Friday, July 29, 2016

Teaching goals for English Language Arts

These are our current objectives for English language study with our children. These are very high level goals, but we feel it is important to write these down so that they can guide our detailed choices. Ultimately, our hope is to guide the kids to be independent learners.

Strong Readers (in-bound communication)
  • Able to use reading as a tool for further learning. This means they must:
    • Enjoy reading
    • Have a large vocabulary and tools to build their vocabulary
    • Good comprehension and tools to analyze what they are reading
    • Familiarity with sources of information
  • Read a wide variety of material: topics, authors, styles, forms
Writing (out-bound communication)
  • Able to communicate ideas clearly and effectively (writing and speaking)
    • Comfortable with the mechanics of writing: vocabulary, spelling, grammar, punctuation, physical writing and typing
    • Learn a writing process: research, brainstorming, forming ideas, organizing ideas, drafting, revising
Conduit for other content
We will make use of language activities that also teach them:
  • History: having data of history to learn from the past, ideas of historiography and perspective
  • Science: technical jargon, tools to understand and develop scientific frameworks
  • Philosophy and comparative religion: what are the great questions, different perspectives, forming their own values and understanding those of other people
  • Current events: understanding the current context of their lives
Develop skills that support other language learning
  • Grammar frameworks
  • Methods for learning vocabulary
  • Motivation

Wednesday, July 27, 2016

A talk for parents about math at our school

Today, P led a session for the other parents at the school. We wanted to share the material and some links for those who weren't able to attend.

  1. What does it mean to be good at math? What are we trying to achieve?
  2. Key concepts we are using in teaching: Concrete-Pictorial-Abstract, Concept Progressions
  3. Examples
  4. What should parents do at home?
Good at Math
To start, P asked the parents, what does it mean to be good at math? Some of the answers:
  • can add, subtract, multiply and divide
  • calculate quickly
  • able to estimate 
The range of opinions was good to see. Many ideas fell within the traditional answer: being good at math means being able to calculate precisely and quickly. We were especially pleased to hear skill at estimating as one of the ideas.

Our additions:
  • Thinking logically. For example, if a certain thing is true, what else is true? 
  • Looking for patterns and relationships; forming connections with other things they know.
  • Asking questions about what they see, especially investigating structure
  • Persevering
Admittedly, these are necessary for many other subjects. Math is a particularly good place to develop these skills because there is much greater objectivity and right/wrong are often clearly distinct. In this domain, the power of reasoning and independent thought is stronger than the power of authority.

Process of Learning Math
For this discussion, we focus on two key concepts in the way we teach and study math: (a) the Concrete-Pictorial-Abstract modes and (b) multiple models in progression and contrast.

"Concrete" means using physical objects. For example, a pile of 15 beads can be a concrete representation of the number 15. Taking 2 beads in the left hand and 3 in the right hand, then combining them is a concrete experience of adding 2 and 3.

In this mode, children are able to see, touch, move, examine, smell, and hear mathematics.  

"Pictorial" shifts to pictures on the page or board. For example, a picture of a room showing a vase with 2 flowers and another vase with 3 flowers can lead us to identify 5 flowers all together.

In this mode, children are able to see, construct (by drawing themselves), obliterate (by crossing out), and add color (by coloring, naturally) the mathematical objects.

"Abstract" is where we shift to symbols. For example, 2 + 3 = 5. This is a sequence of 5 symbols that don't offer any clues to their own meaning.  

In this mode, children are able to imagine and to move beyond physical constraints or necessities.

When new concepts are introduced, they generally go through each stage, starting with concrete, then pictorial, then abstract. This doesn't mean that abstract is superior, however. The ability to move back and forth, to give specific examples, to draw diagrams, to demonstrate a concrete model is also very important.

Multiple Models
Complementing the three modes, we also try to have multiple models, different ways of seeing, new concepts. There are two great resources that nicely illustrate this.  First, the models of multiplication posters from Natural Math:

4 of 12 models at Natural Maths
For the discussion, P gave examples of the equal groups model (sets per each), repeated addition, array, number line, and area.

I strongly encourage you to take a look at all 12 models in their poster, so here's the link again:

The second resource is Graham Fletcher's series of progressions videos: Addition and Subtraction,
Multiplication, and Division.

For this talk, we presented abridged versions of the content in the multiplication and division videos. For your ease and viewing pleasure, here they are.

and the division video:

What to do at home
1) Ask questions
Parents can relax about being the source of knowledge. Don't worry about "teaching" or having the right answer. Instead, develop habits to cue thinking and their use of problem solving strategies:

  • How do you know?
  • What pictures could help us?
  • What do you notice? What do you wonder? This works especially well if the parent serves as scribe writing down the kids' ideas.
"How do you know?" does three things. First, it is one way to escape from the child's questions "is this right?" Remember, the power of reasoning is stronger than the power of authority. We want to reinforce this by side-stepping calls to authority.

Second, it is a cue to get them thinking about their own thought process, which aids learning.

Third, this opens a potential discussion about different ways to attack the problem. Comparing and contrasting multiple strategies is a powerful tool for deeper learning.

"What pictures could help us?" is a cue to move between the Concrete-Pictorial-Abstract modes. If available, go to Concrete by asking about objects or physical models that are related.

"Notice & wonder" is a deep topic. One key idea is that, by serving as the scribe, we parents demonstrate that we care about the ideas that the kids have, we know they can contribute to solving the problem. This also gets us listening and understanding their perspective.
Notice & Wonder also involves skills that the kids will strengthen with practice, starting with superficial or (mathematically) irrelevant ideas and eventually moving on to thoughts about patterns and structure.

 For more about notice and wonder, please take a look at Annie Fetter's talk

2) Talking numbers to develop number sense
Two examples of number sense. Say we bought 8 bags of snacks and each bag cost 17 baht:

  • I know that the total cost is not 1000 baht based on understanding order of magnitude.
  • I know that the total cost is not 137 baht based on the pattern that all multiples of 8 are even
Like learning a language, number sense takes practice, it requires frequent exposure, and is built up by drawing children's attention to numerical and mathematical ideas.

Specific activities to do at home include estimating and measuring (length, time, weight, volume, etc).

Make math a part of everyday life by asking questions about what you see around and asking them to find examples of the concepts they are currently learning.

3) Play games
We play a lot of games at school and ask the kids to play them at home with their parents as homework. The games don't go stale when we move on, parents can play old games again. Some games are very calculation heavy. These are great opportunities to flex the Concrete-Pictorial-Abstract muscles.

Other games (or explorations) are much more about strategy. These are a great place to practice the other questions, especially "what do you notice?" and "what do you wonder?"

Sunday, July 24, 2016

Math Teachers at Play #100 - (Blog Carnival)

Wow, the 100th Math Teachers at Play! Such an honor to put together this milestone edition. Thanks to Denise Gaskins for creating and managing this great resource. Let her know if you are interested in hosting in the future.

I asked the 3J's for observations and facts about 100:
  • It is written with a 1 and two 0s
  • Square number (10 x 10)
  • It is a sum of two squares 64 + 36
  • It is a sum of two primes in several ways: 97 + 3, 89 + 11, 83 + 17, 71 + 29, 53 + 47
  • 1100100 in binary
  • 100 has 9 factors, which sounds like a lot, but is not an anti-prime
  • 100 is the start of a 26 term Collatz Sequence: 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 (little program for Collatz play)
Number Gossip revealed that it is a practical, powerful, happy, odious number. Hmm, a jolly dictator?

Here's a very familiar 100 in our daily life:

Unfortunately, this 100 makes far fewer appearances:

A 100 puzzle

To get started, here's a tricky puzzle that, semi-famously, has been a Google interview question. This phrasing comes from Good Riddles Now:
There are 100 prisoners lining up to go to jail. Each prisoner is wearing a hat that is either black or white. The prisoners don't know their own hat color, just the hat color of those in front of them in line (the first prisoner in line can't see anyone's hat and the last prisoner can see everyone's hat except their own). Starting from the back, one of the guards asks each prisoner what color their hat is. If they are correct they get to go free but if they are wrong they go to jail. 
If the prisoners get to discuss a plan, how can at least 99 of them be saved?
Talking about math
I blended ages here because I've noticed a lot of great cross-age pollination. The little ones are often very engaged and surprisingly insightful on topics that seem much more mature, while elementary conversations often touch some deep concepts.

AO Fradkin and her daughter discussed "how long is 3 minutes?" Hard to get deeper than the nature of time and how our perception depends on context (or does it?)

A view of Fermat's role in Fermat's Last Theorem from Mathematical Enchantments that nicely touching the fact of changing tastes in mathematical research. What mathematical preferences do you and your kids have?

Some tidbits for the Math is Everywhere meme:

  1. Using big data to analyze story arcs.
  2. Life through a Mathematician's Eyes has two posts on What a Mathematician should see in Amsterdam and Visiting Amsterdam like a Mathematician
I can never resist an icosahedron picture and there's a nice one in the first Amsterdam post:

Enjoy a joke anecdote about mathematical precision from Curiousa Mathematica. I like to tell elementary kids these kinds of jokes and see how they respond. It is a delight when they get it, but I also like to laugh at their deadpan expressions when they don't understand. 

Finally, a great way to start talking about math is to do math where you'll be seen doing it. This might require an emergency math kit (from Solve My Maths.)

Crafts and Constructions
Generally, these are accessible to young children, but also have some deep mathematics that offer something for any of us to explore.

Something our kids recently made: Flextangles craft activity (3d flexagons)

We love optical illusions and Sugihara's Illusion is a fantastic one. This post gives an explanation, then the next provides a printable. Agamographs are an accessible craft with a similar idea (instructions on Babble Dabble Do).

Benjamin Leis writes about an eye-catching decomposition and recomposition puzzle to start some exploration: hinged polygons.

Kira Zelbo offers up a free booklet that introduces isometric dot paper for drawing and visualizing three dimensional forms: Spatial Learning. Education Realist recently documented his experience using isometric grid paper with his class in Great Moments in Teaching.

Elementary Explorations and Middle School Mastery

Denise Gaskins (have you heard of her?) stimulates a discussion of favorite puzzle books in her review of Lilac Mohr's Math and Magic In Wonderland.

AO Fradkin again, helping some early elementary kids in discovering the triangle inequality.

John Golden serves another ace with his Wimbledon Game.

An old post from Sue Downing caught my attention: these place value cups are a great idea and make me think (fondly) of combination bike locks.

Addition Boomerang is a Mathpickle activity we played recently with our 1st-4th graders. When can you make 100? With that target, of course we had to include it in this MTaP!

Math Minds has ants-on-the-brain in 100 hungry ants. Trust me, it is better than ants in the pants.

Second graders explore proofs based on a geometric investigation: Squarable Numbers

Another post from Sue Downing that came in handy recently: "Is that all there are?" Multiplication facts. Now that you know the multiplication facts aren't scary, get some practice by playing with 
anti-primes (Numberphile). We got a lot of value out of seeking the anti-prime (er, highly composite number) that follows 24.

Number bracelet investigation. Our school and family have looked at this several times. Another great extension is to work in bases other than 10. Another way to understand it is working modulo 10 (or whatever integer you choose to set to 0).

Baseball, with its large collection of player and team stats, offers a natural entry point for mathematical conversations. Mashup Math walks through some examples in Mathematics of Major League Baseball. For a related take on this idea, see Mathpickle's Introducing Stats to Younger Children.

High School Adventures

Mrs E shares a lesson plan focused on getting kids to analyze and critique advertisements at Mrs E Teaches Math. An important life skill on its own and Mrs E sees it as a helpful bridge into writing proofs.

What happens when you are living inside a word problem? Math in real life.

I don't think this is a recent addition from Dan Meyer, but his Finals Week three-act seemed more fun to do during summer, away from the normal stresses of the school year.

There are some elementary and sophisticated ways to think about divisibility rules. In this post Curious Cheetah hits a bunch at one time.

"Stoichiometry is the math of chemistry," according to Amy Roediger. Here is her explanation and a discussion of teaching methods, parts 12, and 3

Manan Shah contributes his approach to dealing with Annoying Function Notation. Can you make sense of these contrasting pairs:

Also, make sure to check out Manan's curation of the latest Carnival of Mathematics.

Po Shen Lo and Mike Lawler: a great combination... and Mike gives us two posts with PSL (first and second)!

Step into some difficult probability, statistics, and forecasting with Big Thompson Flood.

Singapore Maths Tuition walks us through a vector calculation to find the foot of a perpendicular from a point and to a line.

Ben Vitalis has a constant stream of interesting challenges, most accessible to algebra students: Odds equal Evens.

Puzzling Recreations

Math Arguments makes a surprising re-appearance to post a nice probability dice from Ben Orlin.

Lisa Winer (star of the 99th MTaP) talks about her plexer puzzles. Aside from being fun, these are a good place to practice Notice & Wonder.

Our family has recently become fans of the Futility Closet podcast, especially their lateral thinking puzzles.

Reminder: one great thing to do with any puzzle is CREATE YOUR OWN!!!

Teaching Tips

Which one doesn't belong (WODB) is a great format for a rich discussion. has a really nice collection of mathematical WODBs. I was reminded of this resource by this delightful WODB of WODB from John Golden.

Joe Schwartz continues a MTBoS theme of getting fixes for worksheets.

Amy Roediger put together a good collection of resources as she prepared to lead a
coding camp. A Recursive Process has a further discussion of the links between coding and math, including some more resources.

The folks at the Mind Research Institute have put together a summer reading list of 9 Enlightening Summer Reads for Math Teachers. The list mixes sci-fi and books about teaching. If you don't know the Mind Research Institute, they are behind/linked with ST Math, an on-line elementary grades math system that I really like. I had the opportunity to trial their system years ago and recently went through their free demo.  Now, if only someone there would be willing to get in touch and tell me how I could subscribe for my kids to use .....!

Twitter Math Camp 2016 was just held and there are a lot of math teachers blogging about their experiences. If the posts I've gathered above aren't enough for you, I suggest hitting J Fairbanks's blog 8 is My Lucky Number for 10 (wow!) posts about the convention and further links.

Beast Academy and Dreambox (reviews)

Conflict of interest statement: I do not have a current or pending financial relationship with Art of Problem Solving, but I have several friends on their board and have had direct contact with several other people there. We purchased and currently own all of the books I review below.

I have no relationship with Dreambox. We tested the program using their free trial and then paid for a 6 month subscription.

Beast Academy

What is it?
Beast Academy (from Art of Problem Solving) is a book series with 10 "guidebooks" and 10 parallel "practice" books targeted to 3rd, 4th, and fifth graders. Note that the first of 4 books for fifth grade has only recently come out and they are planning to extend to 16 x 2 books covering 2nd to 5th. We do not have either 5A or 5B yet.

We have read all 8 books from 3a to 4d; J1 and J2 have gone through practice books 3A - 3C.

While these are not math exercise apps, I'm going to borrow some of the elements I've used in past app reviews. One key point I want to emphasize for both books and apps: the way you use them can also determine their benefits or costs.

To drive this point home, let's start with Tracy Zager's Big Three criteria (see here):

1. No time pressure: Neutral since this is really up to you, parents.
Do you set a timer when they start a page of practice or a question? Do you require a certain amount of time spent on math practice? While the books do not suggest or impose a sense of time pressure, there are story segments involving math competitions that imply speed is important.

One time element that is and has always been great about physical books is that they sit around. This means they are available and tempting. Almost every day, there will be someone flipping open one of the BA guidebooks, even J3 for whom the material is too advanced right now.

2. Conceptual basis: yes (pass)
The books introduce models, contexts, and conceptual ways of considering problems and techniques. 

3. How are mistakes handled: again, this depends on you and your kids
My approach is to go through the problems and select ones to discuss. I don't use the answer key, so I do the problems myself. This means we have three categories of questions to discuss (a) answered correctly and I found interesting, (b) answered incorrectly, (c) answered correctly by the kid, but I made a mistake. 

Also, I am very positive in how I talk about mistakes. The key message is that these are actually the best learning opportunities and create a chance for us to understand our own thinking.

Preliminary summary: whether Beast Academy (or any printed material) passes the thresholds depends on how  you plan to use it. If you want to deviate from Tracy's guidelines, either adding time pressure or incentives based on minimizing mistakes, you probably should think carefully about whether that's wise.

The good

For my kids, the stories and themes in the guidebooks hit the right tone. They are engaging and funny, with a humor that is occasionally silly or corny. An extended quote from The Princess Bride certainly wins some extra points as well. More bonus points for becoming, via malapropism, the source of J3's current catch-phrase, "I get it: pointillism!"

For me, the organizing theme of the material seems to be "ideas you encounter when playing with math." In some cases, the exercises create "aha moments," like when J1 realized he didn't always have to calculate side lengths of a polygon to use his knowledge of its perimeter in a challenge. In other cases, like calculating (n+1) x (n-1) there are interesting patterns to notice and connections to make.

I'd note that the workbooks are absolutely essential as there is a lot of material that is introduced in the context of exercises. I think these books are excellent, well selected, well sequenced, with enough repetition to facilitate mastery and enough variation to avoid boredom. In fact, I really enjoy doing the problems myself.

Overall, we find the practice books an especially good source of cues for quick (5-15 minute) math conversations.

The Bad
Any worksheet-based system is weak in generating exploration and deeper investigation. Beast Academy partially addresses this by including open-ended games and an occasional investigation. While nice, this point remains a weakness. I don't want to belabor this point, since it is not a unique problem with Beast Academy. Indeed, I think it is a universal issue with static educational material.

Unfortunately, the only solution I know is to involve a human guide. Fortunately, I am able to play that role, asking their thoughts about interesting problems, helping them form connections with earlier or other material, getting them to follow useful side-branches or to continue more deeply into a particular area.

Eventually, of course, we hope to develop enough mathematical habits of mind that the kids will do these things on their own. Realistically, I don't think that will happen until they are well clear of any elementary age material!

The Ugly
I don't see any fatal flaws in Beast Academy.

Grand Summary
If you can use the material the way we do, I highly recommend Beast Academy.
If you can't or don't feel comfortable engaging as your kids' mathematical guide, these books are probably still one of the best options. Just don't set up a timer and demand perfect answers to all the questions!


Dreambox is a math facts, basic skills system. It has material from pre-school through high school. We have spent a lot of time with the elementary grade material and a little sampling of the high school content.

Dreambox was one of Tracy Zager's positive examples in her app post, so we already expected it would pass these three criteria. After spending so much time with the system, though, we've seen that not all activities within DreamBox completely satisfy the checklist:

1. No time pressure
Some activities do include time pressure. For example, there are a family of "games" around multiplication automaticity where a collection of calculations stream across the screen. This really does raise the stress level for kids.

In a slightly different form, there are other activities involving virtual manipulatives that require the student to do something using the minimum number of moves. Like the time pressure, this seems to create confusion where the kids can get something right, but still get it wrong.

2. Conceptual Basis
I mostly concur with Tracy's original assessment. Almost all activities have a conceptual component. The timed calculations mentioned above don't, so those get a double demerit.

3. How errors are handled
Again, mostly agree with Tracy. However, there are some activities where, for a minor mistake, one is required to redo a number of manipulations, rather than fix the earlier work.

The good
The underlying math curriculum here is solid, if basic. The clear strength of this system is the pictorial representation of manipulatives offering models that build number sense, reflect operations, and show place value. In the early years section, where we have been spending most of our time, almost every activity is based around one of the manipulatives.

The other thing Dreambox does well is present a sensible progression for the different activity streams. I think this works especially well for J3 who is going through much of the material for the first time. As she encounters a new formulation, she will study it for a while and then there is a clear moment when she has figured out the new complication.

I'll give two examples. For J3, there is an activity to replicate a number bead pattern and then click the number of beads in the arrangement. Her primary tool is to count the beads one-by-one. In the most recent module, she gets a short view of the arrangement and then it is hidden (it can be revealed again, if you choose). This is forcing her to build new skills, either memorizing the arrangement to mentally count or a more advanced counting technique.

For J2, one of the place value exercises involves grouping items into pallets (1000s), cases (100s), boxes (10s) or loose items (1s). The current module asks him to consider multiple different ways to pack a given number. For example, 1385 items could be packed in 1 pallet, 3 cases, 8 boxes, and 5 loose items, or 13 cases and 85 loose items (among many other options).

One other strength of DreamBox is the email feedback to parents. Christopher Danielson recently noted this in a post: Parent Letters.

The Bad
I have seen three areas of weakness with Dreambox: the way mathematical tasks are presented, the pace of adaptive adjustment, and the absence of rich tasks. I'll talk about each of these in turn.

The theme gives an irritating appearance of choice. For example, in the early elementary section, the kids can play with dinosaurs, pirates, pixies, or animals. Under each of these, they have a further choice about what story to explore. Those choices, at least, lead them to different narratives and animated sequences.

At that point, all of the stories involve finding missing items. Users then see another choice asking where in 6 map regions they want to look for the missing items, but this isn't really a choice as there are no differences between regions and they will have to go through each region eventually.

Similar to Prodigy Game, the math tasks are presented as an annoyance to be overcome, the cost the student has to pay to move on with the story. Again, I find this creates unfortunate subtext to the mathematical experience.

Second, the adaptive adjustment is very slow, if it actually exists. In their FAQ, I see that they get questions about how to increase the challenge level, so this seems to be a common experience. Part of the problem is that they intentionally start students with material below their grade level.

Finally, the tasks in Dreambox are basic. While they may present a challenge for a new learner, as J3 is experiencing, they should eventually become so easy that they are boring. In some way, this feels like learning to solve math class tasks without having to develop or use any mathematical habits of mind.  Further, the thrill and fun of playing Dreambox lies in unlocking the animated stories and collecting tokens, not in doing math.

For J1 and J2, this thrill has worn off after about 2 months with the system.

The ugly
Nothing in Dreambox is a show-stopper.

Properly understood as a basic curriculum substitute or source of practice exercises, Dreambox is a solid application. Just don't make the mistake of thinking it will either foster a love of math nor deeper mental habits.

*Update* A quick comparison with ST Math
I was sitting on this review, partially written, for a long time. One thing that got me to finalize the review was going through the demo challenges on ST Math with J2. We had previously tested ST Math many years ago with J1 and it was really good. Once again, this is what I saw with J2: really cleverly presented scenarios that gave us good models for the math and a really fun user experience. After playing for about 30 minutes, J2 said: "this is a lot more fun than DreamBox."

If I can get a subscription, we'll test it more extensively and write a review to see whether that really holds up.

Wimbledon game

We have been playing the game Wimbledon from John Golden. J1 and J2 absolutely love the game and J3 has enjoyed pretending to play as well. Definitely try it out!

Below is a session report sharing some of our experiences with the game, including some alternative rules (aka mis-reading).

Our basic play
For the serve, we allow the following options:
(1) serve a single card from the top of the deck
(2) serve one or two cards from your own hand

We also allow returns that have the same value, if the largest card in the combination is higher. For example, 8 + 2 can be played on top of 7 + 3.

When playing doubles, we followed tennis conventions: one player serves the whole game, return of serve alternates. For the return of serve, the designated player must return on their own. For other returns, either of the partners alone or in combination can play cards for a return.

Having gone back to John's original post, I now see that we played with the inverse rules for aces. On the serve, we counted them as 11, all other times 1. We didn't distinguish between aces played from the hand or served from the deck, those were all 11s. That formed a strong advantage for the servers, while making aces essentially worthless for all other players.

With three players, I had the kids play as partners and I played with a ghost partner. The ghost partner would contribute cards randomly. When the ghost played cards that weren't large enough to be legal plays, we considered that an unforced error and awarded the point to the other team. While the ghost was able to hold serve for one game, it was a big disadvantage. An alternative for three players would be to have the ghost partner with the server and for everyone to take turns serving. This would put the server advantage (with our "house rules" for aces) against the ghost disadvantage.

A modification
In our play, we have found that the 10 value cards and aces (on serve) dominate game play. Here are two ideas to address that:

  • Assign face cards values 11 for Jack, 12 for Queen, and 13 for King. Ace, on serve, can have a value 14.
  • Allow players to combine as many cards as they like. This would probably work best with our "Further extension" rules below. A possible sub-variant is to only allow gradual escalation where the players can step from single card plays to 2 card plays, from 2 to 3, etc, but could not jump from a single card play to a 3 (or more) card play.

My instinct is that the variation we will like the most is to differentiate the face cards and allow gradual escalation for multi-card plays.

Further extension
Now that we've gotten comfortable with the basic and doubles games, we are considering a more complex version. The idea we are considering is to somehow limit the players' abilities to refresh their hands by redrawing so that burning a lot of cards will have a cost.

This is the rule modification, written for a 2 person game:

  1. Create a draw pile for each player with 15 cards.
  2. At the start of the game, each player draws 5 cards into their hand.
  3. Points are played as in the normal rules
  4. At the end of a point, the players refresh their hand up to 5 cards from their draw pile
  5. If a player runs out of cards in their draw pile, they cannot draw additional cards to refresh their hand.
  6. If both players run out of cards in their draw pile, then shuffle the pile of face-up played cards and give each player a new draw pile with 10 cards.
  7. Repeat step 6 as often as needed
The idea of this variation is to thematically mimic the idea that one player could push too hard, too fast, and get tired out relative to the other player. Strategically, our idea is that this will also create a tension between dumping your own low cards and letting the opponent dump.

Friday, July 8, 2016

Humbling Improv

Last night, I was part of an improv comedy show. Probably my first time performing on stage for ... at least 25 years.

How did I do? A lot of room for improvement. Not at all a surprise, but what really sticks with me is that the mistakes I made were in the absolute basics of improv:

  1. Listen to the other players
  2. Accept
  3. Who where what
Messing these up is common for beginning players and comes from a desperation to get laughs. I thought I was immune. I didn't think I was desperate. Heck, I didn't even care about the audience reaction.

But still, I made those mistakes and played badly. The experience was an interesting opportunity for better self-understanding and a vivid reminder to focus on the basics.

It was a great chance to fail and learn.

Wednesday, July 6, 2016

Am I dreaming?

Three Js playing cards together. Kid-initiated, kid-managed, no parents involved in any role:

Then, the game devolves a bit, based on excitement about a new set of library books that mommy has gotten:

Monday, July 4, 2016

Evens/odds and a quick update

Early years math seems to put a strange emphasis on even and odd numbers. Recently, a friend asked whether there was a point to this. Maybe it is just one of those little bits of terminology that we are asked to memorize for no reason?

By chance, this was something I had started considering about a month ago. It did seem strange that we spend so much time on this simple way of splitting integers. I wondered if it was worth the attention. From that point, my awareness was raised and I started noticing where it occurs and ways it links with more advanced concepts and future learning. My conclusion is that even/odd is surprisingly deep.

First, it is a simple version of concepts that will be developed further. For example, the alternating (starting with 0) even, odd, even,odd, even... is an illustration of a pattern. They will soon see other alternating patterns, then more complicated patterns and 2d or 3d patterns.

For another example, evens are multiples of 2, odds are numbers with a non-zero remainder when dividing by 2. This leads to understanding other multiple families, division, and division with remainder.

Second, the even/odd distinction is helpful for improved understanding of different calculations. For example, the observations that even+even = even, while odd + odd = even, etc. These can be used to help self-check their calculation and also will form early experiences with algebra. Similarly,
even x odd vs odd x odd reinforce understanding of multiplication. Again, this gets broadened for multiples of 3, 4, 5, etc.

Third, there are a lot of more advanced results that are easiest to prove by parity arguments. Sometimes we are working with a set of things that are even and the key observation is we can pair them up. Other times, we have a set that is odd and the key insight is that, when we pair them, one must be left over.

Recently, with the J1 and J2, we were looking at some constrained ways to put the numbers 1 to 25 on a 5x5 checkerboard. They were able to prove that some versions were impossible simply because 25 is odd, so there are more odd integers in 1 to 25 than even integers.

Lastly, there are techniques in computer science that involve even vs odd. This comes up pretty naturally because of the essential use of binary.

Some games

Recently, J1 and J2 have gotten hooked on some classic card games. In particular, we've been playing a lot of 3-handed cribbage. It is a nice way to do some simple addition practice and build intuition about probability. Probability is now getting even more share of mind: in the last couple of days we started playing poker together. This was actually inspired by some of our reading together.

We are reading the Pushcart War.

In one scene, there is a poker game. Of course, the J's insisted that I explain the game and were eager to try it out. J3 was the huge winner tonight, while I busted out. Oh well.