## 3. Construct viable arguments and critique the reasoning of others.

How are **high school **teachers helping their students construct viable arguments and critique the reasoning of others?

How can students be helped to:

- understand and use stated assumptions, definitions, and previously established results in constructing arguments
- make conjectures and build a logical progression of statements to explore the truth of their conjectures
- analyze situations by breaking them into cases
- recognize and use counterexamples
- justify their conclusions, communicate them to others, and respond to the arguments of others
- reason inductively about data
- make plausible arguments that take into account the context from which the data arose
- compare the effectiveness of two plausible arguments
- distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is

*The CCSS states:*

Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

What has happened to the role that “proof” plays in the high school geometry class? In Texas, it appears that the new end of course assessment does not directly assess aspects of proof. Yes, I suspect there are components of logic such as identifying the inverse, converse and contrapositive of a given statement but the traditional two-column proof has gone by the wayside.

Consistent with the saying that what is not assessed is not taught, the concept of proof, whether as two-column, paragraph style etc. is not or will no longer be taught. Proof is consistent with the CCSS statement above as concerns a method for distinguishing correct logic from flawed reasoning. Proof brings concepts together. Proof is a tool for helping students to think of their math more systematically. Using proof as a part of a math class is, I believe, a good teaching practice when taught as a part of a larger unit or, proof could be used as a framework upon which to construct a unit.

A proof, such as one of many for the Pythagorean theorem, could be a capstone on a unit of factoring for instance. I would like to know what the current thinking is as concerns the use of proof in the high school level geometry class is or other math classes at the secondary level for that matter.

Thank you.

David

To me proof is part of a continuum. Dave Coffey and I got to thinking about the reasoning process standard as:

- Making Sense

- Making Conjectures

- Making Arguments

If students haven’t had much experience with the first two forms of reasoning, making arguments is going to be hard. For me, making proofs has got to start with making arguments: is this true sometimes, always or never? Why? Then going from argument to proof is a matter of tightening up. Well what about if… You said it’s because of this, but why is that true?

I agree that there is a continuum of learning. Please understand that I am not suggesting that we jump students into a challenging proof without necessary background work. All three reasoning processes that you stated are not isolated from each other. In fact, a combination of any of the three can be used at the same time. The development of the argument requires first, determining the question or hypothesis to be asked or tested. To sufficiently state the question to be proved can be a challenge by itself. This is not necessarily an easy task and requires the making of conjectures, and checking to see if ideas make sense. With this in mind, we can develop a strategy for the proof. Will it be deductive? Will it be inductive? How will we know when our proof is finished? Yes, it is helpful, initially, that a background in basic logic needs to be in place. This is an exploratory process for the students which requires higher level thinking. These are general problem skills that can be applied to almost any math class.

In a separate issue, a concern of mine is when I receive feedback from former students. Often, at the start, these students are placed in large classes. During their first year they are taught much of their math through the use of proofs. At the start they struggle a bit but, for the most part they manage. Are our students at a disadvantage if we don’t teach proof? In short, I believe that the reality is that our state mandated tests drive our curriculum. This has implications for students going on in their math. If our state does not find a way to assess specific aspects of proof, then techniques specific to proof may not necessarily be taught.

I don’t view proof as just a “simple” teaching technique. The idea of proof can provide a framework for the use of higher level thinking skills together with allowing students to make deeper connections with previously learned skills. Yes, logistically there are challenges to assessing proof in a state wide assessment. Yet, as previously mentioned, it is my opinion that if a concept or set of skills it is not assessed then it is not a priority and most likely will not be taught.

Thank you,

David Williams

I like the point about thinking about what the students need later in life. Like the literacy teachers, though, that involves more than just writing in a genre. You have to teach reading comprehension in the genre, too. Helping students make sense of mathematical writing is a great and challenging objective. I don’t want to not teach proof – it’s a cornerstone of mathematical literacy.

It’s hard to motivate, though, without it being a student conjecture. For me the need is tied up with sometimes-always-never. “It always does that!” How do you know? “Look how when we drag this it’s always equal.” But we can’t check every possibility, or how would we even know if we did? Now we need a proof.

Of course for mathematicians a proof is a last step, also. They spend a lot of time understanding how something works before getting around to a proof. And then they care about the aesthetic… something we usually don’t touh on with students.

I definitely agree with you about assessment. I’m not crazy about testing proof on tests, though. To me it’s more a matter of a writing assignment, that I want students to be able to revise and edit based on feedback.