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:
Symmetry games:
Composing and decomposing shapes:
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???
I’d love to hear your thoughts. Leave a comment here or on Twitter (@MarkChubb3).
Thinking Mathematically 
Thank you for the post. This year I have a few VERY struggling students in my classroom. While working with rekenreks, one of my students would keep counting one by one each bead. I order to show 16, he would count 10 beads one by one, then continue 6 more on the other rod, then count them all together again. Next time he would count 10 one by one again, as if the number of beads might have changed in the past minute. I teach grade 3. I ran a quick diagnostic test that I start with dot cards subitizing, and he kept trying to count and was not seeing groups. When I put 10 bingo chips on the desk and then took 4, he was unable to tell how many I have in my hand. I came to the conclusion that we need to go all the way back to subitizing to build the foundation.
I’ve been also following Christina Tondevold @BuildMathMinds who has done a lot of work around early number sense development.
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There’s a story about the eastern fool Nasrudin:
Two men were quarrelling outside Nasrudin’s window at dead of night. Nasrudin got up, wrapped his only blanket around himself, and ran out to try to stop the noise.
When he tried to reason with the drunks, one snatched his blanket and both ran away.
‘What were they arguing about?’ asked his wife when he went in.
‘It must have been the blanket. When they got that, the fight broke up.
What’s needed next might not show up in the test.
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More tangential thoughts…
I read somewhere recently – where was it? – that blind students learn arithmetic well. (That sounds too simplistic now that I write it, but it was something along those lines.) Which makes me think we should maybe talk about spatial more than visual. Obviously there’s a big overlap and much of the way the sighted approach the spatial is through the visual, but there’s also the way the body moves through space, and the way we handle objects.
I think also of the idea of a memory palace – a way of remembering things by positioning them in a familiar space. It’s said to have begun with the ancient Greek Simonides, who remembered all those present at a feast by where they were sitting around the table.
https://en.wikipedia.org/wiki/Art_of_memory#Origins_and_history
Spatial thinking and memory are really important for animals to locate food and threats in a complex environment. It’s easy to imagine how they could have become coopted into mathematical learning, and why they could be one of our most robust resources for thinking mathematically.
I think to of how detective work is a kind of descendent of hunting: looking for clues, traces as we move through the physical environment.
Notes for a blogpost maybe…
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My son was totally blind from birth, which is pretty rare. He was always really good at telling the time from digital time. Even when he was little I could say, It’s 11:37 and he would say it is 23 minutes to 12. Or similar. Now he is an adult we are having some interesting talks about number. He is also autistic, and a calendar counter, so goodness knows what is happening in his brain.
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… Quantitative visualization …
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Continuing thoughts after getting home…
I noticed a lot of students who struggle to move into operations and seem to be having difficulty with grasping a concept of place value, and after interviews with them they often have difficulty with subitizing, recognizing and extending visual patterns, etc. I did not have a large enough sample in my career to discard confirmation bias, but I think there is a correlation.
As Simon mentioned, spatial thinking is connected to memory. Is it also possible that attention deficit or problems with short term memory function would affect the spatial ability and in turn slow down the development of number sense?
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Just like what has already been mentioned above, there seems to be a connection between spatial visualization and math,but I think it’s worth thinking about the role of language in early learning. I’ve noticed that students often have an innate sense of what’s going on with concepts but there is a real disconnect in being able to communicate with words. When teachers started telling me that their first grade students didn’t understand phrases like UNDER or NEXT TO, I wondered how this would impact math learning. I’m still wondering,
When I think back on my own education i wish that I would have had more exposure to math as relationship-thinking. Just now I have begun to grasp, that some numbers form squares (it was like an epiphany when I realized that square numbers could actually be represented by squares) while some quantities can only be arranged in rectangles, and others, the primes, just make towers. Seeing numbers as quantities, and quantities as orderly groups, is something I wish I had been introduced to in first or second grade. Actually seeing quantities in this way, like seeing 8 as four rows of two, and how that relates to what 6 looks like, seem like this sort of embedded understanding would go a long way, especially when multiplication then factoring enter the curriculum.
I still have trouble visualize reflection and rotation. Can anyone do this easily?
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I think this kind of spatial awareness comes from a much broader curriculum than just mathematics. Young children need loads of experience playing and making things with large and small objects, lots of cutting out and sticking. Playing games where you move things around. Making a mess and mistakes.
I’m sure the list is much bigger. Coding, like Scratch and controlling a real robot or turtle. Visual and performance arts obviously, as they are so spacial. Malke Rosenfeld has shown me how dance is part of this.
I had a teaching assistant once, from a developing country where he hadn’t had much play in his education. I asked him to cut me a certain size rectangle from a sheet of card. He came back with the right rectangle. But what amazed me was when I looked at the sheet, I saw he hadn’t used the edges of the card; he’d cut the rectangle entirely from inside the sheet. It struck me that we usually assume this kind of knowledge but it’s dependent on lots of experience.
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And the kind of experience you’re giving, Paula, creating books with students and all that other work with paper is such an integral part of that.
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I had a similar experience working with teenage refugees who had spent most of their lives in camps on the Thai border. They could not tell me how many faces a cube has, and had real problem with 2 D representations of 3D shapes. I took along my son’s blocks for them to play with and it made a big difference.
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Really like this post Mark. Just had a conversation with a Trustee in my board and would have loved to be able to use your hook at the beginning to get them thinking (and planning on borrowing the hook when I work with the collaborative inquiry groups I work with).
Just wondering about the connection between decomposing and composing shapes and decomposing and composing numbers. The ten frame visual or base 10 visual (or even array model visual) is so important when students are playing with numbers. Thoughts around making this connection explicit with teachers and students?
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It really is a great way to help us all think about our own assumptions. As for composing and decomposing numbers, Dr Cathy Bruce gave a great talk last year with regards to her own research. Her students had spent quite a lot of time composing and decomposing shapes in a lot of different ways. Other schools in the area had focused their attention on developing number skills in their students. Each of the schools did the same beginning and final assessments (which were entirely about number). Surprisingly, the students at the schools focusing solely on shapes did better than the students who focused on numbers on any question related to algebraic reasoning (flexibility with numbers).
So, the question is what are the experiences that will help our students spatialize the number system (rekenreks, 10 frames, number lines…), and how do these experiences help our students to think more flexibly with the numbers they are using? Still lots to learn about in this area for sure!
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Great stuff. I just lead a PD today w/ middle school math teachers, stressing the importance of concrete and spatial mathematics experiences to help students build a foundation of visual/spatial memories they can draw upon as they incorporate new mathematical ideas. Interestingly, the special education teachers in the room exhibited noticeably more head-nodding agreement with these ideas. Personally, I have always looked for visual/spatial connections between mathematics concepts and remember being surprised as a new high school math teacher when my students didn’t know things like “square numbers make squares”. I have learned a LOT since then through my fascination with how mathematical understandings develop. Still so much to learn!!
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