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Wyyyb committed Feb 2, 2025
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Expand Up @@ -268,15 +268,14 @@ <h1 class="title is-1">32B Models Comparison</h1>
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<h1 class="title is-1">Case Study</h1>
<h1 class="title is-1">Case Studies</h1>
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<p>
Here we showcase a detailed example comparing the performance of CFT with the original model. This case demonstrates how CFT helps improve mathematical reasoning abilities.
Here we showcase detailed examples comparing the performance between three models: CFT model (Qwen2.5-Math-7B-CFT), the original model (Qwen2.5-Math-7B), and SFT model. These cases demonstrate how different training approaches affect mathematical reasoning abilities.
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Expand All @@ -287,73 +286,154 @@ <h1 class="title is-1">Case Study</h1>
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<h3 class="title is-4">Problem:</h3>
<p class="has-text-justified">
Sam is carrying a 2 gallon bucket of milk to his house from the barn, which is 3 miles away from his house. However, the bucket has a leak in it. For each mile he walks, there is $\frac{2}{3}$ as much milk in the bucket as at the beginning of the mile. How many gallons of milk will be in the bucket when Sam gets home?
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<h3 class="title is-4">Detailed Solutions:</h3>
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<h3 class="title is-4">Case Study 1: Milk Bucket Problem</h3>
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A problem involving sequential calculations and fraction operations.
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<h4 class="title is-5">Problem:</h4>
<p>Sam is carrying a 2 gallon bucket of milk to his house from the barn, which is 3 miles away from his house. However, the bucket has a leak in it. For each mile he walks, there is $\frac{2}{3}$ as much milk in the bucket as at the beginning of the mile. How many gallons of milk will be in the bucket when Sam gets home?</p>
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<h4 class="title is-5">CFT Model Solution</h4>
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<p><strong>Step-by-step solution:</strong></p>
<ol>
<li><strong>Initial Amount:</strong> 2 gallons</li>
<li><strong>After First Mile:</strong><br>
$2 \cdot \frac{2}{3} = \frac{4}{3}$ gallons</li>
<li><strong>After Second Mile:</strong><br>
$\frac{4}{3} \cdot \frac{2}{3} = \frac{8}{9}$ gallons</li>
<li><strong>After Third Mile:</strong><br>
$\frac{8}{9} \cdot \frac{2}{3} = \frac{16}{27}$ gallons</li>
</ol>
<p class="has-text-weight-bold">Final Answer: $\boxed{\frac{16}{27}}$ gallons (Correct)</p>
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<h4 class="title is-5">SFT Model Solution</h4>
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<p>[Content to be added when provided]</p>
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<p class="has-text-weight-bold">Final Answer: [To be added]</p>
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<h4 class="title is-5">Original Model Solution</h4>
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<p>The model attempts to track milk reduction over three miles:</p>
<ol>
<li>Initial amount: 2 gallons</li>
<li>Uses $\frac{2}{3}$ reduction factor</li>
<li>Makes calculation errors in sequential multiplications</li>
<li>Arrives at incorrect final fraction</li>
</ol>
<p class="has-text-weight-bold">Final Answer: $\boxed{\frac{8}{27}}$ gallons (Incorrect)</p>
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<h4 class="title is-5">Key Observations:</h4>
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<ul>
<li>CFT model demonstrates superior step-by-step reasoning and calculation accuracy</li>
<li>Original model shows conceptual understanding but makes calculation errors</li>
<li>The comparison highlights the effectiveness of critique-based training in improving mathematical precision</li>
</ul>
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<h3 class="title is-4">Case Study 2: Geometric Area Problem</h3>
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<h4 class="title is-5">Problem:</h4>
<p>As shown, a square is constructed on the hypotenuse of a right triangle whose legs have lengths 3 units and 4 units. What is the area of the pentagon $ABCDE$, in square units?</p>
<p class="has-text-grey">[Note: The figure shows a pentagon formed by a right triangle with legs 3 and 4 units, with a square constructed on its hypotenuse]</p>
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<h4 class="title is-5">CFT Model Solution (Correct)</h4>
<h4 class="title is-5">CFT Model Solution</h4>
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<p>To solve this problem, we need to calculate the amount of milk left in the bucket after Sam walks 3 miles, given that the bucket leaks such that the amount of milk remaining after each mile is $\frac{2}{3}$ of the amount at the beginning of the mile.</p>

<p><strong>Step-by-step reasoning:</strong></p>
<p><strong>Step-by-step solution:</strong></p>
<ol>
<li><strong>Initial Amount of Milk</strong>: The bucket starts with 2 gallons of milk.</li>
<li><strong>After the First Mile</strong>: The amount of milk left is $\frac{2}{3}$ of the initial amount. So, after the first mile, the amount of milk is $2 \cdot \frac{2}{3} = \frac{4}{3}$ gallons.</li>
<li><strong>After the Second Mile</strong>: The amount of milk left is $\frac{2}{3}$ of the amount after the first mile. So, after the second mile, the amount of milk is $\frac{4}{3} \cdot \frac{2}{3} = \frac{8}{9}$ gallons.</li>
<li><strong>After the Third Mile</strong>: The amount of milk left is $\frac{2}{3}$ of the amount after the second mile. So, after the third mile, the amount of milk is $\frac{8}{9} \cdot \frac{2}{3} = \frac{16}{27}$ gallons.</li>
<li><strong>Calculate hypotenuse:</strong><br>
Using Pythagorean theorem: $\sqrt{3^2 + 4^2} = 5$ units</li>
<li><strong>Calculate square area:</strong><br>
Area = $5^2 = 25$ square units</li>
<li><strong>Calculate triangle area:</strong><br>
Area = $\frac{1}{2} × 3 × 4 = 6$ square units</li>
<li><strong>Total pentagon area:</strong><br>
Area = Square + Triangle = $25 + 6 = 31$ square units</li>
</ol>
<p class="has-text-weight-bold">Final Answer: $\boxed{31}$ square units (Correct)</p>
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<p>Final Answer: $\boxed{\frac{16}{27}}$ gallons</p>
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<h4 class="title is-5">SFT Model Solution</h4>
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<p>[Content to be added when provided]</p>
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<p class="has-text-weight-bold">Final Answer: [To be added]</p>
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<h4 class="title is-5">Original Model Solution (Incorrect)</h4>
<h4 class="title is-5">Original Model Solution</h4>
<div class="content has-text-justified">
<p>To solve this problem, we need to determine how much milk remains in the bucket after Sam has walked 3 miles, given that the bucket loses $\frac{1}{3}$ of its milk for each mile he walks. This means that after each mile, the amount of milk left in the bucket is $\frac{2}{3}$ of the amount at the beginning of the mile.</p>

<p><strong>Step-by-step reasoning:</strong></p>
<p><strong>Step-by-step solution:</strong></p>
<ol>
<li>Initially, the bucket contains 2 gallons of milk.</li>
<li>After the first mile, the bucket will contain $2 \times \frac{2}{3}$ gallons of milk.</li>
<li>After the second mile, the bucket will contain the amount from the first mile multiplied by $\frac{2}{3}$, which is $2 \times (\frac{2}{3})^2$ gallons of milk.</li>
<li>After the third mile, the bucket will contain the amount from the second mile multiplied by $\frac{2}{3}$, which is $2 \times (\frac{2}{3})^3$ gallons of milk.</li>
<li><strong>Calculate hypotenuse:</strong><br>
Using Pythagorean theorem: $\sqrt{3^2 + 4^2} = 5$ units</li>
<li><strong>Calculate triangle area:</strong><br>
Area = $\frac{1}{2} × 3 × 4 = 6$ square units</li>
<li><strong>Calculate square area:</strong><br>
Area = $5^2 = 25$ square units</li>
<li><strong>Critical Error:</strong><br>
Incorrectly subtracts triangle from square instead of adding</li>
</ol>

<p>Final Answer: $\boxed{\frac{8}{27}}$ gallons</p>
<p class="has-text-weight-bold">Final Answer: $\boxed{19}$ square units (Incorrect)</p>
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<div class="mt-4">
<h3 class="title is-4">Key Differences:</h3>
<h4 class="title is-5">Key Observations:</h4>
<div class="content has-text-justified">
<ul>
<li>The CFT model maintains precise step-by-step calculations, showing each intermediate fraction.</li>
<li>The original model has the right approach but makes calculation errors in the final steps.</li>
<li>CFT demonstrates better ability to track and compute sequential fraction multiplications.</li>
<li>The original model's error shows how small mistakes in mathematical reasoning can lead to significant differences in the final answer.</li>
<li>While all models correctly apply the Pythagorean theorem, their approaches to combining areas differ significantly</li>
<li>CFT model correctly recognizes that the pentagon is the sum of the square and triangle areas</li>
<li>Original model makes a critical conceptual error by subtracting instead of adding the areas</li>
<li>This case highlights how CFT improves geometric reasoning and spatial understanding</li>
</ul>
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