productive struggle vs. thrashing blindly

Experiments in Learning by Doing

The trick is to choose a goal just beyond your present abilities; to target the struggle. Thrashing blindly doesn’t help. Reaching does. (Coyle, 19 pag.)

What if we teach how to reach? How might we offer targeted struggle for every learner in our care?

SMP-1: Make Sense of Problems and Persevere #LL2LU

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

SMP-8: Look for and Express Regularity in Repeated Reasoning #LL2LU

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

Math Flexibility

When people believe their basic qualities can be…

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The Equation of a Circle

Great geometry lesson

Easing the Hurry Syndrome

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.


Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we…

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CAS with Unit Rates

Using CAS to allow students to investigate unit rates opens up a world of options due to the device recognizing words as variables.  Teachers can actually enter one cup of sugar makes 2 cookies in fraction form on a TI-Nspire CX CAS and get the results shown below.

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Asking questions that promote deep thinking about the topic gets students into the discussion.

  • Why are these true?
  • What stays the same?
  • What changes?
  • What other ratios would work?
  • What other ratios would NOT work?

Asking the right questions is key to starting effective discussions, but what then? Multiple representations help students visually see what the math is doing which in turn leads to better understanding and skill mastery.  Look at the following situation:

Joe can mow 7 lawns in 4 hours.  How many lawns can he mow in 3 days?

Now look how the CAS can investigate this problem using Numerical form, a table, and a graph.

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This allows students to see relationships between “doing the math” and “using the math.”  See how long it takes the students to realize that the data is in hours but the problem is in days.  The number 3 is nowhere in the data which will generate questions from students.  This opens up a broad range of “teachable moments” which is what teachers love to see. The students want you to explain instead of you begging them to listen. Now let’s throw in some geometry.

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This introduces using the unit rates for unit conversion.  Although unit rates are a 6th grade skill, students can see how solving equations will play a part even though they haven’t mastered that particular skill.  Ask the students to investigate how the device got that answer.  You may never have to teach “cross multiply” again. 🙂

10 Yeah But’s I hear in Education with Possible Solutions!

Here are some helpful ideas to overcome come excuses for not using technology.

Inside Education, Outside the Box!

“Progress is impossible without change, and those who cannot change their minds cannot change anything.” By George Bernard Shaw

Over the years, I have heard a lot of what I call “Yeah but’s”….these are the excuses/arguments educators make that usually have a fixed mindset verse a growth mindset. As a leader (you don’t need to be in a leadership position, to be a leader) you need to be able to navigate around the yeah but’s; here are a few I have heard with some possible solutions.

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1. “Yeah But…I don’t have the time to integrate ________ (fill in the blank technology, PBL’s, data analysis etc.)” Solution: Try to find a way to make it so they are saving time, see the value in it and how it connects to the curriculum. For example: start slow by offering to create a Project or Problem Based Learning (PBL) for them that…

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CAS with Percentages

As I think about important skills for students, percentages stand out to me as an important skill for both elementary and middle school students that is not understood completely.  The students memorize the algorithms we set before them but never really reach the mathematical understanding of why and how we use them.  Using the TI-Nspire CX CAS handheld and some teacher preparation, students can delve into the why’s and how’s of percentages.  For example, start the lesson by putting the following slide on the board or send to student handhelds.

Percent Pic 1

Ask students the following questions:

  • What stays the same?
  • What changes?
  • Why do you think the last one is false?

Using a Quick Poll in the TI Navigator system or a cooperative learning strategy, facilitate student discussions on their answers.  The students can use their devices to investigate other percentages to see if their theories hold true. Teachers can use the activity Solving Percent Problems from the TI Math Nspired website as a follow up activity or intro activity for the lesson on using percents to solve problems.  As the unit progresses other investigations and discussion starters could look like this:

Percent Pic 3

Ask questions like, “What do you notice about the numbers?”, “What is the relationship between the numbers?”, “What conclusion can you make based on this pattern?”.  Also show some other percentages such as the slide below asking similar questions.

Percent pic 2

Of course as in any good unit of study, opportunities for practice and hands-on applications are needed throughout the unit to master the skill but getting students motivated to understand the math is the first step.  Investigations such as these will allow students to delve deeper into the math instead of skimming the surface with algorithms only.

Use More CAS

When I was first introduced to a CAS device, I thought it was a great tool for my AP Calculus class.  The more I used CAS activities I realized it would be a great tool for all math.  I believe student investigations are key in deeper understanding of mathematics.  CAS devices such as the TI-Nspire CX handheld and iPad app give students the opportunity to investigate the why’s and how’s of math.  If I write the steps on the board, the students only half believe.  If I give them technology and pose a “dilemma” such as why does ½ + ⅓ = ⅚, then the real learning begins.  I created this blog to share my experiences with and thoughts about using CAS in all grade levels.