Standards in Mancala: How do Common Core and NGSS Standards relate to Mancala play and study?
Mancala as an anchor phenomenon for mathematic and engineering/design play opens up a number of opportunities to connect to common core and NGSS standards. In this piece I'll share how we've seen Mancala play relate to standards and beyond. I make a distinction between mancala free play and mancala study play. Free play is just two children playing mancala for fun, using an agreed on rule set and not trying to solve any challenges or investigations as part of the play. Mancala study play is when students are attempting to solve investigations or challenges and use tools like notation or charts to help them in their investigations. Study play means they are still playing, the experience of fun is at the heart of their motivations for the investigation. Guard this quality of fun and keep it thriving in your club because it is in the fun that we find the learning!
Before we dig into this, I want to offer the suggestion that you experiment with the idea that you are not using mancala to teach these standards, but rather your students are using the skills and practices in these standards to get really good at mancala and maximize the fun of the game. See what happens when children can harness their academic learning to fuel fun!
· CCSS.MATH.CONTENT.K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
During free play with the game (playing just for fun without added study elements like notation or other challenges) This takes place during board set-up as students count to an agreed upon number for each hole, 3, 4, 5, 6 stones, etc. It also takes place at the end of turns when children count up how many more stones they got that round and at the end of the game when children count out how many stones they got.
During study play, this happens in every way described above, and also takes place as children experiment with different first turn strategies where they will count how many stones they win using a given algorithm. You will also see counting and writing numerals as children use notation to write down the total stones they won each turn. When students attempt things like the Mancala Bot Challenge, you will see them use conditional logic tied to exact quantities of stones in a given position. For example "If there are two in B, pick B first."
· CCSS.MATH.CONTENT.K.CC.B.5 Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
During free play, you will see this standard met over and over again as children count up their winnings at the end of a game and will often count above 20, as there are usually a total of 48 stones to be won in the game. You will also see this as students look at individual holes and count up their total before taking a turn, or counting stones in a "jackpot" hole that is full of stones. As children become more competent players, their subitizing skills (the ability to recognize a quantity as a whole group without having to count) will improve and they will be able to recognize familiar arrangements of stones and their quantity. When we provide 2 colors of stones, this process is accelerated and students can immediately see the kinds of number combinations that make up larger numbers.
· CCSS.MATH.CONTENT.K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
This happens throughout free play in two ways: First students will compare quantities of stones in their winnings to see who has more, or if there is a tie. Second, students may compare quantities of stones to the number of moves that number will make. This is essential for gaining extra turns in both Round and Round Mancala and Capture/Khala and Sungka. Students eventually learn to recognize the 1,2,3,4 strategy where if there is 1 stone in position A (1st position), two stones in position B (2nd position) etc. they know to pick them to get extra turns. For assessment, meeting this standard is visible in behavior as students lament or become excited when they see these opportunities ruined or made possible by an opponent's move.
· Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.
This happens repeatedly throughout free play and study play as children compare quantities of stones, but more interestingly this comes up as students begin to observe and compare the length of turns in Round and Round. Students begin observing duration or how many laps around the board a particular decision gives you. Students will also compare sizes of piles of stones and try to guess how many stones are in that pile.
CCSS.MATH.CONTENT.K.MD.B.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count
This is especially visible when teachers provide multi-colored mancala stones and students are given opportunities to notice and compare what they won. When teachers or players assign a point value to each color (eg. 1pt/red, 2pts/blue 5pts/green etc.) students will naturally group stones according to color and then students can also practice skip-counting as they sort their stones and tally their points.
Using multiple colors also connects to supporting number sense and knowledge of what combinations of numbers add up to larger numbers. Using two colors is great for observing ways to make a given low number like 4 or 6. Using more than two colors can make this less visible but opens up other possibilities mentioned above.
For what it’s worth, Mancala Club activities also bring up 1st grade math standards as well:
CCSS.MATH.CONTENT.1.MD.C.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
This comes up in mancala study play, where we use charts to describe first turn strategies and their outcomes, and compare those outcomes. In this phase, students discover that there are multiple pathways to get certain scores on their first turn, and we can find longer or shorter turns that result in the same score. This can be graphed from the chart of first turn strategies. Students also check one another’s algorithms (strategies) to see if they work. In this way they are interpreting data and checking it for errors.
Mancala and NGSS Engineering Practices
This section really only relates to mancala study play, rather than free play. Mancala games are in a sense a built system that we are inviting children to study and modify. In doing so they are engaging in engineering practices similar to what game designers or software engineers engage in. We’ve seen these engineering practices support a thorough exploration of mancala games and how strategies can be represented, and how the game can be modified to maximize fun. These practices are not designed to be isolated from one another, but instead are intended to be combined in iterations as part of the process of investigation. Some practices will be more heavily leveraged in the course of deepening students’ knowledge and enjoyment of mancala.
1. Defining Problems: Initially, we will define the first problem, “What is the best first turn strategy for Round and Round Mancala?” but expect children will define their own problems as the club progresses. During the Mancala Bot challenge, children define their own problems or challenges for their mancala bot to solve, for example, “Design a mancala bot that beats Mr. John’s Random Bot.” Other possible problems students may generate is “what is the best 1st turn strategy for Kallah (capture mancala). Or “What rules can we change in Round and Round Mancala to make first players and second players have the same advantage?”
2. Developing and using models: Through using and refining our game notation system, and through the Mancala Bot challenge that builds off of the notation system, children develop and use models to represent mancala game play and eventually strategies. We see students add conventions to notation to communicate things like board states or if they got a jackpot. Rather than teachers providing all possible commands in our Mancala Bot “computer” language, students will create and name commands to add to this “computer” language. In this way they develop models of mancala moves and formally represent their decision-making. Ultimately the mancala bot is a simulation of their strategy, written in a language that other players (mancala bots) can follow and critique.
3. Planning and carrying out investigations: The heart of this club is a series of investigations around mancala. The challenge of identifying and representing a best first turn for first and second players, and the process of trying multiple rule variations and comparing game play experiences are examples of investigations the children will engage in through this club. When children explore designing their own mancala games, playtesting is an essential investigation that is part of the process. Designing a mancala bot is an investigation where students try to distill their approach to playing their game into simple written instructions a friend or parent (the mancala bot) can follow. We’ve also seen students invent their own investigations like “how do I create notation to show a board state?” Or “When the board is like this, what is the best move?”
4. Analyzing and interpreting data: During the First Turn Challenge we use a simple mancala notation system to help children track their play and game decisions, and include in this notation system a count of how many stones they finish their first turn with. This data can be shared through mancala strategy charts on the wall that will allow students to compare effectiveness of different strategies.
5. Using mathematics and computational thinking: Mancala games lend themselves to computational thinking as children count and estimate number of stones and decide moves to make on their turn. Complex computations occur as students evaluate possible moves, and a set of fractal patterns emerge through the strategy known as “walking the stones.” A second level of computational thinking inherent in this club is programing of mancala bots to represent what children understand about their strategies, and debugging bots as they playtest them.
6. Designing Solutions: The quest for the ultimate first turn results in multiple solutions that students evaluate through comparing stone counts at the end of their turn. Other criteria may emerge as students identify strategic benefits or limitations of how stones are laid out at the end of a turn. One exciting discovery is that different series of moves can result in the same score at the end of the first turn. The mancala bots themselves are solutions to particular questions like “Design a bot that can beat Random Bot.” When children explore designing and playtesting rule sets, they are designing solutions to make the game more fun.
7. Engaging in argument from evidence: The main evidence children will be referring to here will come from stone counts at the end of first turns and at the end of games. Board positions will also be evidence that children might use to compare different strategies.
8. Obtaining, evaluating, and communicating information: At the end of each club session, children share their discoveries, whether it’s a new strategy, a new way to play mancala, their experience with a rule set, or a challenge they beat. A really fun literacy activity that connects to this is writing letters to future players to give them advice on how to play or to pose challenges to another classroom. In addition to comparing strategies, the mancala bot activity also gives children a chance to evaluate programs and identify “bugs” or errors in the programing logic. The act of designing your own mancala game and writing out the rules requires careful communication so that the game can be understood by players.