Perhaps you have seen the Facebook post by an angry mother who is upset about her daughter’s Common Core-based math problem. There’s a larger lesson here, but it’s not about the Common Core.
Click here for the Facebook post by Larisa Yaghoobov Settembro
Carole read 28 pages of a book on Monday and 103 pages on Tuesday. Is 75 pages a reasonable answer for how many more pages Carole read on Tuesday than Monday?
Yes, 75 is a reasonable answer because 103-28 = 75
-1 [pt] Estimate 100-30 = 70
Right math lesson, wrong question
Other blog and news sites have taken this on, for and against the question. I think this is a horrible question, but not because the exercise of rounding is worthless. It’s a poor assessment question because,
- The answer, “75” is both accurate and reasonable.
- It doesn’t lead the student into a larger methodology or purpose. Either the student missed the purpose of this exercise or the teacher didn’t set that expectation.
Aligning Assessment Purpose with Expectations
This question seems to me to be assessing student understanding of the word “reasonable” more than the mathematical process of rounding. If the teacher had already defined a rounded answer as a “reasonable” one, then the student got it wrong.
However, had the teacher just thrown this question out there, how possibly could that child have known to round when faced with the word “reasonable” on an arithmetic problem?
Clarification of what is excepted of the child is due here.
Unfortunately, many teachers engage kids in processes without telling them why it’s important, where they’re going next, and how to apply it forward.
Kids need context, so when they say that they like lessons to be “broken down” what they’re asking for is clear statement of purpose, context, and relevancy. Don’t assume anything, teachers!
And parents, when teachers are vague or opaque, clarify it for your child.
The Problem here is that the Question stopped short of a fuller lesson
This should have been a 3-part question:
- Use rounding to create an “approximate” answer.
- Use your rounding technique to develop a mathematically accurate answer.
- Explain why or why not the approximate rounding answer is “reasonable.”
As a history teacher, I spent a lot of time teaching simple mathematics to 9th and 10th graders who struggled with dates and time. It was always a good teaching moment when looking at dates.
Rounding as History math
If I were to ask students, “How many years ago was 1492 from today?” they’d go running for a calculator. So I instead asked them, “About or approximately how many years ago was 1492 from today?”
My goal, just as that of the Common Core rounding problem, was to help students learn how to look at dates and make approximate statements on how long ago it was or the time between them.
We would start with why that’s important, anyway. Okay, so Columbus first sailed to the Americas in 1492, which launched a series of outcomes that led to today’s world. We want to 1) develop an understanding of how close or far from us that was in years; and 2) investigate both time and change. So, we will:
- calculating approximately how many years ago 1492 was
- measure that in centuries
- count how many family generations that makes
- use rounding to calculate an accurate measurement of how many years ago Columbus first reached the Americas.
After letting them try out their own methodologies, I would offer the process I like to use:
I’d start with defining a century as a common time unit:
- every 100 years is a good tape measure for time in history.
- 100 years = a century
Then I’d use rounding to approximate our two dates to their nearest century:
- Today = 2017 = about 2000
- Columbus = 1492 = about 1500
- I can do that math more easily now: 2000 – 1500 = 500
- Therefore 1492 was about or approximately 500 years ago
(Here we can work on how many centuries and family generations ago it was –for generations, we’ll assume 20yrs per generation = 500/20 = 25 generations –> wow, only 25 generations, that is, parent –> child, was not that long ago, was it!)
Now let’s use our rounded answer, 500 years ago, to make a mathematically correct answer:
- Since 2017 is 17 years further away from 1492 than 2000, we need to add 17 years to the rounded total
- Since 1492 is 8 years further away from 1500, then we need to add 8 years to the rounded total.
- Therefore, 17 + 8 = 25 years to add to the approximate answer:
- 500 + 25 = 525 years = (2017-1492)
To reinforce this lesson, I would work with another date from the 20th century, say a parent’s birth year, 1974. Same process works, but with the one little switch that 1974 is closer to 1492 than 2000, so:
- We already know that 1492 is 8 years further away from 1500, so we will need to add 8 years to the rounded total.
- But, 1974 = 26 years closer to 1492 from 2000
- Therefore we have to subtract 26 from our total (as opposed to adding the 17 years to 2000 for the year 2017)
- So, we will be adding 8 and subtracting 26:
- = +8 + -26 = -18 (or, 26-8=18)
- We next subtract 18 (or add -18) from our rounded total:
- = 500-18 = 482 years between 1492 and 1974.
The lesson could then be enhanced by working with 1975 as the round instead = 475 years. So we have -1 for 1974 and + 8 for 1492 = -1+8= 7; then 7 +475= 482 years between 1492 and 1974.
Teachers, Don’t Cut Short the Lesson!
What have we learned:
- Approximations and rounding is a valuable tool.
- We can use approximations to quickly ascertain accurate math answers.
- Rounding can help us think in different time or math units such as centuries and generations.
So you see, my problem w/ the Common Core question is not that rounding is an invalid exercise but that rounding must be taught as a distinct skill and why it is useful.
Teaching should always be clear about purpose and be applied and extended for new knowledge.