# Effectiveness of proofs in secondary education

I'll have a department meeting in about 10 days and I want to bring the subject of proofs up. While most teachers do proofs in the blackboard, I want to argue that we should put problems to prove in students's exercises sheets. I don't think it is enough to show them proofs, they have to do some themselves.

However, this is mostly based on intuition and my own experience with learning. Is there research that backs up my statements, so that I can discuss them with other teachers, showing that proofs not only help deepen their understanding of mathematics but also helps them with other tasks in mathematics, including computations?

I'm also interested in research that might contradict my intuition, in which case I'll change my beliefs for the better.

• This might be useful: mathoverflow.net/questions/51891/… – Stan Shunpike Jan 18 '15 at 1:34
• How old are the students in question? – Stan Shunpike Jan 18 '15 at 1:34
• Within the range 14-18. They're highschool students, which lasts three years in Brazil. – Mark Fantini Jan 18 '15 at 10:42
• @MattF. Added a line. – Mark Fantini Jan 18 '15 at 15:02
• [Because I don't give research results, I don't post this as an answer.] In general: Whether to let students do proofs on their own or merely show them the proofs depends on the didactic function of the proof: 1. Defend the proposition and thus make it acceptable to students 2. Rigorousity 3. Knowledge of reasons for the proposition 4. Enabling students to proof in general 5. Enabling students to redo the proof on their own 6. Enabling students to a proof/computation/other action in the same field of math 7. Fulfilling the curriculum. So don't be dogmatic but pragmatic on this issue. – Toscho Jan 18 '15 at 16:00

These books will be of interest to you:

Theorems in School edited by P. Boero

Teaching and Learning Proof Across the Grades, ed. by Despina Stylianou et. al.

My subjective sense is that the majority of researchers in math education agree with your point of view: learning proofs requires hands-on practice with writing proofs, and not only improves the ability to write proofs but also supports an understanding of the interconnectedness of mathematics and supports the development of computational skills as well. For example the US elementary and secondary math teachers' trade guild, the NCTM, has a statement stressing the importance of proof:

Instructional programs from prekindergarten through grade 12 should enable all students to--

recognize reasoning and proof as fundamental aspects of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and proofs;
select and use various types of reasoning and methods of proof.


See here.

No links to research, but too long for a comment. This is an alternative approach, which doesn't have the word "proof" in it, but will create proofs as part of it. Proofs can sometimes (not always) be just a symbolic game, but abstraction is motivated, tested and has wide application.

tldr; they are already proving things every day, we need to teach the process of abstraction rather than just proof.

Proofs that will be accessible to your students are just the algebra version of abstraction. Abstraction of patterns is applicable across high school curricula.

First you need to recognise the proofs that your students do in every lesson. Every line of algebra deduction is a proof that the original expression or equation implies the new expression. In geometry, when they draw 2 identical circles which intersect, they have proven that those points of intersection are equidistant from the centres of those particular circles.

I'm guessing what you are looking for are proofs of more general results. You are looking for abstraction. Once you acknowledge the proofs they are doing you can teach how to generalise them.

For instance, students can "prove" several special cases of a more general proof that you want to encourage, using simple algebraic deduction.

$$(x+2)(x-2) = x^2 + 2x - 2x -4 = x^2 -4$$

$$(x+3)(x-3) = x^2 + 3x - 3x -9 = x^2 -9$$

They have proved 2 things already! But it looks like there is a pattern in their proofs. Maybe they can make a more general proof? They can introduce a variable where a constant was.

$$(x+n)(x-n) = x^2 + nx - nx - n^2 = x^2 - n^2$$

This equation would be the start and end for a traditional proof, but using abstraction it is just the mid-point. They have now proven several fairly concrete equations and one more general equation in a way that is motivated and easily accessible.

We now need to test the applicability of the abstraction and maybe extend it. For instance, at this point they probably see this only as a proof for speeding up expanding brackets (students often see equations as implications or rewritings rather than bidirectional equivalence). They can abstract this further by flipping the equation around to prove that certain quadratics can be factorised.

$$x^2 - n^2 = (x+n)(x-n)$$

They can test it for $n^2 = 25, 49$. Irrational numbers are extra for experts, $n^2 = 10$. See how far they can take the general proof.

They have now proven how to factorise a monic quadratic of a certain form. As $x$ is so often used as simply an independent variable of an expression, they probably don't realise that it is generalisable further. So next they can experiment by replacing $x$ by another expression.

They now have:

$$(u+v)(u-v) = u^2 -v^2$$

After all of this they have completed possibly pages of proofs, some very specific, some more general. They have further investigated the meaning of the proof to actually understand how it can be used.

I personally don't see much point in writing down simple proofs, but the process of abstraction and application of these abstractions is key in mathematics, algorithms, and across most school subjects. The above described process will definitely deepen understanding and lead to higher order thinking.