Wednesday, May 20, 2020

Playful Mathematics Carnival 138

Welcome to the Playful Mathematics Carnival 138, the edition with its choice of theme song? (Techno or punk. I vote for the Misfits, I guess.)  

I was frankly a bit surprised at how interesting 138 is. 138 is a sphenic number(the product of three distinct primes from the Greek for "wedge shaped"), but is special among the sphenics for being the smallest product of 3 primes, such that in base 10, the third prime is a concatenation of the other two: (2)(3)(23). What is the next such special sphenic? (Some of the pictures here were made with some Brent Yorgey inspired GeoGebra or David Mrugala inspired GeoGebra.


138 is the sum of four consecutive primes (29 + 31 + 37 + 41); which is the previous and the next? 138 the sum of 2 successive primes; which? And not only is 138 the average of twin primes, it is a number such that 6 times 138, is the center of twin primes, 827 and 829. Is there another number like that?

I had never heard of Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms in exactly one way. 138 is an Ulam number... what are the two terms which make it? Which is the first counting number not an Ulam number?

Obviously, this month's edition is in the time of social distancing. So it has produced some stand out creative home mathematics. Nikky Case made an interactive explanation how the epidemiological models work. Eva Thanheiser wrote a post about numeracy in the pandemic time. Here's an applet if you're trying to model a gathering... or a classroom?

Denise Gaskins, the initiator of this here carnival here, has a post for those new to it on how to homeschool math and Peter Rowlett at the Aperiodical wrote about playful math at home

Whilst there you might check out the 2020 Lockdown Math-Off, with some really accessible entries this year. And you can submit until they run out of entries or they're allowed outside.


Lily Cole
Math Art
Evelyn Lamb collected some of my favoritest math away from school resources. Cited there, Annie Perkins' #mathartchallenge (Twitter) is maybe my favorite thing ever. Here's the home (blog) for it. Brings together so many amazing projects from so many amazing artists and mathers. Like, Paula Krieg's origami firework on Day 53. Several have compulsed me to try to make things, like the Hitomezahi stitching from Day 14. Annie's going until she hits the magical 100! 

Clarissa Grandi's #Maydala challenge has filled Twitter with lovely images.
Clarissa herself
Janet Annetts

Japleen Kaur











Miss Bowkett


Paula Krieg

More mathy math
Henri Picciotto put together two amazing posts looking at wallpaper symmetries, and a catalog of pattern block Wallpaper symmetries; part I and part II. (Challenging, I think, because the blocks themselves are so symmetric.)

Luca Moroni pointed out Raffaella Mulas' self-illustrated article on Wild Mathematics. Quick mind bending read or just ponder the pictures.

ImagePaula Krieg also pointed out this classic puzzle that Dr Olsen points out can be made from origami.

Karen Campe covers a wide variety of puzzles, from the jigsaw to the Catriona Shearer.

Simon Gregg started this thread about duck tiles, which continue to delight. He also published a sweet book on pattern blocks. If you just want the pdf, let them know and they'll donate the book to a local school.

Pat Bellew (of On This Day in Math) chases a white rabbit, running around My avorite Theorem and the arithmetic triangle. (I'm hoping that catches on over Pascal's someday.)

Justin Time
Not sure how to categorize this, ironically, but Justin Aion's post on alignment charts and teaching is amazing.



Games 
Denise Gaskins, the initiator of this here carnival here, has a War meta-post with variations from preK through HS. I use several of these in a variety of classes. 

Sam Shah collected many activities and games you can do over videochat with a class. Nia (@ihartnia) put together a Google doc of quarantine games: tinyurl.com/quarantinegamez 

Dan Finkel updated his Horseshoe Math game with a theme song!

Ben Orlin shared 6 intriguing pen and paper strategy games.


Carole Fullerton covers a silly game called Penguins. Kind of a computation bingo variant.

Kent Haines' kids are chips off the ol' block, having invented Math Ball.

My retired colleague Char Beckmann developed many math games with our students and they've all been collected here. Materials, instructions, and video demonstrations.

This past semester I got to supervise three preservice elementary teacher/math majors in their senior projects to make math games. Char Beckmann's project! Our emphasis was to make instructional games that you could play with materials on hand or with minimal printing. Sam Bosma made a fun Guess Who variant for multiplication and division, Multiply Who. Maggie Eisenga developed Choose Your Path, a hit the target game with playing cards with some very cool discrete elements. Grace Gay increased the math content in Quixx, a dice game with some subtle strategy, to make Rolie Polie Operation Olie. All are upper elementary-middle school, and have how-to-play videos.

In Memoriam 
Playful math lost two giants recently and I wanted to close with some remembrances.

John Conway was an all time great mathematician. Siobhan Roberts, his biographer, wrote two pieces for the New York Times, a memorial and a personal memoir. (Her book is excellent.) Sunil Singh said if you watch just one bit of Conway talking, watch this. James Propp is an unabashed groupie. Ivars Petersen shared a couple recollections. Pat Bellew remembered his impossible knot and Quanta covered its solution. Matt Baker remembered some of his lesser known results. I can't pick a favorite, but some of the most fun I've had with students is the rational tangle.


Don Steward was an amazingly creative and generous middle school maths teacher in England. Brilliant problem poser and visualizer and entirely low tech. Colleen Young wrote about some of her favorites. Jo Morgan wrote about her collaborations. Steven Cavadino shares an irrational triangle. Many people shared their favorite Don Steward task on Twitter. He did a sweet analysis of a Keith Richardson-Jones drawing that inspired some GeoGebra from me. Don's local paper wrote up a note on his importance to the community, and of course you should check the blog of the man himself, which he made arrangements to keep available and free.

On that somber note, we close the carnival.  Be safe, be kind! 

But on your way out... maybe check last month's Playful Math at Life Through a Mathematician’s Eyes or their recent post, 10 math movie recommendations, and look for next month's at Math Mama Writes, or check her post about Pythagorean Triples for an online math circle.  The carnival's homepage is at the site of Denise Gaskins, the initiator of this here carnival here. Contact her for your chance to host!



Rolie Polie Operation Olie, a Math Game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Grace was fascinated by several games, but especially Quixx, a dice game. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.

GUEST POST by Grace Gay

Rolie Polie Operation-Olie is a quick-playing mathematics game played with dice! It is a spin-off of the family dice game Quixx, in which it is a simple game to play but each decision is crucial. There is no downtime in between your turns, so there is a lot of catch-up as you always have a chance to gain from each and every player’s roll. 

If you have kids or students who are working with their basic operations of addition, subtraction, multiplication, and division, then this game is a fun way to test their computations! The object of the game is to score the most points by filling in as many boxes in the three columns as possible while avoiding penalty points. Each player takes a score sheet and something to write with. Before playing, there is one basic rule that is similar to Quixx, which is boxes must be filled out from top to bottom in each of the columns. If you choose to skip any boxes, they cannot be filled out afterward.

Take a quick look at this scoresheet. The score sheet has three columns. One “up” column, in which the values that you will fill in the box will be in increasing order; one “down” column, in which the values that you will fill in the box will be in decreasing order; and one “choice” column, in which you are able to choose whether you want an increasing or decreasing column. In the sixth row of the columns, there is a 12 that is filled in. In order to fill in any boxes past the 12, you must fill in the 12. Lastly, in the last row of the columns, there is a lock symbol. In order to lock a row, you must have filled in at least 6 boxes and rolled either the attached 36 or 0. The “Operation column” is a place for you to write the operation that you computed in order to derive the number in the box.

A how-to-play: the first player to roll a 6 takes on the role of the “active player.” The active player would then roll all four dice. They now have two play options: 

  • The active player announces the first two dice that were rolled. All players may then (but are not required to) use any operation in order to combine the two dice to find a value that they can fill out in one of their columns.
  • The active player (but not the others) may then (but it is not required to) take one of the other dice together with one of the first rolled dice and combine them, using any operation in order to fill in a box with the number corresponding to this found value in one of their columns.

Similarly to Quixx, there are penalty boxes, which must be crossed out if, after the two actions, the active player doesn’t fill out a box of at least one number. Each penalty box is worth -5 points at the end of the game. The non-active players do not take a penalty if they choose not to cross out a number.

Once all players are ready, the player to the left becomes the new active player and re-rolls all four dice. Then the two actions described above are carried out again, one after the other. 

Now, If you wish to cross out the number at the very bottom of a column (up 36, down 0, and choice either 0 or 36), you must have first filled in at least six numbers in that column above, including the 12. If you cross out the number on the bottom, then also cross off the lock symbol directly next to it. This indicates that the column is now locked for all players and numbers of this column cannot be crossed out in the future rounds.

If a column is locked during the first action, it is possible that other players may, at the same time, also cross out the number on the bottom of the column and lock the same column. These players must also have previously filled in at least six numbers in that column. Also, the cross on the lock counts toward the total number of crosses marked in that column when you are scoring.

The game ends immediately as soon as either someone has marked a cross in their fourth penalty box or as soon as someone has locked two columns. Beneath the three columns is a table indicating how many points are awarded for how many crosses within each column (including any locks marked with a cross). Each crossed out penalty is worth five minus points. Enter your points for the three columns and the minus points for any penalties in the appropriate fields at the bottom of the scoresheet. The player with the highest total score is the winner.

I hope you enjoy this fast-paced mathematics game! 



Handouts: gdoc & pdf

John's Postscript: This game is pretty complex on first approach. But the strategy is subtle, and repeated playings
have a good variety. So don't give up with this one! I actually like Grace's over the original. More streamlined, and
the extra operations add interest.

In addition to their own games, the teachers selected an already made math game to promote for classroom use.
Grace selected one of my all time favorites, Fill the Stairs. I have a post on it here, and Joe Schwartz has
an amazing one.




Choose Your Path, a Math Game


This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Maggie wanted a game that used a graph/network as a playing board, and tried several options until coming up with this. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.
GUEST POST by Maggie Eisenga

Choose Your Path
In my capstone with Professor John Golden I got to play and research math games. The big project of this class was to create our own math game that teachers would be able to use in their classrooms and/or parents could use at home! I created a game called “Choose Your Path.” This game is designed to help elementary students (focused more on 3rd grade but can work for other grades as well) work on their fluency in integer equations. I wanted to add strategy to my game as well, which you will see as I explain further.

The materials for this game are simple. You will need a deck of standard playing cards, small household/classroom items, and a scratch piece of paper and pencil if students need to check their answers. This is a two player game, but students can also work in teams against each other. I found it to work better as two individual players, but either works! The goal of the game is to make an equation with the cards you choose to equal the designated number in that round.

To set up the game, shuffle the cards and lay 12 cards, face up, in a 3 X 4 fashion and lay the rest of the deck face down in the middle of the two players. This picture is how it will look. 

Have the students pick their household item, this will be their game piece.

To determine who goes first each player will draw a card from the deck and whoever has the bigger number will go first. In this game K=13, Q=12, J=11 and Ace=1. 

The players will start at a corner of the “board”,” but they cannot start at the same corner. The player who goes first will pick up 2 cards from the deck and they get to decide what operation they want to use to get the first designated number and this can be any operation. That player then gets to decide where they want to go on the board. They only move one card at a time and can go up, down, left, and right, but not diagonal. They then pick up the card they were just on and keep it and replace it with another card from the deck. 

The next player then gets to move their piece and do the same thing. This continues until one of the players believes they can make an equation with their cards to get the designated number. If they can, they discard all the cards they used in that operation and pick up the two cards that were used to make the designated number. They may keep any cards they didn’t use. The player who won that round then gets to pick up another two cards from the deck and choose any operation to get the next designated number. Whoever wins 3 out of 5 rounds wins the game. 

The strategy behind the game is choosing which cards the students want to use for their equations. Having them use any operation is a good strategy as well.

In addition to all of that, if a player draws 2 cards of the same suit they may have an extra turn in that round and may use it whenever they would like but it has to be in
that round and if you land on the same card another player is on, you can kick the other player off and they will have to start their next turn at a corner of your choosing. For more difficulty, have the players win a round by using 3 or more cards and using 2 or more operations when getting the designated number. For example, they may use 3 cards but then it has to have 2 different operations. Not just adding for instance to make it simpler, use only adding and subtracting and take out the bigger numbers in the deck meaning the jack, queen, and king.

Whenever I played this with students they really enjoyed it and loved the challenge it brings! Being fluent in integer equations, I believe, is very important for students to have as they continue in their math education and this game is a good way to practice just that!
Here are the standards that involved within the game: 

  • CCSS.MATH.CONTENT.2.OA.B.2 Fluently add and subtract within 20 using mental strategies 
  • CCSS.MATH.CONTENT.3.OA.A.1 Interpret products of whole numbers 
  • CCSS.MATH.CONTENT.3.OA.B.5 Apply properties of operations as strategies to multiply and divide. 
  • CCSS.MATH.CONTENT.3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division


John's postscript: while this is similar to many computation games, there are several interesting elements that make the play more fun. One, it naturally levels as the operations the players use to make the target connect to the operations they'll use with the cards they pick up. It's turn-based, taking out the speed element to a lot of these games. And being limited to the cards they can get to helps even the playing field, with some chance involved in those replacement cards. I think there is a lot of opportunity for strategic thinking.

In addition to their own game, each teacher chose an already created math game to promote for classroom use.

Integer Solitaire

We looked at so many math games as a class and individually, but I found that the one I am about to talk about, is very beneficial for students who love a challenge and need a little more integer practice. It works for middle school, possibly late elementary (if they are ahead), and even high school students who want to be more fluent in solving integer equations. This game was created by Kent Haines back in February of 2016.

All you need to play this game is a deck of cards and a small white board and marker. If you do not have a white board, a piece of paper and pencil would work just as well! The board will look like the picture to the right. 

Students can work by themselves or in pairs. I recommend using pairs because it helps build teamwork and in this game it is nice to have a partner you can bounce ideas off of. The student will draw 18 cards at random. The black cards will be positive integers and the red cards will be negative. In this particular game Ace=1, Jack=11, Queen =12, and King=13.

The goal of the game is to have the students use their 18 cards to somehow fill in
the 14 blanks on their board to make 4 correct equations. If the students finish early have them start over and pick 17 cards to make it more challenging.

Overall this game can be a challenge because they could get 3 correct equations but not be able to make a fourth. However, students are persistent to win so they will keep trying. For this reason, the fact that you don’t need a lot of supplies, the range of students who can play this is large, and because it is really good practice and fun for students who need to work with integers are all of the reasons I believe this game is a great math game for students.
A couple standards that are involved within this game are:
  • CCSS.MATH.CONTENT.2.OA.B.2 Fluently add and subtract within 20 using mental strategies. 
  • CCSS.MATH.CONTENT.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Cf. Kent's original post on Integer Solitaire

Tuesday, May 19, 2020

Multiply Who, a math game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Sam came into class with the idea for her game. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.

GUEST POST by Sam Bosma.

Learning to Go Forwards to Go Backwards

Many of us remember ‘learning’ multiplication in second or third grade. I remember ‘learning’ multiplication in 3rd grade through the use of flashcards. It didn’t matter what was happening behind the scenes just as long as I got the right answer/ could memorize the answer.
Too often with multiplication, we see children simply memorizing that, for example, 3 x 8 is 24 and not really understanding the ‘why’ behind how it works. The problem with this memorization is that once students move on to division, we see them getting very quickly. This makes sense since how can we expect students to go backwards (the undoing of multiplication) if they can’t even move forwards (multiplication).
In the game I have invented, Multiply Who?, my attempt is to have student thinking more deeply about multiplication instead of simply memorizing. In this game, students will have to find patterns and think about the components of each number in order to ask the most strategic questions in order to guess an opponent’s number.

If you are interested in learning more about this game, watch the video down below or click here for the instructions and rules.


John's postscript: I got to see one of the playtests of this game with a class of 5th graders and was impressed. The format of the game works really well for 2 on 2 playing. And though Sam put the sheets in sleeves, you could play directly on a printup as well. And the game is highly adaptable to level, by modifying the sheet. But the best thing to me is that this could definitely lead to students making up a sheet for playing and really getting to do the math.

Handout: gdoc, pdf

Make It, Take It
In addition to making their own game, the teachers made a video for a game of their choice that they'd like to see in a classroom. Sam chose Make It, Take It, a money game I made up a long time ago for a teacher that wanted a money game for 2nd year.



Again, highly adaptable. It's nice how just the progression of play gets players to think of new combinations.

Handout: docx, pdf

Tuesday, April 28, 2020

A Mathematics for Teaching Reflection

Mediated Field Experience is the term my colleague Esther Billings is using for our preservice elementary math education course that takes place in the schools. The schools have made a classroom open to us, and we teach weekly. What makes it mediated is they use and modify lessons the profs provide, and we observe them and give feedback. When it works out, we also have a student assistant who observes and gives format using a short reflection heuristic. This is a bit of a contrast to many of their school experiences as preservice teachers before their internship, where their time with students is unstructured or less supported. 

This semester both of my sections taught two 3rd grade classes, usually repeating the lesson with their other group. Pairs of college teachers worked with 2 to 4 kids, grouped by the teachers, for 45 min. Before they taught, I did a number talk with the whole class for 10 minutes. (Problem strings, quick images, numberless problems, story problems, etc.) Here's our lessons from this semester.

There is a flipgrid of many of these students talking about their experience, and why they think other students should choose this over the regular college course (which is still an option).

This semester, despite the interruption, I hit on a simple end of semester reflection that surprised me with the connections that these teachers made. I've had trouble giving the principles for effective math teaching much life in an academic setting (or the even harder to communicate high leverage practices). So I thought I would share a representative sample.

I asked: "Read these 8 principles for effective teaching of mathematics. For each, give an example of when you as a teacher did, we as a class did, or read about a teacher doing that principle. (list)" Everything in the bullets is a direct quote from a preservice teacher.

Establish mathematics goals to focus learning

  • As a class, we accomplished this task through the content journals that were assigned throughout the semester. The content journals were focused on content standards which stated clearly the goals of each day we spent together in the class. By explicitly stating the standards in text, the instructional goals were shared with each of us. This not only allowed us to have a clear understanding of what we were going to be learning, but also allowed us to understand the content in a more concise way. We then put this principle to work by using the content journals to determine our understanding of each of the goals. For each of the content journals, we were given the standard that pertained to the classwork and classroom learning we had done and asked to show our understanding of that standard.
  • Everytime my teaching partner and I meant with the students we were working with before we did our warm up activity we would tell them the goals for the day. We never would say by the end of the activity that they should have full understanding or grasp on the topic, but rather we would say such things as, our goal for today’s activity is to think of addition/subtraction, place value, etc. strategies.


Implement tasks that promote reasoning and problem solving

  • As a teacher I have done this multiple times with my lessons. When working with groups of students, not everyone has the same way of getting an answer so I make sure that everyone has a chance to say how he/she solved the problem. If they all have the say way of getting to it I would ask them if they knew another way to solve it or how else they could get the same answer (I would ask this even if they didn’t all take the same path of solving)
  • An example of this is when we created the wordless story problems. We also read about the teacher in Chapter six [Tracy Zager's Becoming the Math Teacher You Wish You'd Had, our text] who gave problems with multiple entry points
  • We did a lot of story problems with one of our groups because we could see their thinking better. A lot of the times they would disagree on what the correct answer was so they would have a discussion with each other for how they got their answer. This helped them show their solution and problem solving through their work.

Use and connect mathematical representations

  • A time I used this as a teacher was when my students and I were doing a number line on a white board, we started at a certain number and I had them added the same number to the prior number, taking turns in a circle. After going around a few times I gave them a number to get to and they had to figure out different ways to solve it. They subtracted the number they ended with from the number I gave them, they added by 10s or 20s until they got close to the given number and figured out the remainder. One of my groups of students even made the connection that adding the number 7 three times is also multiplying 7 by 3. 
  • I saw this principle most while we were learning about teaching fractions as a class. We learned about teaching fractions by making sense instead of just teaching algorithms which don’t strengthen student understanding. We learned to make connections for solving to find if one fraction is smaller than another by seeing how close each fraction is to easier fractions such as ½. There is also an activity where each part of a fraction is a cut up circle, so ⅛ would be shown as a circle cut into 8 equal slices. Then students are able to layer the fractions on top of each other to compare and find out questions such as which is larger. This also helps them see that the larger the denominator, the smaller the individual piece / the smaller the fraction will be.
  • I am brought back to the video that provided an example of representing fractions through a context. Each group of kids were able to come up with their own method of describing how the subs were split up. I remember commenting that there are so many different ways to understand and compute fractions. I find myself struggling to find the best way to solve a problem. I understand the importance of illustrating different ways to solve a problem as a teacher. This allows for students to freely represent their thinking in a way that works with their mindset. And that is what occurred with this particular example.
  • In lesson 8, we focused on practicing multiplication using representations and connections. We played a game where the students drew numbers and made rectangles with the dimensions of those numbers, and multiplied them together to find the area of these. During the game, Joaquin discovered that “multiplication is just like addition” and so I had him explain to the other students why he thought that, using the rectangles as representations. 


Facilitate meaningful mathematical discourse

  • Reading the Becoming book I really liked the story from chapter 13 the teacher Ann’s story with the Powerball. It was exciting to see all of those kids questioning and looking at a whole new view of a question not even the teacher hadn’t thought about before so it brought a lot of meaningful conversation into their classroom even before class was actually started.
  • A specific example of this happened during lesson 7 when we briefly showed our students a picture of a piece of toast with blueberries on it and then they had to tell us how many blueberries were on the toast. We had the students share how many they counted and how they did. They both counted 12, but differently. They talked amongst each other about the ways they had counted, and each acknowledged that both ways were effective, but decided that one's student way was more effective. Count the number of rows and times it by the number of columns.
  • This reminded me of when after we met with our third graders for the day we would share stories about what happened that day. I really think this was helpful because if it was a Tuesday and my group had a rough day with the lesson but another group shared how they had a successful day, I could use their ideas for my lesson on Thursday so it would be more successful. 
  • This is something my partner and I did a lot to keep the students engaged, to build relationships, and to listen for reasoning. During many of our lessons we asked our students to share outloud how they thought about something or their approach to a problem. We encouraged them to share their thinking and to used it as a way to talk about new approaches and different strategies. There was many times that one student answered a problem a different way than another student. We wanted our students to talk to each and learn from each other. Sometimes we can learn the most from our students or from our peers. 


Pose purposeful questions

  • I think our numberless story problems are a great example of this, I had never had experience with them as a student but I love them and when I am a math teacher one day I will definitely use them. The reason I like them so much is because it slowly introduces the idea of the problem and the context of the story before throwing too much at  them by giving them everything all at once. 
  • An example of something I did as a teacher to fulfill this principle was asking the question “how do you know that?”. I had listened to Professor Golden ask us this question in class, so I made an effort to use it when working with the students. This question is simple yet it invites the student to consider their thought process and how they arrived at their understanding. This question is purposeful because not only does it allow the teacher to have a greater understanding of the student’s understanding based on their reasoning, but it also gives the student the opportunity to self-assess and consider their own thought processes and reasoning when learning mathematics and interacting with problems. 
  • Professor Golden's number talks with the children to begin each session with them was a great way to get a routine down with students that help them be able to recognize procedures and ideas about math. The number talks were good ways to emphasize all students' thinking and model and represent the various strategies and thoughts students have on the problem at hand. It is also an inclusion and safe space to talk about wrong or right answers, modeling that mistakes are okay and that struggle to find multiple strategies will come with time.
  • I enjoyed the Notice and Wonder that was encountered in our MTH 222 classroom. From Homework to the discourse we had in class/with our tutees, questions were facilitated in this pattern. It was especially incremental with our Numberless stories! The basic structure of Notice and Wonder follows from ch. 7 of our Becoming Textbook: 
    • Give students an image or scenario without a question. If you like, you can use problems from your curriculum and obscure or delete the question.
    • Ask them, “What do you notice?” (You might want to use think-pair-shares at each step to increase engagement and thoughtfulness.) Record their noticings so students can see them. 
    • Ask students, “What are you wondering?” Record their wonderings.
    • Ask students, “Is there anything up here that you are wondering about? Anything you need clarified?” Pursue any follow-up questions.
    • Once students have had ample time for noticing and wondering, you can either reveal a question you’d like them to solve or have students come up with a question by asking, “If this story were the beginning of a math problem, what could the math problem be?”
  • This reminded me of when after we met with our third graders for the day we would share stories about what happened that day. I really think this was helpful because if it was a Tuesday and my group had a rough day with the lesson but another group shared how they had a successful day, I could use their ideas for my lesson on Thursday so it would be more successful. 

Build procedural fluency from conceptual understanding

  • By picking apart both addition strategies and subtraction strategies, both in class and in the homework in articles and in the book- we learned about a lot of procedures and created fluency in those procedures, from understanding how different methods work and the underlying meanings of the numbers and their many properties. By understanding place value, and the numbers, and groupings and factors of numbers, we are better at understanding and applying those concepts together to understand both the procedures and strategies of, say, addition or subtraction, and develop fluency from correlational learning.
  • This was something I noticed about my own learning. I think I came into the class knowing mathematical procedures for a lot of concepts. I left the class knowing why I do a mathematical procedure here, and how I can use my problem solving skills to build on what I already know. For example, I learned a lot about story problems. I knew how to answer story problems but I didn’t understand how they work and really the differences between them. There are a few different kinds and while they focus on different things they are meant to be used as a mathematical tool. Story problems are designed to make students stop and think about what the question is asking them. They are also used to help students visualize a situation and determine the right approach to it. I used to have the mentality that you should find and pull out the numbers and do something with them like add, subtract, or multiply. I now know to stop, slow down, and think first. 
  • One of my students, ____, had a hard time doing addition. We tried a lot of different techniques and ideas with her, but one class she just decided to try a problem using her own technique and she got the answer correct. She was flexible and once she was able to do it her way, she began to understand the ways that we were presenting to her. 


Support productive struggle in learning mathematics

  • This we used in class and myself as a teacher. This was helpful with the number circle because we were doing this with difficult numbers to see if my students and myself were able to find easier or different ways to find the next number. We supported answers and time to think and did not shout out and point out if the students added or subtracted incorrectly. Figuring out problems in your head and by yourself is more productive in learning new ideas/concepts.
  • This was something I had to learn to do. After Ms. Cordy came and talked to us in class. I got a better understanding on how to do it. A productive struggle is so important in helping a student being able to learn a topic that might be hard for them to grasp, you can't just bail them out with their first answer, or “I don’t know” and move on from the topic, you want to make sure that know what they are talking about, or else you could just move on leaving them behind.
  • We gave our students the problem 123/10 to do, and they were discouraged at first due to the big numbers and division, _____ even said “this is too hard, we can’t do this!”. However, we let the students sit and think for a while, trying out many different strategies and giving encouragement when they seemed stuck on an idea. Eventually, they focused on counting up and down by 10s to find the answer, and were debating whether it is 13 or 12. Hannah and I reminded them there can be a remainder and this helped them decide the answer was 12 with a remainder of 3. They all had tried multiple methods of solving, and Hannah and I encouraged them to keep trying until they figured it out, and they seemed to feel very accomplished after doing so. 
  • A way we supported productive struggle when learning mathematics was by not just giving them the answer when they were wrong. We would ask how they got the answer they got and we would discuss it as a class, hearing ideas from multiple students. We also let everyone know that it is okay to not always get the answer right, and everyone can learn from the mistakes they make including the teachers. We also make mistakes and are learning as we go on, this helped not put so much pressure on them to just “be right” but to actually understand the content they are learning and feel comfortable asking questions. 


Elicit and use evidence of student thinking

  • Throughout the semester I kept a lot of evidence of students thinking through photos as well as notes.  I began writing down student thinking on a white board to help myself understand the student’s thought process.  This proved to be an excellent resource to look back at and analyze.  Oftentimes I learn a lot from the students and they showed me ways of thinking that I would never have thought of.
  • Something I think I developed a lot over the semester was to ask better questions to the students. I wanted them to be able to explain their thoughts to me. I would often ask, “how did you get that?” “why?” “is there another way?” I remember constantly saying, “I want to see your thinking.” That’s where I got a lot of feedback from the students. 
  • The logs we kept on each meeting and observations we made about each individual student helped us pick activities that challenged them appropriately.
  • Hannah and I realized that our students in Mrs. Caterino’s class had a hard time subtracting when the number being subtracted had a larger first digit than the first number (ex 32-18). We decided to change some games from addition to subtraction to help add practice of that, and adjusted them so there were more problems like this so the students could work on doing subtraction problems like that. 
  • Since all of our students were on such a variety of different levels we had to develop our lessons to make sure that everyone would be able to try it and be challenged mathematically. So we’d have follow up questions for our student who would fly through problems and made sure our student who was struggling understood what he was doing mathematically and why.
  • We would evaluate students' work from past weeks and what they were knowledgeable on, and use this student's thinking to tailor our future lessons and make them accessible to all the students.

I was pleased to see connections to our reading, our content time, the number talks they observed, and some progress in their thinking and use of these ideas over the semester. I think the principles became real for them in a way that I haven't seen in traditional teacher prep.

Thursday, December 19, 2019

Playful Math 133


Welcome to the 133rd installment of the Playful Math Carnival.

133 is the 6th octagonal number. How many carnivals until the 7th octagonal? What kind of pattern do the 6th polygonal numbers make? (Play with some figural number GeoGebra.)

133 is also semiprime - the product of two prime factors. (Which two?) What is the next semiprime number? 133 is also a Harshad (aka Niven) number, a number divisible by the sum of its digits. It's an unusual stretch where 3 of four consecutive integers are Harshad - which are the other two? There's also a new to me fact: 133 is the number of partitions of 55 into distinct odd parts, which seems equivalent to the number of symmetric Ferrers graphs with 55 nodes. I haven't made sense of this yet, though! 133 is happy in base 10, so a good number for this holiday edition.






Last fact and shout out to Megan Schmidt: 133 is a square spiral corner number! Is that connected to any of the patterns discussed above?
 Literal math playing...

  • Dave Coffey gave a presentation on Math Play with a Purpose at Global Math. He talks about redeeming Bingo, among other things.
  • Speaking of Bingo, two of the groups at this semester's Family Math Nights made actual math games out of a Bingo premise. Meg and Madison made Food Bingo, which makes it about attributes of food. Erica and Claire had Star Bingo which used better number cards and some choice to enhance the game.
  • There was one game that I liked more than the PSTs who ran it! Give Hamburger a try.
  • Probably the breakout and most original game of FMN was You Must Cross the River. Eddie and Climie brought this D&D style game.
  • One of my HS preservice teachers tried to gamify Which One Doesn't Belong... I think Danielle is on to something.
  • Denise Gaskins, the founder of this here blog carnival, shares one of her many Hundred Chart Games.
  • Marilyn Burns shares her two dice sum game, which is a classic for a reason. She shares using it in 2nd and 7th grade!
  • Kent Haines assembled a Holiday Gift Guide for math games that might be too late for shopping this holiday from this post, but you'll want to keep this list.

The 133rd Space Shuttle mission 
was the last (39th) for Discovery. 
They installed the Leonardo Module 
to the International Space Station.

Some playful interactives...

  • NRICH shared a puzzle that is part about area, but made challenging through Cuisenaire Rods. Great lesson.
  • Kevin Forster shared Factris which is a multiplication/factoring version of Tetris. 
  • Scott Farrar made a cool GeoGebra activity implementing Always/Sometimes/Never with quadrilaterals. 

 The C-133 Cargo plane over San Francisco Bay.

Math stories...



Xenon 133 is an isotope 
that is inhaled to study respiration, 
among other medical imaging uses.

Math art to round us out...

  • Isohedral rounded up some of my favorite animated math artists in this post.
  • Very excited that Clarissa Grandi has a math art activity book coming out.  Look at her website and you'll see why I'm excited.
  • Nathaniel Highstein did an Islamic Geometry project with his students. Scroll down this thread to see their work and how he tiled them!
  • Paula Beardell Krieg has been killing it, but if I was picking one recent post it's this one about the gyrobifastigium. You heard me.
  • Simon Gregg tweeted some mathart that turned into WODB and latin squares and more, as only Simon can do. But what better captures the spirit of Playful Math?

Math art from the Public Domain Review.

The previous Playful Math was at Arithmophobia No More and the next is at Math Misery?. Would you like to host? Contact Denise Gaskins, or see the Playful Math homepage. People don't submit a ton of posts anymore, but I enjoy looking back at what I've found helpful from the math ed community and sharing it all together. So many resources and so much fun to be had.

Happy Holidays, or New Decade Blessings, or Sweet Playful Math!