This worksheet had 10 word problems on it. I'm not sure why you would want to estimate; the questions appear to call for precise answers. And why to the nearest ten? |

A small frowny face. |

"Why the sad face?" I asked.

"I don't like this problem," she responded.

"Why not?"

"Well, when I round each of those numbers to the nearest ten, I get 540 and 250. Those are not so easy for me to subtract."

"What would make things easier for you?"

"I could round the 536 to 500 and the 246 to 200 and just subtract 500 - 200 and get 300. That's easy for me because I know that 5 - 2 = 3."

"Then you go right ahead and do that!" I told her.

"But the directions say to round each number to the nearest tens!" She was nervous that the teacher might mark it wrong.

"That's OK," I assured her. "I'll talk with her."

Her answer. Unit issues aside, isn't it better than two question marks? |

This student knew there was something wrong. She expressed her displeasure in the only way she knew how: by drawing a frowny face. I

*love*that frowny face! Keep 'em coming, kids!

Happy you addressed this problem Joe. I recently ran into something similar. One of the teachers I work with had spent a week on rounding! A week! The students were being taught the specific rule for rounding "5 and up round up, 4 and below round down". They were like robots and it had totally took the context right out. It also didn't address why we round in the first place, that being it is usually because of a situation we find ourselves in. For example shopping, when we price items we usually round them to add up quickly. These students in this class where looking at just random numbers and then being asked to apply a rule. These are the dangers of using rules that don't always make sense. In your situation they are using problems that are contextual but are more pseudo-contextual and are applying a rule or being told what place to round off too. My teacher was applying the rule and telling them what place to round off too but with random numbers. Both equally ineffective and boring. With the situation in the class I was in I wanted to demonstrate why you have to be careful teaching them this blind rule. I also wanted to show why its important to add context to the situation.

ReplyDeleteThe school I was at was having a meet the teacher barbeque that night. I asked the students this question. Mrs. P (the principal) needs 673 hamburgers for tonight's barbeque. That is how many are registered at this moment. Estimate how many hamburgers she should buy. Every student accept for 4 students said 670. I put a couple of their responses on the board and let it stew. Then I asked for a person to justify their answer. The student quickly said "Well the 3 is lower than 5 so I would round it off to 670. This student just inputted the rule to get a rounded off answer and never used his reasoning to see that at least 3 people wouldn't get a hamburger that night and that's not including the possibility that more people than expected may come.

Once we chatted about this a student that did round up to 680 shared their justification. Some light bulbs came on in the class when they heard this persons reason for choosing 680 for the fact that they may need extra and that if you round down to 670 some people won't get burgers. It was also an eye opener for the teacher. She saw first hand what was happening to her students. They were throwing sense right out the door for a rule.

Why some people decide to teach rounding as an specific expectation instead of in a context that makes us think about what makes sense for the situation is puzzling to me. I see this often along with situations like you shared. What did the teacher say when you chatted with her? Did she accept the answer of 300? Thanks for sharing Joe.

As always, thanks for your thoughtful response! Your hamburger example really illustrates the point. Maybe the reason that rounding is taught as a specific expectation is because that's how it appears in the standards. 4.NBT.A.3 reads, "Use place value understanding to round multi-digit whole numbers to any place." That's pretty unambiguous and there's no mention of context, pseudo- or otherwise. So teachers think they're addressing the standard by using worksheets like the one I saw, or by assigning problems like the ones you describe, then they check that standard off the list and move on.

ReplyDeleteAs for the teacher, I have to be careful how I address issues like this, kind of a "pick your battles" scenario. I did bring it to her attention, and she did accept 300, but I think the larger point was lost, and I'm fairly certain she's forgotten about the whole episode.

My current mantra: In math, the power of reasoning is greater than the power of authority. I still struggle to find good ways to really teach this (and I do appreciate the inherent irony).

ReplyDeleteFor this specific concept, Beast Academy does a really good job with a sequence of thoughtful exercises around estimation. They hit three points:

(1) given the context, what type of error is acceptable/preferred (overestimate/underestimate, degree of precision needed)?

(2) is estimating actually needed? For example, #2 on the worksheet is probably easy for the kids to calculate exactly.

(3) what method of estimating is most helpful (rounding to nearest 10s, 100s, rounding to another friendly number, rounding one value but not another, etc)?

In the spirit of fixing resources, this worksheet becomes stronger if that single sentence at the top is removed. However, the questions still seem strange because we are given such precise data. For example, question 5: did Sarah really count exactly 167 dimes? Possible, but feels unnatural to me.

Question 1: Wow! The park workers are planting over 800 trees in a single day. That raises so many questions for me.

Thanks Joshua. I'll be sure to check out Beast Academy. I agree that removing that sentence at the top would make the worksheet stronger. And you were able to articulate what else was bothering me: that the questions are strange because the data is so precise. So it suggested precise answers, not estimates. I wonder if the kids found it strange!

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