The survey was posted on Twitter. I've left the survey open, in case you want to experience it first-hand, but I'm probably never going to go back to look at it.
Can you take a 2- to 7-minute survey for me? What images look like subtraction pictures to you? #MTBoShttps://t.co/WF7oBILnb7 pic.twitter.com/XNTOC6Bv87— Kendra Lockman (@klockmath) July 29, 2017
The survey asks the same question, Does this picture show the operation 9–2? about six different pictures, and asks people to explain why they answered the way they did. Notice that the platform I used for collecting the survey was Desmos Activity Builder, so the results are stated with the participants being called "students."
One thing that I found myself thinking repeatedly, as I was reflecting on my own thoughts and bouncing them off of the responses given, was that so many of the people who responded are involved in math education, and I think that many of them were answering this survey as if they were evaluating it for potential use in the classroom. That's totally understandable, as I did reach out as a math educator, and it's so easy to view these things under the lens of "would this be clear to the students I teach." But that wasn't my goal. I was really wanting to get inside the head of the person answering the question, not how they thought it would play out inside the heads of students. But, maybe with math educators, those two things are impossible to tease apart. Anyway, on to the results.
Image 1 (Screen 2)
Yes
Since it is fixed and cannot be cut up, I see it as "there used to be 9 and two left.
JOHANNES KEPLER
I can see 9 frogs but it looks like two are "different" or meant to be taken away.
KELLY MILLER
There are 9 frogs total and two are crossed out, or "taken away".
ÉMILE BOREL
9 frogs 2 go away
GALILEO GALILEI
9 frogs shown, but 2 crossed out.
JULIA ROBINSON
Removal is a common way of thinking about subtraction. This image indicates removing 2 from 9
KATHERINE COLEMAN JOHNSON
You are cancelling out two frogs making it 7.
HENRI POINCARÉ
There were 9 and 2 are eliminated. Or 9 and don't count two, so just 7.
SRINIVASA RAMANUJAN
Nine frogs pictured in total, but two have been zapped by the evil red x's.
HEE OH
I see 9 frogs with two crossed out to suggest take away.
WEN TSUN WU
Taking 2 away from 9. It's clear to me.
ÉVA TARDOS
Now I'm looking at seven frogs uncrossed out.
JOAN BERMIN
There are 9 total frogs, with 2 of them crossed off. That leaves 7 frogs that aren't crossed off. 9-2=7
HERMANN MINKOWSKI
Nine frogs to start. The X indicates two frogs are being taken away.
AL-KHWARIZMI
Originally there were 9 frogs, but 2 got crossed out
LENORE BLUM
Total 9 with 2 x"ed" out = subtracttion
JOSEPH LIOUVILLE
"take away" two frogs
CHRISTINE LADD-FRANKLIN
It shows taking away
GRACE CHISHOLM YOUNG
There are 9 frogs and 2 are crossed off showing that we aren't including them.
ALICIA BOOLE STOTT
You start with 9 and the X's represent taking away 2.
JOHN WALLIS
There were nine frogs originally and two are crossed out
CHARLES LEWIS REASON
I can see 9 frogs and the two red "X" I assume mean take-away.
SOFIA KOVALEVSKAYA
They're crossed out -- taken away
ANDREY KOLMOGOROV
There are 9 frogs and two are crossed out
OLGA TAUSSKY-TODD
Not proud of this, but this makes me think "There were 9 frogs and I stepped on 2 of them." 🙈
ELANA PISCOPIA
Nine frogs take away 2
FLORENCE NIGHTENGALE
I imagine that there were 9 frogs to start, but something terrible happened to the last 2, so there are 9-2 left
DIANA TAIMINA
Crossing out two means two frogs are gone. So yes, it's subtraction to me.
CHRISTIAAN HUYGENS
Removal of 2
AUGUSTIN CAUCHY
I see 9 frogs and 2 were stuck out.
DIEGO RODRIGUEZ
There is an X over two frogs, which makes them different from the others- separate.
MARJORIE RICE
Start with nine. Take away two.
MARYAN MIRZAKHANI
Nine frogs are visible, but two are marked in a way that has removed or deleted as connotations.
GERTRUDE BLANCH
9 frogs, then two are crossed out =7
KATHERINE JOHNSON
9 frogs are there, with two frogs crossed out, there are 7 frogs remaining.
JOSEPH FOURIER
removal of 2
SHIGEFUMI MORI
Definitely shows subtraction as "take away" mode. Shows original quantity, action of removed quantity and new quantity.
COURTNEY GIBBONS
It. Works if u. R there to explain the exes mean take away
EMMY NOETHER
There are 9 and now two are crossed out.
PETER DIRICHLET
I don't love it because I still had to slow-count to 7, but technically it works
MARIA CHUDNOVSKY
It is one way to represent 9-2 in the context of nine frogs take away two frogs
ARISTOTLE
Start with nine, two crossed out---taken away or subtracted
MANJUL BHARGAVA
Nine total frogs, two left to jump in a pond. I crossed out the two that went swimming
ALAN TURING
There were 9, you killed 2...
DORIS SCHATTSCHNEIDER
The red x looks like you are marking them out. They were there, but now they do not belong as part of the group. "They've been ghosted."
WILLIAM HAMILTON
Streichung impliziert eine Subtraktion
JACQUES HADAMARD
9 total, 2 "taken away"
LOUISE HAY
Nine frogs. Cross out two. Seven left
VICTOR NEUMANN-LARA
There are 9 frogs but 2 don't count
DOROTHY VAUGHN
Took two away
MARIA AGNESI
I see nine frogs. Two of the frogs are crossed out leaving seven frogs
DIOPHANTUS
There are nine with two crossed out - the children would need to understand crossing out is the same as subtraction
ARTHUR CAYLEY
There are 9 frogs to start and 2 are being removed
RICHARD DEDEKIND
Red x's clearly denote cancelling something
SOPHUS LIE
It shows 2 being taken away from 9
ALFRED TARSKI
Clear cut, unambiguous to my eyes.
PIERRE DE FERMAT
Can see all 9 but 2 were taken away
JOHANN BERNOULLI
There were 9 frogs, you removed two.
ERNST KUMMER
there are 9 frogs and 2 are crossed off
HIPPARCHUS
There were 9 and 2 got crossed off.
HENRI LEBESGUE
This is the image that most textbooks show for such an operation, so that is the familiar meaning to me.
ÉMILIE DU CHÂTELET
There are 9 frogs, but 2 are crossed out.
kind of, but not easily
MARY ROSSThe 'story' of the situation isn't terribly clear here. If , for example, there was a row of nine frogs then two,were covered by flowerpots ( for example) I could see separation. If there were 9 lily pads ( for example) and 7 frogs I could see comparison. I can interpret this image, but it feels as if I have to already know And understand subtraction rather than it being a 'constructive' image. Maybe because it is static/ finished. Also, the number of frogs isn't easy quick to discern/ recognise in this layout, although I recognise the importa
GEORG CANTOR
I think for younger kids, removing the 2 ( like moving them below the original group or having am arrow to show them heing taken away is clearer
ELBERT FRANK COX
The 2 are still exactly where they started.
EUPHEMIA LOFTON HAYES
the frogs are still there even though crossed out. kids need to know the meaning of the red X
CARL JACOBI
There's a lot going on.
NGÔ BÀO CHÂU
My immediate reaction was to count the regular frogs then add the two that are crossed out.
JEAN TAYLOR
The subtraction part was clear. However, I had to count to 9 to comfirm how many frogs there were. It would be clearer if the frogs were grouped for subatizing.
JOHN VON NEUMANN
I'd see it more clearly if this were structured like a P-P-W problem or comparative subtraction.
Image 2 (Screen 3)
Yes
MARY ROSSYes, blue frogs are described as two fewer than green frogs here
FELIX KLEIN
9 green frogs minus the 2 more green frogs equals the blue frogs
HERTA FREITAG
There are 2 fewer blue frogs than green frogs
GOTTFRIED LEIBNIZ
Comparing
MARIA CHUDNOVSKY
This is showing me the difference by showing seven frogs. I suppose it could actually be showing 9-?=7.
CARL JACOBI
The two seperate groupings make it easier to identify the problem.
SOPHUS LIE
Yes, you can see how 2 fewer than 9 is 7
JOHANN BERNOULLI
You're comparing the two lines of frogs, and can see the difference as the gap.
No
KARL WEIERSTRASSIt looks like 7 from another team joined them.
ISMAIL MUSTAFA AL-FALIKI
To me, that looks like 9+7.
GEORG CANTOR
It shows the difference of 9 and 7 which feels like an important distinction
ÉMILE BOREL
9 green frogs. 7 blue. Looks like addition
JULIA ROBINSON
This shows nothing unless there is some compelling reason to believe a blue frog cancels a green frog. The directions could also work with context, but either way it is a stretch. Even if blue/green or left/right are opposites this image indicates 9 + (-7) not 9 - 2
HENRI POINCARÉ
I see 16. There is nothing that is obviously telling me that one counteracts the other.
SRINIVASA RAMANUJAN
The image doesn't convey subtraction-ness to me because the left-facing frogs are distinct from the right-facing frogs. And to me, subtraction means "taking away."
WEN TSUN WU
Looks more like comparing
ÉVA TARDOS
Not to me. I see 9-7 ... maybe.
HERMANN MINKOWSKI
If the blue frogs were red, I'd be inclined to say 'kind of.'
AL-KHWARIZMI
It looks like 9+7 since there are nine green frogs and seven blue ones
LENORE BLUM
9-7???
JOSEPH LIOUVILLE
I don't immediately *see* subtraction, but primed to look for it, can think about "the difference between 9 and 7", which suggests 9-7, not 9-2.
GOTTLOB FREGE
There's nothing to tell me to subtract or add
ALICIA BOOLE STOTT
Hmmm, not sure I see this as a good representation.
JOHN WALLIS
To me it shows two separate sets and nothing getting taken away. If I thought about integers and zero pairs I could make an argument, but it seems like a reach to do that.
ANDREY KOLMOGOROV
There are 9 frogs facing right There are 7 frogs facing left
FLORENCE NIGHTENGALE
Without context, I don't know if I should add to find the total, consider how many more green frogs there are, or some other question
DIANA TAIMINA
It looks like adding the green frogs and the midnight frogs to find the sum.
DIEGO RODRIGUEZ
There are two completely different groups of frogs.
MARYAN MIRZAKHANI
There is a row of nine frogs and a row of seven frogs, but they are different looking frogs and so seem like different things.
KATHERINE JOHNSON
Nothing s crossed out, and there are more re than 9 frogs shown.
INGRID DAUBECHIES
These are totally different frogs. I want to add them. COURTNEY GIBBONS Too unclear
EMMY NOETHER
I see 16 frogs.
MARY CARTWRIGHT
This looks like 9 green plus 7 blue
EUPHEMIA LOFTON HAYES
this looks more like 9 + 7
MANJUL BHARGAVA
Not a fan of this model. Am I adding the second batch of frogs? Why are they in a different direction? I'm noticing/wondering too much here.
WILLIAM HAMILTON
Grüne Frösche kann man nicht mit blauen Vergleichen Don't compare apples and oranges
DAVID BLACKWELL
It seems to show 9 & 7.
LOUISE HAY
There's 16 frogs. I don't give a damn what color they are!
VICTOR NEUMANN-LARA
It looks like 9 - 7 to me. There are 9 but 7 are "cancelled out"
MARIA AGNESI
I'm not sure the purpose of the blue frogs, yet.
NGÔ BÀO CHÂU
I'm not sure what my first reaction was but it wasn't to subtract.
ARTHUR CAYLEY
It looks like there are two different kinds of frogs that should be added
RICHARD DEDEKIND
In my algebra brain, those aren't the same things. They're frogs, but they're not the *same* frogs, like apples and oranges. It could be dependent on the age group though.
PIERRE DE FERMAT
Seems to show like terms more than anything
ANDRÉ WEIL
Looks more like adding 9+7 to me
HIPPARCHUS
It shows 9 +7
HENRI LEBESGUE
You would need some context for this to be understood as subtraction. It could be addition.
JEAN TAYLOR
It looks like addition. Same issue as before with the long lines of frogs.
JOHN VON NEUMANN
This shows 9-7 in comparative form.
kind of, but not easily
JOHANNES KEPLERSince you had the 9-2 in the statement I could make a connection - seeing just the image I might think 9+7 or 7+_=9. This image has more interpretations IMO.
KELLY MILLER
If there was some explanation about the 2 kinds of frogs, maybe.
GALILEO GALILEI
My first interpretation of the blue frogs was as "negative" frogs. Also, this could easily be thought of as 16 frogs.
KATHERINE COLEMAN JOHNSON
I can see a green frog cancels out a blue frog but without explanation that could be difficult to understand
HEE OH
I see a comparison of the number of green frogs and number of blue frogs. There are 2 more green than blue. I'd tend towards 9-7=2, but 9-2=7 is related.
JOAN BERMIN
If you have a group of 9 frogs and you remove 2 frogs, you will have a group of 7 frogs. Thus, 9-2=7. But since the frogs are different types, it's not really clear that you're looking for THIS subtraction. I think this picture is a much better illustration of 9-7=2 because that would answer the question of how many more frogs are in the top group than in the bottom group.
CHRISTINE LADD-FRANKLIN
Nothing explicitly indicates that you want the second line to be considered in relation to the first, or that we don't want the total. Also, if there is subtraction, the first I would consider is what is the difference, or 9-7, not 9-2
GRACE CHISHOLM YOUNG
The blue frogs represent the answer, but not the operation as obviously.
CHARLES LEWIS REASON
I guess I can see how this could be viewed as take the number of green frogs (9), and take away two to get the number of blue frogs. But I see it more as an addition problem- the blue frogs are joining the green frogs (maybe because the 9 frogs and from the previous slide/problem, so it seems like the blue frogs were added to the situation).
SOFIA KOVALEVSKAYA
This would fly in the face of what I teach for adding and subtracting: that you have to be adding and subtracting the same things. Changing how they look would really confuse things. My students already don't know which things to look at that matter, and which don't. (Parentheses: when do you "do them first"? Well, not when it's 6 / 2(1 + 3) and "do them" means multiply to the student...
OLGA TAUSSKY-TODD
If the question might read something like, "how many more green frogs than black frogs?"
ELANA PISCOPIA
There are were 9 green frogs. There are two fewer blue frogs. How many blue frogs in all?
CHRISTIAAN HUYGENS
Comparison with difference of 2
AUGUSTIN CAUCHY
By comparison, yes. Many do not interpret the subtraction sign that way though.
MARJORIE RICE
I see a row of nine, then another row with "2 less than 9."
ELBERT FRANK COX
Why are they blue now? Why did they turn? Too many distractors.
GERTRUDE BLANCH
I can see how it could mean 9-2, but to an elementary child, they may be confused by it.
JOSEPH FOURIER
the context of frogs here is odd . . . does it mean living and dying frogs?
SHIGEFUMI MORI
It could be used to show subtraction as difference but because these numbers are so close (9 and 7) I'm more likely to see this as 7+2=9
PETER DIRICHLET
It's better, but it's kind of distracting with the different shapes, directions and colors
ARISTOTLE
You'd have to know to compare the two sets.
ALAN TURING
It highlights the difference between 9 and 9-2 (not sure if that makes sense)
ARCHIMEDES
The last two frogs need to be X'ed.
DORIS SCHATTSCHNEIDER
This shows 9+-2 or 9+7. I get it with the frogs facing different directions, but there are two group so distinct figures. Why would I take subtract 7 blue frog from 9 green frogs? That's like subtracting apples from oranges. (I wonder what people would say if there was no color?)
JACQUES HADAMARD
As a student, with no question or prompt, I would add.
DOROTHY VAUGHN
maybe better if frogs were all the same; that is, here's what's left over after two frogs left
DIOPHANTUS
The frogs have changed. Will the children understand this?
ALFRED TARSKI
I see it, but I wonder if that's because I know it is supposed to. I don't think it's obvious at face value.
ERNST KUMMER
there are 2 frogs "missing" in the array at the bottom to show the 2 and 9 frogs on the top showing 9
ÉMILIE DU CHÂTELET
To me it shows more clearly that there are 2 less blue frogs than green ones.
Image 3 (Screen 4)
Yes
MARY ROSS
Green frogs are described as 7 more than blue frogs here
FELIX KLEIN 9
Green frogs minus the 2 blue frogs equals the difference
HEE OH
I see comparison b/w green and blue frogs. How many more green frogs than blue. Situation equation seems to be 9-2=7
JOAN BERMIN
There are 9 frogs in the first group and 2 frogs in the second group. If you want to know how many more frogs are in the first group than the second group, you'd do 9-2 and get 7 frogs.
LENORE BLUM
9 to right - 2 to left
ELANA PISCOPIA
Comparison situation There are 9 green frogs. There are two blue frogs. How many more green frogs are there than blue?
GOTTFRIED LEIBNIZ
How many more?
CHRISTIAAN HUYGENS
Difference
JOSEPH FOURIER
I guess it could mean two frogs died? Still somewhat confusing as the 'living' ones are still there.
CARL JACOBI
Seperate groupings prove easier
VICTOR NEUMANN-LARA
Yes - there are 9 frogs but the 2 frogs going the other way cancel 2 out
MARIA AGNESI
A zero pair is created when 2 green frogs and 2 blue frogs are circled. Green frogs represent positive 2. Blue frogs represent -2. 2+ -2 equals zero leaving seven frogs
SOPHUS LIE
Yes, this shows the difference between 9 and 2 is 7
JOHN VON NEUMANN
Yes, this would be a good example of finding the difference between two groups.
I see 9 and 2.
ISMAIL MUSTAFA AL-FALIKI
9+2
GEORG CANTOR
Again, these layered images feel like they illustrate difference. To me this is the difference of 9 and 2
ÉMILE BOREL
Same as slide 3
JULIA ROBINSON
See previous answer. This shows 9 + (-2).
SRINIVASA RAMANUJAN
Same reasoning as slide 3 of 9
WEN TSUN WU
Again, looks like a comparison.
HERMANN MINKOWSKI
Same as previous answer--make the blue frogs red.
AL-KHWARIZMI
It seems like 9+2 since there are more frogs
GOTTLOB FREGE
I could add or subtract or multiply or divide
ANDREY KOLMOGOROV
There are 9 frogs facing right There are 2 frogs facing left
FLORENCE NIGHTENGALE
Same answer as previous
DIANA TAIMINA
Same as the previous slide.
DIEGO RODRIGUEZ
Two different groups. You are not taking frogs away.
ELBERT FRANK COX
Same as last one, but worse. I'd say that's 9 - 7 if anything.
KATHERINE JOHNSON
This looks like 9'green frogs facing one way and 2 blue frogs facing the other. There are more than 9 frogs with nothing crossed out.
INGRID DAUBECHIES
No. Totally different looking things.
SHIGEFUMI MORI The blue frogs look like they're being added like a part part whole problem
COURTNEY GIBBONS
Doesn't make sense
EMMY NOETHER
11 frogs
MARY CARTWRIGHT
9green plus 2 blue
MARIA CHUDNOVSKY
This would be more like 9-7 to me with two being the difference. I could see it as 9-?=2
EUPHEMIA LOFTON HAYES
Looks more like 9 + 2
MANJUL BHARGAVA
Again, what's up with the blue frogs? I think this shows addition more than subtraction.
ALAN TURING
Maybe if you had circled the 2 in each line?
ARCHIMEDES
It's not clear.
WILLIAM HAMILTON
Don't compare apples and oranges
DAVID BLACKWELL
It shows 9 & 2.
LOUISE HAY
11 frogs.
NGÔ BÀO CHÂU
Same as the last one but actually less likely to make me want to subtract.
DIOPHANTUS
The frogs have changed and seven have gone not two.
ARTHUR CAYLEY
Again, looks like 2 different kinds of frogs to be added
RICHARD DEDEKIND
See previous response.
PIERRE DE FERMAT
Still like terms
HIPPARCHUS
9+2
HENRI LEBESGUE
Again, context would be needed. Could be addition.
JEAN TAYLOR
Same as slide 3.
Similar reply to last image
KELLY MILLER
Same as last one
GALILEO GALILEI
If students understand that subtraction is equivalent to adding the opposite of the "subtrahend".
KATHERINE COLEMAN JOHNSON
Again without explanation I may tend to just add the frogs together instead of realizing the blue and green ones cancel out
ÉVA TARDOS I have to assume that the blues are being taken away or superimpose the question "how many more frogs are facing right."
JOSEPH LIOUVILLE
See previous comment.
CHRISTINE LADD-FRANKLIN
Slightly better than the previous problem, but again nothing that explicitly says you are interested in the difference rather than a total. Maybe this is related to people's tendency to focus on positive space over negative space.
GRACE CHISHOLM YOUNG
There needs to be some interface between the green and blue frogs to explain what is happening with then.
ALICIA BOOLE STOTT
It shows subtraction as being the difference of 2 amounts. I think there is a leap to this being sub without something showing that what we're focusing on is the 7 extra green frogs. Maybe circle them? I suppose without context we might not know if this is 9-2=7 Or 9+2=11.
CHARLES LEWIS REASON
I could a teacher asking, "How many more green frogs are there than blue frogs?" So you'd subtract two of each (zero pairs) to get an answer of 7. But again this looks more like addition to me.
SOFIA KOVALEVSKAYA
A little mre easily than the previous *if* this is connected to "adding negative integers and I've learned that the blue guys are facing the other way.
OLGA TAUSSKY-TODD
I'm second guessing myself 😂 I can see a kid thinking this represents 9 hops right and 2 hops left, mt thinking about the difference from in the frogs.
AUGUSTIN CAUCHY
Same answer as before regarding comparison.
MARJORIE RICE
The row of two can be described as negative units and make zero pairs with two green frogs.
MARYAN MIRZAKHANI
There is a row of nine frogs and a row of two frogs, but they are different looking frogs and so seem like different things. However, it can show subtraction in the sense of there being 9-2 more green frogs than blue frogs.
GERTRUDE BLANCH
If you cancel out the two frog pairs in the beginning you get 7.
PETER DIRICHLET
Same reasons. I would put them in squares of easily calculated areas
ARISTOTLE
Again only if you knew to compare the two sets
DORIS SCHATTSCHNEIDER
Again, these are different object and color. They are not like terms and there's nothing to indicate one was part of the group, but now is not. Kinda strange how when you stack numbers and put the operation sign, it is clear. But without the operation sign, I see add.
JACQUES HADAMARD
Same as previous
DOROTHY VAUGHN
I'm imagining the different type frogs pairing off and hopping away
ALFRED TARSKI
See answer to previous slide.
JOHANN BERNOULLI
You have to understand the dark frogs as negative frogs, or know commutative straight away.
ERNST KUMMER
I guess it could, I wouldn't normally look at it this way but there are 9 frogs in the beginning and the bottom line shows 2
ÉMILIE DU CHÂTELET
I can figure out how many more green frogs by creating a 1-1 correspondence between the 2 blue ones and 2 green ones, and counting the green frogs remaining - lotsa work.
Image 4 (Screen 5)
9 things, 2 of them are frogs
HEE OH
Could be "I need 9 frogs for my collection. I already have 2. How many more do I need? Some students might solve as missing addend prob but 9-2 seems to match.
AUGUSTIN CAUCHY
This feels like a start unknown problem.
EMMY NOETHER
Here are 7 boxes.
NGÔ BÀO CHÂU
More of 2+? But that's good for equivalency.
SOPHUS LIE
I'm thinking there were 9 spots available and 2 frogs jumped on spots so now there are only 7 spots available.
JOHANN BERNOULLI
'I had 9 frogs; I'm hiding 9, how many can I still see?'
No KARL WEIERSTRASS
I think it shows 9-7.
MARY ROSS
This feels like How many more to make 9 So is additive Recognise that this is comparison/complementary addition, but the frogs concretise this, so it doe
KELLY MILLER
It looks more like 2 + (7 what?)
GEORG CANTOR
I don't really see the logic of thos diagram ( teying to see through the eyes of soneone learning subtraction)
CAROLINE SERIES
To me, it shows 9-7.
ÉMILE BOREL
Doesn't show nine of same thing
JULIA ROBINSON
To me this shows 2 + _ = 9
KATHERINE COLEMAN JOHNSON
Not sure what the empty boxes me or the 2 frogs. I found this the most confusing
SRINIVASA RAMANUJAN
To me, this image conveys "2 + seven unknowns"
WEN TSUN WU
Looks more like 2+_=9
HERMANN MINKOWSKI
Kinda looks like 9-7. The empty rectangles represent frogs that were removed.
LENORE BLUM
Confusing
JOSEPH LIOUVILLE
This could maybe be 9-7.
CHRISTINE LADD-FRANKLIN
The frogs are the objects, the dashed boxes strongly imply that those things are missing. If you are wondering about communicating an idea clearly (as opposed to encouraging all interpretations or out-of-the-box thinking) I think this is most clearly 9-7
GOTTLOB FREGE
This shows addition imo
ANDREY KOLMOGOROV
I think it shows 9-7
FLORENCE NIGHTENGALE
I'm quite thrown off by the empty boxes. Again,cant guess without context what the question is
ELBERT FRANK COX
That's more like 9 - 7 again
KATHERINE JOHNSON
It looks mre like 9 - 7.
JOSEPH FOURIER
this one seems confusing
SHIGEFUMI MORI
It looks like 9-7--like those boxes represent removed frogs.
COURTNEY GIBBONS
Unclear if you should put something into the boxes
MARY CARTWRIGHT
To me it looks like 2 plus boxes
MARIA CHUDNOVSKY
To me is represents ?-2=7 or 9-?=2. It is similar but the context is different to me.
ARISTOTLE
Feels much more like 2+?=9 missing addend
EUPHEMIA LOFTON HAYES
looks more like 9 - 7
MANJUL BHARGAVA
Nope. It looks like frogs are missing and that's not really associated with subtraction.
ALAN TURING
What was in those boxes???
WILLIAM HAMILTON
2 frogs and 7 boxes
DOROTHY VAUGHN
Feels more like 9 - 7
MARIA AGNESI
I see
1. Addition. Such as 2+ what number equals nine.
2. 9-7 = 2
DIOPHANTUS
It looks like you have taken seven away
ARTHUR CAYLEY
Looks like 2 plus ___ is 9; maybe would be different if the frogs were also in boxes
ERNST KUMMER
I don't really see the 9
HENRI LEBESGUE
I feel fairly confident that my students would interpret this to mean 9-7.
ÉMILIE DU CHÂTELET
What goes in the empty boxes?
I thin I this more of 2+_=_ or 2+7=_
ISMAIL MUSTAFA AL-FALIKI
Looks like 2 + ? = 9 which is kind of what you're asking but not exactly.
GALILEO GALILEI
Maybe, but it could also be an addition problem or maybe even a pattern continuations (frogs forever!)
HERTA FREITAG
There are 9 frogs.. only 2 are green. How many are blue
JOAN BERMIN
This picture might fit with the question "If I already have 2 frogs and want 9 frogs total, how many more frogs do I need?" It's 7, and 9-2=7. I'd see this more as an equation to solve, though. 2+x=9, which would be solved by subtracting 2 from both sides, giving the 9-2 operation. But that feels like a stretch.
AL-KHWARIZMI
To me it looks like 2+ what equals nine, or how many frogs need to go in the empty boxes.
GRACE CHISHOLM YOUNG
Eh, it would work better for me if there were 2 white boxes and 7 frogs.
ALICIA BOOLE STOTT
similarly to the previous screen I can see all the parts of the problem but I'm not sure I see the act of subtraction... at least not a form of sub I can think of right now.
JOHN WALLIS maybe with explanation, but appears more like something is to be added in or a pattern created
CHARLES LEWIS REASON
Maybe; if the purple boxes are covering frogs that were originally visible, and you wanted to see how many frogs would need to leave if you only wanted 7 frogs. This doesn't look like an obvious math problem to me. It's unclear what's up with the purple boxes.
SOFIA KOVALEVSKAYA
I might teach it for "two plus what is nine?" but ... I'd want to use something consistent for the nine I need...
OLGA TAUSSKY-TODD
Overthinking? "I have 2 frogs - how many more until I have 9?"
ELANA PISCOPIA
Part part whole There are 9 frogs in all. Two of them are green. How many other frogs are there?
DIANA TAIMINA
How many more to make 9. Or how many to fill up the empty cages. You need 7, which is the same as 9-2 but it seems to show a different thing.
CHRISTIAAN HUYGENS
Difference again I suppose
DIEGO RODRIGUEZ
I see this as 2 + ___ = 9. You will get the same result, but the context does not match the operation.
MARJORIE RICE
I see a row of nine things. Two of one thing and 7 of the others. This could show a relationship of 2,7, and 9. However, this image to me feels more like a ratio with the different units in the same row.
MARYAN MIRZAKHANI
This feels more like an addition picture (2+7) to me, but I can make the jump from that to subtraction.
GERTRUDE BLANCH
9 rectangles, minus two (replaced by frogs?)
INGRID DAUBECHIES
Might be hard to realize at first that the boxes are missing frogs.
DORIS SCHATTSCHNEIDER
This shows 2 + x = 9. You will need to use subtraction (or just knowledge of addition) to determine the answer.
LOUISE HAY
Bit of a stretch. I can see it if I really try to. Not inuitive but I see the add on concept
VICTOR NEUMANN-LARA
I guess there were 9 empty boxes and you've taken 2 away, but by adding frogs
RICHARD DEDEKIND
I think of "-" as something missing. This reads more as 9-7 to me.
ALFRED TARSKI
It seems to indicate 9 - 7 rather than 9 - 2 to me.
PIERRE DE FERMAT
2+x=9 is more what I see
ANDRÉ WEIL
Need to take a second to interpret the boxes
HIPPARCHUS
This seems to show 2 + ? = 9
JEAN TAYLOR
This looks like 2 plus something equals something . More of a missing addend problem than direct aubtraction. I'd still like to see smaller groups within the objects so I don't have to count by ones to know there are 7 boxes.
JOHN VON NEUMANN
I see this as a missing addend problem.
Image 5 (Screen 6)
The arrow indicates that I was at 9 and decreased by 3.
JOHANNES KEPLER
Start at 9 a move "back" or take away 2. Where are you [on the numberline]
KELLY MILLER
A jump of 2 backward from 9
ÉMILE BOREL
Starts at 9 backs up 2 (goes lower)
JULIA ROBINSON
Moving left on the number line, with proper development, is a fairly natural way to think about subtraction.
KATHERINE COLEMAN JOHNSON
You can see the visual of going back 2 in the number line
HENRI POINCARÉ
I would have liked to see the jump to 9 first.
SRINIVASA RAMANUJAN
To me, this is a standard, number line representation of starting at 9 and taking away 2. The arrowhead indicates direction (movement in the negative direction, which is also known as subtraction). But this is based on my familiarity with this representational technique.
HEE OH
Starting at 9, take 2 away.... 2 less than 9...
WEN TSUN WU
Perfect
ÉVA TARDOS
Maybe I'm just familiar with the number line representation of subtraction.
JOAN BERMIN
You start at 9 and move 2 to the left, ending at 7. 9-2=7.
HERMANN MINKOWSKI
A classic picture of subtraction: start at 9, take away 2.
AL-KHWARIZMI
You started at nine and went back to the left two spaces
JOSEPH LIOUVILLE
Start at 9, back up 2 spaces is 9-2.
CHRISTINE LADD-FRANKLIN
We are going back 2 from 9.
GRACE CHISHOLM YOUNG
Traditional.
ALICIA BOOLE STOTT
love number line representations. Start at 9 and go down 2. This could also be 9 + -2.
JOHN WALLIS
I'd rather it started at zero to nine, but shows a nine and two less from the 9 gives 7
CHARLES LEWIS REASON
The arrow starts at 9 and goes to the left (subtraction) two places.
SOFIA KOVALEVSKAYA
Yup. Lots of folks learn about subtraction as "going backwards on the number line.
ANDREY KOLMOGOROV
Move two units from 9
OLGA TAUSSKY-TODD
Start at 9, go back 2
ELANA PISCOPIA
Take away on a number line
GOTTFRIED LEIBNIZ
Stepping back
FLORENCE NIGHTENGALE
Probably because I'm used to seeing this representation
CHRISTIAAN HUYGENS
Counting down
AUGUSTIN CAUCHY
Subtraction as distance makes sense to me.
DIEGO RODRIGUEZ
I see beginning at 9, and jumping back two places.
MARJORIE RICE
Start at nine and move two in the. What I've direction.
ELBERT FRANK COX
From 9 you moved back 2
GERTRUDE BLANCH
Start at 9, minus 2 (move to the left).
KATHERINE JOHNSON
It might help to show two arrows to the left, each with a length of one unit.
JOSEPH FOURIER
starting at 9, travelling back two. would work better if there were other examples to give contexts of plus means forward, minus backward
EMMY NOETHER
Started at 9 and rolled back two.
PETER DIRICHLET MARIA CHUDNOVSKY
This seems pretty straight forward to me. The difference between as nine take away two is seven. I could also see it as the difference between nine and seven is two though.
ARISTOTLE
But would like to start at zero, go to 9 Then subtract 2
EUPHEMIA LOFTON HAYES
yes but need a fair amount of specialized knowledge about number line conventions to understand it.
MANJUL BHARGAVA
It starts at nine and removes two. Makes sense to me.
DORIS SCHATTSCHNEIDER
This is all on one line and clearly shows starting at 9 and counting back 2 spaces, so it does show the operation 9-2.
WILLIAM HAMILTON
9 steps forward and 2 steps back
JACQUES HADAMARD
Love number lines
CARL JACOBI
Line graphs are always a good option.
LOUISE HAY
Back two or left two on numberline from nine
VICTOR NEUMANN-LARA
You're at 9, then you move 2 in the negative direction
DOROTHY VAUGHN
Subtraction as motion
NGÔ BÀO CHÂU
Two less than nine
DIOPHANTUS
Yes but you could add to the loop minus 2 to show you have jumped back two.
ARTHUR CAYLEY
Yes!
RICHARD DEDEKIND
Start at 9, move 2 in the negative direction.
SOPHUS LIE
Starting at 9, removing 2 to land on 7
ALFRED TARSKI
Yes, but only because I know how a number line works. It's not my favorite representation.
JOHANN BERNOULLI
Yes, though lots of people put the subtraction arrow under the line.
HIPPARCHUS
Moving to the left is subtraction.
HENRI LEBESGUE
Especially with the arrow.
JEAN TAYLOR
This is very clear to me. The number line helps a lot!
JOHN VON NEUMANN
I think this is a good example of 9-2.
ÉMILIE DU CHÂTELET
9 on the number land, go backward 2 to 7
It shows how many units to go from 9 to 7. The distance is two. Well, I guess you could say 9-2, but it isn't obvious.
MARY CARTWRIGHT
Maybe to an older student?
CAROLINE SERIES
I think an arrow from 0 to 9, then back to 7 is more likely to show 9-2.
LENORE BLUM
Use arrow to 9 first
GOTTLOB FREGE
Show two jumps
ARCHIMEDES
I would show an arrow going to 9 first.
DAVID BLACKWELL
Shows 9 to 7. Which IS 9-2 but still, come on.
Image 6 (Screen 7)
Distance!
FELIX KLEIN
If you see that 9 is the whole and 2 is the part you know
HEE OH
I see this as distance between 9 and 2
CHRISTINE LADD-FRANKLIN
I think most students wouldn't realize that this is showing the difference between 9 and 2, but this is definitely worth exploring.
ALICIA BOOLE STOTT
The difference or distance between 9 and 2.
CHARLES LEWIS REASON
It shows the difference between 9 and 2. Although it looks more to me like 9-7.
ANDREY KOLMOGOROV
2 numbers are left off
OLGA TAUSSKY-TODD
Shows distance
ELANA PISCOPIA
Comparison on a number line
GOTTFRIED LEIBNIZ
How far apart?
CHRISTIAAN HUYGENS
Difference
MARIA CHUDNOVSKY
The difference between nine and two is seven.
DOROTHY VAUGHN
Subtraction as distance
ARTHUR CAYLEY
The difference between 9 and 2
SOPHUS LIE
Difference/distance between 9 and 2
JOHANN BERNOULLI
Not quite so clear, but one way of seeing that is the difference between 9 and 2.
HENRI LEBESGUE
Although context would be needed for student certainty
If anything it's 9-7, but arrows would make the operation clearer
ÉMILE BOREL
No direction arrow...difference is 7
SRINIVASA RAMANUJAN
This picture shows a seven unit line segment that starts at 2 and ends at 9 because my default is to start at the origin and "read" from there.
HERMANN MINKOWSKI
I'd prefer to see a straight arrow go from 0 to 9 (pointing right) and a straight arrow below it go from 9 to 7 (pointing left).
AL-KHWARIZMI
To me it shows a length of seven but I don't know what the operation is
LENORE BLUM
What?
GOTTLOB FREGE
I would say 9 -7 if any
MINA REES DIEGO RODRIGUEZ
I think a better way of showing this would be 2 + ____ = 9, or 9 - _____ = 2
ELBERT FRANK COX
That's the distance from 2 to 9, which is conceptually very different
KATHERINE JOHNSON
It looks like 9 - 2.
JOSEPH FOURIER
where is the whole 9? maybe if the number line didn't continue, then you could have the total bar of 9 units juxtaposed with two removed
MARY CARTWRIGHT
To me this shows a segment of length 7.
MANJUL BHARGAVA
Where's the starting point? This doesn't make sense to me at all. If you had an arrow then I could make some type of coherent explanation.
ARCHIMEDES
It looks more like 9 - 7.
WILLIAM HAMILTON
Nur eine Gerade auf dem Zahlenstrahl der Länge 7
JACQUES HADAMARD
It tells me how many between 2 and 9 which is the same answer as 9-2, but not the same question
DAVID BLACKWELL
I have no idea what that is.
VICTOR NEUMANN-LARA
It's more like 9-7 because you move 7 backwards
DIOPHANTUS
You have drawn a line. Why that long? How would children understand what has been done here?
I can see 2+7=9 so I can get to it.
JOHANNES KEPLER
More like what is the distance from 2 to 9? could be 9-2 or 2+_=9
KELLY MILLER
Yes in that 9-2 can be thought of "the distance between 9 and 2", but I feel like this is less intuitive...however it's easier to model 9-(-2) this way when trying to understand why 9-(-2)=9+2
JULIA ROBINSON
First you have to understand subtraction as directed distance. Even then does this picture show 9 - 2 or 2 - 9?
KATHERINE COLEMAN JOHNSON
If students understand it's distance they would realize the answer is 7 but as it stands I think it is confusing
MICHAEL ATIYAH
I think an arrow would help.
HENRI POINCARÉ
shows the distance between, so yes...it would depend on what was done with students previously.
ÉVA TARDOS
It COULD. I think I'm just less familiar with this representation.
JOAN BERMIN
It's relatively easy for ME to see 9-2, but I'd expect students to have a tougher time seeing it. I think of this as a question of the segment's length. It starts at 2 and ends at 9, so the length is 9-2=7. I think students would have trouble seeing this as subtraction because the number 7 doesn't jump out at them like in the previous example. And because there aren't objects to count (like the frogs), it's perhaps less obvious that we're looking at 7.
JOSEPH LIOUVILLE
I see it more as 9-7, although I see how it could be 9-2.
GRACE CHISHOLM YOUNG
It makes sense if you think of subtraction as distance, but as is needs some explanation.
JOHN WALLIS
Shows 9-7 more to me
SOFIA KOVALEVSKAYA
Again depends on how I've presented things. If we've discussed sbtraction as "difference," then this shows that.
FLORENCE NIGHTENGALE
It seems to ask me to find the distance, but I might count up (add) instead
DIANA TAIMINA
It shows the distance between 9 and 2. Again, it is not obvious.
AUGUSTIN CAUCHY
Harder to see the answer as the distance than the end point I think.
MARJORIE RICE
It shows a relationship of 2,7, and 9. My interpretation leans more towards 2+7=9 instead of 9-2=7
MARYAN MIRZAKHANI
This image shows 9-2 but for me requires the mental jumps that 0 to 9 has distance 9, and we've removed 0 to 2 (which has distance 2) to get something of distance 7.
GERTRUDE BLANCH
If you count the spaces in between the 2 and 9, you get 7... but not sure a young child would make that connection.
EMMY NOETHER
I feel like this would confuse students.
EUPHEMIA LOFTON HAYES
yes but not in a way that I would use with children. subtracting the first 2 rather than the last 2 could be confusing.
ALAN TURING
It shows the difference between 9 and 2 so yes but I can't for some reason say yes.
DORIS SCHATTSCHNEIDER
This shows 2+x=9
CARL JACOBI
Because the integers aren't marked it makes it a little more difficult.
LOUISE HAY
Less inuitive but the difference between 9 and 2 represented by the line
MARIA AGNESI
I see: the distance between nine and two is seven
RICHARD DEDEKIND
I'm programmed to think about this in terms of the previous image. It looks more like 9-7 to me.
ALFRED TARSKI
In context, yes. If I think about the length of the segment, yes.
PIERRE DE FERMAT
2+x=9
ANDRÉ WEIL
I see it as 9-7 instead
ERNST KUMMER
It shows the distance from 9 to 2 which I guess could be used as a subtraction model, but I wouldn't think this is a typical or easy to understand model for all students
HIPPARCHUS
Subtraction *is* distance, but it seems more like 9-7 to me.
JEAN TAYLOR
This looks more like 9 minus 7 to me, although I can see how it might also be 0 minus 2.
JOHN VON NEUMANN
I see this as missing addend as well (2 + x = 9). It can be solved by finding 9-2.
ÉMILIE DU CHÂTELET
I still need to count the spaces between 2 and 9.
I will note that a few people mentioned the lack of clarity in the instructions. When I wrote "Drag the point to the left," I meant, "Drag that point over there, the one which live to the left of these instructions that you are currently reading," and not, "See that point over there? Drag it leftward." Yes, it was unclear, and I have now changed the text in case I ever decide to use this survey again.
Thank you for ideating and prototyping this.
MARY ROSS
Thought provoking variation in these images
JOHANNES KEPLER
Would love t hear more about the question and the results!
KELLY MILLER
Nope :) I bet your responses will be really interesting
ISMAIL MUSTAFA AL-FALIKI
Good luck!
GEORG CANTOR
Love all that you do Kendra!
CAROLINE SERIES
Hope this helped. Curious to hear how you are using it!
ÉMILE BOREL
Some of these examples would confuse
GALILEO GALILEI
Nope.
JULIA ROBINSON
I have lots of thoughts on this. dm me if you want to have an extended conversation.
KATHERINE COLEMAN JOHNSON
😊
MICHAEL ATIYAH
Good luck!
FELIX KLEIN
I'd love to see your results. You should post them on twitter!
HENRI POINCARÉ
Interesting survey.
SRINIVASA RAMANUJAN
N/A.
HEE OH
Love these representations!
WEN TSUN WU
Loved the survey. Love math.
HERTA FREITAG
Nope!
ÉVA TARDOS
I don't know how to answer the mathy question. Maybe something more objective re years of university math ... I don't know ...
JOAN BERMIN
I don't really know who you are or why you're studying this, but I found it fascinating. For me, it was hard to answer the question on each slide because I wanted to see the 9-2. On a few of them, my gut response would not have been 9-2, but thinking about the picture for a few seconds made me recognize a way it could represent 9-2. I'm curious to see what results you get as well as how teachers might use these pictures to develop more robust understandings of subtraction.
HERMANN MINKOWSKI
Interesting survey! Just to be picky, in question 8, I thought you wanted me to drag the point leftwards. 🙂 Next, I'd like to see a survey where you make it more open ended and ask us to draw our own representations of 9-2.
AL-KHWARIZMI
There are so many different ways to show subtraction, I'm not sure if I know all of them! Thank you for sharing this with us. I hope it goes well
LENORE BLUM
Where is this going? How do I learn to create these things?
JOSEPH LIOUVILLE
This is a really good idea.
CHRISTINE LADD-FRANKLIN
I am curious about how you are going to use this.
GOTTLOB FREGE
No
ALICIA BOOLE STOTT
That was fun. I'd love to see what other people said about screens 4 and 5. Thanks!
JOHN WALLIS
nope
CHARLES LEWIS REASON
Thank you for setting up an opportunity for me to try to see visual models in ways not most obvious to me!
SOFIA KOVALEVSKAYA
"Drag the point to the left to show how mathy I am?" Is that a trick question? Generally lines get bigger left to right. I consider myself ridiculously mathy so I"d drag the line to the end...
OLGA TAUSSKY-TODD
Hoping you blog the results! I started overthinking.... I think. 😂 Thank you for making me think, though!
ELANA PISCOPIA
None :)
FLORENCE NIGHTENGALE
I'm curious to see what students think, and would be interested in a follow up blog post about the purpose of the exercise!
DIANA TAIMINA
Love the frogs.
CHRISTIAAN HUYGENS
Thanks
AUGUSTIN CAUCHY
Love these representations!
MARJORIE RICE
Interesting queries. And I love the sliding scale with dynamic text.
ELBERT FRANK COX
For the pictorial, maybe put the 9 frogs in a box and then show two of them out of the box. Or, in one image, 7 in the box with two dashed rectangles or frog outlines, and 2 outside the box.
MARYAN MIRZAKHANI
All of these representations were horizontal, and I think some of my answers might change for seemingly equivalent vertical representations.
GERTRUDE BLANCH
Hope this helps!
KATHERINE JOHNSON
This is a good activity. Having students give their thought - for both yes and no - is a good idea.
JOSEPH FOURIER
Thanks for the survey! Super interested in hearing about the results!!!
INGRID DAUBECHIES
This was a cool survey. Thanks for making it.
EMMY NOETHER
Nope :)
PETER DIRICHLET
There are so many interesting ways to show subtraction!
MARIA CHUDNOVSKY
I loved thinking about these problems. I'm sorry if my explanations brought forward more questions than clarity. Thanks for letting me play!
EUPHEMIA LOFTON HAYES
I am thinking about whether the images represent the values in the expression, the operation, or the result....
MANJUL BHARGAVA
Great activity. I enjoyed it!
ALAN TURING
:)
WILLIAM HAMILTON
No
CARL JACOBI
I think this would've been a better way to represent things when I was in school.
DAVID BLACKWELL
Nope.
LOUISE HAY
I think subtraction is hard for kids. I teach upper level HS and don't deal with this much, but looking at how my daughter is taught subtraction, I think it could be a lot better. Bravo for trying to figure out how people/kids see it best!
VICTOR NEUMANN-LARA
thank you!
DOROTHY VAUGHN
I love discussing models!
MARIA AGNESI
Not at the moment.
NGÔ BÀO CHÂU
None, but I'm intrigued to find the results of this survey!
DIOPHANTUS
Interesting ideas. Prefer the use of jumps on the number line.
ARTHUR CAYLEY
No, thank you!
RICHARD DEDEKIND
I think a lot of it is dependent on your experience. As an Algebra teacher, I'm programmed to think in "like terms," etc. I'd love to see what a younger kids would think of this.
SOPHUS LIE
That one picture with the 7 boxes and 2 frogs was really interesting. It felt different from typical subtraction representations and I had to think through it (rather than just identify it).
ALFRED TARSKI
Clever use of Desmos for surveying purposes. I think, with explanation, you could use all of the models for subtraction. Some require more front-loading/scaffolding than others.
JOHANN BERNOULLI
Another way: I use Cuisenaire rods to begin modelling difference by laying a '9 rod' and a '2 rod' alongside each other.
HIPPARCHUS
I wonder what the motivation behind this survey is. :)
HENRI LEBESGUE
I'm not sure if I answered as you might have intended. Maybe a little more direction on what angle you want me to think from.
JEAN TAYLOR
I loved seeing so many possible ways to show subtraction visually!
JOHN VON NEUMANN
Interesting survey. Thanks for sharing. I hope you'll let us know how you plan to use the results.
ÉMILIE DU CHÂTELET
Cool survey. Thanks, and good luck with your data!
FELIX KLEIN 9
Green frogs minus the 2 blue frogs equals the difference
HEE OH
I see comparison b/w green and blue frogs. How many more green frogs than blue. Situation equation seems to be 9-2=7
JOAN BERMIN
There are 9 frogs in the first group and 2 frogs in the second group. If you want to know how many more frogs are in the first group than the second group, you'd do 9-2 and get 7 frogs.
LENORE BLUM
9 to right - 2 to left
ELANA PISCOPIA
Comparison situation There are 9 green frogs. There are two blue frogs. How many more green frogs are there than blue?
GOTTFRIED LEIBNIZ
How many more?
CHRISTIAAN HUYGENS
Difference
JOSEPH FOURIER
I guess it could mean two frogs died? Still somewhat confusing as the 'living' ones are still there.
CARL JACOBI
Seperate groupings prove easier
VICTOR NEUMANN-LARA
Yes - there are 9 frogs but the 2 frogs going the other way cancel 2 out
MARIA AGNESI
A zero pair is created when 2 green frogs and 2 blue frogs are circled. Green frogs represent positive 2. Blue frogs represent -2. 2+ -2 equals zero leaving seven frogs
SOPHUS LIE
Yes, this shows the difference between 9 and 2 is 7
JOHN VON NEUMANN
Yes, this would be a good example of finding the difference between two groups.
No
KARL WEIERSTRASSI see 9 and 2.
ISMAIL MUSTAFA AL-FALIKI
9+2
GEORG CANTOR
Again, these layered images feel like they illustrate difference. To me this is the difference of 9 and 2
ÉMILE BOREL
Same as slide 3
JULIA ROBINSON
See previous answer. This shows 9 + (-2).
SRINIVASA RAMANUJAN
Same reasoning as slide 3 of 9
WEN TSUN WU
Again, looks like a comparison.
HERMANN MINKOWSKI
Same as previous answer--make the blue frogs red.
AL-KHWARIZMI
It seems like 9+2 since there are more frogs
GOTTLOB FREGE
I could add or subtract or multiply or divide
ANDREY KOLMOGOROV
There are 9 frogs facing right There are 2 frogs facing left
FLORENCE NIGHTENGALE
Same answer as previous
DIANA TAIMINA
Same as the previous slide.
DIEGO RODRIGUEZ
Two different groups. You are not taking frogs away.
ELBERT FRANK COX
Same as last one, but worse. I'd say that's 9 - 7 if anything.
KATHERINE JOHNSON
This looks like 9'green frogs facing one way and 2 blue frogs facing the other. There are more than 9 frogs with nothing crossed out.
INGRID DAUBECHIES
No. Totally different looking things.
SHIGEFUMI MORI The blue frogs look like they're being added like a part part whole problem
COURTNEY GIBBONS
Doesn't make sense
EMMY NOETHER
11 frogs
MARY CARTWRIGHT
9green plus 2 blue
MARIA CHUDNOVSKY
This would be more like 9-7 to me with two being the difference. I could see it as 9-?=2
EUPHEMIA LOFTON HAYES
Looks more like 9 + 2
MANJUL BHARGAVA
Again, what's up with the blue frogs? I think this shows addition more than subtraction.
ALAN TURING
Maybe if you had circled the 2 in each line?
ARCHIMEDES
It's not clear.
WILLIAM HAMILTON
Don't compare apples and oranges
DAVID BLACKWELL
It shows 9 & 2.
LOUISE HAY
11 frogs.
NGÔ BÀO CHÂU
Same as the last one but actually less likely to make me want to subtract.
DIOPHANTUS
The frogs have changed and seven have gone not two.
ARTHUR CAYLEY
Again, looks like 2 different kinds of frogs to be added
RICHARD DEDEKIND
See previous response.
PIERRE DE FERMAT
Still like terms
HIPPARCHUS
9+2
HENRI LEBESGUE
Again, context would be needed. Could be addition.
JEAN TAYLOR
Same as slide 3.
kind of, but not easily
JOHANNES KEPLERSimilar reply to last image
KELLY MILLER
Same as last one
GALILEO GALILEI
If students understand that subtraction is equivalent to adding the opposite of the "subtrahend".
KATHERINE COLEMAN JOHNSON
Again without explanation I may tend to just add the frogs together instead of realizing the blue and green ones cancel out
ÉVA TARDOS I have to assume that the blues are being taken away or superimpose the question "how many more frogs are facing right."
JOSEPH LIOUVILLE
See previous comment.
CHRISTINE LADD-FRANKLIN
Slightly better than the previous problem, but again nothing that explicitly says you are interested in the difference rather than a total. Maybe this is related to people's tendency to focus on positive space over negative space.
GRACE CHISHOLM YOUNG
There needs to be some interface between the green and blue frogs to explain what is happening with then.
ALICIA BOOLE STOTT
It shows subtraction as being the difference of 2 amounts. I think there is a leap to this being sub without something showing that what we're focusing on is the 7 extra green frogs. Maybe circle them? I suppose without context we might not know if this is 9-2=7 Or 9+2=11.
CHARLES LEWIS REASON
I could a teacher asking, "How many more green frogs are there than blue frogs?" So you'd subtract two of each (zero pairs) to get an answer of 7. But again this looks more like addition to me.
SOFIA KOVALEVSKAYA
A little mre easily than the previous *if* this is connected to "adding negative integers and I've learned that the blue guys are facing the other way.
OLGA TAUSSKY-TODD
I'm second guessing myself 😂 I can see a kid thinking this represents 9 hops right and 2 hops left, mt thinking about the difference from in the frogs.
AUGUSTIN CAUCHY
Same answer as before regarding comparison.
MARJORIE RICE
The row of two can be described as negative units and make zero pairs with two green frogs.
MARYAN MIRZAKHANI
There is a row of nine frogs and a row of two frogs, but they are different looking frogs and so seem like different things. However, it can show subtraction in the sense of there being 9-2 more green frogs than blue frogs.
GERTRUDE BLANCH
If you cancel out the two frog pairs in the beginning you get 7.
PETER DIRICHLET
Same reasons. I would put them in squares of easily calculated areas
ARISTOTLE
Again only if you knew to compare the two sets
DORIS SCHATTSCHNEIDER
Again, these are different object and color. They are not like terms and there's nothing to indicate one was part of the group, but now is not. Kinda strange how when you stack numbers and put the operation sign, it is clear. But without the operation sign, I see add.
JACQUES HADAMARD
Same as previous
DOROTHY VAUGHN
I'm imagining the different type frogs pairing off and hopping away
ALFRED TARSKI
See answer to previous slide.
JOHANN BERNOULLI
You have to understand the dark frogs as negative frogs, or know commutative straight away.
ERNST KUMMER
I guess it could, I wouldn't normally look at it this way but there are 9 frogs in the beginning and the bottom line shows 2
ÉMILIE DU CHÂTELET
I can figure out how many more green frogs by creating a 1-1 correspondence between the 2 blue ones and 2 green ones, and counting the green frogs remaining - lotsa work.
Image 4 (Screen 5)
Yes
FELIX KLEIN9 things, 2 of them are frogs
HEE OH
Could be "I need 9 frogs for my collection. I already have 2. How many more do I need? Some students might solve as missing addend prob but 9-2 seems to match.
AUGUSTIN CAUCHY
This feels like a start unknown problem.
EMMY NOETHER
Here are 7 boxes.
NGÔ BÀO CHÂU
More of 2+? But that's good for equivalency.
SOPHUS LIE
I'm thinking there were 9 spots available and 2 frogs jumped on spots so now there are only 7 spots available.
JOHANN BERNOULLI
'I had 9 frogs; I'm hiding 9, how many can I still see?'
No KARL WEIERSTRASS
I think it shows 9-7.
MARY ROSS
This feels like How many more to make 9 So is additive Recognise that this is comparison/complementary addition, but the frogs concretise this, so it doe
KELLY MILLER
It looks more like 2 + (7 what?)
GEORG CANTOR
I don't really see the logic of thos diagram ( teying to see through the eyes of soneone learning subtraction)
CAROLINE SERIES
To me, it shows 9-7.
ÉMILE BOREL
Doesn't show nine of same thing
JULIA ROBINSON
To me this shows 2 + _ = 9
KATHERINE COLEMAN JOHNSON
Not sure what the empty boxes me or the 2 frogs. I found this the most confusing
SRINIVASA RAMANUJAN
To me, this image conveys "2 + seven unknowns"
WEN TSUN WU
Looks more like 2+_=9
HERMANN MINKOWSKI
Kinda looks like 9-7. The empty rectangles represent frogs that were removed.
LENORE BLUM
Confusing
JOSEPH LIOUVILLE
This could maybe be 9-7.
CHRISTINE LADD-FRANKLIN
The frogs are the objects, the dashed boxes strongly imply that those things are missing. If you are wondering about communicating an idea clearly (as opposed to encouraging all interpretations or out-of-the-box thinking) I think this is most clearly 9-7
GOTTLOB FREGE
This shows addition imo
ANDREY KOLMOGOROV
I think it shows 9-7
FLORENCE NIGHTENGALE
I'm quite thrown off by the empty boxes. Again,cant guess without context what the question is
ELBERT FRANK COX
That's more like 9 - 7 again
KATHERINE JOHNSON
It looks mre like 9 - 7.
JOSEPH FOURIER
this one seems confusing
SHIGEFUMI MORI
It looks like 9-7--like those boxes represent removed frogs.
COURTNEY GIBBONS
Unclear if you should put something into the boxes
MARY CARTWRIGHT
To me it looks like 2 plus boxes
MARIA CHUDNOVSKY
To me is represents ?-2=7 or 9-?=2. It is similar but the context is different to me.
ARISTOTLE
Feels much more like 2+?=9 missing addend
EUPHEMIA LOFTON HAYES
looks more like 9 - 7
MANJUL BHARGAVA
Nope. It looks like frogs are missing and that's not really associated with subtraction.
ALAN TURING
What was in those boxes???
WILLIAM HAMILTON
2 frogs and 7 boxes
DOROTHY VAUGHN
Feels more like 9 - 7
MARIA AGNESI
I see
1. Addition. Such as 2+ what number equals nine.
2. 9-7 = 2
DIOPHANTUS
It looks like you have taken seven away
ARTHUR CAYLEY
Looks like 2 plus ___ is 9; maybe would be different if the frogs were also in boxes
ERNST KUMMER
I don't really see the 9
HENRI LEBESGUE
I feel fairly confident that my students would interpret this to mean 9-7.
ÉMILIE DU CHÂTELET
What goes in the empty boxes?
kind of, but not easily
JOHANNES KEPLERI thin I this more of 2+_=_ or 2+7=_
ISMAIL MUSTAFA AL-FALIKI
Looks like 2 + ? = 9 which is kind of what you're asking but not exactly.
GALILEO GALILEI
Maybe, but it could also be an addition problem or maybe even a pattern continuations (frogs forever!)
HERTA FREITAG
There are 9 frogs.. only 2 are green. How many are blue
JOAN BERMIN
This picture might fit with the question "If I already have 2 frogs and want 9 frogs total, how many more frogs do I need?" It's 7, and 9-2=7. I'd see this more as an equation to solve, though. 2+x=9, which would be solved by subtracting 2 from both sides, giving the 9-2 operation. But that feels like a stretch.
AL-KHWARIZMI
To me it looks like 2+ what equals nine, or how many frogs need to go in the empty boxes.
GRACE CHISHOLM YOUNG
Eh, it would work better for me if there were 2 white boxes and 7 frogs.
ALICIA BOOLE STOTT
similarly to the previous screen I can see all the parts of the problem but I'm not sure I see the act of subtraction... at least not a form of sub I can think of right now.
JOHN WALLIS maybe with explanation, but appears more like something is to be added in or a pattern created
CHARLES LEWIS REASON
Maybe; if the purple boxes are covering frogs that were originally visible, and you wanted to see how many frogs would need to leave if you only wanted 7 frogs. This doesn't look like an obvious math problem to me. It's unclear what's up with the purple boxes.
SOFIA KOVALEVSKAYA
I might teach it for "two plus what is nine?" but ... I'd want to use something consistent for the nine I need...
OLGA TAUSSKY-TODD
Overthinking? "I have 2 frogs - how many more until I have 9?"
ELANA PISCOPIA
Part part whole There are 9 frogs in all. Two of them are green. How many other frogs are there?
DIANA TAIMINA
How many more to make 9. Or how many to fill up the empty cages. You need 7, which is the same as 9-2 but it seems to show a different thing.
CHRISTIAAN HUYGENS
Difference again I suppose
DIEGO RODRIGUEZ
I see this as 2 + ___ = 9. You will get the same result, but the context does not match the operation.
MARJORIE RICE
I see a row of nine things. Two of one thing and 7 of the others. This could show a relationship of 2,7, and 9. However, this image to me feels more like a ratio with the different units in the same row.
MARYAN MIRZAKHANI
This feels more like an addition picture (2+7) to me, but I can make the jump from that to subtraction.
GERTRUDE BLANCH
9 rectangles, minus two (replaced by frogs?)
INGRID DAUBECHIES
Might be hard to realize at first that the boxes are missing frogs.
DORIS SCHATTSCHNEIDER
This shows 2 + x = 9. You will need to use subtraction (or just knowledge of addition) to determine the answer.
LOUISE HAY
Bit of a stretch. I can see it if I really try to. Not inuitive but I see the add on concept
VICTOR NEUMANN-LARA
I guess there were 9 empty boxes and you've taken 2 away, but by adding frogs
RICHARD DEDEKIND
I think of "-" as something missing. This reads more as 9-7 to me.
ALFRED TARSKI
It seems to indicate 9 - 7 rather than 9 - 2 to me.
PIERRE DE FERMAT
2+x=9 is more what I see
ANDRÉ WEIL
Need to take a second to interpret the boxes
HIPPARCHUS
This seems to show 2 + ? = 9
JEAN TAYLOR
This looks like 2 plus something equals something . More of a missing addend problem than direct aubtraction. I'd still like to see smaller groups within the objects so I don't have to count by ones to know there are 7 boxes.
JOHN VON NEUMANN
I see this as a missing addend problem.
Image 5 (Screen 6)
Yes
KARL WEIERSTRASSThe arrow indicates that I was at 9 and decreased by 3.
JOHANNES KEPLER
Start at 9 a move "back" or take away 2. Where are you [on the numberline]
KELLY MILLER
A jump of 2 backward from 9
ÉMILE BOREL
Starts at 9 backs up 2 (goes lower)
JULIA ROBINSON
Moving left on the number line, with proper development, is a fairly natural way to think about subtraction.
KATHERINE COLEMAN JOHNSON
You can see the visual of going back 2 in the number line
HENRI POINCARÉ
I would have liked to see the jump to 9 first.
SRINIVASA RAMANUJAN
To me, this is a standard, number line representation of starting at 9 and taking away 2. The arrowhead indicates direction (movement in the negative direction, which is also known as subtraction). But this is based on my familiarity with this representational technique.
HEE OH
Starting at 9, take 2 away.... 2 less than 9...
WEN TSUN WU
Perfect
ÉVA TARDOS
Maybe I'm just familiar with the number line representation of subtraction.
JOAN BERMIN
You start at 9 and move 2 to the left, ending at 7. 9-2=7.
HERMANN MINKOWSKI
A classic picture of subtraction: start at 9, take away 2.
AL-KHWARIZMI
You started at nine and went back to the left two spaces
JOSEPH LIOUVILLE
Start at 9, back up 2 spaces is 9-2.
CHRISTINE LADD-FRANKLIN
We are going back 2 from 9.
GRACE CHISHOLM YOUNG
Traditional.
ALICIA BOOLE STOTT
love number line representations. Start at 9 and go down 2. This could also be 9 + -2.
JOHN WALLIS
I'd rather it started at zero to nine, but shows a nine and two less from the 9 gives 7
CHARLES LEWIS REASON
The arrow starts at 9 and goes to the left (subtraction) two places.
SOFIA KOVALEVSKAYA
Yup. Lots of folks learn about subtraction as "going backwards on the number line.
ANDREY KOLMOGOROV
Move two units from 9
OLGA TAUSSKY-TODD
Start at 9, go back 2
ELANA PISCOPIA
Take away on a number line
GOTTFRIED LEIBNIZ
Stepping back
FLORENCE NIGHTENGALE
Probably because I'm used to seeing this representation
CHRISTIAAN HUYGENS
Counting down
AUGUSTIN CAUCHY
Subtraction as distance makes sense to me.
DIEGO RODRIGUEZ
I see beginning at 9, and jumping back two places.
MARJORIE RICE
Start at nine and move two in the. What I've direction.
ELBERT FRANK COX
From 9 you moved back 2
GERTRUDE BLANCH
Start at 9, minus 2 (move to the left).
KATHERINE JOHNSON
It might help to show two arrows to the left, each with a length of one unit.
JOSEPH FOURIER
starting at 9, travelling back two. would work better if there were other examples to give contexts of plus means forward, minus backward
EMMY NOETHER
Started at 9 and rolled back two.
PETER DIRICHLET MARIA CHUDNOVSKY
This seems pretty straight forward to me. The difference between as nine take away two is seven. I could also see it as the difference between nine and seven is two though.
ARISTOTLE
But would like to start at zero, go to 9 Then subtract 2
EUPHEMIA LOFTON HAYES
yes but need a fair amount of specialized knowledge about number line conventions to understand it.
MANJUL BHARGAVA
It starts at nine and removes two. Makes sense to me.
DORIS SCHATTSCHNEIDER
This is all on one line and clearly shows starting at 9 and counting back 2 spaces, so it does show the operation 9-2.
WILLIAM HAMILTON
9 steps forward and 2 steps back
JACQUES HADAMARD
Love number lines
CARL JACOBI
Line graphs are always a good option.
LOUISE HAY
Back two or left two on numberline from nine
VICTOR NEUMANN-LARA
You're at 9, then you move 2 in the negative direction
DOROTHY VAUGHN
Subtraction as motion
NGÔ BÀO CHÂU
Two less than nine
DIOPHANTUS
Yes but you could add to the loop minus 2 to show you have jumped back two.
ARTHUR CAYLEY
Yes!
RICHARD DEDEKIND
Start at 9, move 2 in the negative direction.
SOPHUS LIE
Starting at 9, removing 2 to land on 7
ALFRED TARSKI
Yes, but only because I know how a number line works. It's not my favorite representation.
JOHANN BERNOULLI
Yes, though lots of people put the subtraction arrow under the line.
HIPPARCHUS
Moving to the left is subtraction.
HENRI LEBESGUE
Especially with the arrow.
JEAN TAYLOR
This is very clear to me. The number line helps a lot!
JOHN VON NEUMANN
I think this is a good example of 9-2.
ÉMILIE DU CHÂTELET
9 on the number land, go backward 2 to 7
No
DIANA TAIMINAIt shows how many units to go from 9 to 7. The distance is two. Well, I guess you could say 9-2, but it isn't obvious.
MARY CARTWRIGHT
Maybe to an older student?
kind of, but not easily
CAROLINE SERIESI think an arrow from 0 to 9, then back to 7 is more likely to show 9-2.
LENORE BLUM
Use arrow to 9 first
GOTTLOB FREGE
Show two jumps
ARCHIMEDES
I would show an arrow going to 9 first.
DAVID BLACKWELL
Shows 9 to 7. Which IS 9-2 but still, come on.
Image 6 (Screen 7)
Yes
GALILEO GALILEIDistance!
FELIX KLEIN
If you see that 9 is the whole and 2 is the part you know
HEE OH
I see this as distance between 9 and 2
CHRISTINE LADD-FRANKLIN
I think most students wouldn't realize that this is showing the difference between 9 and 2, but this is definitely worth exploring.
ALICIA BOOLE STOTT
The difference or distance between 9 and 2.
CHARLES LEWIS REASON
It shows the difference between 9 and 2. Although it looks more to me like 9-7.
ANDREY KOLMOGOROV
2 numbers are left off
OLGA TAUSSKY-TODD
Shows distance
ELANA PISCOPIA
Comparison on a number line
GOTTFRIED LEIBNIZ
How far apart?
CHRISTIAAN HUYGENS
Difference
MARIA CHUDNOVSKY
The difference between nine and two is seven.
DOROTHY VAUGHN
Subtraction as distance
ARTHUR CAYLEY
The difference between 9 and 2
SOPHUS LIE
Difference/distance between 9 and 2
JOHANN BERNOULLI
Not quite so clear, but one way of seeing that is the difference between 9 and 2.
HENRI LEBESGUE
Although context would be needed for student certainty
No
GEORG CANTORIf anything it's 9-7, but arrows would make the operation clearer
ÉMILE BOREL
No direction arrow...difference is 7
SRINIVASA RAMANUJAN
This picture shows a seven unit line segment that starts at 2 and ends at 9 because my default is to start at the origin and "read" from there.
HERMANN MINKOWSKI
I'd prefer to see a straight arrow go from 0 to 9 (pointing right) and a straight arrow below it go from 9 to 7 (pointing left).
AL-KHWARIZMI
To me it shows a length of seven but I don't know what the operation is
LENORE BLUM
What?
GOTTLOB FREGE
I would say 9 -7 if any
MINA REES DIEGO RODRIGUEZ
I think a better way of showing this would be 2 + ____ = 9, or 9 - _____ = 2
ELBERT FRANK COX
That's the distance from 2 to 9, which is conceptually very different
KATHERINE JOHNSON
It looks like 9 - 2.
JOSEPH FOURIER
where is the whole 9? maybe if the number line didn't continue, then you could have the total bar of 9 units juxtaposed with two removed
MARY CARTWRIGHT
To me this shows a segment of length 7.
MANJUL BHARGAVA
Where's the starting point? This doesn't make sense to me at all. If you had an arrow then I could make some type of coherent explanation.
ARCHIMEDES
It looks more like 9 - 7.
WILLIAM HAMILTON
Nur eine Gerade auf dem Zahlenstrahl der Länge 7
JACQUES HADAMARD
It tells me how many between 2 and 9 which is the same answer as 9-2, but not the same question
DAVID BLACKWELL
I have no idea what that is.
VICTOR NEUMANN-LARA
It's more like 9-7 because you move 7 backwards
DIOPHANTUS
You have drawn a line. Why that long? How would children understand what has been done here?
kind of, but not easily
KARL WEIERSTRASSI can see 2+7=9 so I can get to it.
JOHANNES KEPLER
More like what is the distance from 2 to 9? could be 9-2 or 2+_=9
KELLY MILLER
Yes in that 9-2 can be thought of "the distance between 9 and 2", but I feel like this is less intuitive...however it's easier to model 9-(-2) this way when trying to understand why 9-(-2)=9+2
JULIA ROBINSON
First you have to understand subtraction as directed distance. Even then does this picture show 9 - 2 or 2 - 9?
KATHERINE COLEMAN JOHNSON
If students understand it's distance they would realize the answer is 7 but as it stands I think it is confusing
MICHAEL ATIYAH
I think an arrow would help.
HENRI POINCARÉ
shows the distance between, so yes...it would depend on what was done with students previously.
ÉVA TARDOS
It COULD. I think I'm just less familiar with this representation.
JOAN BERMIN
It's relatively easy for ME to see 9-2, but I'd expect students to have a tougher time seeing it. I think of this as a question of the segment's length. It starts at 2 and ends at 9, so the length is 9-2=7. I think students would have trouble seeing this as subtraction because the number 7 doesn't jump out at them like in the previous example. And because there aren't objects to count (like the frogs), it's perhaps less obvious that we're looking at 7.
JOSEPH LIOUVILLE
I see it more as 9-7, although I see how it could be 9-2.
GRACE CHISHOLM YOUNG
It makes sense if you think of subtraction as distance, but as is needs some explanation.
JOHN WALLIS
Shows 9-7 more to me
SOFIA KOVALEVSKAYA
Again depends on how I've presented things. If we've discussed sbtraction as "difference," then this shows that.
FLORENCE NIGHTENGALE
It seems to ask me to find the distance, but I might count up (add) instead
DIANA TAIMINA
It shows the distance between 9 and 2. Again, it is not obvious.
AUGUSTIN CAUCHY
Harder to see the answer as the distance than the end point I think.
MARJORIE RICE
It shows a relationship of 2,7, and 9. My interpretation leans more towards 2+7=9 instead of 9-2=7
MARYAN MIRZAKHANI
This image shows 9-2 but for me requires the mental jumps that 0 to 9 has distance 9, and we've removed 0 to 2 (which has distance 2) to get something of distance 7.
GERTRUDE BLANCH
If you count the spaces in between the 2 and 9, you get 7... but not sure a young child would make that connection.
EMMY NOETHER
I feel like this would confuse students.
EUPHEMIA LOFTON HAYES
yes but not in a way that I would use with children. subtracting the first 2 rather than the last 2 could be confusing.
ALAN TURING
It shows the difference between 9 and 2 so yes but I can't for some reason say yes.
DORIS SCHATTSCHNEIDER
This shows 2+x=9
CARL JACOBI
Because the integers aren't marked it makes it a little more difficult.
LOUISE HAY
Less inuitive but the difference between 9 and 2 represented by the line
MARIA AGNESI
I see: the distance between nine and two is seven
RICHARD DEDEKIND
I'm programmed to think about this in terms of the previous image. It looks more like 9-7 to me.
ALFRED TARSKI
In context, yes. If I think about the length of the segment, yes.
PIERRE DE FERMAT
2+x=9
ANDRÉ WEIL
I see it as 9-7 instead
ERNST KUMMER
It shows the distance from 9 to 2 which I guess could be used as a subtraction model, but I wouldn't think this is a typical or easy to understand model for all students
HIPPARCHUS
Subtraction *is* distance, but it seems more like 9-7 to me.
JEAN TAYLOR
This looks more like 9 minus 7 to me, although I can see how it might also be 0 minus 2.
JOHN VON NEUMANN
I see this as missing addend as well (2 + x = 9). It can be solved by finding 9-2.
ÉMILIE DU CHÂTELET
I still need to count the spaces between 2 and 9.
How Mathy Are You? (Screen 8)
Here I asked people to drag a slider to indicate how mathy they rated themselves to be, from "I avoid mathy things when possible" on the far left, to "I do mathy things for fun" on the far right. Because I had dynamic text, the screen overlay looked terrible. But here's a pretty representative grab of several of the responses. All but 3 of the responses were to the right of center.
I will note that a few people mentioned the lack of clarity in the instructions. When I wrote "Drag the point to the left," I meant, "Drag that point over there, the one which live to the left of these instructions that you are currently reading," and not, "See that point over there? Drag it leftward." Yes, it was unclear, and I have now changed the text in case I ever decide to use this survey again.
Thank you for playing. Do you have any thoughts to share on this matter? (Screen 9)
KARL WEIERSTRASSThank you for ideating and prototyping this.
MARY ROSS
Thought provoking variation in these images
JOHANNES KEPLER
Would love t hear more about the question and the results!
KELLY MILLER
Nope :) I bet your responses will be really interesting
ISMAIL MUSTAFA AL-FALIKI
Good luck!
GEORG CANTOR
Love all that you do Kendra!
CAROLINE SERIES
Hope this helped. Curious to hear how you are using it!
ÉMILE BOREL
Some of these examples would confuse
GALILEO GALILEI
Nope.
JULIA ROBINSON
I have lots of thoughts on this. dm me if you want to have an extended conversation.
KATHERINE COLEMAN JOHNSON
😊
MICHAEL ATIYAH
Good luck!
FELIX KLEIN
I'd love to see your results. You should post them on twitter!
HENRI POINCARÉ
Interesting survey.
SRINIVASA RAMANUJAN
N/A.
HEE OH
Love these representations!
WEN TSUN WU
Loved the survey. Love math.
HERTA FREITAG
Nope!
ÉVA TARDOS
I don't know how to answer the mathy question. Maybe something more objective re years of university math ... I don't know ...
JOAN BERMIN
I don't really know who you are or why you're studying this, but I found it fascinating. For me, it was hard to answer the question on each slide because I wanted to see the 9-2. On a few of them, my gut response would not have been 9-2, but thinking about the picture for a few seconds made me recognize a way it could represent 9-2. I'm curious to see what results you get as well as how teachers might use these pictures to develop more robust understandings of subtraction.
HERMANN MINKOWSKI
Interesting survey! Just to be picky, in question 8, I thought you wanted me to drag the point leftwards. 🙂 Next, I'd like to see a survey where you make it more open ended and ask us to draw our own representations of 9-2.
AL-KHWARIZMI
There are so many different ways to show subtraction, I'm not sure if I know all of them! Thank you for sharing this with us. I hope it goes well
LENORE BLUM
Where is this going? How do I learn to create these things?
JOSEPH LIOUVILLE
This is a really good idea.
CHRISTINE LADD-FRANKLIN
I am curious about how you are going to use this.
GOTTLOB FREGE
No
ALICIA BOOLE STOTT
That was fun. I'd love to see what other people said about screens 4 and 5. Thanks!
JOHN WALLIS
nope
CHARLES LEWIS REASON
Thank you for setting up an opportunity for me to try to see visual models in ways not most obvious to me!
SOFIA KOVALEVSKAYA
"Drag the point to the left to show how mathy I am?" Is that a trick question? Generally lines get bigger left to right. I consider myself ridiculously mathy so I"d drag the line to the end...
OLGA TAUSSKY-TODD
Hoping you blog the results! I started overthinking.... I think. 😂 Thank you for making me think, though!
ELANA PISCOPIA
None :)
FLORENCE NIGHTENGALE
I'm curious to see what students think, and would be interested in a follow up blog post about the purpose of the exercise!
DIANA TAIMINA
Love the frogs.
CHRISTIAAN HUYGENS
Thanks
AUGUSTIN CAUCHY
Love these representations!
MARJORIE RICE
Interesting queries. And I love the sliding scale with dynamic text.
ELBERT FRANK COX
For the pictorial, maybe put the 9 frogs in a box and then show two of them out of the box. Or, in one image, 7 in the box with two dashed rectangles or frog outlines, and 2 outside the box.
MARYAN MIRZAKHANI
All of these representations were horizontal, and I think some of my answers might change for seemingly equivalent vertical representations.
GERTRUDE BLANCH
Hope this helps!
KATHERINE JOHNSON
This is a good activity. Having students give their thought - for both yes and no - is a good idea.
JOSEPH FOURIER
Thanks for the survey! Super interested in hearing about the results!!!
INGRID DAUBECHIES
This was a cool survey. Thanks for making it.
EMMY NOETHER
Nope :)
PETER DIRICHLET
There are so many interesting ways to show subtraction!
MARIA CHUDNOVSKY
I loved thinking about these problems. I'm sorry if my explanations brought forward more questions than clarity. Thanks for letting me play!
EUPHEMIA LOFTON HAYES
I am thinking about whether the images represent the values in the expression, the operation, or the result....
MANJUL BHARGAVA
Great activity. I enjoyed it!
ALAN TURING
:)
WILLIAM HAMILTON
No
CARL JACOBI
I think this would've been a better way to represent things when I was in school.
DAVID BLACKWELL
Nope.
LOUISE HAY
I think subtraction is hard for kids. I teach upper level HS and don't deal with this much, but looking at how my daughter is taught subtraction, I think it could be a lot better. Bravo for trying to figure out how people/kids see it best!
VICTOR NEUMANN-LARA
thank you!
DOROTHY VAUGHN
I love discussing models!
MARIA AGNESI
Not at the moment.
NGÔ BÀO CHÂU
None, but I'm intrigued to find the results of this survey!
DIOPHANTUS
Interesting ideas. Prefer the use of jumps on the number line.
ARTHUR CAYLEY
No, thank you!
RICHARD DEDEKIND
I think a lot of it is dependent on your experience. As an Algebra teacher, I'm programmed to think in "like terms," etc. I'd love to see what a younger kids would think of this.
SOPHUS LIE
That one picture with the 7 boxes and 2 frogs was really interesting. It felt different from typical subtraction representations and I had to think through it (rather than just identify it).
ALFRED TARSKI
Clever use of Desmos for surveying purposes. I think, with explanation, you could use all of the models for subtraction. Some require more front-loading/scaffolding than others.
JOHANN BERNOULLI
Another way: I use Cuisenaire rods to begin modelling difference by laying a '9 rod' and a '2 rod' alongside each other.
HIPPARCHUS
I wonder what the motivation behind this survey is. :)
HENRI LEBESGUE
I'm not sure if I answered as you might have intended. Maybe a little more direction on what angle you want me to think from.
JEAN TAYLOR
I loved seeing so many possible ways to show subtraction visually!
JOHN VON NEUMANN
Interesting survey. Thanks for sharing. I hope you'll let us know how you plan to use the results.
ÉMILIE DU CHÂTELET
Cool survey. Thanks, and good luck with your data!
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