I'll give you some math problems. Work on solving them and try to think out loud as much as you can. My goal is to see and understand how you reason with this math. When you hear me ask questions, it's just to help make your thinking visible to me. The more we understand, the more successful we can help you be!
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OK, first problem. 1/3 + 1/4.
Jaylen's Work
1⁄3 = 4⁄121⁄4 = 3⁄124⁄12 + 3⁄12 = 7⁄12
J
7/12.
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Can you walk me through what you did there?
J
I needed a common denominator. 3 times 4 is 12, so I used twelfths. I multiplied the top and bottom of 1/3 by 4 to get 4/12, and 1/4 by 3 to get 3/12. Then I added the numerators.
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That's really helpful. Thanks for explaining your steps.
MCT's Thinking
Thinking
Procedure is solid. Found LCD of 12, converted correctly, added numerators. But correct execution doesn't tell me if she understands fraction magnitude. The conversion to 4/12 and 3/12 will be interesting to revisit.
Hypothesis
FRAC-5a Add & Subtract Fractions pending
Next
Probe fraction comparison
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Here's another question. Which is larger, 1/3 or 1/4?
J
1/4.
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Can you tell me more about how you're thinking about that?
J
4 is more than 3. So 1/4 is more.
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OK. Thank you for sharing that. It helps me understand your thinking.
MCT's Thinking
Thinking
Whole-number reasoning. Comparing denominators as whole numbers. She just converted 1/3 to 4/12 and 1/4 to 3/12. Her own work shows 1/3 > 1/4 but she can't see it. Procedure and understanding are disconnected.
Confirm with number line to check if this is verbal or her actual mental model
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I'm going to show you a number line. It goes from 0 to 1. Can you place 1/3 and 1/4 on it for me?
Jaylen's Work
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Can you tell me about where you placed them?
J
1/4 is bigger so it goes further along. 1/3 is a little less.
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That's helpful. You're doing a great job sharing your reasoning with me.
MCT's Thinking
Thinking
Number line placement matches verbal claim. Not a slip. This is her actual model. She can't locate fractions on a number line through unit fraction iteration. Operating purely from whole-number intuition.
Hypothesis
FRAC-3 confirmed. REP-2 Number Lines & Tables, gap confirmed
Next
Sufficient data. Ready to classify
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OK Jaylen, that's everything I need. Give me just a moment to put this together for you.
Interview Complete
What MCT learned about Jaylen's thinking, and the plan to help
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OK, let me tell you what I see.
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First, that fraction addition? You nailed it. Finding a common denominator, converting, adding. That's real fifth-grade math and you did it right.
MCT's Thinking
Thinking
Starting with strengths. Anchoring what's solid before addressing the gap. She needs to know the building is mostly strong.
Mode
Assessment complete. Communicating results to student.
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Now, when I asked which is bigger, 1/3 or 1/4, you said 1/4. Actually, 1/3 is bigger. Here, look:
Fraction Magnitude
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When you split something into more pieces, each piece gets smaller. 4 pieces means each one is smaller than if you only made 3 pieces. That's why 1/3 is bigger than 1/4, even though 4 is a bigger number than 3. That part, understanding how big fractions are, is something we need to work on.
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Here's what's interesting. Look at your work from the first problem. You converted 1/3 to 4/12, and 1/4 to 3/12. 4/12 is more than 3/12. So your own math already shows that 1/3 is the bigger fraction. The answer was right there in your work. You just couldn't see it yet.
MCT's Thinking
Thinking
Reflecting her own work back. Not teaching, mirroring. The knowledge is inside her procedure. She needs to connect procedural output to magnitude meaning. This is exactly the FRAC-3 to FRAC-4 bridge.
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Remember that building I just showed you? All that green is yours. Numbers, place value, addition, subtraction, multiplication. You built all of that. And the fractions computation floor? Solid too. You earned that today.
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That one red section is the "how big are fractions" piece. It's one section, on one floor. And see those dotted lines above? That's ratios and proportions. The math you're heading into next. It needs that red section to be solid. Fix it now, everything above goes up strong.
MCT's Thinking
Thinking
Connecting the gap to her future. She can see the ratio between what's strong (almost everything) and what needs work (one section). The building makes the stakes concrete without being scary.
Mode
Motivating action. Student sees why the repair matters.
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Here's what we're going to do. We'll start with questions like this:
This week's practice
1/3
vs
1/4
Which piece would you rather have?
More cuts = smaller pieces. The 1/3 slice is bigger.
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We'll also practice placing fractions on number lines, like this. You'll build them by making equal jumps from 0 to 1. A couple weeks of this and that red section turns green.
Number line practice
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You've already built almost the whole building, Jaylen. We've just got one wall to fix.
MCT's Thinking
Thinking
Student leaves knowing: what's strong (almost everything), what needs work (one thing), why it matters (the floors above), and that it's fixable (weeks, not months). Building repair, not building demolition.
Classification
FRAC-5a procedural • FRAC-3 conceptual gap • REP-2 representation gap
Intervention
Fraction magnitude comparison with benchmarks + number line iteration. Target: 2 weeks to solid FRAC-4.
What just happened
MCT conducted a clinical interview in under 8 minutes
It identified a hidden conceptual gap that a test score would miss: correct procedure, broken understanding
It communicated results to the student: what's strong, what needs work, and the plan
Jaylen is a 5th grader who got the right answer. You'll see both sides of the conversation, plus MCT's internal thinking as it analyzes his reasoning in real time.