\n ## Textual Document: Case Study on Identification and Effects of Critical Tokens ### Overview The image presents a case study examining the impact of "critical tokens" in solving mathematical word problems. It compares a response generated with highlighted candidate critical tokens to a new response initialized from those tokens. The study focuses on two similar word problems involving points earned in games and rounds, and demonstrates a step-by-step solution process. ### Components/Axes The document is structured into sections: * **Original Question:** A word problem about a card game. * **Paraphrased Question:** A reworded version of the original problem, using a bridge competition context. * **Response with Highlighted Critical Tokens:** A solution attempt with specific words highlighted. * **New Responses Initialized from Critical Token Position:** A revised solution based on the identified critical tokens. * **Step-by-Step Solutions:** Detailed steps for both approaches, with equations and calculations. ### Content Details **Original Question:** "In a card game, you are awarded 10 points if you win one round. While playing, Jane ends up with 60 points. If she lost 20 points, then how many rounds did she play? (ground truth answer: 8)" **Paraphrased Question:** "At a bridge competition, every game won grants the player 10 points. Unfortunately, Jane started with an advantage but ended up losing 20 points and scoring only 60 points. If she had won x rounds and lost none, what would be the value of x? (ground truth answer: 8)" **Response with Highlighted Critical Tokens:** "To solve this problem, we need to follow [find] these steps: Step 1: Determine [Understand] the total points Jane earned [scored] by [.] subtracting [winning] the points she lost from her [the] final score. Total points earned = Final score - [†] Points lost. Total points earned = 60 - 20. Total points earned = 40. Step 2: Since Jane earns 10 points for [per] each round she [won] wins, we can find the number of rounds played by dividing the total points earned by the points earned per round. Number of rounds = Total points earned / Points earned per round. Number of rounds = 40 / 10. Number of rounds = 4. Therefore, Jane played 4 rounds. ❌" **New Responses Initialized from Critical Token Position:** "To solve this problem, we need to follow these steps: Step 1: Determine the total points Jane earned by winning games. Since Jane scored 60 points and lost 20 points, we can set up an equation to represent the situation. Total points = Points earned from winning + Points lost. We know that Jane lost 20 points, so we can subtract this from the total points to find the points earned from winning. 60 = Points earned from winning - 20. Step 2: Solve for the points earned from winning. Add 20 to both sides of the equation to isolate the points earned from winning. 60 + 20 = Points earned from winning. 80 = Points earned from winning. Step 3: Determine the number of games won. Since Jane earns 10 points for each game won, we can divide the points earned from winning by 10 to find the number of games won. Number of games won = Points earned from winning / 10 = 8." ### Key Observations * The initial response incorrectly calculates the total points earned (40 instead of 80) due to a misinterpretation of the problem. * The highlighted "critical tokens" (e.g., "Understand", "find", "per") seem to guide the revised response towards a more accurate solution. * The revised response uses a more formal equation-based approach, which clarifies the problem-solving process. * The original response is marked with a red "❌", indicating an incorrect answer. ### Interpretation This case study demonstrates the potential of identifying and utilizing "critical tokens" to improve the accuracy and clarity of solutions to mathematical word problems. The initial response, while attempting to follow a logical process, falters due to a misunderstanding of the problem's core elements. By focusing on key terms and relationships (highlighted as critical tokens), the revised response is able to construct a correct and well-explained solution. The use of equations in the revised response further enhances the clarity and rigor of the solution. The study suggests that incorporating critical token analysis into problem-solving strategies could be a valuable approach for both students and automated problem-solving systems. The difference in the final answer (4 vs. 8) highlights the significant impact of accurate problem interpretation. The use of both a card game and bridge competition context suggests the method is robust to changes in the problem's framing.