Discovering Relationships Through Context and Inquiry

Tuesday, March 1, 2016 / 6 comments
This year, the idea that there is a relationship between addition and subtraction was discovered in a Number Talk, a Number String to be more precise.








Following the Number Talk, I wrote the equations off to the side and asked the kids what they noticed about them and recorded what they said.

This moment seemed to me, like an obvious opportunity for an investigation.  So, I recorded their noticings on chart paper so that we could refer to it later when we would be ready for the investigation (about a week later).
As I prepared for the upcoming investigation, I knew that I could do better than I did the year before. 
The idea that there might be a relationship between addition and subtraction wasn't discovered by my students last year, so we are already beginning in a better place.  My students last year understood related facts within addition quite well, but I knew there was more work to be done on how students discovered and investigated this idea of a relationship between addition and subtraction.

Now, how do I help students learn this?  Here's what one publisher's curriculum thinks is a good idea:
This is on a parent letter and introductory lesson that is meant to be sent home at the start of the unit.  There is no opportunity for discovery. It's just plain dissemination of meaningless information.


A piece of homework from the same publisher.

A different publisher's homework

Back to planning, which will likely not be aided by the district provided publisher's curriculum.

To begin... What is my mathematical goal?
I can use addition to help me subtract. 
There is a relationship between addition and subtraction.

What opportunities do I need to provide my students for them to notice, wonder, investigate and eventually prove this?

I turned to the California Math Framework, a narrative of the standards to serve as my foundation for my lessons.  

What do you think about the part where it says, "Instruction should include opportunities for students to investigate, identify, and then apply a pattern or structure in mathematics?"  What would that look like? Would the publisher's curriculum give students those opportunities??? Maybe yours does, but mine doesn't.

Van de Walle says, "Contextual problems are the primary teaching tool that you can use to help children construct a rich understanding of the operations... A discussion of how these two equations [referring to equations in an example] can be written of the same model is an important opportunity to connect addition and subtraction.  This is significantly better than the traditional worksheet activity of "fact families" in which children are given a family of numbers, such as 3, 5, and 8, and are told to write two addition equations and two subtraction equations. Very quickly this becomes a matter of putting the numbers in the various slots without much meaning."

From here, I began planning contextual problems. We began with "Notice Wonder" at the beginning of the lesson, and then came full circle with "Notice Wonder" at the end of the lesson, using the equations that came out in student work.  The student work here that is collected throughout the lesson becomes the direct instruction.




Equations taken from student work.

My tiny first graders have noticed over and over again that there is something going on with these groups three numbers.  I tried to leave it very open at the end of each lesson...  "Huh, that's weird it happened again.  Let's put it on the wall with the other ones."  Many students have moved from "noticing" this relationship to an understanding that it will will always work. Some are still investigating this conjecture.

The following unit will focus on compare problem types. This understanding of the relationship between addition and subtraction will be very helpful in the next unit. Now back to planning... Definitely going to need "numberless" word problems for compare problems.
Here is a little sample of student work on compare problems from my class: https://youtu.be/K9clsjyk7jo


6 comments:

  1. Great stuff Jamie! Tomorrow I'm exploring exponents and square roots using 'clothesline' math with my 8th graders. I'm hoping for some great student revelations like you had. :)
    @mathgeeksrock

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    Replies
    1. Thanks, Lisa! I love using the clothesline to help with number sense. :)

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  2. Compare problems and Put Together/Take Apart (or Part-Part Whole) problems are great to use because they don't involve action and thus lend themselves to a variety of ways to solve problems. I love how you gave your kids multiple opportunities to make sense of structure which is one of the Mathematical Practices. I gain so much from reading your posts! Thank you!

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  3. Compare problems and Put Together/Take Apart (or Part-Part Whole) problems are great to use because they don't involve action and thus lend themselves to a variety of ways to solve problems. I love how you gave your kids multiple opportunities to make sense of structure which is one of the Mathematical Practices. I gain so much from reading your posts! Thank you!

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  4. Jamie, some common uses of the number line and I would say not bad when just working with + and -. I feel there lies a problem when students get to + - and - - problems later and try to use this horizontal number line. I also feel this common method of teaching is out of context of anything a child can relate to, when else will a child use a horizontal number line in real life and jump around the number line?? I would ask you take a look at my Watertank Math method which I developed because my own students had a hard time relating this horizontal number line method to anything they did in real life. I use the operations of a toilet, yes a toilet, to teach all the operations. I feel with this context, which is obviously one of the first things we are all taught as humans to do, is something practical students can use to visualize adding and subtracting integers! I use a vertical number line with my concept and the process of water coming in or leaving a toilet to allow students to visualize adding and subtracting and be able to verbalize the operations, especially when later they see + - and - - operations. Please take some time to check out my concept at watertankmath.com I will be presenting at the NCTM 2017 in San Antonio in April so if you are there please stop by my session. I would love to hear any feedback. Please feel free to email me at jason@watertankmath.com

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  5. this is a fun way for the kids to learn maths and i am sure it is not going to kill the fun of the subject. i can not wait to try it on my class. i am sure they will appreciate it. thanks

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