Topic (follow-up to Friday, 25th February 2011):
What misconceptions are most common in mathematics?
As it happened on Monday, 28th February 2011, 20:30 - 22:00 GMT
8:32 pm
ColinTGraham:
Hello everyone and welcome to today's #mathchat
8:33 pm
ColinTGraham:
The topic is: "What misconceptions are most common in mathematics?" For newcomers, don't forget the hashtag ---> #mathchat
8:33 pm
harrysarms:
#mathchat i am a grade 11 math student and am looking for examples of arithmetic or geometric sequences in real life?
8:34 pm
dweksler:
@harrysarms Glad to have students join in and ask questions during today's mathchat :-) #mathchat
8:35 pm
ColinTGraham:
@harrysarms Hi Harry, some examples of arithmetic sequences were put out earlier today on #mathchat
8:35 pm
dweksler:
@harrysarms Any guesses where to look to answer your question, Harry? #mathchat
8:38 pm
ColinTGraham:
David started us off with this misconception, any others to add? RT @dweksler: That a number bigger than infinity is (infinity +1) #mathchat
8:38 pm
dweksler:
#mathchat So, I'm sort of thinking of what things may grow in number over time - maybe a couple of rabbits, boy and girl?
8:40 pm
dweksler:
@ColinTGraham Thinking that misconception is a possible answer from a 6 year old as they like to ask the question :-) #mathchat
8:40 pm
ColinTGraham:
Things which I've always seen are related to fraction manipulations and exponents: 1/2 + 1/3 = 1/5 or 1/(2^x) = 2^(1/x) #mathchat
8:41 pm
ColinTGraham:
@dweksler Yes, I think I mentioned in last's week discussion of this that misconceptions are part of learning... #mathchat
8:41 pm
dweksler:
@ColinTGraham I most certainly AGREE with that! #mathchat
8:43 pm
ColinTGraham:
The value for us is to try and be aware of what the misconceptions are, & the best way of addressing them... algorithms confuse! #mathchat
8:44 pm
bucharesttutor:
RT @ColinTGraham: Things which I've always seen are related to fraction manipulations and exponents: 1/2 + 1/3 = 1/5 or 1/(2^x) = 2^(1/x) #mathchat
8:45 pm
EarlSamuelson:
division by zero equals zero & that reliance on memorization as opposed to understanding through reasoning constitutes "learning" #mathchat
8:45 pm
ColinTGraham:
@dweksler Your example of growth is a good example of confusion between arithmetic and geometric sequence... #mathchat
8:45 pm
bucharesttutor:
One misconception I was asked by my 4th grader, why is 1/2+1/2= 1 whole and why not 1/4 #mathchat
8:46 pm
dweksler:
@bucharesttutor What happens when you take two halves of an orange or apple and hold the cut sides facing each other? #mathchat
8:47 pm
ColinTGraham:
@earlsamuelson I think the 'need' for everything to have a defined value or place is difficult for many people to 'get over' #mathchat
8:48 pm
bucharesttutor:
Why is any number raised to the power of 0 equal to 1? Actually I need a specific answer pls #mathchat
8:48 pm
ColinTGraham:
@bucharesttutor That's best explored graphically... #mathchat
8:49 pm
chris_1974:
@bucharesttutor what about 0^0. That's when it gets interesting.Use indexlaws. 3^0 = 3^(2-2) = (3^2)/(3^2) = 9/9 = 1 #mathchat
8:50 pm
ColinTGraham:
@bucharesttutor You can also show: 1/2 * 1/2 = 1/4, by cutting one of the halves in half... to show 1/4 can be an answer, but to * #mathchat
8:52 pm
bucharesttutor:
@chris_1974 frankly speaking 0 to power of 0 is undefined as per my knowledge #mathchat
8:53 pm
chris_1974:
@bucharesttutor yes - but its a *great* question to ask studetns #mathchat
8:53 pm
ColinTGraham:
@chris_1974 So this suggests that you need to have explored index laws before you can explore the zero exponent... #mathchat
8:53 pm
chris_1974:
@chris_1974 once they realise 0^n = 0, and n^0=1, it is a real curve ball. #mathchat
8:54 pm
chris_1974:
@ColinTGraham I do. Index laws for +ve integer powers, then to 0, and -ve powers via division. #mathchat
8:54 pm
EarlSamuelson:
@ColinTGraham I attempt to connect algebraic representations to geometric whenever possible and make justifictions from there #mathchat
8:54 pm
chris_1974:
@bucharesttutor once they realise 0^n = 0, and n^0=1, it is a real curve ball. #mathchat
8:55 pm
bucharesttutor:
@chris_1974 yes I did ask this question to a 9th grader and the answer was 0 #mathchat
8:55 pm
ColinTGraham:
Many misconceptions come from overgeneralizing 'facts' eg 2+2=4 becomes 1/2+1/2=1/4 (or 2/4!!) #mathchat
8:56 pm
chris_1974:
@bucharesttutor which begs the question, why not 1? #mathchat
8:56 pm
ColinTGraham:
@earlsamuelson I agree with that approach. There are a lot of misconceptions due to disconnects, and geometry is not done enough! #mathchat
8:56 pm
dweksler:
@ColinTGraham Is it fair to make the comparison with "M" and "W" - same shape but different orientation? #mathchat
8:56 pm
bucharesttutor:
Another problem I have seen is that students don't know whether there exists answers such as "undefined" or "meaningless" #mathchat
8:56 pm
Janshs:
@chris_1974 what's the #mathchat topic/theme this eve?
8:57 pm
ColinTGraham:
@Janshs Common misconceptions in mathematics, Jan #mathchat
8:57 pm
chris_1974:
@Janshs I think its common misconceptions in maths #mathchat
8:58 pm
ruth4916:
-x^2 is not equal to (-x)^2. Repeat,repeat,repeat...#mathchat
8:58 pm
Janshs:
@chris_1974 aha! I have trouble some times convincing that root 2 times root 2 = 2 etc #mathchat
8:59 pm
ColinTGraham:
@dweksler Possibly, spatial awareness is another issue where problems result. Turn fractions upside-down and multiply to divide?? #mathchat
8:59 pm
EarlSamuelson:
@chris_1974 do they realize that n^5=>5 extra factors of n on top and that n^-5=>5 extras on bottom;then n^0 => equal # top/bot #mathchat
8:59 pm
bucharesttutor:
@ruth4916 the placing of the bracket could be very important here #mathchat
9:00 pm
chris_1974:
@earlsamuelson yes - fairly quickly. I get them to do it long hand a few times. They soon work it out! #mathchat
9:00 pm
ColinTGraham:
@ruth4916 Order of operations is the keypoint here, in the absence of parentheses... #mathchat
9:01 pm
EarlSamuelson:
@ColinTGraham I agree completely....geometry is VERY powerful.......knowledge of it can put learners in a very solid position #mathchat
9:02 pm
ColinTGraham:
If overgeneralization is 1 source of misconceptions/error (as in language learning), does the order we introduce topics impact 2? #mathchat
9:02 pm
EarlSamuelson:
@chris_1974 I think that's very effective.....graphing of such things can also be effective.....negative exponent is a reflection #mathchat
9:02 pm
bucharesttutor:
Another question by my S's, what is infinity/infinity, is it 1, is it undefined etc. #mathchat
9:03 pm
chris_1974:
@earlsamuelson agreed, also relating 2^(-x) and (1/2)^x #mathchat
9:05 pm
chris_1974:
@bucharesttutor I'd want to to ask the question, what is infinity #mathchat#gettingphilosophicallydeepnow
9:05 pm
EarlSamuelson:
@chris_1974 Yes!!! very powerful..... #mathchat
9:06 pm
ColinTGraham:
@bucharesttutor Any discussion around zero, infinity and other concepts is bound to be useful. Not enough discussion happens #mathchat
9:06 pm
bucharesttutor:
@chris_1974 they may actually just reply Infinity is a very large number #mathchat
9:08 pm
bucharesttutor:
@ColinTGraham sorry can you be a little clearer about that? #mathchat
9:09 pm
EarlSamuelson:
@bucharesttutor are you referring to limits at infinity?.....might be a good place to introduce that #mathchat
9:10 pm
ColinTGraham:
Does the order we introduce topics impact on misconception? Should we leave exceptions zero, infinity, etc. until later? #mathchat
9:11 pm
ColinTGraham:
@bucharesttutor By discussion, I mean exploring the concept in more detail with Ss, especially when they raise the question #mathchat
9:11 pm
ColinTGraham:
@bucharesttutor Less-confident teachers will use the "because it is" or "that's the rule" instead of discussing the "why" #mathchat
9:12 pm
EarlSamuelson:
@ColinTGraham I think if we open up a "can of worms", we need to be prepared to deal with the fallout out right then and there #mathchat
9:12 pm
bucharesttutor:
@EarlSamuelson exactly we face that scenario in limits but S's turn a blank face when faced with infinity and its kind #mathchat
9:13 pm
ColinTGraham:
@earlsamuelson Which comes back to identifying what "worms" are going to jump out of the "can"... if it's pre-planned! #mathchat
9:14 pm
EarlSamuelson:
@ColinTGraham @bucharesttutor we need to be prepared to discuss why at any time if the topic comes up.......that's teaching #mathchat
9:15 pm
EarlSamuelson:
@ColinTGraham we need to be prepared for all cases........that's part of what makes teaching challenging and fun; anticipate #mathchat
9:16 pm
ColinTGraham:
What 'off-the-wall' activities do you use, or have you used, to point up misconceptions or explore a new area which is a minefield #mathchat
9:16 pm
bucharesttutor:
@EarlSamuelson @ColinTGraham agreed, we need to be proactive. In my case I'd start with limits, continuity, Calculus in that order #mathchat
9:18 pm
mathsatschool:
Joining in for a bit with my common misconceptions #mathchat. x-axis is x=0, y axis y=0.
9:19 pm
mathsatschool:
double the sides = double the area / volume #mathchat misconceptions
9:19 pm
ColinTGraham:
@mathsatschool So good luck with the z axis... #mathchat
9:19 pm
bucharesttutor:
@ColinTGraham I have never tried any out of the book theories to teach my Ss Math as I believe in keeping them simple n straight #mathchat
9:20 pm
mathsatschool:
All probs equally likely (and I was guilty of repeating this aged 12). "It's either raining or it's not!!" #mathchat misconceptions
9:20 pm
chris_1974:
@mathsatschool i watched a collegue squirm recently, was trying to do bog standard enlargement with lowish set... #mathchat
9:21 pm
ColinTGraham:
@bucharesttutor Do you tend to rely on the same or similar approaches or do you mix things up Vijay? #mathchat
9:21 pm
chris_1974:
@mathsatschool comment from student was "its twice as long and wide but you can get 4 of them inside so its 4x bigger". #mathchat
9:22 pm
ColinTGraham:
@mathsatschool Statistics and Probability is probably a misconception in its own right! (says the statistician in me!) #mathchat
9:23 pm
mathsatschool:
A square is not a rectangle? #mathchat misconceptions And a square on its side is not a square, but a diamond!! (Bugbear of mine)
9:23 pm
bucharesttutor:
@ColinTGraham I tend to more or less rely on the same approach but I vary the pace as per the students understanding #mathchat
9:23 pm
ColinTGraham:
@chris_1974 Well if "it" was the area... ok. If "it" was the scale of enlargement... oops! #mathchat
9:24 pm
chris_1974:
@mathsatschool my fave question to sort out square's and rectangles is "Find 2 rectangles where P=A "(numerically) #mathchat
9:24 pm
mathsatschool:
Very common one - 3.456>3.87 (for example) #mathchat misconceptions
9:24 pm
chris_1974:
@mathsatschool have you come across the "sqaure of area 10" investigation. Sorts out "diamonds" too. #mathchat
9:24 pm
ColinTGraham:
@mathsatschool Once you get them to talk about rectangles instead of oblongs and ellipses instead of ovals... #mathchat
9:25 pm
chris_1974:
yup. RT @mathsatschool: Very common one - 3.456>3.87 (for example) #mathchat misconceptions
9:26 pm
mathsatschool:
@chris_1974 Not familiar with that one. Possibly post it? #mathchat
9:26 pm
ColinTGraham:
@bucharesttutor Are there misconceptions or questions which you find it difficult to respond to? #mathchat
9:26 pm
ash_eng:
#mathchat another one that always makes me want to bang my head off the whiteboard is (x+y)^2 = x^2+y^2 @mathsatschool
9:26 pm
TeaKayB:
#mathchat misconceptions: "It looks like the same angle/length so it is."
9:26 pm
bucharesttutor:
@chris_1974 never seen this misconception although I guess I anticipate this in the near future lol #mathchat
9:26 pm
mathsatschool:
How about 'add a zero' for multiplying by 10! #mathchat misconceptions
9:27 pm
dotsmy:
AMEN TO THAT! RT @chris_1974: @chris_1974 once they realise 0^n = 0, and n^0=1, it is a real curve ball. #mathchat
9:28 pm
mathsatschool:
Or 1/3 = 0.3, 1/7 = 0.7 or even 1.7? #mathchat misconceptions
9:28 pm
chris_1974:
@mathsatschool I'll write it up on my blog. May not be tonight, so feel free to subscibe (about to add subscribe box). #mathchat
9:28 pm
ColinTGraham:
Algorithmic subtraction... carry/borrow 1, instead of 10, 100, 1000 etc... #mathchat
9:29 pm
bucharesttutor:
@ColinTGraham yes some questions like what's the diff between a square n rhombus, I know the answer but not rightaway #mathchat
9:29 pm
mathsatschool:
@ash_eng Always! Have modular exam tomorrow and am STILL seeing it this end of y11! #mathchatmisconceptions
9:29 pm
chris_1974:
Can i apologise now for @dotsmy crashing #mathchat!
9:30 pm
ColinTGraham:
@bucharesttutor Well, technically a square is also a rhombus... best to have a set of activities for classifying quadrilaterals #mathchat
9:30 pm
bucharesttutor:
Another question I was asked recently, why is 1 not a prime number? #mathchat
9:30 pm
emth:
Misconception: to convert a decimal into a percent, multiply by 100. #mathchat
9:30 pm
mathsatschool:
Not really a misconception but 'difference'? Well one's odd and one's even, one has a curve in the number, one has 3 digits..... #mathchat
9:30 pm
ColinTGraham:
She's a big girl, she can apologize for herself if she feels the need! RT @chris_1974: Can i apologise now for @dotsmy crashing #mathchat!
9:32 pm
ColinTGraham:
@bucharesttutor Because of the way prime numbers are defined, they have exactly two factors 1 and themselves. 1 only has 1 factor. #mathchat
9:32 pm
bucharesttutor:
@ColinTGraham can you mail me some quadilaterals activities stuff? #mathchat
9:32 pm
mathsatschool:
how about a= 1, b=2, c=3, etc. Or n+1 MUST mean o (or 'm' as I got today! - y11 again...) #mathchatmisconceptions
9:33 pm
EarlSamuelson:
@ColinTGraham regarding algorithmic addition/subtraction......have a look at this http://t.co/tpdAw9D#mathchat
9:33 pm
ColinTGraham:
@emth Related is increased by 200% means two times bigger... percentages are another 'sticky' area... fractions again! #mathchat
9:35 pm
dotsmy:
@ColinTGraham "She" doesn't.(PS Get @chris_1974 to tell you the one about 0.9 recurring being the same as 1. That's my favourite. #mathchat)
9:35 pm
mathsatschool:
No difference between bars touching / bars separate (to be fair, most adults don't understand this too inc.scientists) #mathchat misconcept
9:35 pm
ColinTGraham:
@bucharesttutor My Library tagged quadrilaterals http://bit.ly/if0h6Q these two sites are good starting points... #mathchat
9:35 pm
bucharesttutor:
Another troubled area for Ss, addition n subtraction of unlike fractions #mathchat
9:36 pm
ColinTGraham:
@mathsatschool Good luck on the 7,3,19,5 then... #mathchat
9:36 pm
mathsatschool:
Misconception #mathchat multiplying always makes the answer bigger, dividing smaller?
9:37 pm
chris_1974:
#mathchat @dotsmy makes a good point with his 0.99... If a line is measured as 5 cm (neartest cm) why is 5.5 the upper bound, not 5.49999...
9:37 pm
mathsatschool:
I could be thinking of these when all #mathchat tweeps have gone to bed! They keep appearing in my head!
9:38 pm
chris_1974:
obv I know the answer, but students don;t always get it. #mathchat
9:38 pm
chris_1974:
@mathsatschool this is where a fun little "Always Sometimes Never" lesson leaps to mind. #mathchat
9:38 pm
mathsatschool:
And I say the main solution is to pre-empt them. Prepare for them coming in advance. Make a list. Share with others. As we are :-) #mathchat
9:39 pm
chris_1974:
3 am job! RT @mathsatschool: I could be thinking of these when all #mathchat tweeps have gone to bed! They keep appearing in my head!
9:40 pm
ColinTGraham:
@mathsatschool The key point may be to make a list... with appropriate teaching strategies/approaches. Hasn't someone done this? #mathchat
9:40 pm
bucharesttutor:
@mathsatschool yes we must try compiling all the misconceptions n then offer possible answers #mathchat
9:41 pm
emth:
#mathchat #misconception x * x = 2x
9:41 pm
mathsatschool:
@ColinTGraham I don't know. Would be useful to find out. We could gdocs a list but not re-invent the wheel! #mathchat misconceptions
9:41 pm
bucharesttutor:
This is an evergreen classic lol RT @emth: #mathchat #misconception x * x = 2x
9:44 pm
ColinTGraham:
Does separation of topics or introduction of information 'piecemeal' compound things? What role do open investigations play? #mathchat
9:44 pm
mathsatschool:
Well count me in. #mathchat
9:44 pm
EarlSamuelson:
@emth very good point #mathchat
9:44 pm
chris_1974:
@ColinTGraham @mathsatschool could we create a similar to movemeon? # Tag collated by twapperkeeper, then edited to book. #mathchat
9:44 pm
emth:
#mathchat #misconception (10n)^2 = 100n^2 implies (10^n)^2 = 100^(2n) or 100^(n^2)
9:45 pm
ColinTGraham:
@chris_1974 That's what I was thinking, but using more than one archive solution... TwapperKeeper's been a bit iffy of late :-s #mathchat
9:45 pm
mathsatschool:
Will stop with adding random misconception thoughts for now. #mathchat. Still work to do. (Unless I get itchy and really MUST add!) Lol
9:46 pm
emth:
I would love to share these conversations with my colleagues in university! I will plan a presentation for tomorrow! #mathchat
9:46 pm
EarlSamuelson:
@ColinTGraham I think it CAN compound problems if care is not taken..... #mathchat
9:47 pm
ColinTGraham:
@chris_1974 I was thinking #mathmis fewer characters... @mathsatschool #mathchat
9:47 pm
chris_1974:
@ColinTGraham works for me #mathmis #mathchat
9:48 pm
ColinTGraham:
Anyway, I'll set something up and tweet it out... the more non-maths teachers who tweet, the merrier! #mathmis #mathchat
9:49 pm
bucharesttutor:
Anyway the bottom line is, it wd be wonderful for the all of us to get a copy of the Math misconception in one roof #mathchat
9:49 pm
ColinTGraham:
@earlsamuelson At what stages or with which topics is care especially needed, do you think, Earl? #mathchat
9:49 pm
mathsatschool:
@chris_1974 @ColinTGraham Cool. and 'mis' implies mistakes OR misconceptions (as different entities....) #mathchat (with hashtag!)
9:50 pm
bucharesttutor:
RT @ColinTGraham: Anyway, I'll set something up and tweet it out... the more non-maths teachers who tweet, the merrier! #mathmis #mathchat
9:52 pm
EarlSamuelson:
@ColinTGraham oh boy!!!! That's a loaded question......there are so many areas that we can botch up when introducing #mathchat
9:53 pm
emth:
Handling #misconceptions: drill and practice! Hate me if you want, but it certainly plays a role. #mathchat
9:53 pm
bucharesttutor:
Teach the basics very well, grill them in their minds by surely making them work harder n practice more n more #mathchat
9:53 pm
chris_1974:
being aware of misconceptions they are likely to see - the most common ones for a starter. plan for it. #mathchat
9:53 pm
EarlSamuelson:
@ColinTGraham I think care needs to be taken at all times when attempting to instill thoughtful insights into math #mathchat
9:54 pm
EarlSamuelson:
@bucharesttutor I agree with that.......anticipate problem areas and be proactive #mathchat
9:54 pm
ColinTGraham:
How about "spot the mistake" where students have to identify the misconceptions (if they can!) #mathchat
9:55 pm
emth:
Teachers should make every effort to be aware of and understand the faulty logic behind #misconceptions. #mathchat (like now!)
9:56 pm
ColinTGraham:
@emth What role does exploration play in the practice, or is it repetition of a 'set of procedures'? #mathchat
9:56 pm
EarlSamuelson:
@ColinTGraham that's a good approach......makes the students think more deeply #mathchat
9:57 pm
GrampsyPete:
RT @emth: Teachers should make every effort to be aware of and understand the faulty logic behind #misconceptions. #mathchat (like now!)
9:57 pm
emth:
@ColinTGraham my students hated "spot the mistake" problems -- a sure sign that they exercise the mind in just the right way! #mathchat
9:57 pm
bucharesttutor:
@ColinTGraham spot the mistake would be slightly asking for more as they need to know "concepts" b4 they know the misconception #mathchat
9:58 pm
mathsatschool:
@ColinTGraham ha! Just got the gcse reference. #mathchat #notslowhonestguv
9:58 pm
ColinTGraham:
@emth @bucharesttutor It can be a good diagnostic tool where students don't/can't see the misconception, though #mathchat
9:58 pm
emth:
@ColinTGraham I'm definitely a fan of inquiry/discovery, but motivating my students I find to be the hard part. #mathchat
9:59 pm
ColinTGraham:
@mathsatschool Codebreaker extraordinaire...! :-P #mathchat #notslowhonestguv
9:59 pm
ColinTGraham:
@emth What age group are you dealing with Jared? #mathchat
10:00 pm
emth:
@ColinTGraham I just completed my internship and worked mostly with grade 10 students. #mathchat
10:00 pm
ColinTGraham:
OK thanks everyone for another lively and entertaining #mathchat
10:01 pm
suburbanlion:
@ColinTGraham I like the "spot the mistake" idea. I might have to try that! #mathchat
10:01 pm
ColinTGraham:
Thursday is World Book Day, so we'll be discussing recommended reading either for mathematics or teaching (or a break from both) #mathchat
10:02 pm
EarlSamuelson:
@ColinTGraham Thank yo Colin et.al..........that was very fulfilling #mathchat
10:02 pm
bucharesttutor:
@ColinTGraham agreed a diagnostic approach sounds pretty exciting #mathchat. Challenging Ss with misconception is worth a try.
10:03 pm
chris_1974:
which reminds me - I'm taking my copy of "The man who counted" in to school to read to some classes on Thursday. #mathchat #worldbookday
10:03 pm
emth:
Thanks, glad I could catch the end! #mathchat
10:03 pm
bucharesttutor:
Thanks everyone for a wonderful n insightful #mathchat. Have a good one people :)
10:04 pm
ColinTGraham:
From 10th March, we'll be trying the "How do I teach..." focused on more specific topics. Please add ideas: http://bit.ly/aOhxOU #mathchat
10:05 pm
ColinTGraham:
Don't forget to follow @mathchat to keep up to date with changes. Archive will be online shortly, I hope... #mathchat
10:05 pm
ColinTGraham:
@bucharesttutor Thanks, Vijay have a good night! #mathchat
10:06 pm
chris_1974:
I've just posted "Square of Area 10" in repsonse to tonights #mathchat & @mathsatschool.http://bit.ly/fpNFg5
10:07 pm
ColinTGraham:
@emth By that stage, you're fighting against 10 years of 'established' methods. Ss r more likely to question purpose of 'new' ways #mathchat
10:07 pm
ColinTGraham:
@chris_1974 Ooh, Mr Insta-blog! #mathchat
10:08 pm
mathsatschool:
Thank You. RT @chris_1974 I've just posted "Square of Area 10" in repsonse to tonights #mathchat & @mathsatschool. http://bit.ly/fpNFg5
10:08 pm
bucharesttutor:
@ColinTGraham Thanks again Colin for this wonderful #mathchat. Good Night to you as well.
10:09 pm
emth:
@ColinTGraham Yes, luckily we have gone through quite a curricular reform in Sask. and inquiry is now central to the objectives. #mathchat
10:17 pm
chris_1974:
@ColinTGraham for me it's a good way of archiving ideas, and sharing them. #mathchat