Question

A child discovers that \(2 \times 2\) is the same as \(2+2 .\) He therefore wrongly concludes that \(3 \times 3\) is the same as \(3+3\) What tendency is affecting this child's problem-solving strategies? a. mental set b. divergent thinking c. test bias d. response bias

Short answer

The tendency affecting the child is a mental set.

Step-by-step solution

Understanding the Problem

The problem is to identify the tendency affecting the child's problem-solving strategy. The child mistakenly concludes that multiplication follows the same pattern as addition, based on a single instance where the result is similar.

Analyzing the Child's Conclusion

In the problem, the child observes that \(2 \times 2 = 4\) is the same as \(2 + 2 = 4\). This observation leads the child to incorrectly assume a general pattern that \(n \times n = n + n\) for any number, which is not true.

Identifying the Tendencies - Mental Set

A mental set is a cognitive tendency to approach problems in a certain way, often based on past experiences or what has previously worked. This may lead to a fixation on a specific pattern or method, such as the operations of addition being similarly applied to multiplication.

Confirming the Correct Option

The child applied the specific pattern seen in \(2 \times 2 = 4\) and \(2 + 2 = 4\) to other problems like \(3 \times 3\), showing a fixed mindset. This fits with the definition of a 'mental set' as they are stuck on using this pattern to solve different problems, ignoring mathematical rules for multiplication.

Key concepts

Mental Set

A mental set refers to the tendency of people to solve problems using a familiar strategy, even if a different approach would be effective. It involves the use of past experiences to guide current problem-solving attempts. In the example provided, the child incorrectly generalizes that multiplication follows the same rule as addition, because it occurred once where the outcomes were identical. This illustrates how a mental set can limit one's ability to recognize that a problem requires a different solution strategy. The child is trapped in a fixed mindset, focusing on the pattern they witnessed, rather than the underlying principles. This cognitive bias restricts flexibility in thinking, potentially leading to repeated errors or misjudgments.

Problem-Solving Strategies

Problem-solving strategies are specific methods employed to find solutions to challenges or equations. These strategies can include trial and error, algorithms, heuristics, and insight. In our example, the child uses a flawed approach of directly equating multiplication to addition based on limited observation and fails to apply different strategies that are typically more effective in mathematics. It highlights the importance of understanding various methods and knowing when to utilize them. Effective problem-solving requires adapting one's strategy based on the unique demands of a given situation. This flexibility can help avoid the pitfalls of a mental set and improve problem-solving outcomes.

Cognitive Tendencies

Cognitive tendencies are habitual patterns of thought that influence how we perceive and address problems in our daily lives. These include biases, assumptions, and predispositions that impact decision-making processes. The child in the exercise demonstrates a common cognitive tendency known as overgeneralization, where a single observed instance is used to draw wide-ranging conclusions. Recognizing and adjusting for these innate biases is crucial for accurate reasoning and effective problem-solving. Being aware of how cognitive tendencies can skew our interpretations assists in developing more accurate and holistic approaches to tackling problems.

Mathematical Reasoning

Mathematical reasoning is the process through which individuals use logic and mathematical principles to solve problems. It involves understanding and applying rules and operations correctly. In the child's case, there is a breakdown in reasoning, where they conclude that the operations of addition and multiplication are interchangeable based on insufficient evidence. Effective mathematical reasoning requires not just recognizing patterns, but also verifying that these patterns hold under different conditions. This skill is critical for achieving accurate and reliable results in mathematics and everyday problem-solving. Strengthening mathematical reasoning involves practice, exposure to diverse problem types, and learning how different mathematical principles interrelate.