Multiplication & Fractions Math Games from Denise Gaskins (a review)

I really like Denise Gaskins's new book: Multiplication & Fractions Math Games (links to paperback edition and accompanying printable.) How much do I like it? Well, I had already written a lengthy review that, somehow, I managed to lose and am now back writing another one.

I'm going to forego my preferred Good (what I liked), Bad (arguable weaknesses) and Ugly (unforgivable sins) because I don't really have anything to say in those two negative categories. Instead, let me just talk about who would find the book useful and why.

Group 1: Parents who feel their own math skills are weak.

Maybe you never really understood what multiplication means or what fractions are? As long as you start with an open mind and are willing to engage playfully, the activities in the book can help you as you help your kids. It starts with models that are visual explanations of the concepts. Gaskins also breaks learning these concepts into comfortable steps that emphasize patterns and relationships, the real ideas that are behind properly understanding multiplication and fractions (indeed, math generally). The sequence of games in each section starts by building familiarity and then fluency (speed) to solidify all of that work.

Group 2: Parents who worry about their kids struggling with these concepts

Anecdotally, these two areas are the first major stumbling point for students in their math studies. As I noted above for parents themselves, the sequencing in the book will help kids develop a strong foundation, beginning by understanding what multiplication means (and what fractions are). Beyond that, playing the games will make these concepts familiar and, I believe, lead them to recognize examples around them in their daily lives.

Group 3: Families who like to play games

Kids (and parents!) find these games fun. I've been field testing math games for the last 18 months and keep seeing how engaged kids get when playing math games. I have played many, though not all, the games in Multiplication & Fractions and strongly believe the games in the book will be winners with most kids.
Now, let's face it, you might not be thrilled with every game. For example, I wasn't so excited by the idea of playing War variations. However, a lot of other games in the book that are strategically and mathematically rich. Also, truth be told, my kids and students have really enjoyed playing multiplication war!

So, there's really nothing weak in this book?

There is only one worry I have about making a blanket recommendation: parents who start with a completely wrong mindset. If you believe in speed over understanding or mathematical gifts instead of effort, then this book is the wrong place to start. Instead, read Dweck's Mindsets and spend time with Jo Boaler's website. Maybe also re-watch Karate Kid (no joke, this is what I'm currently playing.)

A disclaimer, sort of

I'm friends with Denise Gaskins and got a review copy of this book. However, you should understand that we're friends because I'm a fan of her math teaching work and not the other way around. We've never even met in real life and, in fact, live in different continents. I know of her because advocacy of play-based math learning. I admire her because she is one of the best at creating resources that bring this material into the reach of the typical parent.

Vacation Plan: Emotional Skills

This month is a school vacation period for the three Js. One area of focus this month will be on emotional intelligence skills.

Component Skills

We found a nice overview on Psych Central. We talked through the first four with J1 and J2 to start the month:

1. Self-awareness: (a) recognize your own emotions and their effects, (b) sureness about your self-worth and capabilities
2. Self-regulation: using a number of techniques to alleviate negative emotions
3. Motivation: tools to manage motivation to achieve goals.
4. Empathy: discerning the feelings behind others’ signals

The article also includes Social Skills as a category, but these seem separate to us.

Around the discussion of empathy, J1 asked how it differs from sympathy. We think that a difference is understanding how other people feel and their perspective (empathy) vs sharing their feeling (sympathy). I admitted that I do not have much sympathy.

A vocabulary list

Underlying many of these skills is a vocabulary of emotions. We found two nice resources for this:

• Emotions color wheel: this is a great visual for the kids. 
• Vocabulary list for greater shading: the idea is to move beyond the standards, happy, sad, angry, to get more shading and nuance. Another hope is that, in the moment of analyzing the emotion and comparing with the vocabulary, it will help their self-awareness and provide a point of detachment from the emotion.

Our Focus

J1 chose to focus on skills relating to empathy and sympathy.

J2 chose to focus on skills related to self-regulation.

Daily Schedule

Supporting this skill development and general household organization, we are posted this schedule for the month. You can probably tell that the boys helped write the schedule:

Things to do everyday:

1. 3 pages of Beast Academy and discussion with J0
2. 10 minutes of spelling with P
3. Vocabulary: writing a sentence and 5x words (5 words/ day for J1, 3 words/day for J2) 

07:00 Wake up
          urinate
          put dirty nightclothes in basket
          shower
          get dressed

08:00 prepare breakfast
          eat breakfast
          brush teeth
          poop

12:00 help with lunch
          eat lunch
          clear table from lunch

17:00 help with dinner
          eat dinner
18:30 clear table from dinner
          practice music
          put dirty clothes in basket
          urinate
          bath
          brush teeth

20:00 get in bed
          listen to story
20:30 lights out

Block blobs redux

Last December, we played Block Blobs (notes here). This week, we are trying a slightly modified version for two digit multiplication.

The Game

Materials

• 4d6. Two dice are re-labelled 0, 1, 1, 2, 2, 3 (see notes below)
• Graph paper (we are using paper that is roughly 20 cm x 28 cm, lines about 0.5 cm apart)
• colored pencils

Taking a turn

Roll all four dice. Form two 2-digit numbers using the standard dice as ones digits. Then, use your color to outline and shade a rectangle in the grid so that:

1. the side lengths are the 2-digit numbers you formed with the dice
2. At least one unit of the rectangle's border is on the border of your block blob
3. Note: the first player on their first turn must have a corner of their rectangle on the vertex at the center of the grid. The second player has a free play on their first turn.
4. Write down the area of your rectangle

Ending the game

The game ends when one player can't place a rectangle of the required dimensions legally.

When that happens, add up the area of your block blob. Higher value wins.

Notes

"Counting" sides
The side lengths of the rectangles are long enough that counting on the graph paper will be irritating. Instead of counting directly, they can measure the side lengths. For our graph paper, the link between the measurement and the count is nice, since the paper is very nearly 5mm ruled, so they just double the measure. I think this is a really nice measurement and doubling practice, too.

Dice labels

Other labels could be used on the special dice. We chose this arrangement because of the size of the graph paper. Rectangles with sides longer than 40 often won't fit and we think we will even need some single digit rectangles to allow a fun game length. An alternative we are considering is 0, 0, 1, 1, 2, 2.

As an alternative to labeling the dice with new numbers, you could label them with colored dots and give out a mapping table. For example:

blue corresponds to 3
red corresponds to 2
green corresponds to 2
yellow corresponds to 1
black corresponds to 1
white corresponds to 0

This would keep the tens digit dice distinct from the ones digit dice and allow rapid modification if you want to change the allowed tens digits (just tell everyone a new mapping). Alternatively, you could create a mapping using "raw" dice and even allow more strategic flexibility from the players. I have a feeling that this would be confusing to most kids, though.

Reinforcing the distributive law

To facilitate calculating the area and reinforce the distributive law, you might have the students split their rectangles into two (or four or more) pieces and calculate the partial products. You can further decide whether to ask them to split the sides in particular ways or encourage them to find the easiest way to split to help them calculate.

Our math curriculum

Sasha Fradkin (who writes one of our favorite blogs) asked a question about the curriculum we use. My reply was getting long, so I decided to make it into a separate post.

Do we use an existing curriculum or are we making our own?

We are doing a mix. My wife prefers to have a linear curriculum as a guide and fall-back, in case there wasn't time to plan anything more customized. She currently uses:

• RightStart/Abacus Math: ok, but not exceptional curriculum, highlight is the extensive use of physical manipulatives.  
• Beast Academy workbooks: wrote more extensively about this in a review before. I think these create good jumping-off points for really fun conversations. 
• DreamBox: for consolidation of standard skills, our enthusiasm for this is waning, rather than waxing right now (noted in same review as BA). 

We also use the CCSS math standards as a reference. I periodically check against the standards to see whether we are missing anything. If so, I will go to the Georgia Standards of Excellence, read through their activities for the related unit, and pick a couple that seem fun.

If I were forced to use only a single source, GSE would be my recommendation.

About Georgia Standards of Excellence
As far as K-5 math, this is a really awesome resource with a ton of great activities. For some reason, we find the webpage organization a bit confusing, so here is our quick recipe.  

To get to the great activities, I click the expansion menu in the right-hand box for the grade level of interest, then look at the curriculum map for that grade. I find the topic of interest, then click the link for the unit that covers that topic. The unit doc includes a lot of teacher background, which I mostly skip and focus on the activity descriptions.

Games and explorations
My personal preference is much less structured. I really like games and explorations and spend a lot of time exploring math activities on the MTBoS. Most of what I do with the kids is inspired by something I saw while doing my own play.

That said, there are some sources that are so good, we are essentially going through all of their activities:

1. Mathpickle. Cannot say enough positive about this.
2. Peter Liljedah's numeracy activities. Now that I notice them, I bet his good problems, card tricks, and resources pages will all have gems as well. 
3. NRICH and Wild Maths.

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

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.

Agenda:

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.

C-P-A

"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:
http://naturalmath.com/2013/09/12-models-of-multiplication/

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?"

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 1, 2, 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. WODB.ca 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 .....!

TMC16
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.