I’ve been spending a lot of time lately observing students who struggle with mathematics, talking with teachers about their students who struggle, and thinking about how to help. There are several students in my schools who experience difficulties beyond what we might typically do to help. And part of my role is trying to think about how to help these students. It seems that at the heart of this problem, we need to figure out where our students are in their understanding, then think about the experiences they need next.
However, first of all I want to point out just how difficult it is for us to even know where to begin! If I give a fractions quiz, I might assume that my students’ issues are with fractions and reteach fractions… If I give them a timed multiplication test, I might assume that their issues are with their recall of facts and think continued practice is what they need… If I give them word problems to solve, I might assume that their issue is with reading the problem, or translating a skill, or with the skill itself……. Whatever assessment I give, if I’m looking for gaps, I’ll find them!
So where do we start? What are the foundations on which the concepts and skills you are doing in your class rest on? This is an honest question I have.
Take a look at the following quote. How would you fill in the blank here?
Really, take a minute to think about this. Write down your thoughts. In your opinion, what are the foundational skills of math? Why do you believe this? This is something I’ve really been reflecting on and need to continue doing so.
I’ve asked a few groups of teachers to fill in the blank here in an effort to help us consider our own beliefs about what is important. To be honest, many of the responses have been very thoughtful and showcased many of the important concepts and skills we learn in mathematics. However, most were surprised to read the full paragraph (taken from an explanation of dyscalculia).
Here is the complete quote:
Is this what you would have thought? For many of us, probably not. In fact, as I read this, I was very curious what they meant by visual perception and visual memory. What does this look like? When does this begin? How do we help if these are missing pieces later?
Think about it for a minute. How might you see these as the building blocks for later math learning? What specifically do these look like? Here are two excerpts from Taking Shape that might help:
Visual perception and visual memory are used when we are:
- Perceptual Subitizing (seeing a small amount of objects and knowing how many without need of counting)
- Conceptual subitizing (the ability to organize or reorder objects in your mind. To take perceptual subitizing to make sense of visuals we don’t know)
- Comparing objects’ sizes, distances, quantities…
- Composing & decomposing shape (both 2D or 3D)
- Recognizing, building, copying symmetry designs (line or rotational)
- Recognizing & performing rotations & reflections.
- Constructing & recognizing objects from different perspectives
- Orienting ourselves, giving & following directions from various perspectives.
- Visualizing 3D figures given 2D nets
While much of this can be quite complicated (if you want, you can take a quick test here), for some of our students the visual perception and visual memory is not yet developed and need time and experiences to help them mature. Hopefully we can see how important these as for future learning. If we want our students to be able to compose and decompose numbers effectively, we need lots of opportunities to compose and decompose shapes first! If we want our students to understand quantity, we need to make sure those quantities make sense visually and conceptually. If we want students to be flexible thinkers, we need to start with spatial topics that allow for flexibility to be experienced visually. If we want students to be successful we can’t ignore just how important developing their spatial reasoning is!
In our schools we have been taking some of the research on Spatial Reasoning and specifically from the wonderful resource Taking Shape and putting it into action. I will be happy to share our findings and action research soon. For now, take a look at some of the work we have done to help our students develop their visual perception and visual memory:
Relating nets to 3D figures:
Constructing unique pentominoes:
And the work in various grades continues to help support all of our students!
So I leave you with a few questions:
- What do you do with students who are really struggling with their mathematics? Have you considered dyscalculia and the research behind it?
- How might you incorporate spatial reasoning tasks / problems for all students more regularly?
- Where in your curriculum / standards are students expected to be able to make sense of things visually? (There might be much more here than we see at first glance)
- How does this work relate to our use of manipulatives, visual models and other representations?
- What do we do when we notice students who have visual perception and visual memory issues beyond what is typical?
- How can Doug Clements’ trajectories help us here?
- If we spend more time early with students developing their visual memory and visual perception, will fewer students struggle later???