Number Munchers is a DOS game developed by Minnesota Educational Computing Consortium (MECC) in the 80’s to teach elementary schoolers to identify factors and multiples, as well as to evaluate algebraic expressions to make equality/inequality comparisons. You can play the game here. Players move around a 2D grid, munching the numbers that meet the level’s criterion, while avoiding the “Troggles” (monsters) wandering the grid. You must munch every number (or expression) that matches the level’s requirement to beat the level, and you lose a life if you either munch a wrong number, or run into a Troggle. Levels are untimed, so as long as you avoid the Troggles, accuracy matters more than speed. Before starting the game, you choose a mode: Multiples, Factors, Primes, Equality, Inequality, or Challenge. In Multiples and Factors, each level will display a condition like “Multiples of 5”, or “Factors of 20”. In Primes, every level has the same condition of munching only prime numbers.
Equality levels have conditions like “Equals 10”, where each square on the grid has an algebraic expression like “5+5”, instead of a single number. The Inequality levels are similar, except the prompts will look like “Not equal to 5”, “Greater than 10”, or “Less than 16”. In Challenge mode, every level is a random mode.
Learning Objectives
- Students will be able to identify multiples of 2-10
- Students will be able to identify factors of 2-20
- Auxiliary goal: Students will recognize that some numbers are only divisible by 1 and themselves
- Students will be able to identify prime numbers < 40
- Students will be able to evaluate two term/factor expressions of numbers < 40
- Given a two term/factor expression and a number for comparison, students will be able to determine whether the number equals the expression and if not, which is bigger.
Students playing a given mode can be expected to improve in the corresponding objective. So students playing “Multiples” mode will likely get better at answering the question, “Is X a multiple of Y?”, where X < 40 and Y < 10. Similarly, students are likely to improve their accuracy at performing addition, subtraction, multiplication, and division, depending on the mode and their strategy (multiplication is used in every mode, division is helpful in every mode, but can sometimes be re-framed as multiplication, addition and subtraction are only practiced in Equality, Inequality, and Challenge modes. It is reasonable to suppose that these improvements to arithmetic fluency and to divisibility judgements would transfer outside of the game to either paper tests, or even real-world scenarios like money handling where the need for arithmetic might organically arise. This is because the kind of in-your-head quick thinking required to munch the correct numbers mirrors the way we do such simple arithmetic problems in our heads outside of the game. Paper tests are the most similar to the in-game task insofar as the way the expressions are presented in-game looks like how analogous questions could be asked on a paper test. Every level could be cleanly mapped onto a series of questions about whether any of the following expressions fit the level’s pattern. So students playing “Multiples mode” will likely improve their ability to answer questions like “Is 12 a multiple of 3?”, or more broadly, “Which of the following numbers are multiples of 3?”.
Notably, where memory is concerned, the game is more about recognition than recall; students using a memorization strategy (like memorizing the multiplication tables) usually need only to decide whether a given answer fits the description; they do not need to produce their own answers. This means students playing the game may not see less improvement in their ability to answer the question, “Name 10 multiples of 3”, than their ability to identify 10 multiples of 3 from a bank of potential answers. The Equality and Inequality modes are exceptions to this, as students need to simplify expressions to compare them to the level’s requirement. Additionally, students may use arithmetic strategies to identify numbers to munch, and this could improve their recall and arithmetic fluency even on the simpler levels.
Prior Knowledge
Players must be able to read numbers, and be able to perform addition, subtraction, multiplication, and division of a one-digit number with a two-digit number. For example, in Multiples mode, if the goal is to find “Multiples of 5”, players should be able to multiply 5*3, or divide 15/3 or 15/3 to confirm that 15 is indeed a multiple of 5. Additionally, it they should understand that a prime number means a number whose only factors are 1 and itself. This way, when playing Primes mode, they can check whether 17 is a prime, using the fact that primes are only divisible by 1 and themselves, by seeing whether figuring out whether 2, 3, 4, etc. go into 17. Players must additionally have sufficient computer literacy to navigate their character with the arrow keys (or WASD) and use space to munch.
That might be bullshit, though. All of the modes can be approached with a variety of strategies that have varying requirements for prerequisite knowledge. Take #2: identification of prime numbers. You don’t strictly need to know what a prime number is to be able to recognize one. The game gives you correctness feedback every time you munch. It is possible to figure out which numbers you can munch in Primes mode by brute force and memorization, in which case you wouldn’t need to know what a prime is going into the game. You might start to figure it out as you played however.
Similarly, for Factors mode, it’s easy to say that you need division to identify which numbers are factors of a given number, but students could instead approach the problem as multiplication by thinking something like “3*5=15, so 3 is a factor of 5; munch!” These factors muddy identifying exactly what prior knowledge is necessary to learn from the game, and the issue is further complicated by the fact that each learning objective can be independently pursued by sticking to a given mode. TL;DR; the above skills should be enough, but you may not need all of them, depending on which learning objectives you’re pursuing and which approaches you use.
Mechanics
You move around with the arrow keys, and hit spacebar to munch a number or expression. You gain points when you munch correctly, and you beat the level if you munch every correct number or expression. You lose a life (1 of 4) if you munch a wrong number/expression, or if you run into a Troggle. Each Troggle type has a different movement pattern which stays the same throughout the game. Squares on the grid will randomly become safe zones that vaporize Troggles. Every 3 levels, gameplay is interrupted by a goofy Road Runner style cutscene, where a Troggle tries and cartoonishly fails to hit the number muncher with a rock, dynamite, etc.
One interesting consideration is that Factors mode contains a number of levels where you need to find all the factors of a prime number, without telling you that there is anything special about these numbers, or calling them prime. Something like 3 out of the first 10 levels in Factors mode will ask you to find all the factors of a prime, where the only munchable numbers will be 1 and the given prime. This seems to attempt to implicitly build students’ intuition about what a prime number is, without naming it. It is reminiscent of Zombie Division in this regard (Zombie Division also teaches students to make divisibility judgements and presents primes without explicitly naming the concept). The Primes mode does offer explicit feedback for identifying why numbers are not prime, though it never defines exactly what a prime is:
Dynamics
The Troggles are not very threatening, and there is no time pressure, so generally the game goes as quickly as the player chooses to move. You can move in a deliberate grid, getting every answer in a given row, column, etc., or you can erratically jump around munching any number that you spot as correct. Beyond the order in which you munch, there is no prescribed strategy for how to decide whether a given number is munchable. The relationships tested in the game can generally be recognized from a number of angles (like using multiplication or division), and the game doesn’t differentiate between these thought processes at all. This freedom to do the task in a way that suites the player focuses the core experience around the identification of numbers and expressions that match the given pattern, which is well aligned to the game’s goals.
Aesthetics
The inclusion of the enemies and randomness with which they and the safe spaces appear, makes the game feel somewhat frantic, even though you could play quite slowly without penalty. Often you are left with the sense that you must have gotten all the numbers already, how could you have missed one?! This makes the game feel like a bit like a scramble to search the whole grid for the one thing you missed while making sure not to bump into any enemies. I’d call the aesthetic a scrambling treasure hunt. The treasure hunting feeling (finding the right numbers/expressions) definitely supports the core learning objectives of pattern recognition.
It’s unclear whether the frantic scrambling supports learning, however. It probably decreases accuracy, but it does boost engagement. The game would not be fun without being pursued by the enemies, because it would be too easy. So in this sense, the frantic feeling may support learning simply by encouraging learners to play.
Learning Science Principles
Timely feedback: Players receive immediate, just-in-time feedback on every munch. Either it is correct, and they get points, uninterrupted, or it is incorrect, and they receive an error message explaining their mistake (this pauses the game). This supports learning by allowing students to iterate quickly. It explains their mistakes and confirms their correct actions so they know what went right and what to improve.
Segmentation: The silly cutscenes break the game’s pace, allowing you to rest and laugh in the face of the foolish Troggles. I’m not confident in this point, but I think this segmentation bolsters learning by decreasing the cognitive load, and allowing learners a break when they need it.
Fading Scaffolding (by increasing degrees of freedom): The levels increase in complexity over time by presenting more complex expressions or larger numbers to work with. This furthers the students learning by increasing the cognitive load required of them once they have mastered lower levels.
Final Analysis
Number munchers works well, as an engaging pastime and an educational tool. Each mode focuses on a single learning objective, and these objectives are well-aligned to the core mechanics of identifying expressions that match the level’s requirement. Excluding a timer in favor of enemies adds tension to support engagement while avoiding prioritizing speed over accuracy. The aesthetics serve to incentivize meeting the learning objectives by making finding the correct expressions feel like a treasure hunt, and by using a frantic flight from enemies to create tension that drives engagement. The relatively narrow focus of the learning objectives facilitates a tight delivery that targets growth very well.
The biggest ambiguity about the game’s effectiveness centers around the multitude of approaches that can be brought to bear in a given mode (flexibility in the dynamics). Students may or may not need to use and improve their arithmetic fluency to succeed on the simple modes of Multiples, Factors, and Primes, as these tasks can largely be completed with recognition. However even the recognition strategy requires students to improve their ability to identify Multiples, Factors, and Primes, and in this sense, the game succeeds in teaching its learning objectives so long as a player persists in playing. The more advanced objectives of arithmetic fluency are indeed fostered by the more difficult modes, which can’t be approached with pure recognition. These fluency objectives can even be developed on the easier modes, depending on the player’s strategy. Number Munchers provides a robust and engaging learning experience that has stood the test of time.
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