6th Grade Mode and Range

[Julie Joseph]

We are discussing 6th grade SP standards.  Do we need to include mode and range in 6th grade instruction?

[Bill McCallum]

Mode is not a topic in the standards. As for range, it is certainly implicit in the notion of interquartile range, which is in 6.SP.5c. You can’t calculate the interquartile range without first calculating the range, and the former is probably a better measure of spread.

[Joann Barnett]

Good day Bill.

I am thankful for this blog because it allows teachers to practice precise mathematical language when constructing arguments and critiquing the reasoning of others. 🙂

With that said, I will attempt to be as clear as possible.

The reason I have posted in this thread is to ask you if the idea of the mode is indicated in 6.SP.2? “Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.” The Arizona math standards included the following statement in examples for 6.SP.2: “Students consider the context in which the data were collected and identify clusters, peaks, gaps, and symmetry.” Is a peak the same idea as the mode?

When describing the overall shape of the data displayed in a dot plot, the use of the word “peak” certainly has meaning for students, while the word “mode” is another word to memorize. I am in no way “married to the mode” but just curious if the meaning of the mode is still contained in the common core in terms of the overall shape of the data, while at the same time correctly disconnected as a measure of center.

Thanks for all you do.

[Bill McCallum]

Thanks Joann, I like your characterization of the blog, and your question is a good one. On the hand, the concept of mode is not in the standards. So we don’t have exercises where students have to remember the definition or where they have to pick the mode out of artificially constructed data set.

On the other hand, the peak (or peaks) of a data distribution are certainly parts of its overall shape, as you suggest, and one can imagine situations where you might ask students to look at a data distribution and ask them questions about that. For example, you could look at an age distribution and ask which age group has the most people in it. However, you can do this without making a big deal about the vocabulary item; it’s simply a matter of looking at a data distribution and being able to answer questions about the situation it represents.

[Jim]

 You can’t calculate the interquartile range without first calculating the range, 

Actually, calculating range is unnecessary for calculating IQR.  Sure, it’s easy to do if you are calculating the IQR at the same time, but that’s not the same thing.

[Jim]

The publisher’s criteria explicitly mentions mode on page 9 even though the standards do not.  The question Joseph asked originally remains:

“Do we need to include mode and range in 6th grade instruction?”

McCallum writes:

 

“you can [ask which age group has the most people in it] without making a big deal about the vocabulary item; it’s simply a matter of looking at a data distribution and being able to answer questions about the situation it represents.”

Sure, agreed.   But if a test asks, “What’s the mode of this data set?” a student who knows how to identify which age group has the most but who was not taught the vocabulary term “mode” will be lost.  Vocabulary matters.

[Bill McCallum]

[I’ve rewritten this answer a few times since I posted it a couple of hours ago.]

The mention of mode on page 9 of the publisher’s criteria comes under the heading: “Materials do not assess any of the following topics before the grade level indicated” (emphasis added). That is, a criterion for alignment is that materials avoid assessing topics not mentioned in the standards. The table mentions mode in Grade 6 because measures of center are introduced in Grade 6, and curriculum writers might include mode as a measure of center. But it should not be construed as suggesting that mode must be introduced in Grade 6. Rather it is saying “assess mode in Grade 6 if you must but certainly not earlier.”

In short, my answer is no, you do not need to introduce mode in Grade 6. Of course, the assessment consortia might not agree with me.

The middle school curriculum before the Common Core was full of trivial, meaningless questions about small, artificial, and context-free data sets. You know the sort of question I mean: “What is the mode of 1,2,6,6,6,6,6,23?” Who on earth cares what the mode of this set is? Or the median or the range? Summary statistics serve to describe large data sets. There is no need for such statistics for the set 1,2,6,6,6,6,6,23. Everything you need to know about that set you can see just by looking at it (if indeed you need to know anything at all).

A curriculum could meet the Grade 6 standards on Statistics and Probability by working with large data sets arising from real contexts, using technology to plot them and compute their summary statistics. Students should be able to answer statistical questions, display data graphically, choose appropriate summary statistics and interpret them in terms of the context. That’s what the Grade 6 standards say.

Vocabulary should serve understanding, not replace it. A test question that fails a student who meets the Grade 6 standards as described above but forgets which is the mode and which is the median is a bad question. We can’t stop bad test questions, but we can avoid letting them drive the curriculum.

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