Spatial Structuring: a Public Service Announcement

I'm knee deep in writing an article about an area lesson I taught to rising 4th graders in summer school a couple weeks ago and, as usual, something totally unexpected and profound has emerged as I take a closer look at student work. I wanted to do a quick post today as a shout-out to the importance of spatial structuring for students who are being asked to think about multiplication in terms of arrays and area.

In the lesson, students were given two tasks related to area. The first task was to find different rectangles with an area of 12 square units, where students were given 12 square units to build their rectangles and then asked to draw pictorial representations on a square grid. The second task was to determine the area of a table given only enough square units for two side lengths and to draw a representation of their thinking on a blank sheet of paper (for an amazing read on the power of a blank sheet of paper, check out Tracy Zager's blog). There was a startling correlation between how a student used the square grid to show their rectangles and the way they drew their representation of area on the blank sheet of paper. What do you notice about Student A's work? Student B's?

Student A 
Student B

Here's what I noticed:

  • Student A shaded each of her 12 square units individually in the first task
  • Student A drew each square individually in the second task to show length and width of the table
  • Student B outlined the dimensions of the rectangle she built in the first task and shaded the area within those dimensions
  • Student B used straight lines to structure rows and columns to show total area of the table

These observations have given me a few things to think about. First, Student A's strategy of one-at-a-time shading and structuring of square units suggest to me that she has not made the connection between arrays and area yet. I love this graphic for this idea:

Progression from arrays to area models (Source: Eureka Math)

Content note: when students first start organizing equal groups into arrays, they may count each item or square one-by-one without attending to the rows and columns. With appropriate instruction and experience, they learn to apply the structure of the array to add repeatedly and eventually to multiply rows and columns to find a total number of objects in the array. In 3rd grade, students are introduced to the idea of factors as dimensions of area and products as total area of rectangles and composite figures. Understanding the structure of arrays is foundational for understanding the area model.

I interpret Student A's strategies as evidence that she is thinking about number of squares vs. squares as the unit that makes up a larger figure (squares make up columns/rows, columns/rows make up a rectangle). I observed her using trial and error to build different rectangles with her square units. This almost-4th-grader needs some more experience with tiling and partitioning rectangles into rows and columns before we can expect her to use that structure to represent her work.

Student B seems to have a better spatial structuring foundation. She and her partner approached the first task by thinking about factors of 12 and then building their rectangle from their square units (in fact, they wanted to draw their representation on the grid paper before building it). Understanding that they needed lengths 6 and 2, tracing the outline of the rectangle with these measurements, and then shading the resulting area of 12 suggests these students have had more spatial structuring experience than Student A. In addition, Student B was able to use the row and column structure to show how she calculated the area for the surface of the table in the second task even though she didn't have enough square units to cover the whole thing. Clearly these two students have had very different instructional experiences. 

So why the PSA? According to the K-6 Geometry Progressions, the concept of spatial structuring is foundational for transformational geometry, and development is clearly based on experience. The progressions (especially the 2nd grade section, starting on pg. 10) give some powerful commentary on this idea, and suggest foundational activities like building a floor or wall from blocks, playing battleship, copying and creating designs on grid paper, and designing tessellations to build spatial structuring in our students. I would love to hear some ideas or experiences you've had related to these ideas! 


Comments

  1. Ooh, I love a good noticing about student work! It's so fascinating to me how the more I look at student work, the more I learn about math, and the more I learn about math, the more I see in student work!

    For example, I *might* have noticed (eventually) that one student colored in each of her 12 squares while the other shaded the whole rectangle. But I wouldn't have known that could be connected to their thinking about the relationship of arrays and area. But now that I see it, I will be looking to notice more ways in which students show what they have and haven't experienced connecting arrays to area.

    One thing I'm wondering as I read about your students' work on this task is "who was this task perfect for?" because I don't know the answer to this question and I've been thinking about it a lot since reading Andrew Gael's post about measurement: https://thelearningkaleidoscope.wordpress.com/2015/12/03/sharing-reflecting-with-a-little-help-from-our-friends/. At first I was like, "were his kids ready to notice and wonder about this image?" and Christopher Danielson reminded me, "if they immediately saw that the blocks shouldn't be spaced out, and notice what you want them to notice, then did they even need to see that image at all?" Whammo! Now I'm thinking a lot about what it means to be ready for a task... Is sometimes being ready meaning ready to get it really wrong in order to learn from it?

    Was this task perfect for Student A because she had a chance to grapple with making rectangles out of 12 squares, and may have experienced the relationship of 12-ness, rectangular-ness, and array-ness? Was it beyond what she was ready to think about because she couldn't "see" the rest of the table that she outlined two sides of with squares? What about Student B? Was it perfect for her because she was ready to generate factor pairs of 12 and could see the relationship between areas and arrays? Was it not enough of a challenge for her because she already "sees" area and array as fundamentally the same and so there wasn't much to grapple with? Was the task perfect for both of them because Part 1 challenged Student A and Part 2 challenged Student B?

    I'm also wondering: What could each student learn from encountering the thinking of the other student? Might thinking about Student A's work help Student B connect back to the idea of each tile being 1 unit of area? Might Student A have more success with a tricky task like "How many 1" square tiles would it take to tile just the edge of a 6" by 9" table?" because she wouldn't get tripped up by adding 6 + 9 + 6 + 9, she would actually model it? Might thinking about Student B's work help Student A make the connection between rows and columns of objects and filling up the whole table's area?

    Thanks for sharing this!!!
    Max

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    Replies
    1. Wow, Max! Thanks for your thoughts. I love what Andrew Gael did with that Notice/Wonder. Such a great example of deleting the textbook - Dan Meyer would be so proud.

      I think one of the marks of a good task is when the answer to the question "who was this task perfect for?" is "everyone in my class." Without knowing these students (this was a summer school class and I was just a visiting teacher), I felt that there were ample entry points, and I love your wonderings about how the task "fit" each of the students. Next time I try this, I want to use some 5 Practices magic with this template from my colleague, Jamie Garner (https://mavenofmath.files.wordpress.com/2016/05/5-practices-lesson-planning-updated.pdf), so I can make sure students get the chance to encounter the thinking of others, like you said.

      Thanks, again! I value your input!

      Christine

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    3. I was just reading your response, Max, and was going to say that it was a task perfect for both students *if* the 5 Practices approach was used to debrief the different strategies, but then Christine said it for me! I also agree with Christine that there are multiple entry points to ensure a greater level of success for all.

      What I wonder is will the use of a good debrief be enough for Student A to see the structure of Student B's work, or will the conversation need to be supplemented with more hands-on experience for Student A to build that concrete level of understanding?

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  2. I work with secondary students, and I have seen more than a few middle and even high school students who lack the sense of spatial structuring you describe. Your insights help me understand how to support students' experience in elementary with that initial learning, as well as how to engage older students in experiences to build those missing connections. thank you!

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