# When is “Doing” Enough?

“Our focus should be on the relationships between conceptual and procedural knowledge.” -Jon R. Star

So I have come to that subtle realization that just because a student can algebraically work through an example, doesn’t necessarily mean that they possess the concepts I would like them to know. That is, I am slowly realizing why so many educators are concerned that students can not only show their work, but also explain their work; and why “conceptual understanding” and “big ideas” have moved to the forefront of mathematics education. This blog is a walk through my thoughts for limits in first-year Calculus and is somewhat based around the ideas that Barry Garelick & Katharine Beals discussed in the Atlantic last year.

**My Experiment and Hypothesis
**

In my first-year calculus class, I wanted to get a sense of whether or not my students understood conceptually what a limit was. We had discussed them somewhat informally using the definition that “as x gets close to a, f(x) get arbitrarily close to L” and not in an epsilon-delta kind of way. They came up naturally as a way to help us answer the question “How do we define 2^pi?” in which we observed two one-sided limits (over the rational numbers) converging to a decimal number somewhere around 8.82. That is, their first work with limits came as observing one-sided limits in chart form. From here, I connected limits to graphical representations using Desmos and sliders, and came up with a few introductory rules (such as when we are allowed to use the substitution rule). After this, we worked on more of the algebraic rules: factoring, conjugation, vertical & horizontal asymptotes.

By the end of all of this, I was curious as to how they were thinking about and visualizing limits. I came across this paper, borrowed a few of their questions, and created a Desmos lab to try to help me determine this. I hypothesized that my students would be very good at the algebraic portion – that is, given a limit, I figured they would be great at working through this kind of question and getting to a final answer. However, I didn’t think that they would be as keen conceptually – that is, I thought they would have some major misconceptions about how to think about a limit (for instance, they would think that a limit is an approximation). So, let’s dive in and compare how their answers on the first test (algebraic) compared/relate to their answers from the Desmos limit lab activity (conceptual).

**Test Results**

The test results weren’t as good as I had originally hoped. This may be to a few factors – not enough studying or poor study habits on behalf of the students, testing being a stressful situation for the students, not assigning or covering enough questions on my behalf, the questions being too challenging on my behalf. Overall, the limit question was out of ten marks and had five parts. The mean and median mark was 4.5/10, with the highest mark being 9/10 and lowest being 0.5/10. The first three parts were standard, (a) being a conjugation [2 marks], (b) being an analysis of a horizontal asymptote [3 marks], and (c) involving factoring and analysis of a vertical asymptote [3 marks]. The last two questions were particularly challenging, with students requiring some knowledge of the asymptotes of arctan(x) and ln(x) to be successful [1 mark each]. The first three parts were definitely done better than all five parts combined; however, I did not parse these results.

So, at best, I can say that algebraic simplification of limits was average from a whole-class perspective. With that said, it might be interesting to parse down to individual student responses and analyse these to make some comparisons/contrasts. I have picked out three students that did exceptionally well with algebraic simplification of limits (some responses shown below). Let’s label these students as Student A, Student B and Student C. I am curious if exceptional knowledge of the procedure for solving limits is equivalent to a excellent conceptual knowledge of limits.

**Lab Activity Responses**

**Question:** How do you feel about the following statement “A limit is a number past which a function cannot go.”

Student A: Depending on the function, the limit is a number the function approaches at a certain value of x.

Student B: True.

Student C: A function can approach the limit from either side of the limit.

**Interpretation**: This statement is false. Student A seems to be on the right track – that this statement depends on the function and x-value we are looking at. Interestingly, he/she has drawn what looks like a horizontal asymptote at y = 5. Is the student trying to make a connection based on the wording of the question, or because the limit is being viewed as related to an asymptote? Perhaps they are showing an example of a function that is getting arbitrarily close to y = 5, but does not pass this number? For Student B, it is hard to tell what he/she is thinking, especially without a drawing. They may be thinking about how the limit is not concerned about what is happening at x = a (based on a previous response not shown). Student C has given a statement about one-sided limits, perhaps thinking that if the limit approaches from both sides that the function must necessarily go there. Further questioning would be needed for all students to really understand their thinking.

**Question**: How do you feel about the following statement “A limit is a number that the function value gets closer to but never reaches.”

Student A: A limit is a number that the function value gets closer to, but may or may not reach, depending on the function and x value chosen.

Student B: True if a cannot equal x such as with an asymptote.

Student C: True because the function is undefined at the limit.

**Interpretation**: This statement is true, as the limit is unconcerned about what is happening exactly at x = a. Student A is perhaps equating f(a) with the limit of f(x) as x approaches a, which is a common misconception. Student B thinks the statement is true, provided that f(a) does not exist. However, the statement is true in general, even if f(a) is defined and is equal to the limit of f(x) as x approaches a. Making this disconnect is perhaps the most challenging piece conceptually with regards to limits. Student C perhaps is thinking of a removable discontinuity (hole); however, is falling into a similar thinking compared to Student B.

**Question**: How do you feel about the following statement A limit is an approximation that we can make as accurate as we wish.”

Student A: A limit may either be an approximation or an exact value, depending on the function and x value chosen.

Student B: False.

Student C: True.

**Interpretation**: The statement is false. Without any more information, we cannot make any judgement on the answers of Students B or C – they may only be best guesses. As for Student A, there is no separation of the limit from the approximation procedure used to illustrate the limit (often we try to illustrate the limit as the upper/lower bound of a sequence of values). One common misconception is to view the limit as a process of approximation when the limit is actually the upper/lower bound of this sequence of approximation values.

**Question**: How do you feel about the following statement “A limit can fail to exist at a certain point.”

Student A: True

Student B: True

Student C: True, when there are two points for the same x value

**Interpretation**: The statement is true. Student A and Student C have insightful responses. I believe Student C is thinking about a piecewise function where the two one-sided limits are not equal (even if the wording is a bit off).

*Analysis and Conclusion*

Overall, there does not appear to be much correlation with students success algebraically with limits and students conceptual understanding of limits. However, when we parse down to a more individual level, selecting a few students who show great success algebraically, we note that a couple misconceptions began to arise: (1) viewing the limit as an approximation, and (2) believing that the limit is somehow connected to f(a). There may be some confounding factors at play here. I am teaching calculus for the first time and my own explanations may not be as clear as a seasoned teacher. Also, the wording of the lab activity questions were taken directly from the paper above, and could have been difficult for first-year students to comprehend – more translation into easier-to-work-with statements could have bee done.

In conclusion, we have to ask ourselves what is it that we want of our students? Do we want students who are only able to work algebraically through the problems without any sense of the conceptual underpinnings? Do we want students who are conceptually strong, but lack algebraic skills? It is obvious that we want neither of these. So how do we attain students who show both algebraic finesse and conceptual understanding? We have to be respectful that algebraic proficiency and conceptual understanding pave a bi-directional road. One helps build the other, and we should spend ample time in each *neighborhood* (that’s a calculus joke). In fact, our students may learn better if we are able to combine the best of both approaches, without falling into the trap of promoting one type of knowledge over the other.

Don’t beat yourself up if they lack deep conceptual understandings right from the start. These things take a while to absorb. It might be a couple of years before they really get it — it was for me, and I am quite good at Maths.

If you insist on pushing tricky theoretical stuff like limits until they get it totally right from the start you will find that it won’t help. They are a thing to return to several times, each time getting a bit better.

(Which is not to say that misconceptions shouldn’t be dealt with, just that deep understanding of difficult concepts is not something you can teach. It is only something they get with experience.)

So how do we attain students who show both algebraic finesse and conceptual understanding?Time, practice and clear explanations.

Those that need it will have calculus in later years. You don’t have to teach them absolutely everything perfectly. It pays to remember that.

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Katherine and I were talking mostly about lower grade math, though the example used in the article was a percent problem from middle school. Problems so simple that they defy explanation are a bit different than the concepts of limits you’re talking about here.

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Now that you mention the different contexts I totally agree. And I do agree that a percent problem done the way you have shown is a bit “overkill”. There are certainly other transfer questions we cpuld ask to see if the student conceptually understands the odea of a percent. I was hoping to use your article to spur my thoughts on how much time I want to spend on algebraic fluency and how much time I want to spend talking to the concept. I feel like the amount of time spent speaking to a concept should increase as we move further through our mathematics career.

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The degree of explanation differs depending on mathematical maturity and level of expertise. Certainly college freshmen are better able to articulate reasons than students in 1st through even 6th grade. That said, I tend not to believe in the popular belief that there are “math zombies”. When I was in algebra 1, I clung to certain modes of solving distance/rate problems in such a fashion that those sworn to the math zombie theory would say that that was what I was exhibiting. But when I entered algebra 2, I had matured enough to be able to analyze the problems better and not cling to my guiding principles like “Oh if they’re driving towards each other you add, and if they’re driving in the same direction you subtract”. Which isn’t totally wrong, and it did serve to get me through the night. And it is getting some of my current students through the night, though I insist on explaining what’s going on in such problems. Some get it, some don’t. Some of those who don’t, will eventually get it as they mature. And some won’t.

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Thank you for the excellent thoughts! I could not agree more with trying to find a middle ground between conceptual understanding and algebraic skills.

One thing I wonder about is the progression the curriculum uses to expose students to the idea of a ‘limit’. In grade 11, students first encounter an asymptote from the graph y = 1/x. The students graph both the horizontal and vertical asymptotes. This leads to many misunderstandings in grade 12 because the students believe that “you can not cross an asymptote.” This is a reasonable assumption if the only graphs you have even been exposed to have never crossed their asymptote.

In grade 12, students begin to graph functions such as f(x) = x / (x^2-9) which cross the asymptote for small values of x but approach the asymptote for large values of x. However, the emphasis continues to be on ‘large values of x.’

It seems to me like students are not given ample time to explore a ‘limit’ and apply it as x aproaches a. Given that in grade 11 students learn about piecewise functions, maybe an SLO could be added that explores the behaviour of a function piecewise around the sub-function changes.

Given how foundational limits are to Calculus, we need to ensure that the Pre-Calculus curriculum is not instilling conceptual errors that have to be ‘unlearned’ at the university level.

-Jehu Peters

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I would agree with you Jehu that there are probably some disconnects between college/university mathematics and high school mathematics. I might venture a guess that much of it has to do with the informal language we use to talk about the mathematics we teach. For example, in Calc I we often describe the limit in terms of function-values (as x –> a, the function values get arbitrarily closer to L). Rather than saying this aloud several times, we might shorten it to “as x gets closer to a, f(x) gets closer to L”. I can start to see where the “limit is an approximation” and “limit is always equal to f(a)” misconceptions come from! I do agree with you that some varied examples in the younger years would be helpful to tackle the misconceptions, since seeing the same style of question repeatedly without facing the misconception often leads to “learning” misconceptions as you have pointed out. But all in all, learning is very messy and I am not sure of any particular ways to go about doing these things 🙂

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Given how foundational limits are to Calculus,To the theory of Calculus, yes. To actually doing it, not so much. There are millions of engineers out there using calculus who couldn’t find a limit.

How an internal combustion engine works is foundational to making a car. But not to driving one.

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Im not really sure which of my students will become engineers and which will go into further mathematics since I am teaching first year college level. Similarly for Jehu who teaches at the high school level. We don’t really know who will need this concept in the future… Certainly there are other more important topics, however I will need more convincing to agree with your laissez-faire stance on the concept of limits.

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