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Many students take calculus or algebra courses in high school, then later take college courses of the same name. There are various reasons for this, but in most cases the students in a college calculus class are not already experts in calculus. Students in a college algebra class are not already experts in algebra.

However, in these students' minds, they "already did all this in high school." This disengages them from lectures, activities, and homework.

What is the most productive way to fix this misconception?

  • I've tried to directly address the misconception on the first day. This simply alienates students and puts them on the defensive. From then on, they are always trying to prove you wrong, which is completely counterproductive.
  • I've tried letting poor grades speak for themselves, but often students have trouble connecting the poor grades to a lack of understanding. All errors look like "stupid mistakes" to someone who thinks they already know the material, and again directly addressing this causes a seemingly-counterproductive adversarial relationship.
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You might be interested in this other question: matheducators.stackexchange.com/questions/852/… –  JPBurke May 1 at 16:01
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Excellent link. I would delete my question, but I'd like answers that acknowledge the fact that these students in fact know very little. That question was focused on dealing with a student who apparently did already know a lot. –  Chris Cunningham May 1 at 16:50
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Over the past few years many of David Bressoud's Launchings posts have dealt with this issue, some directly and some indirectly. I was going to find some that are especially relevant, but it seems he's been posting more frequently in the past year or two, so there's quite a bit more to look through than I first realized. I think he began talking about this a few months, maybe a year, before he began his tenure as MAA president. –  Dave L Renfro May 1 at 17:03
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Here is a list of Bressoud's posts I found just now that is easier to look thorugh. The April 2007 is the post I was thinking of when he first started writing about this issue. He wrote a 7-part sequence of posts in 2010 that probably gives the most comprehensive treatment he's posted. –  Dave L Renfro May 1 at 18:11
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Put them in a situation where their lack of knowledge results in real or simulated discomfort, loss of money, negative affects on others, etc. –  Michael Martinez May 1 at 20:00
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9 Answers 9

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Let me suggest two things.

First: I think a lot of the misjudgments about how much students know stem from having an incomplete or wrong idea about what the learning objectives for the course are. For example, calculus students think that the sole objective of calculus sometimes is learning how to differentiate polynomials by hand. That's part of it, but there are many other things you want students to be able to do, both computational and conceptual. If you lay these things out for them in the form of unambiguous, action verb-oriented learning objectives and then make those objectives clear to students, maybe they will see that what they know (or think that they know) is just a small subset of what needs to be known.

Example: Here's my learning objective list for the section on the Chain Rule (section 2.5 in Matt Boelkins' Active Calculus book):

  • (Review) Apply all the derivative rules from Sections 2.1--2.4 with fluency.
  • (Review) Identify composite functions (that is, functions of the form (y = f(g(x)))) and identify the “inner” and “outer” functions.
  • State the Chain Rule and explain how it works in English.
  • Identify situations where the Chain Rule should be used when taking a derivative.
  • Use the Chain Rule to differentiate a simple composite function, for example a composition of a polynomial and a power function (e.g., $f(x) = (x^2 + x + 1)^{1/2}$).
  • Use the Chain Rule to differentiate a composite function involving two (more complex) functions.
  • Use the Chain Rule to differentiate a composite function involving three or more functions.
  • Use the Chain Rule in combination with other rules from earlier in this chapter.
  • Use the Chain Rule to differentiate a composite function in which at least one of the functions in the composite is given as a graph or a table of values.

I run calculus in a flipped classroom model and as such, students get these learning objectives when they are assigned the pre-class exercises to do prior to the class meeting on the chain rule -- so the students know what the learning objectives are. Students from a garden-variety AP Calculus course in high school probably have seen 2/3 of these objectives and are fluent with maybe 1/2 of them. What they need to notice is that there are some objectives that are not simple, like the last couple. Just knowing what needs to be known is a prerequisite for noticing that you don't really know it all yet.

Second: Use lots of formative assessment during class to give nearly-live updates to students on what they actually know -- short individual exercises, group quizzes, clicker questions, etc. that are aimed at common tasks and common misconceptions. Students say that they know more than they do because for many of them knowledge is a feeling rather than an objective fact. Give them data about what they actually know and their feelings should fall into line. And it will give you more info about how to teach better to meet their needs

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+1 Yes. That's very sensible - I think the more clearly we state the content the less they assume about it. –  AndrewC May 1 at 18:46
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This is a great idea. I've been building review sheets for exams for a long time; it seems I could follow your lead and hand out (for example) the review sheet for exam 2 immediately after exam 1 instead of the week before exam 2. –  Shay May 2 at 15:29
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Allow me to contribute from the student perspective. I've taken College Algebra classes three times at three different levels and schools due to unusual circumstances. To make things even more interesting, I started out as an English Lit major, and later became a computer programmer.

First, you have already realized that confronting the students in an aggressive manner on the first day sets you against them, and challenges them to "prove" that you're wrong. If you walk over to the English department you may overhear a saying popular in creative writing classes. Don't tell me, show me. One of the most common mistakes in amateur writing is to tell the reader how a person feels, rather than showing the emotion through action and description.

I would submit that telling a math student that the high school class taken does not give them a complete understanding of algebra is a "tell me" rather than "show me" approach. The difference between knowledge and wisdom lies in the ability to apply knowledge to relevant situations at hand. Thus, the person who truly understands algebra is the one who can apply it successfully. In the understandably academic environment of college, I believe that sometimes gets lost.

During my first college algebra class, I decided to look for a tutor and found a junior who was pursuing mechanical engineering. He impressed me in the first tutoring session by drawing out pictures of various engineering principles and how the equations and expressions in my textbook were useful for describing these actions. It was the first practical application anyone had ever shown me, and it completely rocked my perspective of mathematics. This guy understood how to apply the concepts, not just how to memorize or regurgitate them.

Consider providing scenarios that can be solved by a correct understanding of the math rather than only providing equations to solve in familiar formats. Many students who memorized formulas will find that they struggle to apply their rote memorization to real world situations. Likewise, you may also discover a select few who genuinely do grasp the meaning and value of the principles they learned in high school classes.

In fact, I would propose that the value of the high school class they took is that it provided enough context and common language to begin having the discussions you will initiate at the college level. If the only conversation occurring is how well the student has memorized a particular fact or method, then both student and teacher have missed out on valuable interaction. In my opinion, your greatest opportunity is to affect the student's worldview and perspective of math, which is also where the greatest gap exists between teacher and student. I am proof that perspective can still be changed at the college level.

If you are passionate about your craft, and you know how to apply it, then show them why it matters and what can be done with it. That alone should be enough for anyone who is humble to recognize their own limitations. Much like the difference between sharpshooting and hunting, algebra is a mathematical tool that in and of itself has little value until it is applied in some way for a useful purpose, usually to produce information that can be acted upon. It is the surprise of seeing a master do something previously unrecognized as possible or relevant that makes a novice realize and admit how little they really do know about a subject.

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"In fact, I would propose that the value of the high school class they took is that it provided enough context and common language to begin having the discussions you will initiate at the college level." I like how you said this. –  JPBurke May 2 at 11:23
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I was unofficially advising a student the other day who told me "I am struggling, but I know more than my friends in the class." This was the excuse she used for not attending meetings of a popular study group. How could she increase her grasp of the subject if her friends couldn't help her?

An answer to this question (if not the answer, as I believe there will be multiple possibilities) and to yours is to find some way to:

  • Increase interaction
  • Increase opportunities for skills to be presented

I told this student that she should be involved with the study group even if she would be the one helping all the other students. You are no doubt aware of the benefits to learning that lie in the act of instructing, but I was more interested in the possibility that her self-assessment would be challenged if she must use her knowledge to interact in a group on-task.

The key is that students like this probably need opportunities to:

  • See what their peers know, for comparison
  • Discuss what they know, to explore their knowledge and strengthen what they do know
  • Encounter the edges of what they know

I can't say whether these students need a hard confrontation with the edges of their knowledge, or a slow dawning in a comfortable context in which they can encounter skill limitations, see a need for effort in that direction, start to address where to focus their efforts, and finally improve their skills and understanding.

Part of the key here is, as you said, the belief is "in their mind." But human activity is out here in the world where we can observe it. We have to get math out of their minds, in a way.

To address this practically:

When I told the student that "going over what you know in the presence of others will help your friends, but most of all it will help you strengthen what you know" this was the truth, but I hoped it could do more for her if her self-assessment is not accurate. On the basis of that, she is going to meet with the study group. So, one course of action is to convince a student that a tutorial role in a social context helps them. This is the truth, and it is not incompatible with their beliefs about themselves, but could lead to a change in those beliefs.

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Some ideas that may help

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Help them understand the point of any difference in approach

If the emphasis turns to proof or derivation from more fundamental principles, you can make the analogy of "you know how to drive a car, let's learn how the engine works". They can argue that being a better driver makes you a better mechanic and visa-versa but they can't argue that knowing one means you know the other. This analogy exemplifies the link and the difference. Calling it all "motoring" isn't helpful in the first instance.

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Name topics or even whole courses to more clearly indicate new content

A shift in direction often causes tension in the students, where they felt on solid ground with applying memorised rules in predictable contexts, but are very uncomfortable questioning why or delving into how, especially if this seems needlessly complicated to them.

In this case it's important for them to understand they're studying something other than what they were before (the engine rather than the driving) Flagging this up with language and explaining what's happening reduces the upset that students feel when the thing they're good at morphs into a different beast and also the overconfidence that comes from the familiarity of some of the answers.

I'd advocate calling topics like differentiating by taking limits things like "differentiation under the hood", "the truth behind differentiation" or similar phrases emphasising slightly more esoteric content. The more you can make clear what more you're expecting than their existing knowledge, the better you have communicated the content.

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Acknowledge prior learning where appropriate

Taking the same calculus with limits example, you can satisfy the students' desire to have their existing knowledge recognised by saying "but of course you know what the final simplified form of the derivative will be from the rules you memorised at high school, so you'll be able to quickly and easily check your final answer with high school math".

Where the content is recap, making this clear can also lend more credibility to the claims of newness elsewhere.

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Examples from my experience lecturing in the UK

Taking some examples from UK undergraduate courses with which I'm more familiar, and where the topic is rather radically different to what they learned at high school, you could call highlight this by calling Algebra 1 "Abstract Algebra 1" or even just "Sets, Groups, Rings and Fields 1".

(I used to work at one of the UK's top mathematics departments where we ran a similarly titled course in the first term for undergraduates with the specific aim of helping them to make the transition to university mathematics. There was some resistance at first, including to the name, but it was an effective course.)

At the end of the course you can then admit that at university we collectively call Group Theory, Ring Theory, Field Theory etc "Algebra", and say that it's strange that we use the same name, but you can see there's a sort of link there somewhere.

"Calculus" could be (as appropriate) "Multidimensional Calculus" or "Applied Real Analysis", as specific as you can be, again flagging with some words that there's something very new about this.

~

(edited version of answer to correct out some assumptions I'd made based on UK rather than US undergraduate courses)

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This suggestion of renaming courses would be very unpopular among those who have to deal with issues involving transfer credit, with future employers of the students, etc. –  Dave L Renfro May 1 at 17:06
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@DaveLRenfro: In my experience, they'll have those issues anyway. No two universities will have exactly matching course names -- and even when the names match, the content often doesn't. Of course, if your department does follow some uniform course naming scheme coordinated on a higher level, then you can't implement this directly, but if not, this is worth suggesting. –  Ilmari Karonen May 1 at 18:25
    
@DaveLRenfro I taught at one of the top four university maths departments in the UK for a while, who took this approach with their first undergraduate algebra course. Transfer is rare in the UK, and I value education over administration anyway. In any case you just need to do it for the first unit that's very different in flavour. Once the students know how different the topic is, you can tell them what we normally call it. –  AndrewC May 2 at 5:49
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I once took a doctoral-level course called Functions of One Real Variable thinking it would be quite easy. It turned out to be the most difficult course I had ever taken. –  Joel Reyes Noche May 2 at 10:52
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@AndrewC: The original poster was discussing material covered in high school. Renaming a college algebra course (covers quadratic equations, factoring, graphing polynomials, exponential and logarithmic functions, etc.) that covers high school topics to something that suggests abstract algebra is involved is not going to happen. Also, Chris was talking about basic college math courses (college algebra, calculus), whose enrollment (at least in the U.S.) is over 90% non-math majors (probably 99.5% for college algebra courses), so "employers of maths graduates" is not really an issue. –  Dave L Renfro May 2 at 14:08
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You mentioned that

I've tried to directly address the misconception on the first day

You should try to start with a complex and practical example, not just in order to see the reaction of your students but also to create gossip like "that professor was doing some weird stuff so I was wondering if I was in the wrong place"

Regarding the next one

I've tried letting poor grades speak for themselves

You can do your lecture more dynamic try to interact with them with simple and tricky questions, but this doesn't have the same effects if they aren't willing to talk so maybe you should try something that is very useful in RPG games, which consists in give a small reward after some effort involved, this could mean "extra points" that are going to be managed for you so they can assure their good marks.

Try to keep them interested on what you're teaching and showing them. One can think that giving and update from time to time is strange on a lecture, but in this case I can bet that they will be impressed with the crazy things that are going on developing with basis in some basic theory, of course that this comes along with advanced topics, but this is just a thinking.

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The quickest way to reframe this misconception is to early and repeatedly acknowledge/accept that your students already know a lot and together you are going to use that understanding to take them to the next level. In fact you reassure them that it is good they have a solid base on which to build. (Just be sure not to scare any students with a more shaky understanding)

You then need to confront them with reality, not by head butting them but by reviewing, you quickly review what you think they know, constantly testing they do actually understand the basic principals, and reassuring them that it is good they already know most of this stuff, until you hit the point where they become unsure. You then change your language so that this is what they are going to learn and you give them a summary of what you are going to teach them. Depending on the size of your class and the variability in them, you may acknowledge that some students already know this stuff, others are going to have to work really hard and catch up.

I like to do this both at the start of a semester in a more general way and as we begin each new topic, it is almost a joke between me and my 'regular students'.

It is always possible that some of your students actually do know that you are teaching them, you then have to find something to do with them, let them move on, use them to teach the others, or expand the curriculum so they do learn, I guess this depends on their and yours reason for doing the class.

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Personally, I address the issue in the following way.

Assume for simplicity that the textbook has an initial chapter with preliminary material. I tell them "How do you know if you are ready for the class? You should be able to take exercises from that chapter, and you should be able to get the correct answer, on your own, without asking for help, without looking at the text, and in a reasonable amount of time".

I use this same idea for them to assess whether they are ready for a test.

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You asked for a productive way to fix the misconception. Here's one: Nobody really knows it! Always incorporate enough content to reflect this. Try to construct a series of problems that moves from the very basic to the extremely tricky (think Putnam Exam level).

The point is, any course is just an introduction to a topic. Just because I teach Calculus doesn't mean I know it. This is the real misconception that we need to debunk.

Regarding the "stupid mistakes", just because you can throw a ball doesn't make you Nolan Ryan.

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Thanks for this answer -- I think it is surprisingly underappreciated, votes-wise. It would certainly be a challenge to successfully communicate this idea to students, though, so it probably will not be my first approach to the problem! –  Chris Cunningham May 5 at 11:13
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Hmm, from a student's perspective there are a few things I want to point out. I don't mean to be attacking here, but you say that the students believe their mistakes are trivial, aka, "stupid mistakes." I am curious how you have ascertained that they were not stupid mistakes? This is something that can often be difficult to determine if a student does not show all or almost all of their work (and thought process) outside of speaking individually with every student (which is often impossible in larger intro courses).

Another thing is that you may have students that have genuine understanding but poor memories. In this case what is going on is that they understand, but because they understand they are perhaps not devoting enough practice to have the formulas and proofs ready in their heads when they need them. This might sound like a lazy pupil, but there are many demands on a student's time from all directions and the first thing to take a back seat is going to be the one they feel most confident in. Some have suggested that they should be able to solve problems without the book, that does not necessarily test understanding. I'm someone that has quite a poor memory, in fact I failed memorization heavy subjects like Biochem. the first time around because of it, but my understanding is strong and I have done well enough on homeworks to carry me through and pass despite poor exams scores due to my bad memory in many math courses.

In my opinion the best things you can do is to have as much in class practice as possible with immediate feedback as another poster suggested. Also, really try to schedule your office hours for directly after classes so that students that didn't get problems that were just done in class can get further explanations and help while the problems is still fresh in their mind.

I'm not really seeing how they can think they understand but not understand that they don't unless you're giving them problems that are more ambiguous than you realize (given your bias as someone that already knows the intended solution). My most fair professors have been those that would evaluate your answer based on your stated assumptions as long as the assumptions are not counter to the problem statement (aka in a somewhat ambiguous case you can pick one and go with it without being penalized). To be a little more clear, a lot of profs I have met think they're being very clear when they are in fact being quite ambiguous and it would only be clear if you considered the context of recent material covered (but math is not subjective and profs should not make these assumptions, they should be explicit). That may sound reasonable to an instructor, but theoretical problems can be solved many different ways and just because I'm in topology doesn't mean I shouldn't solve a problem with a proof from modern algebra. The topology proof may be easier, but the modern algebra one shouldn't be marked incorrect just because it wasn't what the prof had in mind. I've had profs mark many things like this incorrect, even if the proof was from the same course and just not the one they were thinking of. This is partially because in order for them to know my proof was correct they would need to put a good deal of thought into understanding what I did in my proof, which would take ages to do for everyone.

Sometimes (often old timer profs) will not accept that my proof is correct, they are unable to articulate a reason that it is not correct, but claim it is wrong because it isn't how they solved it, or how their text did. I have encountered many other cases where after speaking with the professor directly they would look at it seriously and return points (if I was in fact correct, otherwise they would show me where I went wrong, which is something I really appreciate). Knowing one way works isn't good enough for me, I also want to know why the other way doesn't work.

I think you may get more helpful answers if you're more explicit about what things they think they know and do not. Do I understand trigonometry, yep, you bet I do. Can I solve a calculus problem without using references that is only made doable by hand by knowing a relatively obscure trigonometric transform, nope, no chance in hell I'm going to remember that in say my modern algebra class when I haven't had a calc class since my freshman year (I was done Differential Equations while in high school and took Partial Differential Equations freshman year. Needless to say you don't remember these trig function when they randomly show up (in 1-2 proofs the whole semester in modern algebra, but they crush you out of the blue, and then occasionally out of nowhere in physics courses) 2-3 years later. Of course you remember the basic transforms, but the more exotic ones, nada, and I see no reason for profs to expect students to have this knowledge after such a long time, but they expect it anyway and if you don't know, then they're all, "but you said you took a course in trigonometry." Yeah... three years ago you jerk.

I know this may have seemed personal, but the idea is to try and give you a little perspective on the frustrations that even very bright students have in courses. I know that a few years after finishing courses it is easily forgotten what it's like to have a full 18 credit course load with two majors (where one actually required more courses than was technically allowed by the university's policy) and being expected to have magically had time to re-memorize everything you did in every previous course in the subject dating back as far as five or six years to high school courses by the time you're graduating.

For all I know your students are idiots that really don't get that they don't know something... but keep all this in mind just in case they aren't. Also, I know I said before to have office hours right after class, but also discuss what other time would be good with your students so that maybe once a week they will be at a different time for those with a course right after yours. Mmmk, I think I hit all the major points there. Please excuse the scattered train of thought, neither of my majors were English.

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Thanks for your comments. Regarding "I'm not really seeing how they can think they understand but not understand that they don't unless [narrow situation]" -- I recommend looking into the Dunning-Kruger effect. Cheers! –  Chris Cunningham May 4 at 15:55
    
Aye, I wrote hastily and did not fully mean for that to cover every base. I later stated that "For all I know your students are idiots," which while I am not using "idiot" in a formal sense, probably applies to those in the bottom 12% noted in Dunning and Kruger's studies. Interestingly they note that top performers can be less accurate, but biased in the other direction. I wonder if this stems in part from their perceived peers. That is, there is likely self segregation related to intelligence. So bright students are comparing themselves with other very bright students and dull with dull. –  Student's_View May 4 at 17:33
    
I was in your situation (even more -- took PDE's, Herstein algebra, complex variables, and others at a university 30 miles away while in high school), but I found that my command of elementary math returned (and strengthened) while grading papers and tutoring as an undergraduate and teaching in graduate school and beyond. Nonetheless, I tried to be aware of this, and the hint for the extra credit in a quiz of mine posted here will give you an idea of how I tried to deal with this by teaching students "how to fish". –  Dave L Renfro May 5 at 15:04
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