Scaffolding Instruction Through the CRA Approach

When I was an instructional Numeracy Coach, one of the common trends I noticed was that many teachers were not providing concrete experiences for students to learn mathematics and were relying heavily, if not solely on abstract teaching methods.  Although our district curriculum resource came with tubs of manipulatives for students to use, many teachers had never even opened the packages and they sat collecting dust in cabinets and book rooms.  This led me to reflect on why that might be and to do some digging to determine how to get these resources in the hands of students.

It occurred to me that many teachers simply didn’t know how to use the manipulatives themselves to scaffold learning and so they simply taught the way they knew how.  Another reason was that they felt it took too long to get out the manipulatives and put them away so they opted for less mess and management.  Both of these were understandable reasons and as an educator, I knew that I had to provide experiences for teachers to see the value in these powerful tools.  I was on a mission to model how to use  manipulatives when I went into rooms for coaching, but knew I wanted to provide them a resource to refer back to in the end.   I created several tools such as resource sheets for tools for mathematics and scaffolding documents for addition and subtraction and multiplication and division, but these tools never got to the heart of what I wanted to provide for teachers.

It wasn’t until I met with Jennifer Ritter at Republic Public Schools that I realized how I would begin organizing these resources for teachers.  Jennifer shared a trouble-shooting document that some of her teachers and specialists had begun for a third grade standard organized with the CRA method of number acquisition; Concrete ->Representational ->Abstract.  I loved the idea of organizing standards this way and have been pondering for months how to best organize the examples into a resource for teachers.

See below for a modified version of what Jennifer and her team started.  I hope to continue to collaborate with them to finish this project, but would love to open this project up to all interested in contributing.

This project is going to take time and lots of it.  The videos will be linked over time, but I plan to start by creating ideas for each area at each grade level.

The process will look something like this:

  1. Create a Google Slideshow for each grade level that includes each standard.
  2. The standard will have the unwrapped expectations for the Missouri Learning Standard as well as a table with Concrete, Representational, Abstract columns.
  3. Under each column will be ideas for scaffolding students using manipulatives and diagrams.  For now I will be simply putting the unwrapped standard under the abstract column, but I will eventually replace this with stems for the grade-levels that do not have those provided in our Item Specifications for Missouri.

Please feel free to offer suggestions and revisions if you use these documents by emailing me or replying below so that we can make these valuable resources for teachers.

Here is an example of the first grade resource I have started for our teachers:

The great thing about Google Slides is that there is an option to export individual slides as .png images or a .pdf so I can link them directly to our other standards documents.

Please realize that this is a work in progress and will be refined over time.  I wanted to get this out there though in case anyone was looking to create a similar resource.

Feel free to contact me via email at mscastillosmath@gmail.com or on twitter to @MsCastillosMath.

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Doodling the C’s – Getting Started

This is so awesome!

Experiments in Learning by Doing

How do we practice Information Age skills?  Which of the C’s do we actively engage with, share in the-struggle-to-learn with others, and intentionally insert into daily practice?

Creativity and innovationCommunicationCritical thinking and problem solvingCollaboration, …

At Trinity, a small cohort of faculty meet at either 7:15 a.m. or 3:30 p.m. to learn more about sketch noting.  We call it #doodling #TedTalkTuesday (or #TEDTalkThursday).  We meet, watch a TED talk, and doodle.  We share our work and offer each other feedback.

But, how do we differentiate for faculty unavailable at these times? In other words, how can we leverage technology to learn and share together?

Challenged by members of the Trinity Faculty to exercise creativity and critical problem solving,  I have started developing the following prototype to attempt to offer a solution to this identified need.

Screen Shot 2014-11-02 at 2.58.06 PM

At the end of these eight 75-minute sessions, participants should…

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How Do I Change Math Class Tomorrow?

mscastillosmath

As Dan Meyer put it, “math class needs a makeover.”  It breaks my heart to hear students say things like, “I’m not good at math” or “I hate math!”

In my opinion, one of the first things that needs to change is the focus on correct answers and the need for speed.  This leads students to believe that the only way to be good at math is to be right (quickly).  The other practice that needs to be eliminated is the language we use as teachers such as “that’s right” or “you’re so smart.”  

You’re So Smart

You’re so smart is one of the most damaging things you can say to a child.  What is smart?  What do we value and what are we showing them that we value?  Being smart needs to mean learning as a means in itself.  Or rather, the journey of learning.  This is not…

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Rethinking Geometry Instruction: How to Think Holistically About Standards

Each year I continue to see teachers frustrated when they get their middle of year testing results back for their class because Geometry is often one of the lowest strands.  Over the years I have puzzled about why this is but have realized that most textbooks (ours included) waits to teach Geometry until the last couple of units.  So many teachers who rely heavily on the textbook do not teach Geometry until right before the state test.  Often this means students get told about geometry instead of getting the opportunity to experience it through hands on application and discussion.

So why do we wait until the last minute to teach geometry?

I would like to propose that we teach geometry throughout the year by incorporating geometric ideas into instruction of other standards.  I would argue that too often we teach skills in isolation and don’t connect the mathematical ideas.

In Kindergarten, instead of giving students random objects, have them count shapes such as triangles, squares, rectangles, circles and as they develop the idea of cardinality have them confirm their count by labeling the objects.  For instance, if they count nine triangles, they would say “there are 9 triangles.”

Provide multiple types of triangles so that they can develop the idea that triangles come in all sizes and orientations and that counting does not have to be an object of a particular size or congruency.

Put out squares and ask students to get 18 squares or lay out circles and say how many circles are there?
Give students a geoboard and ask them to make a shape with 6 sides.  This is a great conversation starter and gives another context for counting while offering the opportunity to discuss shape names and properties.
When asking students to compare objects, ask them to compare a group of circles and a group of squares and ask which one has more?
Or provide a low floor, high ceiling task like this:
I have some shapes.  The total number of sides of all of my shapes is 21.  What shapes do I have?
Build arrays of multiple shapes and then ask how many sides are in the array? How many corners (or vertices).  Or build an array out of 3-D objects and ask how many faces are in the array.
Use the geometric subitizing cards by Graham Fletcher to have number talks about attributes.

Have students engage in conversation to solidify understanding and practice vocabulary with WODB sets like this one:
Look at your grade level standards for geometry.  Are there opportunities to teach geometric concepts through operations and algebraic reasoning?  Data?  Fractions?
Leave comments here or on twitter to share your ideas!

3 Act: On a Roll

Students really struggle with perimeter and when I saw this Lego tape in the store, I knew I wanted to make a task with it.  The only problem was in what context?  We went to Legoland a few weekends ago and I saw that they had put Lego tape around their bulletin boards to store mini-figs and decorate.  I was sold!  This would be a perfect addition to my son’s building area.  I have added some extension activities at the end:)

This task best supports CCSS 3.MD.D.8.

Act 1:

What do you notice?  What do you wonder?

How much tape will it take to go around the entire board?

Give an estimate that is too low and one that is too high.

Act 2:

Act 3:

Extension:

This extension can incorporate conversion and address CCSS: 5.MD.A.1.

How much of Lego tape was left?  How much of each color?

2 inches green, 13 inches blue, 13 inches grey, 36 inches red.

3 Act: The Pay Out

I have to admit, Coin Dozer is my kryptonite!  I took my boys to Incredible Pizza the other night and happened upon this Wizard of Oz style coin dozer.  I couldn’t help but use the tokens we received as a math lesson because after all, we were doing the math as they fell!

This task may need a little set up by asking students about going to the arcade and telling them that this game gives you tokens and cards instead of tickets.  Then you can trade them in for tickets.  Each item is worth a different amount of tickets.

This task could support 5.OA.A.2 for order of operations or 5.NBT.B.5 or even 6.EE.B.6.

Act 1

    img_0719.jpg

What do you notice?  What do you wonder?

How many tickets is this worth?  Give an estimate that is too high and one that is too low.

Act 2

There were already 355 tickets on the card.

img_0737.jpg  img_0730-1

Act 3

So a total of 1035 tickets + 355 tickets = 1,390 tickets!

 

3 Act: The Arcade (Part 2)

I couldn’t help but use the awesome footage I got at the Arcade for a follow up task.  Use this after students have completed the first Arcade task. This task could be used for CCSS 4.OA.A.3 or 5.OA.A.1 (MLS 4.RA.A.2 or 5.RA.B.3).

So you figured out how many tickets they got…what did they buy?  The only setup you need is that there were two boys and they split the tickets equally among them.

Act 1:

Act 1 (1).png

Act 2:

If you want to dial this activity in a bit, you could add that they bought six items between them.  You could further narrow the focus by saying they both bought the same items…but what’s the fun in that???

Act 3:

There are three different reveals here.  Choose the first to perform another calculation and answer the additional questions.  The second and third image show the total cost of the items they purchased.

act 3 (1).png

act 3.1 (1)act 3.1b

Did they buy what you thought they would?

How much did they spend?  Did they have any tickets left?  How many?

Were you correct?  Was your answer possible?  Are there other combinations they could have bought?

(You couldn’t see the ball on the left so I had to add one in using Google Drawings – that’s the reason it looks out of place!)