Learning mathematics is not just about knowing, understanding, and applying its concepts, principles and all the associated mathematical procedures and algorithms. It’s not just even about acquiring the capacity to solve problem, to reason, and to communicate. It is about making these capacities part of students’ thinking habits. It is only then that one can be said to be mathematically literate.

The test for example that solving problem is no longer just a skill but has become part of students thinking habit is when students are doing it without the teachers still having to ask “Can you explain why you solve it that way?” or “Can you do it another way?” Those should be automatic to students.

“A habit is any activity that is so well established that it occurs without thought on the part of the individual.”

Here’s is a list of important mathematical habits of mind that I believe every teacher should aim for in any mathematics lesson.

**Habit #1: Searching for Patterns**

Students should develop the habit of

- generating cases and generalizing patterns
- looking-out for short-cuts that arise from patterns in calculations
- investigating special cases, extreme cases from patterns observed

**Habit #2: Reasoning**

Students should develop the habit of

- explaining the positions they take
- providing mathematical evidence/justification for the conjectures or generalizations they make
- testing conjectures by generating cases both special and extreme
- justifying why a generalization will work for all cases or for some cases only

**Habit #3: Solving and posing problems**

Students should develop the habit of

- always looking for alternative solutions to problems
- extending problems and solutions to more general case
- solving problems algebraically, geometrically, numerically
- asking clarifying and extending questions

**Habit # 4: Making connections**

Students should develop the habit of

- Linking algebra, number, geometry, statistics and probability
- Finding/devising equivalent representations of the same concept
- Linking math concepts to real-world situation

**Habit #5: Communicating mathematically**

Students should develop the habit of

- using appropriate notation and representation
- noticing faulty, incomplete or misleading use of numbers

**Habit #6: Reflecting and self-directing learning**

All these are only possible in an environment where students are engage in problem solving and mathematical investigation tasks.

If you want to know more about mathematical thinking, the books below are great read.

This is great. I especially like how you differentiate between these processes as skills vs habits. You then put together a nice concise list of habits. If you’re interested in a longer list, take a look at a draft of my own habits of mind list.

Thanks!

Avery