Why some multiplications do not make sense?

 

The Confounding Nature of Multiplications: When Results Just Don't Feel Right

Mathematics is a field of precision and logic, yet sometimes certain multiplication results can feel puzzling or unsatisfying. Have you ever looked at results like 17 multiplied by 3 equaling 51 or 19 multiplied by 7 yielding 133 and felt that something just doesn't add up? You're not alone in this feeling. Let's explore why some multiplication results might not make sense and what psychological and mathematical factors contribute to this phenomenon.

The Discomfort of Multiplication

1. Lack of Familiar Patterns

   • Uncommon Results: Unlike more familiar multiplications like 73=21 or 35=15, which we encounter often in early education, results like 173=51 or 197=133 are less common and don't follow the easily recognizable patterns we are used to. This unfamiliarity can create a sense of discomfort and confusion.

2. Cognitive Load

   • Mental Strain: Multiplying larger or less common numbers requires more cognitive effort. The brain has to work harder to compute and verify the results, leading to a feeling of mental strain. This increased cognitive load can make these multiplications feel less intuitive and more perplexing.

Psychological and Perceptual Factors

1. Expectation and Consistency

   • Mismatch with Expectations: Our brains thrive on consistency and predictability. When a multiplication result doesn't match our internal expectations, it can feel off. For example, seeing 19*7=133 might initially seem incorrect because our brains are more attuned to simpler and more familiar multiplications.

2. Numerical Aesthetics

   • Aesthetic Preferences: Some numbers and results are perceived as more aesthetically pleasing than others. Multiplications that result in irregular or complex numbers, like 133, can lack the aesthetic appeal of simpler, more symmetrical results like 64 or 36.

Mathematical Factors

1. Prime Numbers

   • Complexity of Primes: Numbers like 17 and 19 are prime numbers, meaning they cannot be divided evenly by any number other than 1 and themselves. Multiplying primes often results in less familiar and more complex numbers, which can feel less intuitive and harder to verify mentally.

2. Lack of Multiples

   • Non-Multiples: Multiplications involving non-multiples often result in less common numbers. For instance, 173 or 197 don't produce results that we encounter frequently in basic arithmetic, making them stand out as unusual and harder to reconcile mentally.

Educational and Cultural Influences

1. Educational Emphasis

   • Focus on Simpler Multiplications: In early education, emphasis is often placed on learning and memorizing simpler multiplication tables (up to 12). Less attention is given to more complex multiplications, which can make these results feel less familiar and more difficult to grasp.

2. Cultural Context

   • Shared Experiences: Cultural and societal norms influence our collective experiences with numbers. Multiplications that are not commonly used or emphasized in daily life can feel alien and harder to accept as valid.

Individual Differences

1. Personal Associations

   • Unique Perceptions: Individual experiences and preferences play a significant role in how we perceive numerical results. Personal associations with certain numbers can influence whether a multiplication feels right or not.

2. Cognitive Styles

   • Variations in Processing: Different cognitive styles mean that people process numbers and calculations in varied ways. What feels intuitive to one person might feel confusing to another, based on their unique cognitive approach.

Conclusion

The perplexity surrounding certain multiplication results is a complex interplay of familiarity, cognitive load, mathematical properties, and individual perception. While some results like 173=51 or 197=133 might not make immediate sense or feel satisfying, understanding the underlying factors can help demystify this phenomenon. Recognizing that these feelings of confusion are shared and rooted in the intricacies of human cognition and mathematics can provide comfort and clarity. So next time you encounter a baffling multiplication, remember that it's all part of the fascinating world of numbers and the human mind's interaction with them.