Higher Level Thinking in Mathematics

I want to share a few of my recent favorite strategies for getting math classes thinking more deeply about concepts.

Let's get them really wondering about what makes the math work!

The first one, writing in sentences, they really do not like (sorry, kids!) but it works so well and is SO important.  The other ideas are ones that the students really do enjoy and request.  I love to see a class really engrossed in trying to figure something out and ENJOYING IT!!

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Writing Prompts

Writing in complete sentences is the key to incorporating writing in math.  This is always a struggle.  You will have to tune out the complaints that "This is math class; Why would we have to write in sentences?".

When questions are worded the right way, you can succeed with getting your students to write out real answers.  Here are some samples (from my Pre-Algebra Question Pack) - Try writing a few test questions or Warm-Ups that sound like this:

Turn the Tables

Flip your questioning around for a few minutes, and ask students to supply you with the information that is usually given.  This works a little like Jeopardy, where you offer what is typically the final answer.  I like to ask for TWO DIFFERENT questions to keep students thinking hard.  You can do this in a whole class setting or even use this type of problem on a quiz.  Sometimes I include extra criteria to guide the responses and keep students from taking the easy way out.  Here are some samples:

• Given the final conclusion that triangle ABC is congruent to triangle DEF, write two different sets of possible given information that would lead to this result.

• Write two different expressions containing like terms that can each be simplified to 3x.  Give each of your expressions at least three terms and include both positive and negative coefficients.

• Give two different numbers that will have both a 3 and a 7 in their prime factorization.

• Write two different equations that both have the same solution.
  

"Always," "Sometimes," or "Never" True?

free template

Here are some sample statements to give you an idea.  This type of activity works for any subject.

Is the statement always true, sometimes true, or never true?

• "A noun is followed by a verb."
• "An atom that has lost one electron is an ion."
• "The product of two numbers is greater than both original numbers."

This works well with partners. They will make sure to test every case they can think of to see if a statement can sometimes be true.

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This strategy is the absolute best that I have seen for getting students really thinking and discussing a concept.  I ask individuals or pairs to determine whether statements are "Always True", "Sometimes True," or "Never True."  This type of questioning goes so much farther than a standard true/false question.  High school teachers, textbooks, and tests do this sometimes, but I like to incorporate this at the middle school level as well.  Getting kids used to this type of thinking early is so beneficial!

I developed some activity sheets with statements.  To make it a little more fun, I added coloring.  Students color each circle according to the directions and end up with a pattern for quick checking.  Classes really seem to love this, and the higher level questioning and thinking that you can hear as they test cases is amazing! 

Here is a link to the pre-created puzzles in my store.

If you would like to make your own, here is a FREE template to download.

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inquiry-Based Learning

Remember that one of a teacher's most valuable tricks is.....

... the PAUSE...

Get your students intrigued, and then sit back and wait.  Do not always give the answer.  At first they may be surprised that you are withholding information, but resist the urge to always tell.

When students figure out a concept for themselves, they remember it better, and they understand it better.  Try some of the wordings below:

"There is a formula for surface area of a cube, but I am not going to tell you what it is.  Now that you know what the term "surface area" means, use the nets in front of you and develop a formula."

"There is a formula using distance, rate, and time.  You need "d," "r," "t," "=," and one basic operation.  Think it out with your partner using this word problem about driving at a rate of 25 miles per hour.  See if you can write a formula that works and seems reasonable.  Then explain to the class."

"Now that I have explained zero pairs to you, use the manipulatives on your desk to model the integer addition and subtraction problems on the worksheet.  Then write a set of rules for adding and subtracting integers based on the patterns that you observe."

Read more about inquiry strategies using the links below.

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