## Screenshot: Case Study on Critical Tokens in Problem Solving ### Overview The image presents a case study comparing two responses to a math problem involving point calculations in a game. It highlights critical tokens (color-coded keywords) to analyze reasoning errors and correct solutions. The problem involves determining the number of rounds/games played based on point gains/losses. ### Components/Axes - **Title**: "Case study on the identification and effects of critical tokens" (top blue banner). - **Original Question**: - Text: "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**: - Text: "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)" - **Responses**: - **Response 1** (incorrect): - Steps with color-coded tokens: 1. "Determine [Understand] the total points Jane earned scored by [. ] subtracting [winning] the points she lost from her [the] final score. Total points earned = 60 – 20. Points lost. Total points earned = 40." 2. "Since Jane earns 10 points for [per] each round she [won] wins, we can find the number of rounds she played by dividing the total points earned by the points earned per round. Number of rounds = 40 / 10. Number of rounds = 4." - Final answer: "Therefore, Jane played 4 rounds." (marked with ❌). - **Response 2** (correct): - Steps with color-coded tokens: 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." 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." 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 = 80 / 10 = 8." - Final answer: "8" (marked with ✅). ### Content Details - **Color-Coded Tokens**: - **Red**: "Determine" (Step 1), "points earned" (Step 2). - **Purple**: "Understand" (Step 1), "winning" (Step 1). - **Blue**: "per" (Step 2), "won" (Step 2). - **Green**: "games won" (Step 3), "10" (Step 3). - **Ground Truth Answers**: - Both questions have a ground truth answer of **8**. - **Annotations**: - ✅ (correct) and ❌ (incorrect) symbols next to final answers. ### Key Observations 1. **Critical Token Identification**: - Tokens like "Determine," "Understand," and "won" are highlighted to emphasize logical steps and potential errors. 2. **Reasoning Errors**: - Response 1 incorrectly subtracts lost points from the final score (60 – 20 = 40) instead of adding them to isolate points earned from winning. - Response 2 correctly sets up the equation (60 = Points earned – 20) and solves for the correct value (80 points earned from winning). 3. **Structural Differences**: - Response 1 uses ambiguous phrasing ("scored by [. ] subtracting [winning]"), while Response 2 uses precise algebraic reasoning. ### Interpretation The case study demonstrates how critical tokens (keywords) guide the identification of reasoning flaws. Response 1’s error stems from misapplying arithmetic operations (subtracting lost points instead of adding them), while Response 2 correctly isolates variables using algebraic principles. The color-coding of tokens helps trace the flow of logic, revealing how minor phrasing differences ("per round" vs. "per game") and operational errors (subtraction vs. addition) lead to divergent conclusions. This underscores the importance of precise language and step-by-step validation in problem-solving. **Note**: No numerical data trends or visualizations (e.g., charts) are present; the focus is on textual analysis and token-based reasoning.