Provide ample opportunities for students to practice problem-solving and critical thinking skills.  Whenever most math teachers see a teaching tip encouraging more opportunities for problem solving they immediately think about word problems.  Word problems are good and serve a purpose in increasing critical thinking, but there are other way to promote critical thinking and problem solving in math classes as well.  Here are a few ideas to consider.

Desmos Marbleslides

Students explore transformations of functions (translation, reflection, stretch/shrink) in a game-like setting.  Students must manipulate the parent function in such a way as to capture the stars on the screen when balls are launched from a specific location.  This activity is great for exploring how the graph changes when different numbers in different places in the function are changed.  Problem solving is required to discover the correct combination of transformations to successfully capture the stars.  Tasks are designed to start with simple transformations and work up to some really challenging problems for students who want to stretch their thinking.  Check out the link to this Marbleslides: Parabolas activity on Desmos (you may need to create a free account if you do not already have one).

Desmos Polygraphs

This is the math version of the game 20 Questions.  Students are paired up with other students in your class who are signed in to the same activity.  The goal is to guess which image (usually a geometric figures or a graphs of functions) the other student chose with the fewest questions possible.  One student is prompted to choose one of the images.  The other student then asks yes/no questions by typing into the question box.  When their partner responds, the questioning student can click on images that would be eliminated based on that response.  The student WINS when they have figured out the other student’s choice.  Then students are re-matched with another student.  This activity promotes critical thinking in that students must notice differences in the details of the images, and then must communicate about those differences using proper vocabulary.    Check out this YouTube video by Brian Wise to see how a game would be played.  Or, click on this link to see a Polygraphs: Exponentials example on Desmos (you may need to create a free account if you do not already have one).

Geometric Proofs

Writing proofs, start to finish, certainly requires problem solving and critically thinking.  However, many students initially struggle with this skill.  Teachers can scaffold the kind of critical thinking necessary by presenting proofs as puzzles to solve.  For a two-column proof, teachers can put each statement and each reason of a proof on separate pieces of paper.  Then give students a “framework” for the proof.  The framework could be set up in three different ways, depending on the level of the student or where the students are in the scaffolding progression.

1.     The framework could include all of the reasons in the correct order and the students must match the correct statement.

2.     The framework could include all of the statements in the correct order and students must match the correct reason.

3.     The framework could just show spaces for statements and reasons and the students must determine the correct order for all of them.

The framework could also be adapted to work for paragraph proofs or flowchart proofs.  However the framework is structured, students are practicing the logical thinking necessary to eventually write geometric proofs on their own, but in a more accessible format that encourages problem solving and critical thinking.  (Note:  The “correct” order should be based on what is logically correct.  That means that for some proofs, different arrangements could be considered correct.)  Check out this example of what a geometric proof might look like.

Trigonometric Proofs

Trigonometric proofs of identities can be set up similarly to geometric proofs, but we do not usually include reasons as part of these proofs.  As such, the separate pieces of paper will only contain the individual steps for changing one trigonometric expression into another, equivalent expression.  Then students are tasked with putting the steps in the correct order.  (Note:  The “correct” order should be based on what is logically correct.  That means that for some proofs, different arrangements could be considered correct.)  Check out this example of what a trigonometric proof might look like.

Error Analysis

When teachers ask students to “do” a problem, students are demonstrating one level of understanding - that they can follow the correct steps to arrive at a correct response.  But we can ask students to demonstrate a different level of understanding when we ask them to analyze a problem that has already been worked out and discover the mistake in the work.  This requires students to use a higher order of thinking to follow someone’s steps and find the error in their work.  We can raise the level even higher if we also ask students to explain WHY that step is an error, WHY it does not work, and HOW the problem could be reworked correctly.  One more layer can be added to this type of problem if you tell students that it is their friend who has made the error and they must write the explanation as if they are addressing their friend and they want their friend to actually learn how to do the problem correctly.  This encourages them to write their analysis in a positive, helpful way because it makes the task a bit more personal.  Error Analysis can be especially helpful is the “error” in the problem is something that students often miss.  Students get a chance to see the mistake in “someone else’s” work and will hopefully learn not to make that mistake themselves.  Check out this example of what an error analysis problem might look like.

The key in all of these examples is to get students to think beyond just following the steps of working out a problem.  Give students chances to think differently or more deeply about a problem, and turn it into a puzzle to solve or something to figure out.  They will benefit from these opportunities to practice problem solving and critical thinking skills as they broaden their ability to think mathematically.