I have been having wonderful conversations on Twitter recently with Kassia (@kassiaowedekind), Simon (Simon_Gregg), Mike (@MikeFlynn55), Elham (@ekazemi) around the topic of students making claims, more specifically differentiating between claims and conjectures. I have to admit, I have really just formed my own idea of how I differentiate between the two, so it was nice to hear others’ perspectives around this. I consider a conjecture a noticing they think to be true, more on a case by case basis. A claim, to me, becomes more generalized and then followed with a proof. (I have also had great convo with Malke (@mathinyourfeet) around these proofs w/geometry).
The conversation last night started with Kassia…(Look Kassia, I finally learned to embed tweets:)
@MathMinds Do you differentiate between claim and conjecture in your class? If so, how?
— Kassia Wedekind (@kassiaowedekind) May 3, 2015
Mike gave us a nice perspective of claims based on his work with Virginia Bastable….
@kassiaowedekind @MathMinds a conjecture is an expression of a general claim that students have noticed and suspect is true. — Mike Flynn (@MikeFlynn55) May 3, 2015
@kassiaowedekind @MathMinds But we are using the term conjecture when students are exploring this to see if it is true. — Mike Flynn (@MikeFlynn55) May 3, 2015
@kassiaowedekind@MathMinds Credit to Virginia Bastable for this explanation. This is her area of expertise. I’m just lucky to work w/ her. — Mike Flynn (@MikeFlynn55) May 3, 2015
My students have now started to say, “I have a claim to make” when they notice something happening over and over again. In those moments, I don’t really think about “what” they are calling it because I am just so excited to hear them talking about the patterns and regularities they are seeing. But is what they are saying a conjecture or claim? Does it make it to the claim wall to be revisited and proven? This year being my first work in really having students think about making “claims” beyond just noticings, I have made a “Claim Wall.” Students see things happening in certain cases and I ask them if they can write a statement for “any time we…” to see if they can make it more general. I like Simon’s idea to expand on my wall…
@ekazemi@MikeFlynn55@kassiaowedekind@MathMinds Statements could travel across the wall: ?” -> confident claim -> (triumphantly) PROVEN — Simon Gregg (@Simon_Gregg) May 3, 2015
We all agreed that the proof piece is the difficult piece of going from being a conjecture or unproven claim to a substantiated, generalized claim. I find my students prove over and over again that it “works here and here and here…” but have trouble with the why. It is hard to do, even as adults putting it into words is difficult.
@ekazemi @Simon_Gregg @kassiaowedekind @MathMinds Yes. Very helpful. Here are the guidelines. pic.twitter.com/QQ01IpnmTN — Mike Flynn (@MikeFlynn55) May 3, 2015
What I love most about these conversations is the fact that the next day it continues, but this time with the kids. Simon tweets this morning about a claim that two of his students made while folding paper…
Rod & Samyak’s claim: every even-numbered fold -> square. To prove? @ekazemi@MathMinds@MikeFlynn55@kassiaowedekindpic.twitter.com/EMyr5oNN3N
— Simon Gregg (@Simon_Gregg) May 4, 2015
Which coincidentally would help my students tremendously to think about when proving their claim from Friday’s number talk…
@Simon_Gregg@ekazemi@MikeFlynn55@kassiaowedekind One of my Ss work from Friday.. #mtbospic.twitter.com/I7sf6ez9ar
— Kristin Gray (@MathMinds) May 4, 2015
The coolest part about this claim was that it stemmed from a multiplication of fraction number talk, yet they proof show division. I loved that. Also loved the explanation that accompanied their statement. I did ask them if this was true for taking half of any fraction because they seemed to be just dealing in unit fractions at this point. So is this a conjecture or a claim? I am not sure. How generalized would make it a claim? Could it be “When taking any unit fraction of another fraction…”
Would love any thoughts, conjectures or claims on this…:)
To be continued…
Isn’t it great how embedded tweets help us to tell our stories!
Perhaps we need to help each other fishing for proofs. If this is the sticking point, do we need to find simpler claims and find proofs for those, so that the students, however young have experience of proof? Proof is such a wonderful idea, and something that mathematics has to offer all other kinds of thinking (“What kind of secure knowledge do you have? Look, here, we’ve got proof…”) Here’s a project for us!
I am wondering if we could start with the word “generalizable.” I think that is where I get a but hung up. How broad does it need to be or is it just about any operation we are doing? I see it clearly sometimes like when we were doubling volume. Ss noticed that doubling one dimension, doubles the volume. I then see the generalization as multiplying any factor in a multiplication equation by a number will increase (or decrease) the product that many times as well. However other times it is not so clear, like the claim today. Is how taking half affects the numerator and denominator generalized? I don’t know? Lots of pieces to this project for us!
I really like the idea of introducing proofs to kids at a very young age. For example, there are many beautiful pictorial proofs that may be appealing and understandable to kids quite early on.
I would say that any statement that has a level of abstraction from the particular is a generalisation.
So “Half of ½ is ¼” isn’t a much of a generalisation, algebraically speaking, though of course it applies in general to all sorts of pies, pizzas, flapjacks, cakes, biscuits… “Half of any unit fraction -> double the denominator” is a proper generalisation. “Half of any fraction -> double the denominator” generalises even further. “Divide any fraction by a number -> multiply it’s denominator by that number” would go even further.
This pattern of further and further generalisation is a great thing in itself, isn’t it, though not to be hurried…
I love that progression of generalization way of thinking about it, it puts a nice perspective on it. Thanks Simon!
For me the difference between conjecture and claim depends on whether you think you know how to prove it. Thus, the same statement can be a conjecture or a claim.
For example, a student may say:
I conjecture that the sum of two odd integers is always even. I can’t prove it but I’ve noticed that it’s true for every pair of odd integers I try.
On the contrary, the student may say:
I claim that the sum of two odd integers is always even. Here is my proof. Let a,b be odd integers; then a=2n+1 and b=2m+1 for some integers n,m, and a+b=2(n+m+1), so a+b is even. (This is just to illustrate a point – not saying they’d have to prove it this way).
Even if their proof is incorrect, in my mind it is still a claim until they realize that they don’t have a valid proof, at which point it goes back to being a conjecture. This is just my view of things, but in general, I say define the terms in such a way that will be most useful for you and your students. The key is for them to be well defined!
I completely agree, that is the way I see it too. There is just that weird in-between phase between conjecture and claim. I think I like to think of them as “in-the-works claims” or “claims to be revisited as we learn” and then “proven claims.” Such fun!
And then if several “proven claims” lead to a bigger claim, it can become a theorem!
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