Demanding the How or Why.

With the advent of apps which solve problems for students, what can we do to create deeper understanding and a knowledge of why problems are solved in a certain order.  I've had a student who used an online calculator to factor trinomials. 

It was easy to tell she'd done that.  She didn't show her work and she bombed a quiz I gave her.

I'm starting to ask questions which not 'What is the next step?' but 'How do you know this?"  I'm asking for evidence of them stopping to really think about what they are doing.

My Algebra I students are currently working on solving systems linear inequalities.  I start by asking what type of boundary they will be using in each problem.  Then I ask them to tell me how they know they are right.  This is where they might say its a 'dashed line because the inequality does not have an equal sign'.  The ones who can answer this question are doing much better than those who want to just do it without thinking.

According to an article I read, we do not want to ask simple information type questions such as what is the formula for area.  That is simple recall and can be looked it up in the book.  The author says these questions can often be used to see what the student knows but they do not require higher order thinking.

It is better to ask questions which require students to elaborate, explain or show their thinking.  These types of questions require the student to explain the steps they took to solve a problem or their thinking behind the method they chose to solve it.  Furthermore, give students sufficient time to put together their thoughts so they can explain. Unfortunately, one average, students are only given five seconds to answer.

Require students to connect mathematics to relationships as they answer such as the visual representation of the coordinate plane of an inequality with the actual equation.  Encourage them to reflect on their thinking and justify their choices.  Have them explain why they chose a certain method to solve the problem.

Do not steer the conversation to a desired outcome because the student may not have chosen that way of doing it.  When we funnel the conversation, we ask very specific questions to steer the conversation but save the higher level questions for later and do fewer.  Use questions which blend questioning, reflection, justification and probing.  The questions have students share what they notice while encouraging them to share their thoughts.

Teach students to ask each other more open ended questions such as 'Why do you think that?' or 'Could you have solved it a different way?'.  This technique helps students begin discussing mathematics which helps create a deeper understanding. 

I'd love to hear what you think.  Have a great day.