I asked 37 math teachers why they think kids find proof so hard in geometry. Here were there top four responses:
- Discovering a proof is a non-trivial problem that requires persistence. (11 responses)
- Kids have not yet developed logical thinking or deductive reasoning. (8 responses)
- Students don’t see the point/need/motivation for proof. (8 responses)
- Students don’t have enough prior experience with justification or explanation in math class.” (4 responses)
For a moment, let's take a close look at the idea that kids aren't logical thinkers. It's a common view among teachers, and it's also an assumption that's built in to a lot of textbooks, activities and curricula. It's why the second chapter of your book might involve a careful analysis of "if...then" statements in non-mathematical contexts.
A concern about the logical capacity of kids is also why you might design an activity around the idea that "children are fuzzy thinkers."
This morning at TMC14 I lead a workshop on proof with a bunch of really thoughtful geometry teachers. They had some really good points:
- Kids use logical reasoning to argue with their parents, right? So kids sometimes have the capacity to reason logically.
- When we dug into some math and tried to prove some ideas that were new to us, we found it difficult to offer justifications and proofs. So our adult ability to articulate our reasoning was being stretched beyond comfort.
In short, this means that thinking logically isn't an always thing. It comes and goes, depending on the context. Kids can't reason about school geometry, we math teachers struggled to reason about unfamiliar math.
The instructional implications of this are that we can't hope to improve our students' ability to reason about geometry by improving their ability to reason logically in general. Kids can use logic, sometimes. We need to help them reason in geometry.
(See also: Rebecca Talks Logic With Her Kids)
I always introduce proof in Algebra 1, making students.justify their steps in solving equations. Sure it does generally only involve 4-5 justifications, it does get students thinking about why the math works, and introduces the idea of.justifying.steps.
ReplyDeleteOne thing I find is that kids *can* reason about geometry problems, but they struggle in particular to *generalize* their thinking (and/or to see the need/point of generalizing). For instance: two intersecting lines form 4 angles. If the measure of one such angle is 30 degrees, find the measures of the remaining 3 angles. Would you agree that most students can solve this problem with little or no assistance? If I then try to elicit the idea that vertical pairs of angles are congruent, again it's not a big deal. But here's the kicker: prove that this conjecture is true in general: WOWZERS -- that is a toughie.
ReplyDeleteWhy is this so hard? One non-trivial aspect of this is that, on the one hand, there's working out the missing angles; on the other hand, there's doing a meta-analysis of the thinking that allowed you to find those angles -- that is harder. Also, there's the difference between performing the calculation 180 - 30 = 150 (easier) and 180 - x = y (harder, if only because it's more abstract). Lastly, when working with numbers, you merely observe that 30 = 30 or 150 = 150. When working with expressions, you have to figure out how to "know" that y = w -- i.e. since both are equal to 180 - x. That particular inference is hidden when doing numerical calculations, so in the proof phase it must be brought to the surface, often in the form of the transitive property.
Them's me thoughts!
"on the other hand, there's doing a meta-analysis of the thinking that allowed you to find those angles -- that is harder"
DeletePerfect. It's hard to do meta-analysis, to make the thinking itself visible. This is a crucial piece, in my opinion. All of this becomes easier if the thinking is made visible for kids before they set out to write a formal proof.
How do you help kids notice their thinking? A steady diet of arguing, debating, explaining, teaching and informal discussion and proving that happens before we ask them to formalize it all in a formal proof.
There is absolutely an analogy between what I'm saying and Willingham's work on reading.
ReplyDeleteThere absolutely are general proof skills that cut across mathematics, but think about what it takes to write a proof in number theory. First, you've got to understand the claim and statements involved in the argument, which means you need to know a bunch of stuff about number theory. And then you need access to a bunch of reasons, the justifications. Informal argument is a great way to help us be aware of the sorts of reasons that you have available in number theory, and the reasons you use in number theory are likely to be specific to number theory.
I absolutely believe that there are mathematical habits of mind, and I also believe that there are wide-ranging proof skills. But building these skills is a tremendously difficult task, not the sort of thing that you pull off with a few general reasoning exercises with "if...then" statements in your second unit.
One problem with proof for kids is how it differs from proof for grown-ups:
ReplyDeleteThat which is to be proved is obviously true.
As a sometime geometry teacher (I get another shot at Euclid this coming spring), I have struggled with this.
We asks kids to prove a theorem or a variation on a theorem we already proved. Parroting. Or we give them a pile of sticks that can be made into a house, if they assemble them in the right order, and they are playing "assemble".
Worse, about the second kind of proof, given ΔABC and ΔDEF, and AB = DE, and <ABC = <DEF, and BC = EF, prove triangles are congruent, and there is a diagram there, and it is SO OBVIOUSLY true... why are we doing this? (asks the student, but also asks the teacher)
Ben Blum-Smith at http://researchinpractice.wordpress.com/ had some cool stuff in 2009 - 2011 (dates reflect my reading, not his production).
I attempt to introduce proving as theorem proving, and come up with some easier, but unimportant things for them to prove. In algebra, that squares are multiples of 4, or one more than multiples of 4 (justify using cases). In geometry, often something diagramless feels theorem-y: Prove that the segment formed by joining opposite sides of a parallelogram is parallel to the other two sides.
I also experiment with asking them to show things that aren't true, so they think of hunting for counterexamples as a useful exercise.
I know I don't have the right answer, but the routine that shows up in our books is barely proof at all, and we should be looking for something better.
Jonathan
I was working on a problem a few weeks ago. As soon as I looked at it I was pretty sure I knew the answer, but I wanted to be positive. So I proved it.
DeleteI think that's the sweet spot for proof: 95% sure, but not certain. Too much doubt makes careful proof impossible, since I'm always willing to ditch the proof and go after the alternatives.
How much do we gain from replacing "Prove X" with "Prove X or not X"? A bunch, I'd bet, but there's still more to be done. Kids will still struggle to prove statements about a domain they haven't muddled around much in, and a steady diet of informal arguments are needed to set the stage for formal proof.
This, though, I think I disagree with:
DeleteI attempt to introduce proving as theorem proving, and come up with some easier, but unimportant things for them to prove.
For the most part, I think that people struggle with geometry proofs because of geometry, not because of proofs. Giving kids practice with proofs in non-geometric contexts won't give kids the practical knowledge and reasoning skills in geometry that are the ingredients of proof-writing in geometry.
I introduce proving as arguing, and I push everyone to be more skeptical of our arguments as the year goes on. Formal proof is the end of a journey, not the beginning.
I introduce formal geometry proof about as late in the course as I possibly can. But when we prove things, they are not the textbook proofs, which look nothing like what a non-high school student would prove.
DeleteBefore then we justify, study, and construct a whole lot. We ask the "why do you know this?" and "why do you think this?" question constantly.
My daily lessons, in almost any course, consist in large part of deriving the math we are about to learn. The math is right, because we can justify it, not because I say so. This is habit, this is pattern, not only in geometry, but yes in geometry, too.
But when we finally reach those formal proofs, we're going to do more stuff that looks like proving theorems, and less that asks the kids to prove that the two obviously congruent overlapping triangles are congruent.
Proof is hard because proof is hard. We are asking for reasoning, for process, not an answer. Try that in algebra II, give them the answer and ask them to justify it. I've done it. Kids get angry, frustrated.
I think the difficulty is the proof, not geometry.
Thanks for responding. I learn a lot from these back-and-forths, and I really value disagreement on teaching ideas. So,
Delete"Proof is hard because proof is hard. We are asking for reasoning, for process, not an answer."
I agree that formal proof should show up as late in the course as possible. I agree that we should ask those "why?" questions constantly. I agree that the truth of math shouldn't rest on our authority. And I agree that those "prove the obvious thing" problems are a bit lame.
I should say that I also agree that "proof is hard," to an extent. But I think it's important to dig deeper and have a theory about what makes proof hard. Your theory seems to be that it's reasoning that's hard, and the process is always hard for kids.
My perspective -- informed by the van Hieles and some of the experiments I link to -- is that reasoning is not one thing, and so its problematic to say that reasoning itself is hard. With content that kids feel comfortable with, they are completely able to express their reasoning process. Ask a kid why he bought the shirt that he did and you'll get a nice, coherent chain of reasoning. Ask a kid -- even a very little kid -- to reason about a make-believe context, and you'll hear some nice reasoning.
The instructional implications of this aren't much different than your's, but I think that this calls for a steady diet of quasi-proof throughout the year. Students will be able to reason, informally, about content that they're comfortable with, and it can help create continuity between those "why?" questions and formal proof at the end of the year.
So I think it is not "reasoning" that is hard, but rather "explaining reasoning," especially in a formal way.
DeleteEven if our exercises focused on questions such as "are these lines parallel?" and didn't ask for a formal, written "why" part, more of our kids would find success, more easily.
But the course we teach as geometry is the descendant of the only exemplar of an axiomatic system that kids encountered k-12. As such, it includes inherited demands on explaining reasoning, in sequence, that children encounter nowhere else, in school, or in life.
Your shirt example does not require a deductive chain of statements, and seems more analogous to asking for a list of reasons that WWI could not be avoided - reasoning, explaining, but without the sequential chain.
I don't think you agree with that part.
But I'm glad you point out that our different ideas bring us to a similar place: we can help by asking children to explain their reasoning in many places and settings before they encounter "geometry proof". (I set foundation for the "why" questions in algebra. They are hard, but ease the transition).
Jonathan
Consider the difference between the following two scenarios:
ReplyDelete#1. (Traditional) Prove that the diagonals of a rectangle are congruent.
#2. (Alternative) Draw four quadrilaterals and measure their diagonals.
A. Do any of your figures have congruent diagonals?
B. Is it even possible to make the diagonals congruent? Draw a figure whose diagonals are exactly congruent.
C. Now consider "the set of quads with congruent diagonals." Do the elements in this set have anything in common? If so, what?
Now draw four parallelograms.
A. Do any of them have congruent diagonals? Is this even possible? Try to draw one that does indeed have congruent diagonals.
B. Now consider "the set of p-grams with congruent diagonals." Do the elements of this set have any essential features in common? If so, what?
Now we have a tough challenge: it would appear as though the set of p-grams with congruent diagonals are actually all RECTANGLES. How can we be sure that this is indeed true? Try to construct an argument to prove this conjecture.
There are two levels at work here which should be made explicit to students if we are to succeed at this:
1) Verification. On the basis of examples, we strongly believe that we are after "the set of rectangles." We want to be sure a) that there are no "alien cases" which satisfy our condition on the diagonals, but which are not rectangles, and b) that there are no "degenerate rectangles" which fail to have congruent diagonals. Our proof will help with both of these aims.
2) Explanation. Even if you are 100% convinced that the set of p-grams with congruent diagonals is "all rectangles and only rectangles," it still remains to answer -- why? What exactly is it about right-angled p-grams that make their diagonals congruent, whereas non-right-angled p-grams don't enjoy this special feature? Writing a proof can point to us to (perhaps otherwise hidden) structure in both the logic and the geometry of the situation, and this can provide insight into the topic of study, can be interesting, challenging, and fun. Let's get started! :-)
This is good stuff, James. This sort of progression ensures that (a) kids know what we're talking about by the time we get to proving and (b) proving is satisfying some need, the need for explanation.
DeleteI'd like to add that informal proof might be an important step here. We want an explanation? Then offer an explanation, in your own words, in any way that you know how! Imagine a skeptic -- how could you convince her that these really have to be rectangles?
Then the formal proof comes at the end of the process, as a careful record of everything that we discovered, justified and explained.