A recent tweet from Mr. Hamada led to my thinking about skills and concepts in mathematics. I think both are important. Students need to be able to perform the skills. In addition, students need to understand the concepts *behind* the skill. Without the concepts, I’m not sure how valuable the skill is. How will they apply the skill, let alone remember it without understanding the concept?

I can’t imagine that many people would say that only one or the other is important^{1}. Yet, how does what we do in the classroom reflect what we value? What do we typically assess? Skills.

Now don’t get me wrong, skills are good. Skills are necessary. But if we never ask them about the concepts, what message are we sending about what we value?

I’m doing my best to try to reinforce both in my daily classroom activities and in assessments. In one of my preps, we’re learning about some basic statistics (mean, median, mode, and range – which they’re mostly done before, mastery level on day one varied greatly though) in addition to standard deviation and normal distributions (which is new to everyone).

I recently gave a quiz^{2} on standard deviation. I wanted to assess both their ability to calculate standard deviation and their understanding of the meaning. Here are the questions:

*Calculate the standard deviation of the following set of numbers: 20, 23, 32, 34, 16*
*Make up a set of data with five numbers that has the same mean but a smaller standard deviation than the set of numbers in problem #1.*
*John was calculating the standard deviation of a set of data that ranged from 3 to 18. He determined that the standard deviation was 16. Do you think his answer is reasonable? Why or why not?*

I think I did a decent job of writing questions^{3} that addressed both concepts and skills. The answers to the third question told me that the vast majority of my students have a decent grasp of what standard deviation means. Although we still need to work on how to word these responses. We have time though, they’re only freshmen.

^{1} I may be totally wrong on this. I often am.

^{2} I grade by “total points” yet somehow they take something called a “quiz” more seriously than an “assignment”. I still don’t understand this.

^{3} Writing decent assessments/questions is something I’ve had to learn (am still learning) on my own. I wish it had been emphasized more in my education classes.

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I love your concept questions. My question to you: when you put these questions on a quiz, have you explicitly talked about them in class before? Or do you just throw them on and see where the kids land – in terms of how well they’re able to grapple with ideas on the spot?

I’m teaching a number of non-accelerated math classes this year, and I have been putting on some definite concept questions, but I make it a point to have talked to them about this or that nuance or understanding beforehand, instead of making a question for them to be freaked out by on the fly.

I’m just curious about the context…

Sam

Good question

Sam. The second question was very similar to one we had done in class. The third one… not directly. We talk a lot about what is reasonable (and why) and do a lot of estimating, but we hadn’t explicitly talked about that question.By this point in the year, they’re getting used to extending ideas. Or at least being expected to do so. In the beginning of the year, I grade the freshmen pretty leniently on these types of questions (or not at all). I step up the expectations as we go. The “non-accelerated” kids can handle it. I’m really proud of the work they’re doing.

Just passing by.Btw, you website have great content!

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Making Money $150 An Hour

I really like that last questions. I think my stats students would have to really think about it. The “concept question” that I’ve used for this is:

You have two sets of data:

LIST A: 6, 6, 6, 6, 6, 6,…. [1000 numbers, all 6]

LIST B: 3,5,4,5,3,3,4,…. [1000 numbers, a mixture of 3, 4, 5 in random order]

Which list has a larger mean? [A, B, or it’s impossible to tell]

Which list has a larger standard deviation? [A, B, or it’s impossible to tell]

(They usually get the first question right, but I see more mistakes with the second question.)

Oh I like those Ξ, just in time for our unit assessment. Thank you for sharing them!