Exponents Misconceptions

Discussion in 'General Education' started by SF_Giants66, Sep 19, 2014.

  1. SF_Giants66

    SF_Giants66 Cohort

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    I want to start out saying these last two semesters have been eye opening for me, and I have been so fortunate that many of these students have been making me work in these lessons I've been teaching, as my initial worry is that my lectures would be boring because many of the lessons I've taught seemed really easy, that the first time with an Algebra 1 class I actually entertained the thought that the teacher told the students to ask me a lot of questions to make me really do some explaining. Of course, I know that didn't actually happen.

    An issue I've been dealing with in both semesters is the students no matter how gifted they are in math have been dealing with misconceptions, and needed to be reminded of exponential rules such as x^m*x^n=x^m+n. Also, another issue is frequently students seem to confuse multiplication with exponents.

    Yesterday, a group of students when on a practice problem where the answer was 3 to the 10th power immediately said the answer was 30, not realizing that it isn't the same as 3 times 10.

    Is this a common problem that people have noticed in math?

    Mostly, what I've been saying to students is correcting them asking them how they came up with that answer, and asking a question such as, "Is 3 to the 10th the same as 3 times 10?" Usually it doesn't take anymore than just asking this, then they usually end up doing something such as smacking themselves in the head and fixing it.

    Is there a trick or anything that usually gets them to stop doing this, or do they just end up getting it in time with more practice?

    Also, I thought exponents in 8th grade were a review topic in Common Core based on what they previously learned. Do they do any conceptual lessons in earlier grades to get them to learn the basics and dynamics of exponents that upper grades teachers can refer back to and clear up misconceptions?
     
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  3. Pashtun

    Pashtun Fanatic

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    I would definetly think more time on task is needed.
     
  4. Peregrin5

    Peregrin5 Maven

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    I don't know if you've ever experienced this, but most kids, after leaving a class seem to proceed to forget everything they ever learned in that class. (I know it's happened to me frequently.)

    In order to do it again, many times the students need to completely relearn the material. This is normal.

    Students simply don't use the material they learned over the summer or they didn't practice enough with it the first time around (due to having to move on to cover all of the material or whatnot).

    Students can retain things better when they're frequently asked to cold-call it and test their learning, using formative assessments and what not.

    But yes, it's normal for students who should know a certain concept, claim that they have never in their lives encountered that concept (even when you know for a fact, their previous teacher had taught it to them).

    Your job as a teacher is to catch those misconceptions, address them (probably by breaking down the exponents into their products or using other methods to prove it to them), and reassess whether those misconceptions were solved.

    Just saying they got it wrong, and telling them the correct answer isn't always a deep enough intervention to change their misconceptions (which often, are deep misconceptions)
     
  5. agdamity

    agdamity Fanatic

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    I teach exponents in 5th grade with common core. It is common for them to want to multiply the base by the exponent. It just takes repeated review. Exponents are not a major skill in my grade though, and they are mostly used to notate powers of ten.
     
  6. Rockguykev

    Rockguykev Connoisseur

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    <<Is there a trick or anything that usually gets them to stop doing this, or do they just end up getting it in time with more practice? >>

    The trick is to not let them use tricks. Same reason I don't let kids "cross cancel" while they are in Algebra 1. I have no problem teaching the shortcuts but for quite some time I'd make sure they didn't skip the step of expanding from 3^10 to 3 x 3 x 3 x 3 etc.
     
  7. gr3teacher

    gr3teacher Phenom

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    Try connecting with a few situations where they've all seen exponents... square units and cubic units.
     
  8. orangetea

    orangetea Connoisseur

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    This is very, very common! They just need a lot more reinforcement and practice!
     
  9. Upsadaisy

    Upsadaisy Moderator

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    Very, very common. Make them write out 3 multiplied 10 times enough and they will start to remember.

    I think you misstated the rule about exponents with a common base - when two exponents with a common base are multiplied, you keep the base and add the exponents. Make them write this one out, too. They will see how many times they are actually multiplying the base.
     
  10. mmswm

    mmswm Moderator

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    If you want to get technical, it might be wise to clear up the misconception that exponentiation is repeated multiplication. While repeated multiplication is a handy way to compute positive, whole number exponents, that's not what the operation actually does. It's not quite a scalar, but a scalar on a curve is the best simple explanation.
     
  11. Upsadaisy

    Upsadaisy Moderator

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    That's not simple.
     
  12. readingrules12

    readingrules12 Aficionado

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    SF_Giants66,

    We start teaching exponents in 5th grade. I see the mistake that you state a lot. We spend a lot of time with small math manipulatives to show the large difference between 2 x 5 and 2^5. Also, I require them to show what 2^5 represents for awhile. They have to write out 2 x 2 x 2 x 2 x 2. Eventually, I allow them to not write this step when they have had a lot of correct answers. This extra step helps a lot to reduce the mistakes. They don't mind doing this as I give them less problems to do, in order to give them more time to take the time to show this and any other extra steps.

    I am sure nomatter what we do, there will always be some students who go too fast, or don't think enough while doing their math problems, and will say 2^5 =10.
     
  13. SF_Giants66

    SF_Giants66 Cohort

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    Here is a new one I saw last week. We taught powers of powers. The book showed the example of say 2^5 4 times. (2^5)(2^5)(2^5)(2^5). They know from basic exponential rules they could add 5 four times and get 20, but I introduced them a new rule of (a^m)^n= a^m*n, so what some of them did is wrote it out individually and multiplied them together instead of adding them together after I explicitly stated that writing them out individually was only if they weren't using the short cut.

    The problem with student teaching is that you only teach one lesson here and there, so you don't really get to know the class to see where they are and what kind of practice they need. I think maybe with more practice they'll get it.
     
  14. MrsC

    MrsC Multitudinous

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    Until students have a strong understanding of concepts, using shortcuts won't make sense to many of them. They either won't use them or will use them incorrectly. I have found, that many students are able to come up with the shortcuts on their own after lots of exposure to and practice with the concepts.
     
  15. dgpiaffeteach

    dgpiaffeteach Aficionado

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    In most student teaching experiences, you teach every day all day after a certain point. I had seven weeks of observing and gradual takeover. Then I had eight weeks of teaching all day. Is this student teaching or a field experience for a class?
     
  16. SF_Giants66

    SF_Giants66 Cohort

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    The Common Core textbook had those rules written in them. They had the examples of them being individually multiplied together while adding the exponents with the common base.

    The purpose of the shortcut was for after the scaffolding with the long way. I thought since exponents began several years earlier, that they would have used them enough times that they would have remembered. They seemed to get it near the end though. This was the third lesson on exponents in the week, which they already covered and did several easier homework problems on exponents, so I wasn't introducing them to exponents for the first time of the year or anything.
     
  17. MrsC

    MrsC Multitudinous

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    The powers of powers was a new concept though?
     
  18. SF_Giants66

    SF_Giants66 Cohort

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    It's a student teaching field. It's not clinical internship. That's next semester assuming they place me with a teacher (I'm kind of on their watch list for suicidal and depression remarks last semester, but that's another story).

    I taught a lesson on factoring polynomials in Algebra 1 last semester, and the kids were very smart in math, yet many of them still didn't remember the multiplication rules for exponents. I think maybe by then they hadn't used them regularly.
     
  19. SF_Giants66

    SF_Giants66 Cohort

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    That's correct. It wasn't the rules that I was wondering why they didn't know right away, it was the fact that a few students thought 3^10 was 30, which I didn't expect to have to remind them of that. They figured out themselves where they went wrong after I asked them to explain how 3 to the 10th is 30 though.
     
  20. kcjo13

    kcjo13 Phenom

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    What kind of background did your CT give you?
     
  21. SF_Giants66

    SF_Giants66 Cohort

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    There were the first two lessons that were covered on reviewing exponents, and exponents with a common base. I planned and designed the lesson myself on Smart Notebook based on the textbook. I was the only one of my classmates that taught a lesson so far this semester, because I was eager to jump in.
     
  22. kcjo13

    kcjo13 Phenom

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    No, that's not what I asked.

    When you talked with the CT about the lesson you were planning, what did he or she tell you about the background knowledge the students have? What did he or she recommend you teach? What concepts, objectives, or standards did he or she want you to teach?

    I'm not asking what the textbook said, or what CCCS or whatever set of standards your school and state follow a said, I'm asking what the real, professional teacher, the one responsible for these kids, told you to do.

    Also, as a follow up, what kind of feedback did that person give you when you were finished?
     
  23. SF_Giants66

    SF_Giants66 Cohort

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    She wasn't done filling out the forms. She will give them to me next week. The teachers the last two semesters really didn't tell me anything about the classes. She said she was a bit concerned in an e-mail because I hadn't observed this class yet before teaching a lesson, but I wanted to get one in, and her other class was the inclusion class which isn't really doing much that requires new lesson planning, as so far we have only observed them practicing problems the teacher has been writing on the board.

    I told her I would be happy to teach her inclusion class as well, but I'm not clear exactly on what they are learning, because I have only seen reviewing so far. So we have to talk about that more.

    Last semester, they allowed us time in between classes to talk with the teacher, but this semester, there is one class period after the next with no time in between to talk about anything. It makes planning difficult.
     
  24. Shanoo

    Shanoo Habitué

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    Jumping in like that is hard. As a teacher, I would also be concerned about someone else planning a lesson in the middle of one of my units without ever having observed my class, planning with me or, at the very least, shown me the plan beforehand. If you don't know the kids, it's really hard to get a feel for the room, for their needs, for how they learn.

    As a general rule, unless I've specifically taught something to them myself, I assume that they don't know it. And, if I *have* taught it to them myself, I assume they've forgotten it (even if I taught it to them yesterday). My kids always get a short refresher from me about the concepts they need to know going into the lesson and, if I find most of them have forgotten, it turns into a mini lesson on the fly before jumping into what I had planned.

    In my room, the kids dictate the pace.
     
  25. MrsC

    MrsC Multitudinous

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    Agreed. My students have a 3 day weekend as we have a PD Day tomorrow. The first 5-10 minutes on Tuesday will be spent reviewing last weeks concepts.
     
  26. kcjo13

    kcjo13 Phenom

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    Forgive my ignorance, but I'm still having a hard time understanding the situation.

    What I'm trying to grasp is how you arrived at this topic to teach. You said you assumed they had learned this, this, and this. How do you know that? Did the teacher entrust you to teach a totally new concept?
     
  27. SF_Giants66

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    The topic was the next unit in their lesson. They hadn't learned powers of monomials before, but they had learned exponents previously. Also, multiplying and dividing exponents.

    Ideally, observing a class and seeing where they are at, communication with the teacher are what I would have liked, but that just hasn't happened. However, I asked for permission to teach a different class period and she said okay. We don't really have any time to talk other than by e-mail, and she doesn't answer them all.

    All I really have to do by the end of the semester is teach 3 lessons, and one is out of the way already. I don't wanna say I don't care, but I'm not really interested in bending over backwards to try and get open communication and previously observing the class and the students when our schedules are not set up for such. I skipped my science class last week to teach this math class. The science teacher really didn't give us any kind of schedule of what lessons would be covered, and I told her I wasn't all that interested in teaching any science lessons anyway, but I think I'll tell her I'm interested in doing at least one.

    In this class, we really have the freedom to do whatever we want as long as we have 3 documented lessons having been taught. The problem with science is I really only plan on teaching science if I can't find a math job anywhere, and I can't plan a science lesson at the spur of the moment the way I can a math lesson, and she hasn't given us anything to go from.
     
  28. SF_Giants66

    SF_Giants66 Cohort

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    As far as assuming they don't know it, ideally they really should. Eighth grade math requires pre-requisite knowledge in order to pass the previous grades. I can see them forgetting some fundamentals, but assuming they don't know anything would be a bit tough I think.

    However, I actually didn't have a problem in past lessons with them not knowing certain things, I just didn't necessarily thing to cover them until they needed to be clarified. For example, in polynomial factoring, I thought 8th grade Algebra 1 students would have used exponential multiplication and division several times throughout the year, but many forgot the rules, so I just reminded them of them. No big deal.
     
  29. MissScrimmage

    MissScrimmage Aficionado

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    Agreed - never assume the kids know something until you have taught it.
     
  30. kcjo13

    kcjo13 Phenom

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    Ah. I see now.

    So you don't have the time to adequately prepare yourself for teaching a lesson, and you don't want to be bothered with what the students in the class know anyway, and your objective is to get your three lessons in by the end of the semester - yet you are genuinely interested in why they might be mistaking 3^10 as 30.

    Makes perfect sense.
     
  31. SF_Giants66

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    Nope, as I said I'd actually love to know what is being taught, but there is little time for communication. Since I got the answer I was looking for I just ask we leave it as this and discontinue the insults and condescending remarks.
     
  32. kcjo13

    kcjo13 Phenom

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    I think that's a GREAT idea.
     
  33. SF_Giants66

    SF_Giants66 Cohort

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    Yes, thanks for providing the example of patronizing too.
     
  34. kcjo13

    kcjo13 Phenom

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    Oh...so we're not really done? Or are we? Silly me, still confused!! :rolleyes:
     
  35. AdamnJakesMommy

    AdamnJakesMommy Habitué

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    Sf,
    What grade is this?
     
  36. Shanoo

    Shanoo Habitué

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    All of this is a huge issue for me. As a student teacher, you *need* to be bending over backwards for such things. Why? Because that's part of the job. I have kids this year that I've never taught before. I'm teaching a full assignment, performing all of my out-of-class duties, helping on committees, am 32 weeks pregnant and I STILL found time to track down their teachers from last year, the resources teachers, and guidance counselor to get information on most of them. Otherwise, I'd be setting them up to fail.
    Really, all you need is five minutes. Or a quick email. Send your ct your lesson plan and ask for feedback.

    Also....I don't care how well you know your content. As a student teacher, you shouldn't be planning ANY lesson (math or otherwise) on the fly. Perhaps that is part of the problem.
     
  37. gr3teacher

    gr3teacher Phenom

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    What SF is calling student teaching, most of us would call field experience. What he's calling his clinical internship, most of us would call student teaching.
     
  38. vickilyn

    vickilyn Magnifico

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    Is that the same as a practicum in NJ? One full semester? Thanks!
     
  39. gr3teacher

    gr3teacher Phenom

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    Based on what he's said, I assume so. I think he's doing the internship second semester (if he is approved to do so).
     
  40. vickilyn

    vickilyn Magnifico

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    Thanks - just wanted to make sure I am on the same page as everyone else!
     
  41. SF_Giants66

    SF_Giants66 Cohort

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    This is the 3rd field experience course. I don't have the clinical practice until next semester.
     

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