Friday, February 25, 2022

Early El Math Games

As my preservice teachers have had the opportunity to work with a K/1 classroom this year, I've been thinking a lot more about early math games. Mostly I'm trying to tie these to the components of number sense. 

Number Sense

In our class we discuss these as: 

  • one-to-one correspondence - as learners count, they have one (and only one!) number assigned to each object being counted.
  • hierarchical inclusion - (worst name candidate) the idea that a number contains smaller numbers. If you have 6 you also have 5, etc.
  • subitizing - visual recognition of quantities. Perceptual subitizing is immediate recognition of quantities, most commonly up to 5 or 6. Conceptual subitizing is visual chunking of a collection into smaller groups that can be perceptually subitized.
  • cardinality - the center and core. Recognition of numbers as quantities, a characteristic of a collection that doesn't change with rearrangement. Kids can have most of these other concepts but still not have assembled them into cardinality.
  • magnitude/comparison - both being able to directly compare quantities, and identify relative size - like locating where 7 is between 5 and 15.
If possible, my favorite thing for many of these games is for kids to have number cards which they have a hand in making. Similar to Tiny Polka Dot cards, which are a great commercial version. The idea is to make four suits, 0 or 1 to 10, where the suits are different representations of the numbers. Ten frames, symbols or shapes organized into patterns, randomly placed or groups of shapes to encourage subitizing, etc. You can have numerals or tally marks or number words if that's something you want your learners working on. I tend to prefer cards that involve counting and supportive structures. I used to have my own cards I'd print, but the opportunity for creativity, ownership and doing mathematics is strong with kids making the cards. (Not to mention some sneaky assessment.)

Once you have the cards, familiar games create terrific mathematical opportunities. Go Fish and Memory/Concentration create counting opportunities, and set up future games using those structures, like 10s or equation Go Fish or Concentration.

General Educational Game Advice
Many traditional games have a rule that when you're successful, you go again. I recommend against this because it increases wait time for other players, works against catch up, and can discourage the kids we want most to engage.

Similarly, I try to avoid games that emphasize speed, or require correctness to score and advance. I love for games to be an opportunity for collaboration and discussion, not a stand in for a quiz.

Divvy Up (Counting, Hierarchical Inclusion) Materials: Number Cards

Put about ten objects in the middle for each player. Using your number cards or dice, a player flips over a card and takes that many objects from the pile. Then counts up how many they have total. If appropriate, can have a score sheet where they write down that number. Game has two winners - one who takes the last object, one who has the most things.

Optional, arrange the 10 objects in two rows of five to sneak in some 5s structure and complements of 10.
Variation: if there are not enough to take, you have to pass. Encourages comparison, but can make the end take a while.

More or Less (Comparison, Strategy)
Materials: Number Cards

Idea: instead of War, which is not bad, in the math game sense, try this game. Draw 3 cards and teams take turns. The team whose turn it is chooses more or less. Both teams choose a card and hold it face down, then reveal. If more was chosen, the larger number wins, if less, the smaller. If it's a tie, you chose a 2nd card from your hand with the same rule.

More Together (Counting on, addition, hiearchical inclusion, decomposition)
Materials: Number cards mixed up in four piles.

Two teams: each turn over a card. Who has more? Then the teams turn over their 2nd card. Who has more together?

If learners are ready to count on, can just count from the first number. (6,5) Had 6, then 7, 8, 9, 10, 11 - pointing to pips on cards. If students would benefit from counting out blocks for how much (6 for this, 5 for that, count together), use blocks. Can introduce counting on here, too.

A tie? Flip over one more. No need for an overall winner, just who wins each turn.

Staircase  (Counting, counting on, hiearchical inclusion)

Materials: optional gameboard, a lot of stacking cubes and a die.

Play: roll a die, and build a stack of that many cubes, then roll another (or reroll) and add that many, with the two summands in different colors. Put them on your team’s track on the sum. If you already have that number, that’s okay, put it on the same space. Winner is the first to get three spaces in a row (make a staircase). Some students lay them down, some stand them up. Variation 1: If the three step game is too short, play to four or five steps. Variation 2: if you roll a sum you already have, you can choose to remove the same sum from your opponents’ board. (Increases interaction.) Variation 3: Playing with number cards 1-10. If you get a 1 or a 2 first card, you must take another. Otherwise it’s your choice. Bigger than 12 is a bust, you lose your turn. Probably best with a four or five step win condition, and can be combined with variation 2 as well. Lots of opportunity to notice and wonder. Notice the different ways to get the same sum, wonder how much you have together, notice that 2+5 is the same as 5+2, ask what you hope to get on that second die roll…

How many behind? (Decomposing, hiearchical inclusion, part part whole stories) Materials: 10 (or 12!) unifix cubes.
Show and count how many cubes in the stack. Now put the whole stack behind your back, and bring 1 cube out front. Ask: how many cubes behind my back? Next time, keep 1 behind your back, then show the rest. (If your partner’s there, have them go.) Learners and teachers take turns being the hider. If you want, you can always start with the same amount shown in front, or let people show a different number, then hide some behind. If the learners haven’t got the one less idea, try that one a few more times.

Big Three (Magnitude) Materials: deck of number cards. Idea: Players start with 3 face down cards. On your turn, draw a card from the deck or the top card of the discard pile. Replace one of your face down cards with it. No peeking! The goal is to find the biggest cards you can. The card you replace is then discarded, even if it was a high card. When someone thinks they have the biggest cards, they call “Last Turn” and everyone else takes one more turn. Players add up their cards to see who has the Big Three. Option: need more challenge? Play Big Four!
(Riff on Rat-a-Tat-Cat, a great commercial math game.)

Moving to Story & Operation
As kids have started to acquire number sense, we move into stories that provide the context for operations. The Cognitively Guided Instruction Framework, based on research analyzing how children acted out elemental math stories.
  • Join. One quantity, increasing over the story. Unknown could be the start, the change or the result.
  • Separate. One quantity, decreasing over the story. Unknown could be the start, the change or the result.
  • Comparison. Two quantities, related by the difference between them. Unknown could be the referent, the difference or the compared quantity.
  • Part Part Whole. Two quantities that are part of a group. Unknown could be either part or the whole.
  • Grouping. A number of groups, each group with a number of things, and a total. If the total is unknown, it's multiplication; if the number in the group is unknown, it's fair share/partative division; if the number of groups is unknown, it's measure/quotative division.

Comparison Game
Materials: number cards, especially if you have organized ones like dice face, hashmarks (if those are good for your kids), or ten frames. Plus 50-60 unifix cubes. Both players flip a card and build a stack that tall. Compare the stacks. Count the difference and take it off the taller stack. The player with more scores the difference. First player to 20 scored cubes wins. If it’s a tie, no score. Afterwards be sure to describe the score as 8 is 3 more than 5, or 5 is 3 less than 8. You could write down 5+3=8 (or 8-5=3 if they seem familiar with subtraction and super-comfortable with addition number sentence already.) Transition to them writing the number sentences and saying which is how many more than the other. If they are able to find the difference without counting blocks, make sure to have them describe their thinking. If they need challenge, don’t put the stacks together as they try to figure out how much more and less.

Making a Difference Materials: unifix cubes or counters about 30, number cards. Play: Both players have three cards. Choose a card to play. The lower card scores how many blocks it takes to make it equal to the other card - let the learners know that low cards are better.. If students can do with just numbers, that’s fine. But at least the first couple plays, build both numbers and count up how many cubes to make the difference. The person with the lower card scores those blocks. If it’s a tie, you have to play a second card from your hand. Draw back up to three cards. Winner is the first player or team to 12 cubes.

I feel like this is a place where games have made an inroad. But still, there's plenty of fun to be had.

10s Go Fish and Concentration Make 10
Pretty self explanatory. Remember to not let kids take extra turns. Both games I like to have kids score by counting their 10s.

Double Time (Doubles and counting on)
Materials: a game track, which can be numbered. 1 to 40 or 50 makes a good length with number cards, 30 is okay with dice. Bonus if you color or design the track in alternating spaces, to hint at the counting by 2s connection.

Play: students roll one die and move that plus the same. First to the finish line wins. I like to have students write down what they rolled and how far they went. 3+3=6, etc. If the track is numbered, you can start sneaking in some questions like 'Oh, you're on 24 and moving 8? Where will you end up?' For students working on counting on, this game provides lots of practice, since you don't start with 24, 1 is 25.

Ten Penny Game (Fives structure, sums to 10)
Have two ten frames out, the blocks, and some pennies or chips for scoring. Put a penny on the tenth spot of each. Players take turns rolling a die, and adding that many blocks to one of the ten frames. If they fill up the last spot, you get the penny as a point. Clear all the blocks and put on a new penny. There will be lots of opportunities for counting, counting on, and using the fives structure. "How many on this ten frame? How many more to fill it?" Are good questions here.

Cover All (Addition, decomposing)
This is the classic math game Shut the Box.

Cover All gameboard, but really all students need is a track from 1 to 10.
Play: roll two dice, and cover up any combination of numbers that add to the same amount.

With some kids, blocks help. If they set out how many they rolled, they can break them up in different ways. Consider questions to ask: what would be a good roll? What numbers might be harder to cover? What are different ways to split up our roll? (Helping them realize they have a choice.) What really makes this game a classic to me is that it really generates problems. Not how do you make 10, but how do you make 10 if I already used 7, 6 and 5. Is it even possible?

Dice Squares (adapted from Illustrative Math)
Materials: Gameboard, dice. This is a clever variation on dots and boxes. Roll two dice and fill in an edge next to that number. The player who puts the fourth edge on a box scores it! Mark with your symbol (X or O) or initials. 

Play with your students, thinking aloud at how you get your sums. For most of the kids, counting on would be a good strategy. 3 & 5, 5 -> 6,7,8. If students could benefit from using manipulatives to count, have them take as many as each roll, then find the total.

Make Your Own

Notice how simple some of these are? Really, some of these tiny math games are just born from thinking what do I want learners experiencing, and then adding dice or cards. Competition is fine - and a reason to engage for some learners, but try to avoid rewarding speed and correctness. Add in a representation (cards or the gameboard or a manipulative) and you probably have a classic in the making. (Then send it to me!) The easy wrinkle to add to the strategy and thinking required is to add choice. Much like More or Less above is basically War - with two layers of choice added in. Instead of flip a card, have a hand of two or three and choose one. Try to make choices real choices though. In More or Less, the choice of more or less makes the choice of the card much more significant.

Give Me More

Just two resources to end.

  • One of my favorite YouTube channels is Michael Minas, who makes up tiny math games with his kids and then demonstrates them. A lot of good games, but what's better is the spirit of invention.
  • Jenna Laib has a few easy, high leverage games. She writes about making games and then shares her favorites. We've used Number Boxes a lot this year, from 1st to 5th grade, just altering for what content the kids are thinking about. (Really, just read everything she writes.)

Just this week we were using ___ x ___ – ___ with a trash can ___ with 3rd graders. I wanted it not to be just who gets the biggest numbers, so added in the subtraction. I like having a trash can because it adds some choice, which gives even kids who have all their facts something to think about. There is so much thinking you can see and assessing you can do even just watching kids play these, and if you get to play with them... forget about it!

Game on!