Can we see the “sturdy rubric describing the beginning, middle, and end of great application problems?â€

Sure. The following rubric arose (without much coercion!) as we tried to resolve the different instructional outcomes between these two variations on the same theme.

What did we miss? Where did we overreach?

**Beginning:**

- engage the students with multimedia â€” pictures, videos, sound.
- the students come up with the question.
- the students make predictions â€” â€œgive me a guess.â€
- the students establish a range around their answer â€” â€œgive me a wrong answer. give me an answer thatâ€™s too high, thatâ€™s too low.â€
- there isn’t information on the first image.
- â€œannounce the problemâ€™s constraints quickly and clearly.â€
- ask questions that lend themselves to guesses: “how long? how many? how heavy? how far? how fast?”
- try to translate questions that are harder to guess into questions that don’t change the objective but which lend themselves to guesses: “what is the area? what is the circumference?”

**Middle:**

- ask: â€œwhat information do you need to solve this?â€
- ask: â€œhow do you know that?â€
- ask: â€œwhy?â€ â€œhow?â€ â€” even on right answers.
- encourage students to explain their reasoning to other students.
- ask students to collaborate â€” â€œwhat do you think about jerold did?â€
- ask: â€œhow would that help you?â€ after they tell you certain information is necessary.
- ask: â€œwhat
*isnâ€™t*necessary to find the answer? what information*donâ€™t*we care about?â€

**End:**

- ask students to summarize the process.
**sequel technique #1**: change a variable. eg. change the height of the water tank. change the number of sides of the base. make it a hexagon or a dodecagon. change the rate of flow.**sequel technique #2**: turn the answer into a question. at first we asked, â€œhow many tickets are on a roll with a particular diameter?â€ now: â€œwhatâ€™s the diameter of a roll that has 1,000,000 tickets?â€- ask: “does the answer make sense?” â€” have them compare their answer to their ranges.
- show, don’t tell, the answer â€” ie. the label said 2000 tickets; the timer said 8:12.
- discuss sources of error.

## 6 Comments

## Chris Riesbeck

July 29, 2010 - 9:52 am -In the beginning, for WCYDWT developers, and at the end, for students, ask “what else could you do with this?” What will/have we learned about solving what other problems?

## blaw0013

July 29, 2010 - 7:52 pm -It is very interesting the confluence of what I would name curriculum and pedagogy that define a “great application problem.” It seems lots of conversation about correcting the woes of math education are about one or the other, hugely ignoring the interrelatedness of the two.

## Debby Smith

July 30, 2010 - 7:30 am -@blaw0013: Good point, there is much overlap between the two. This rubric is not so much a measure of the design of the problem, but of that plus the playing out of the process of wrestling with the (well-designed) problem. As Dan took us through this process, I was continually impressed with the quality and impact of his questioning. Note all the “asks” in the middle portion.

The power of quality questions–from the driving opening question, to helping students craft good questions, to asking challenging questions throughout, to ending with key summary questions–is common across the curriculum.