An Abstract Understanding of Concrete Things

Recently Kyle Pearce blogged about concreteness fading.  It got me thinking about a session at the Math Recovery Council annual conference with Dr. Robert Wright on the topic of settings and distancing the setting.  I am constantly reflecting on the best ways to move my students forward in their math thinking and it seems to me that this is an area in which I need to be more purposeful.  I decided to read a little more, ask a few more questions, and this blog is my reflection on all that I am thinking and learning about these topics, focusing more specifically on the role of manipulatives in my classroom.

The concept of concreteness fading as described both in Kyle Pearce's blog and in this article by the Learning Scientists is about the progression of moving students from concrete to representational to abstract.  The idea of this makes sense to me.  I feel that too often I introduce a manipulative, like integer tiles, for example, only to take it away too soon.  I allow the students to use them, but if it isn't specifically in the directions many students struggle through without them thinking they should be able to work abstractly.  This is where many kids will latch on to rules and memorization in order to work abstractly with something they don't fully understand.

What stood out to me in Kyle's blog was this quote, "We must understand that concrete manipulatives are still more abstract than using the actual items in the quantity being measured."  Just the other day I was working with a student and using cubes to represent something.  It made perfect sense to me and I realized as we were working that this student did not connect the cube with the item from the problem.  I need to keep in mind that it is not just a 3 part continuum from concrete to abstract, but there is a continuum within the concrete category.  There is so much to be purposeful and thoughtful about when selecting manipulatives and concrete examples.  I don't think I do enough to modify and adapt the manipulative choice if a student is not able to connect with it.  Plus Kyle's blog really helped me think about choosing manipulatives with the end result in mind.  Where do I want the students and what manipulative/tools get them there?  Perhaps it will be different if I am focused on adding mixed numbers with a distributive strategy versus a fraction great than one strategy.  I love the vertical number line for proportional relationships that Kyle shows to then connect to the table that shows the numbers vertically.  It's a simple change that makes a big difference! (It was both an aha moment and a face slap when I saw it. How did I not think of this?)

I also started to think about this process in reverse.  By 7th grade, many of my students have memorized rules or tricks and can solve problems abstractly.  The issue is that they lack the depth of understanding.  They can work abstractly but don't understand the number relationships enough to show what is happening concretely.  They are unable to mathematically justify what they are doing and it affects their ability to transfer their understanding to new problems.  This video of Ruth Beatty (that Mark Chubb shared with me) speaks to this.

In the constructivism definition that is given in the video, the word concrete is being applied to the students understanding.  It is not just about the concrete example or having something concrete to manipulate.  This video changed my thinking about the goal I am trying to reach with students.   I need to reverse the concrete to abstract model and work with what my students already know to help them understand more concretely.  Sometimes as a middle school teacher it can feel like I have to start from the beginning with these concepts, but I need to keep in mind that the starting point for my students is their abstract understanding.  

So how does this model of concreteness fading compare with what I know about distancing settings?  This paragraph is taken from the book Teaching Number in the Classroom.  Distancing the setting is part of progressive mathematization (which is the progression of learning and thinking in terms of mathematical sophistication).  

I am not sure if there really is a difference between concreteness fading and distancing the setting, however, the definition laid out here seems to include more.  When planning a setting, it isn't just thinking about the concrete representation, but it's selecting what verbal language and number range are appropriate, deciding how to move students through the concrete section of the continuum while interweaving the representational and abstract.  I know I have talked about concreteness fading as a continuum, but it is also cyclical so students are not just moving from concrete to abstract and done.  After giving it much thought I feel that distancing the setting is something that is done within concreteness fading.  There is one student who comes to mind with all of this.  We have worked to build strategies and his multiplicative reasoning and yet unless directed his default is to count on his fingers.  It is by purposefully distancing the setting that I can help him progress to naturally select more sophisticated strategies and thinking. 

In thinking about the examples Kyle laid out, I feel that those are the first steps in lesson planning.  I need to map out the models and progression through them.  Once that is done I need to be thinking about how to distance the setting.  Screening and flashing are great strategies.  Showing students the concrete and then covering it/part of it or showing it only for a brief moment.  This will challenge students to start creating that mental picture.  They don't have to create a mental picture themselves yet.  It is given to them, then covered, and they continue the problem with that picture in their head.  

Since I feel that distancing the setting is something that fits inside of concreteness fading, I wanted to go to Kyle's middle school example in his blog and consider ways to distance the setting as I move students from concrete to representational.  

The original picture prompt is really a screened image.  I can see part of the problem but 4 of the boxes are not pictured.  

My thoughts about distancing the setting for this example are all over the place and I have typed and retyped many times.  So many options and thoughts running through my head.  I am going to try bullet points:

• For some students, this number range will be too high.  They will need enough cubes for all 7 boxes.   
• Some students may need boxes to put the cubes in.  Connecting them together may not be concrete enough.
• If a student has boxes and cubes inside of them (or cheerios to look like donuts), one way to distance the setting is to have cubes in the box and then close the box.  This could also be recreated by using cardstock that covers the cubes, acting as a box. (which I would totally print the donut logo onto)
• Closed boxes could be added, one at a time for students who need to think additively and the through discussion and activity we could get to some multiplicative reasoning. 
• Using boxes that show the outer donut (one row and one column) would be a way to distance the setting to help students use additive or multiplicative reasoning over counting strategies.
• As we move to the number line in Kyle's visual example, I am loving this number line picture from Sara VanDerWerf's NCTM presentation.  I am now thinking about how to add donut visuals to the number line and table to help students connect the concrete ideas to the representations.

There is a lot to think about with just this one example.  It is a little overwhelming to think about doing this every day in my classroom.  I think, however, that once I get into the habit of thinking about these things (and dedicating time to think and reflect on it)  the ideas will flow and I will see how ideas for one concept transfers to other concepts.  It will be a progression and continual goal in my teaching.  The main questions floating through my head right now are this:

• Do I really offer my students enough strategies to understand concretely?  
• Is my teaching limited by my limited knowledge of "tools"?  
• How do I build my concrete understanding so I can help my students?