with laptops, so students do not have iPads.
I loved Mathy Cathy's blog about her first day, and think her use of Tackk is incredible. Since I am just getting familiar with Notability, I decided to use that but make something similar. I love the website https://www.canva.com/ and made this to display for students coming in:
Next, I am going to show students part of their homework, which is Who I Am...who can I give credit to for this? Mathy Cathy is also is doing this, so I felt good that I was doing the same. I knew my students would be confused about the self-portrait, so I like her addition in blue, but don't print that one out for the students!
Next, I am going to go over my syllabus. I know I've blogged about them before, but I can't remember where. I also am not sure who to give credit to here, so let me know if this was your idea! The "live" version has links to their text help and classroom google.
I know it's a lot of stuff to cover in 30 minutes! I am not going over the entire syllabus in one day.
We gave summer packets in the math department this year for the first time, so then I plan on showing students the second part of their homework, which is to fill out this google form on the questions they did not understand the most. I can look at the pie chart before class and see what I really need to go over in class. I already have one response and class didn't even start yet! I love Honors!
I am going to show them my extra help times so there is no question--I also have this posted on both of my doors.
It seems really cool and I'm excited to do it. You prepare a sheet of paper cut and folded in such a way that it looks easy to create, but it actually is not. You tell students they can walk around and look at it, but can't touch it and you give them paper and scissors and tell them they must recreate it, but they only get one shot. At the same time, you walk around and write down the things they are saying, like, "I can't do this!" "This is impossible!" etc. You write what they are saying on stickies or on the board and then eventually stop them and ask them how they felt.
Tom writes, "I then refer back to the language used during the activity and how the language we tell ourselves quickly becomes our own best friend or worst enemy. If students can start to reflect on the language they use about themselves at a (sic) young and choose a better path it is a fair bet their lives will be productive and fulfilling."
I know I will be short on time, so I will probably follow up with it in my classes the next day.
Thursday/Friday (blocks so only meet class once):
In Problem-Solving, I first need to share how cool my workbook cover is, in the shape of our crest and designed by a student who graduated last year.
I am going to start with the "National Math Salute," which I saw at FAU Math Day last year,
and its reveal...
I think they will love this because it seems so easy but it's almost impossible :)
Then we will do the topics on my first unit: 1-5-4-2-3 card game and Angela Duckworth's talk on grit that I posted about here. They will also get their first problem set assignment, due back in a week.
In Pre-Calculus, based on the questions that they check marked, I will go over whatever topics they need, and will have them go up to the board to practice problems. This is all we will do because we have only a 60-minute special schedule today, and they have a quiz on their summer packet Monday, and I know I may need to go over the activity and/or finish the syllabus.
In Algebra 2 Honors, I will do the same thing as in Pre-Calculus, but we have 30 more minutes. So I am going to do the 1-100 numbers activity from Sara V. I have seen this a lot, most recently from Sarah Carter, and because she said we have to do it, I'm gonna do it :)
So that's it for the week...whew! It seems like a lot, but I'm going to give it a try. Better to be overprepared than underprepared :)
Looking for an algorithm for how to solve the Rubik's Cube? It's here. However, it's not a "quick learn." But it's a gritty learn. You will need the tenacity to do it all, but you got this! If I can do it, I promise you, you can too. My suggestion: do it in parts, and practice that part over and over again until it becomes ingrained in you. Then move on to the next part.
These directions have been tested out by students...good luck! I hope they work well for you! And if you have any suggestions or struggles, please email me at lisa.winer@saintandrews.net and I will try to get back to you in a timely fashion!
I have blogged about how I used the Rubik's Cube in my problem-solving class here and created a mosaic here. I have gotten great help from http://www.youcandothecube.com/cube-mosaics/ and Dan Van der Vieren, and the video also displayed at the bottom of the page (not mine). My work is a combination of the video and Dan's help. This is NOT to help you solve the Rubik's cube super fast, but it is a way for you to solve it in anywhere around 3 minutes. Any suggestions are welcome. Here is a direct link to download the directions, but I also made it visual below, page by page. Any changes will be on the direct link, though, since the pictures are static. Do you have the grit it takes to solve it? If you made it this far, then of course you do! Good luck!
-------------------------------------------------------------------------------------------------------------------------How to solve the Rubik’s Cube!!
Top Layer
Start with the yellow center and form a white cross around it
Rotate so that middle of a side cube matches the color coming off of a white cube
Rotate that face 180o so that the white part of the cross is now touching the white center.
Repeat steps two and three until you have the white cross with a white center. Note that all of the sides have center cubes that match the white cross outer color.
Play around to get the white corners in the right spot. The 4 white corners that you need to get in the right spot must be in the bottom row. You are in a good spot if the white cube is directly under the spot where you want it on the top layer (facing you.) Turn the bottom row twice in either direction to move the corner out of the way, turn the right side down, move the bottom twice in the opposite direction to get the white corner in place, and turn the right side up. If instead, the white corner was facing the right side and not you, turn the bottom once counterclockwise, turn the left side down, bring the bottom back clockwise, and turn the left side back up. Finally, if the white part of a corner cube is facing the bottom, you need to get it so that it is on the bottom row rather than the bottom itself. To do this, turn the left or right side down (where ever the corner misoriented), twist the bottom clockwise twice to move it out of the way, and then turn the left or right side (whichever you moved first) back up. Now the white face should be in the bottom row, and you can repeat the steps from the beginning of step 5.
Second Layer
Turn cube upside down and hold the cube so that you have a left side, a right side, and a top.
Find a cube in the left or right face (top middle) that needs to be moved to the middle layer of the corner column. You may need to rotate the top so that the color lines up with the middle cube.
In my case, I am moving the top middle cube to the right. So you will use this sequence: TRtrtlTL, where capital letters rotate clockwise and lowercase letters rotate counterclockwise.
If instead, you want the top middle cube to move to the left, you use tlTLTRtr
In some cases, you will not have a cube to move from the top, but will have to “pop out” one that is in the wrong place. Just repeat one of the sequences above to move a cube into the spot and then the one in the wrong spot will be out.
Keep repeating this by turning the cube and ensuring all of the top middles are moved…you will know this is accomplished when the bottom two layers are complete or when all of the top middles have a yellow in them.
Keep the cube upside down.
You can memorize the sequence by noting that the 3rd and 4th moves undo the 1st and 2nd, and same for the 7th and 8th undoing the 5th and 6th. Doing this over and over again will result in your brain “knowing what to do.”
Third Layer (4 stages)
Stage one: Getting the yellow cross on top. If you already have this, skip stage 1.
There are several possibilities that you can have. Do this move until you get the yellow cross, but be careful to hold it the correct way: LRTrtl
-a middle yellow and no other yellow touching it (hold any way)
-a yellow V, which you must hold the way I am (left side is to the left, right side is to the right)
-a row including the middle (it doesn’t matter if there are more yellows in the corner). You must hold it the way I am. (Same as above)
-a yellow cross, which is the ending position…it doesn’t matter if there are yellow corners or not.
Stage two: Houses
This is a house.
You can have a house in two ways…across from each other or adjacent. The goal is to get 4 houses, so if you already have 4 houses, skip stage 2.
-You must have one house on the right, like below. You can twist the top layer to ensure you find your two houses.
-From here on out, you will have the cube face you (not its corner). THIS IS A NEW WAY TO HOLD THE CUBE...we no longer talk about counterclockwise and clockwise. Hold it like this:
Stage three:
Goal: to get corners in the right spot (not necessarily the correct orientation). (Sometimes this step can be skipped if all corner cubes are in the correct place but not the correct orientation)
The picture below has a corner NOT in the right spot.
-To start, the top right corner has to have the correct cube; it can be incorrect or correct orientation. (IF YOU DON’T HAVE A CORRECT CUBE: If none are in the right place, proceed with the algorithm below, and then after the first time, one should be in the right place.)
Stage four:
All corner cubes are in the correct place but not the correct orientation. For example, the corner shown is in the correct place (corner of red and blue and yellow), but it is not oriented right. This stage will put all the corners in their proper location, just not necessarily oriented.
-Start with a cube that has an incorrect orientation and face that side toward you.
-The mantra is “Down, Left, Up, Right” meaning Right side down, bottom turn left, Right side up, bottom turn right. Do this until the cube on top is in place. Usually takes 4 times.
-The top corner should be in the correct place now and may look like this:
It’s OK that the white cubes are where they are! Just be sure to have finished the mantra (meaning, don’t forget to do the “over” after the corner cube is in place.)
-Now rotate the top over until a new corner cube is in the old corner place…this will likely be over a white color cube. Here is my new cube over the two white cubes from before.
-Repeat “Down, Right, Up, Over” two to four times, and the upper right corner should be in correct orientation now.
-Rotate the top over again until the new corner cube that is not the correct orientation is in the same right corner spot again, and repeat until you solved the cube!
Today my Problem Solving Class unpacked the 225 Rubik's Cubes shipped from http://www.youcandothecube.com/cube-mosaics/ and began the task of creating our school crest. Last week, a student used Photoshop to turn a picture of the crest into a mosaic, and after spending time adjusting some of the squares that did not turn out quite right, we were ready! I asked Diane from the company for a bit of help, and she immediately responded and helped me to figure out what to do. My class and I divided the mosaic into 25 sets of 9 cubes. Each student (9 altogether) were assigned 2 or 3 sets of the 9 cubes, and their task was to make the top face look like the one portion of the 9 cubes...so for example, if you remember the opening of the Brady Bunch, one student had to first figure out the face of "Marcia", then "Carol", and this continued till they did the whole group of 9...then they placed them on the butcher paper I had on the floor.
Here is how we divided up students, and we looked at the original just to make sure we fet good about the colors.
It was pretty amazing to get 225 Rubik's cubes all solved...it was quite satisfying see them all! But we had to mess them up to get them to match the picture.
We had no idea how long it would take. Our class is 1.5 hours long, and I definitely thought we would need at least two classes to complete our task. But about an hour in, I new we could do it...it started getting really exciting as we had only a handful of cubes left. We finished with about 10 minutes to spare!
We may do one more, depending if we can find a piece of student artwork that works well with the mosaic program. Next year, we will use 400 cubes to get an even more accurate design.
I think this project really promoted teamwork, collaboration, and grit. We talked about it for months, so finally doing it was very fulfilling for students.
I am working on writing up directions to solve the cube--it will not be directions to make you the fastest, but rather to make you remember how to solve it and show other students...I am collaborating with the teacher who skyped with me and taught my class and me how to solve a Rubik's Cube, Dan VDV. He was so patient and helpful, and I hope to do the same for the #MTBoS! I have written about how solving the cube helps to develop grit here.
It all started when my 17-year-old son came home from his summer teen tour and had some much needed down time. The normal teenager would probably just sleep or watch videos, but AJ decided he wanted to solve the Rubik's Cube. I remember playing with one as a child and could get one face complete pretty easily. But in "those days," the directions were all on a folded piece of paper (the horror!), and I did not have the GRIT needed back then...at least not with this puzzle. But he watched a video over and over again, and by the next day, he had it down. (As I am writing this, I hear the clicks from him solving his cube...it's a good "brain break" for him between homework assignments.)
Over same summer, I was planning for my new Problem Solving Seminar, and I thought I would make solving the Rubik's cube an assignment for the class around Thanksgiving. I thought it would be a good time for kids to practice...and perhaps get encouragement from their families over break. I decided I would have them watch the video my son learned from, and I would facilitate, but that I did not have to really solve it...after all, it was an assignment for them, and maybe I didn't really have to (gulp) solve it?! It felt really daunting to me, and yet, I knew my kids could do it. I just didn't necessarily want to--which I know does not make a whole lot of sense right now...but it somehow did to me then.
I asked the bookstore to stock Rubik's cubes...the only thing they needed to purchase for the course, and after avoiding lots of "when are we going to solve the Rubik's cube?" questions, we finally watched the above video together last week. I broke down the first few steps as such:
The white cross on top with yellow in the middle
The white cross flipped to the bottom with white in the middle and a partial matching T
The entire white face with one full layer (top) complete
The entire second layer complete
As we were watching the video and pausing A LOT, I noticed that many of the students were having trouble visualizing. Ironically, one of the top students in the class could not follow the directions at all at first. But most kids were still very interested...solving the Rubik's Cube is like a fun party trick to pull out-out of nowhere, so most were determined. Some asked me to share the video via Classroom Google, so they could watch at their own pace, which I did.
Oddly, I was able to see how to do the first three of the four steps, above, pretty easily. I say oddly because my spatial reasoning is my weak point as a math teacher. You could spin me around in my own driveway, and I will get lost. But I guess having played around with the cube a lot as a kid, I could do these steps fairly quickly. So once I realized my kids needed help, and I could help them, I started to want to solve it myself. But it was not until then that I felt the need to solve it. They needed my help in explaining it, and that I could do. But I didn't know how far I could get...that second layer blew my mind.
Then luck happened. I got an email from the Mathematical Association of America (MAA) highlighting an article about Dan Van der Vieren (known to students as Mr. VDV), a teacher who wrote his undergrad thesis on the Rubik's cube. I immediately followed him on twitter and asked if he was willing to skype with my class...and he said yes! And then the magic happened.
Via skype, and with pictures like the one to the left, he showed us how to get the entire second layer complete. This involved an algorithm. My students struggled through this (as did I), making mistakes and having to redo it all over again, but once they finally got it on their own the texts with pictures started pouring in...on a Saturday night?! And it coincided with me getting that layer complete as well. We felt so accomplished!
Mr. VDV, is skyping with us again on Wednesday (the day before Thanksgiving break) to get us to the next stage. This, more than anything else that I have ever taught, is teaching the kids tenacity and grit and stick-to-it-tiveness. I do have a student who wants to give up. I hope more than anything else, I can encourage him to stick with it and solve it. He will learn more from that, I think than anything he has learned in my class. If he learns how to do it, which will come not only from my helping him but also from his willingness to learn from his mistakes, I will feel like I have done my job.
Mr. VDV has tweeted with me regularly, sharing pictures like these to help me help my students--so incredible. I am so thankful to him...funny this is the case right around Thanksgiving.
Lastly, Mr. VDV has told me that his class has made mosaics with Rubik's Cubes, and now, of course, we HAVE to do this...next semester. I can't wait. He told me to register at http://www.youcandothecube.com/, which I did, and someone got back to me Sunday morning! They will ship all the cubes to you for the mosaic making; all we have to do is pay for shipping back.
I am looking forward to my post when we actually create this. Another challenge, Mr. VDV told me, is to make our school logo as a mosaic. There is an app for that! The possibilities are endless.
I am not there yet...I haven't solved the puzzle fully. I'm 2/3 done, but I can do the entire 2/3 from memory...by Thanksgiving break, I hope to have it fully done, along with the rest of the students in my class. I finally am learning about the grit I have been talking about to my class. And it feels great.
It's easier to stay "in the box." My cat, Tiki, thinks so at least. But I say, let's help students step out-of-the-box and take some risks. Let's teach them to learn some grit and begin to think critically and not just through rote memorization. Students will rise to the occasion when you expect them to.
I have caused a little bit of dissent in our department over the years...insisting on putting a few out-of-the-box questions on tests and quizzes. Some colleagues have firmly stood with me, saying a test should not just be a "worksheet," but others have been worried that their students would not be able to solve the problem(s) and would therefore do badly on the test.
Most of these colleagues that originally were skeptical of putting problems on a test that were not directly taught in class have come around to the idea of giving them problems that are not in the textbook. I used to say that this type of question "separates the A students from the B students," but that's not the reason I do it anymore. It's to try to get ALL students to think critically and not just regurgitate what I have given to them. It's more about teaching students to problem solve rather than to teach them how to memorize isolated facts and then just spit them back out. I have found that at first, students are nervous about these kinds of problems, but the payoff is great..they learn the power of grit, and they feel fantastic when they get these problems right. Therefore, my review sheets are often harder than the test, because I put several questions on them that cover the same topics, but are asked differently or require several ideas to be combined into one.
Today I came across the article Why our Smartest American Students Fail Math. It is a fantastic read, and I strongly recommend that you read it. Basically, Carol Lloyd, exec editor at http://www.greatschools.org/, writes that our top students who go to college for a STEM (Science, Technology, Engineering, or Math) tend to drop the major because they get terrible grades in these courses. They are not learning problem solving or perseverance in their high school courses and, as Richard Rusczyk, former Math Olympiad winner and founder of the awesome website artofproblemsolving.com stated, “These were kids who had never gotten anything but 95s and 100s on their tests and suddenly they were struggling and were getting 62s on tests and they decided they weren’t any good [at math].” Lloyd writes, "Indeed, traditional math curriculum is to teach discrete algorithms, a set of rules that elicit a correct answer, like how to do long division, say, or how to use the Pythagorean theorem. Then students 'learn' the material by doing a large quantity of similar problems." She continues that Rusczyk says the result is that "students are rarely asked to solve a problem they are not thoroughly familiar with. Instead, they come to think of math as a series of rules to be memorized. The trouble is kids don’t necessarily learn how to attack a new or different kind of equation."
This is what I have been saying, too. We cannot just give students a worksheet, have them memorize all of the topics, and then test on the same material. What does that teach them? I think it teaches them to remember things just for a test and not how to critically think in high school. Students need to fight through problems and talk through them with others. They need to hear all the different approaches others have to solving the problem. I spoke about grithere, and my problem solving students are learning about grit daily. I gave the following problem to them on a problem set:
I was shocked at how many students did not know how to do this. There are three Calculus students, three Pre-Calculus students and one Pre-Calculus regular student in the class, and they did not even know how to start it. Finally someone said, "Oh, it's like a composition of functions problem?" Then they got it. The next week, I gave them this problem on their problem set:
This time, they got it right away. But it is important to give different questions to ALL students, not just those in a Problem Solving course. They begin to recognize the type of problem that it represents, not the exact one on a worksheet.
My Mu Alpha Theta students, which I wrote about here, are exposed to new and different problems each week at our club meetings. I feel like the exposure to these problems is definitely making them better thinkers. Spending an hour on math that is not following the curriculum is SO COOL, and I would love to hear from others of you who are either an advisor to a math club or are just thinking about it...and if you are thinking about it but aren't sure, JUST DO IT! Not only is it great for the kids, but it is also great for you--it truly makes my "cup runneth over."
Attached are two review sheets I gave my Algebra 2 Honors classes that had some problems on it that were out of the ordinary, but were related to what we were learning.
These are from famat.org. I know the font is pretty old on the first worksheet. It's because I am using the oldest tests from the '90s in class, as we are using the most recent ones during Mu Alpha Theta practice. In math club, I have students work in pods on these worksheets, and it's fun to see them do this when they get the question right:
