Mindset
notion of mindset I think if any of you are more expert than I am which is likely about the idea of mindset that was introduced by Carol Dweck please don’t hesitate to interrupt and clarify more effectively what I’m about to say but a lot of people have found that her work very helpful and when I’ve been thinking and talking with other people about error rates and simple ways of helping people learn beginning algebra the mindset idea keeps popping up so here’s a quick very quick summary of it one is that in general Carol Dweck and her colleagues suggest that people can be characterized as having either a growth mindset or a fixed mindset and a fixed mindset is a constellation of attitudes or beliefs behaviors that and and I I don’t think anybody wants to claim they know why or how someone is in this category but the idea is if you have a fixed mindset you’re quite reluctant to accept error in encountering new challenges the assumption is that you’re you have certain kinds of abilities capability skills so forth that are good fits for some kinds of things so when you encounter something seems easy comfortable and so forth on the positive side then that’s a good match for your situation your abilities and so forth but if you encounter something that’s difficult then it’s not a good match and the sort of implication from this way of looking at things is that you probably should avoid or and probably do avoid things of that kind because they’re just not going to work out well for you on the other hand the growth mindset is a constellation of things again that is more in the has that sort of style of welcoming challenge welcoming learning opportunities where error is a common experience because the person believes based on whatever the experience or other things that the making mistakes is a usual ingredient or element of some activity that involves learning something new that can very likely be valuable fun and so forth and so the mistakes that are made are not deterrence to going in the direction of encountering them but rather just something that indicates you’re in a new area that may be particularly valuable helpful and so forth so they’re quite different and what I would like to suggest if I can move this there we go that you can think of those two mindsets as a little bit related to some ways of thinking about learning with respect to error rate
<TAKE A BREATH>
and again I’m gonna I’ll take a breath and see if anybody wants to clarify anything I’ve been saying about mindsets before I go forward I’m looking at the chat don’t see anything I don’t care anything I hope I’m not alone out here in the internet right now somebody at least indicate that you can tell them you’re still here oh great thank you data okay so let me then the
Three Error Rate Paradigms
next part is these three paradox the
1. Error Rate as Attainable Goal
first one I think I was sort of the good and old and that is to think about error rate as an attainable goal that is associated with quality of learn and that somebody can achieve a better error rate associated with new learning by doing certain things like improving study habits learning more about the subject before you begin the subject learning having the right background on content background having developed study habits and practices that are seem to be successful in learning that the person can be helped by others teachers tutors and so forth or do it on their own to improve their own abilities to learn in certain ways and that by doing that which is not nobody ever thinks of that as a rapid process but by doing that the person the learner is likely to reduce error rate as part of new learning a particular new set of learning activities for some new topic or subject ok is that any and I say that’s an old good one because there are some people for whom there’s a sort of a very different attitude which is if you make mistakes that’s because you’re incapable incompetent were undesirable to join the part of the world in which I’m an expert but I’m talking about a lot of people in education from higher and other parts of Education where there is a commitment to helping people learn better and a belief that that can happen and that’s a paradigm that looks at error rate as something to improve by making an effort and using certain recommended practices
2. Error Rate as Measure of Current Quality
second an even older error rate and I think from at least my point of view and so there’s not quite so admirable is that the error rate is proud primarily a measure of the current quality of learning so that what really is doing is it’s the grade it’s more like grading in the sense that how your your percentage of error in whatever task you’re doing associated with learning is if it’s high that means you’re not doing very well if slow means you are doing very well and that’s all it tells you doesn’t tell you how you’re going to do doesn’t tell you how you did it tells you how you’re doing right now and as I was saying for some in some kinds of things like some the the rumor that there have been lecturers and so-called difficult subjects who would say the freshman what gear left with your right you know that one of the three of you is not going to be here by the midterm or after the midterm is you’re going to do so poorly and the rate at which you make mistakes as you go along is kind of a way of indicating that and that’s just the way it is and that’s gonna be your fate yeah then
3. Error Rate as Dynamic Learning Tool
I want to suggest a new paradigm and this is to me most appealing and it’s taken me a long time to figure out how to say this and I hope this is going to be more comprehensible than previously and that is to think of error rate in learning as a tool itself that the error rate is a tool for dynamic improvement and that is by thinking of the error rate associated with some sort of learning activity as something that can be controlled or influenced by the learner and by the person or people or system helping the learner and it’s not to be seen as a some people as not its value is not in being summative or giving a some kind of evaluation of the quality of things as at the moment but rather that by going back to the mindset way of looking at things if you think maybe people really aren’t at either extreme of mindset are either the fixed mindset entirely or the growth mindset entirely maybe people a lot of people I think probably are somewhere along a spectrum of what how much challenge they want how much how high an error rate is good for them under certain circumstances and as I say error rate I’m using a visit through a portland toad big turn to encompass other things like not just like difficulty level which is not the same as error rates challenge rewards and all that stuff but error rate is sort of a indicator or something for for the challenge level anyway what I’m saying is
the new paradigm is you can think of that as something you can change think of it as suppose somebody is for some topic and like say algebra for example they’re they’ve had take someone who has failed beginning algebra once or twice and is now taking it to the third time because it’s a requirement for a certain major in college that person probably doesn’t have much of a tolerance for error rate in that subject and so the way in which that person is introduced to the subject yet again could be modified by an instructor or by a system to make sure that person has a low error rate you could give the person many examples make sure that you go increase the difficulty very slowly that maybe you provide hints and other things so that the person is getting correct giving correct responses a lot of the time as opposed to the alternative and then you can think of other situations where some people would find that embarrassing or insulting or irritating they want to move along faster and they don’t won’t feel that they’re moving fast enough if they aren’t having some difficulty when they with what’s going on one of the things I think that helps explain this thing I just about is a notion of
direct or indirect
direct or indirect control of the area and that is that it’s not difficult to imagine a system and approach in which somebody whether it’s the learner or a teacher or the algorithm associated with tool sets things up so that even before the person starts there’s a level of difficulty that as assumed the person’s gonna find either very challenging not so challenging etc but then there’s also once things the person starts learning and doing things in that system the parameters can be adjusted to make it hard or easier error rate going up or down and that can be controlled consciously and directly by the learner by the system or by the guide or teacher and then on the other hand there’s a notion of indirect control that it may not even be visible or perceptible to anyone you may not even know that anyone’s manipulating the error rate but there are things associated with the system their algorithms responding to your behavior that change things depending on what you do and may be set up to change the error rate you encounter the learner encounters to the benefit of the learner
Video Games Use Error Rate as Dynamic Tool
now this sounded to me when I first encountered this with a colleague I wasn’t sure that this really either made sense was very possible or that anybody had ever done anything like this and then I was reminded about our work with online video games and realized that there are a lot of extremely popular video games that do precisely what I was just talking about that when you’re playing in these highly interactive very they said that the video games things happen that make your progress more or less difficult and you may have no idea of why they happen is getting more difficult or less difficult but it’s quite clear that the designers of these games over the last decade have gotten very sophisticated at making the system adjust in ways to keep you going if that’s what they want to have happen so that the difficulty level which I think as I said before is highly related to error rate is it just that dynamically as you go along with the purpose of keeping you engaged
we go back to people learning beginning algebra especially those who have had an unpleasant or very difficult experience before trying again having that kind of adjustment seems to be very appealing
<TAKE A BREATH>
ok I really think I should shut up now and see if we still don’t have David Parker back but I would really like to hear from somebody else so that I know not only have I been heard words but at least so far this is making some sense or better yet you have a something you want to ask or say that the change the way I’m proceeding this is Dale great thanks
Dale: “opportunities for success” & Math Phobia etc.
I just wanted to just add or validate what you were saying about the error rate thing I have called that providing students with opportunities for success which which you lets them know that lack of a better term they’re not stupid in that they can learn so I like what you’re saying in the sense that as soon as we as soon as I start by saying I’m gonna focus on error rate there are some people for whom that sounds it sounds like something I don’t want to do whereas opportunities for success is a phrase that maybe a little more comfortable for people and and I think I think Dale that you’re making a good point especially since we know a lot about the the audience for beginning algebra and re beginning algebra has an awful lot of people who are not very happy to be doing that and feel don’t feel very confident so I like your your phrase better and maybe as we go forward we should I should think about some other term than error rate that might be helpful I don’t know about when if there’s some version of opportunities for success that suggests something about rate of or measure or something that that would be helpful to I don’t know if you have something like that day over go ahead no it’s just that I try to build in whatever I’m doing with students you give them these opportunities because because I teach I teach adults at the at the college level and at the literacy level students who are trying to get their GEDs and they are very leery even more so than most students about the math thing the math phobia in the math anxiety so you know you you’ve got to provide them some opportunities for success so that they know that they can do it it’s that it’s just that basic
<Return to sample ambiguous order of operations 6 + 2h – 5 = >
and see I was going to do this this isn’t offensive let me ask you right now can you see that problem yes okay I wonder if each of you would look at the problem and try to think of two answers for it but let me read it to you it says 6 plus 2h minus 5 equals and then it says if H equals 7 and this is I hope an example from very early algebra and I’d like to you to think about right there right now can you think of two answers to things that six plus two H minus y would be equal to if X H is seven one of which you think is the correct thing and one of which you think somebody might come up with it’s not correct but you you think it would be likely so if you’re willing put two numbers in the chat and you don’t have to tell us which is the correct and which is the other but put down two numbers that you think this six plus 2h minus five might be if H equals seven is that a clear question it’s clear but I can’t think of a wrong answer okay well just put in put one end if you can only think of one put in one Kevin’s is 15 and 20 thank you tails is 15 anybody else willing to put one or two numbers in there Oh minus 1/2 I like that I’m gonna add one more it’s on my money John that was I like that okay so Kevin tell John Cathy me that’s almost everybody okay I believe that see if I can show you this now I didn’t think of John’s one setting the thing equal to zero at ignoring that I said H equals seven I like that I like it because I suspect people do it and that’s really the point of this is that I was trying to find some simple thing that could illustrate some variety of different you know ways of thinking about something and it turns out see if I got this right wasn’t 6 plus 2h yeah if you think about this there any of you want to say what what John you explained how you’ve got your variation the 15 let’s see 15 2 times 7 is 14 6 plus 14 is 20 minus 5 is 15 that seems like it’s probably right and how about Cathy 51 or 13 you want to say or 10 I like you got 3 or are you willing to say a little bit about any one of them the point being the purpose of this is why would somebody say something different than what we think of as the correct one and I’ll tell you why I the 28 is if you do this if you say it’s not letting me type I’ll bet this will work yeah ok I can do 6 plus 2 7-5 28 that’s how I got 28 anybody else played somebody was about to say something yeah I was after I thought about it this is Dale yep after I thought about it I if you don’t follow the order of operations you can I can’t see the example again you can show me the example the can you go back up and the sheep yeah if you if you do the if you do 7-5 first is 2 times 2 is 4 plus 6 is 10 do 7 minus 5 I’m putting and then you did where’d you go after that if you substitute the H that’s a 7 minus 5 yep and then you put the 2 there like that yeah but that’s 2 times 2 times it’s 6 plus 4 okay there’s another one I really had to stretch my brain to do it wrong well if you weren’t in this environment you probably wouldn’t have to stretch your brain you do it wrong or if you have a bunch of people but the point is anybody want to also explain one of the other ones all right can you guys hear me lose Kathy yes Kathy go ahead I was just gonna say that wine are all from mixing up the order of operations because they most of our students I don’t know my students don’t seem to follow them despite having taught it so to get 51 they would do 8 times 7 Oh 8 is 6 plus 2 and then they multiply it by 7 and then they subtract 5 and the 13 is on the state because they do 6 and then they try to add 2 times 7 and then instead of subtracting 5 and they’re thinking they’re taking away one they actually add 1 they somehow often mix up adding negative numbers or subtracting them so I see a lot of errors and mistakes like that so you did I lost I was behind you for a minute it’s did you do for like five – seven instead of seven minus five to get which one today I don’t I can see that 6 plus 2 is 8 times 56-51 I got that one but then Oh No ok 6 plus 2 times 7 so you have 7 minus 5 and then my students problem often is is they make the mistake because it’s 2 times 7 and you’re – seam 1 actually I mean you’re adding 1 actually but they subtract 1 because they see the minus sign they mix up negatives a lot so which answers oh they make that equal to 13 13 books see there’s math mistakes too I mean not only order of operations mistakes but arithmetic math or sure because they they tend to have problems with negatives so this one they were subtracting 7 instead of 5 yeah or it doesn’t even matter well it doesn’t matter they see the negative end so they subtracted even though you’re only subtracting the 5 well no they do – fine well whatever it is ok that’s what that’s what I don’t like about multiple choice because they seem to know all the mistakes that students make and put them in the choices and what when I’m trying to get at is and I have been talking to a bunch of teachers about stuff like this it seems to me that if somebody has not been helped too get this kind of thing correct most of the time without doing a lot of work that they’re going to have trouble with almost everything that comes later and therefore I go back to something like on here math comm where there’s some things for order of operations and well I see what okay what I want to know is what would be a useful thing to do in terms of the kind of quiz practice and so forth to help somebody at the level of problem we were just looking at and that was order of operations and low-level arithmetic calculation and I apologize that I had expected that I’d be showing you at least four or five different online things and talking about features and what made them better or worse but I didn’t have time to do that so I’m now at this stage showing you one and hoping some of you weren’t one or more of you can indicate or tell us point us to some other system that you found
Order of Operations & PEMDAS
helpful to do this sort of thing I still go back to that old acronym please excuse my dear Aunt Sally for the order of operations please excuse my dear Aunt Sally and Nathan’s fun so do you have a Dale I know you taught this kind of thing is there an online thing you found helpful in teaching this I primarily use Khan Academy you know I’m a real fan of Khan Academy yourself but I do but I use I use a lot of different things I used ed helper comm for quizzes and worksheets in helper it helped her and helped her – yes there’s a free version but I use a paid version that gives you the answer sheets and what do you like about this one I like the worksheets I like the the bear the the variation in the worksheets that you can get so here’s order of operations with F helper com see what I mean you can you can you can choose you can use you can choose 3 to 4 numbers 4 to 6 numbers you can do addition subtraction or operations and those kinds of things that’s what that’s what I like about that helper ok when it says free order of operations math games does that mean we’ll get a game probably I don’t use the games too much on here mostly I use it helper format for worksheets I’m feeling like this isn’t one with that I may be today I’m not meeting the expectations you had coming to this which I want to apologize anybody to any or all of you for whom that’s true and make sure that we do I’ll do a follow-up session in which I’ll list ahead of time the online things that we will take a look at in the follow-up session and looks like Eid helper and mass comm r2 are there others that we should list and maybe we’ll just get a little quick list now and then call it a day and I’m
Khan Academy
Khan Academy and Khan Academy is known mostly for lectures but the videos they’re very helpful especially in math and are there also I know you’ve mentioned before there’s some practice kinds of things there as well yes math by subject well now anybody else want to suggest something that we repair and look at the next time – near
hippocampus.com
the weekend hope that somebody else wants to speak that would be great whoever that is hippocampus how do you spell hippocampus yes I’m sitting here and I couldn’t I was quiet because I couldn’t not think of it it just went out of my mind it’s a chai PP and then campus I think that’s it it’s part of I know that the word is part of your brain wait a minute maybe that’s wrong I know that’s part of the brain it’s us it’s a site do you think the word hippo is part of it yes okay and with two peas hippo mass I think so there look simple campus org there you go not as crazy as I thought I was no that’s great okay what do you like about this one do you know it has videos and games and I like their their explanations of things too and look a link to Khan Academy yes they’re all related and there’s our order of operations is that the car I’ll bet that is yeah that’s gonna be the Khan Academy video right so so that’s under even if you go on YouTube
YouTube & Musical Math via Bob Dorough
Dale: I use YouTube a lot for a lot of different things when you go on YouTube and you look for math most of them are Khan Academy videos I think we’re about to stop for a variety of reasons but before we do would anybody like me to play one or more of these videos and Kevin you said do you see one the the guy that did the mathematician Rock and those those videos for kids he just died the other day and I just found out at his death that he was a jazz musician he played piano Bob what’s his name my brains network and see or yeah try multiplication rack multiplication Rock okay I’m going to do it up new one because I like that okay so you two patient schoolhouse from okay they’re your three is a magic number that’s uh single that’s by can you go over to the left it’s his name is Bob something he wrote it but I want I wanted the Bob Dorough there it is Oh where’s Bob Dorough deal ro ugh the three the three is a magic number is a nice short one okay I paid but I played it for my math class last night three tell you what three is the magic number let’s let’s make that our closing today we’ll be playing that thank you for your patience and your contributions I promise the next time we do this it’ll be we will tell you in advance the things that we’re going to look at the specific things but meanwhile I’m going to do one thing I know I’ve learned I have to stop sharing because when I share this if we’re going to play it I have to make sure I turn on the shared computer sound and then share yes this ledge somewhere in the entrance okay kids so my surname take your tips we Canada Canada you’re gonna be six man – you were looking 7:30 yeah do you 36 9 1:24 eighteen times 50 it’s a magic okay everybody’s doing now we’re gonna do that multiplication in where we started three times in this time man we’re gonna cut out on the answer I’ll be listened see 27:24 18:50 ha message 33 and that’s a message