Throughout this year I have been thinking a lot about perspective. It’s why I wrote a post about Spatial Reasoning here, and shared templates for skyscraper puzzles here and why I shared my presentation at this year’s OAME that included perspective here. It’s also why I shared videos like this or this or this.
So why is perspective so important? Mathematically, perspective taking involves us being able to mentally rotate objects in our mind, it includes us composing and decomposing shapes and figures into other shapes and figures… Really, perspective taking is one small piece of what Spatial Reasoning (Read this monograph to learn more) is all about:
Another look at perspective:
Have you ever been in a situation before where somebody is discussing with you their students – the ones you will be getting next year? Was it enlightening or awkward, helpful or fear-inducing? In my buildings, this is the time of year when lots of decisions start to happen for next year. Administrators moving to new schools and new ones coming, new teachers being hired, experienced teachers being given new assignments within the school or in a new school, and of course, students about to be promoted to the next grade. And with change comes uncertainty and anxiety.
In all of this uncertainty we often have opportunities to discuss with others some of our personal thoughts and feelings about our year, those who we work with, those who we teach…… So I thought I would challenge you to think about what your typical conversations look like.
A few things to think about when discussing others in your building:
Do you tend to see the good in others and describe things in a positive light or aim to help others see the potential issues and risks?
Do you tend to talk in generalities or specifics?
How will the other teacher/administrator perceive the messages you have given?
What information would you like to hear from others? How specific and detailed of a description would you like?
Most importantly, where is your line? You know, the line where it’s too much information and is either starting to cause you more stress or where you realize that the information is too negative.
A challenge for you:
What conversations have you had lately to describe to others about any of these changes? I think it can be powerful to hear both positive and negative experiences so we can think about how to navigate through changes.
If you are changing roles, or schools, or even getting new students, what conversations have you had? What might you have done differently?
For the last few months, a team of kindergarten teachers and myself have been working together to deepen our understanding of early years mathematics, spatial reasoning, and the importance of guided play as a vehicle to engage our students to think mathematically. Below is a copy of our slideshow presentation we shared at OAME 2017, and some of the documents we have created over the past few months.
A quick synopsis of our work first:
While our research led us toward Doug Clement’s work about trajectories, and research about spatial reasoning and early mathematics, much of the tasks we actually did with students directly came from the book shown above (Taking Shape) which I can’t recommend enough – if you can, get yourself a copy!
We discussed the quote above to help us realize what actually underpins mathematics success. More details about how the quote ends here.
We shared research showing just how important early mathematics is, and specifically what the kinds of instruction could / should look like to accomplish this learning. Duncan et. al., is a widely quoted piece of research that has led many to realize that early math learning needs to be a focus in schools – even more so than early reading!
We played a few games that helped us stretch our spatial reasoning abilities. The image above was part of our “See It, Build It, Check It” activity (found in Taking Shape). Everyone saw the image for a minute, then was asked to build it once the image was removed. What we noticed is just how difficult spatial tasks are for us!
After we had the opportunity to play for a bit, we dug back into the research about spatial reasoning and the jobs typically chosen based on (high school) spatial ability. Hopefully you noticed something interesting on the graph above!
So, we know how important spatial reasoning is, but the 3 pieces above (taken from Paying Attention to Spatial Reasoning document) might help us realize how important a focus on spatial reasoning is for both our students and us.
In our time together, we learned a lot about the importance of observing our students as they were engaged in the learning. From the initial choices they made, to how they overcame obstacles, to understanding the mental actions that were happening… Observing students in the moment is far more powerful than collecting correct answers!
See the link at the bottom of the page for our connections to Doug Clements’ work.
We also discussed the specific connections between the mathematics behaviours and the learning that happened beyond. In our Kindergarten program document, our students’ expectations fall under 4 frames (see above) so we linked the learning we saw to the program document in a way that helps us see the depth and breadth of the kindergarten program (see document linked below).
We then ended our presentation with a synopsis of what we learned throughout our work together. While the slideshow might be helpful here (I’d love for someone to comment on those slides at the end), the conversations that Sue, Kristi and Kristen had with others have shown me just how valuable it is to spend time learning together. I couldn’t be prouder to be able to work with such reflective and dedicated teachers!
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:
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:
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?
I spent some time with a wonderful kindergarten teacher a while back. The two of us were chatting when a little boy, probably 4 or 5 years old walked up to us with a inquisitive look and asked if it was home time yet. Since it was only about 10 in the morning I smiled at the question. But the teacher in a very calm and candid manner spread his arms out wider than shoulder width and said, “There’s still THIS much time left.“
The child, very satisfied with the response, happily headed over to another activity and continued to play. I, however, was quite stunned at what just transpired. The little boy wasn’t given an answer… or was he?
What was it about the gesture given that helped this child understand that there was still lots of time left in the day? I found myself quite interested in the above conversation for some time until I started thinking about the types of experiences our students have with time as young children.
Clocks like those below don’t have any meaning for most 4 or 5 year-olds. There are far too many numbers or symbols… different ways to read… for many to understand here!
But they have likely had many experiences looking at things like these:
Let’s think about these items for a minute. Children by age 4 or 5 have likely been exposed to a cell phone, a tablet or some type of electronic device that has a battery meter. Without any understanding of percent or without even being able to recognize large numbers, children instinctively know when the device is charged or when it is nearly “dead”. When they watch movies, they can determine if we are currently near the beginning or end based on the progression meter (when they hit pause of fast-forward)… or when downloading, they can tell by how much space is left before they can access the app or game… Even when waiting in a line-up, children can tell if it will be a short time or not without having to count the people in front of them.
Each of the pictures above show our students’ natural ability to determine information based on the relative space provided. This is a great example of how we use Spatial Reasoning to make sense of things.
There is a lot of research about Spatial Reasoning and it’s relationship with mathematics. Much of this idea was first brought to my attention through the article: Paying Attention to Spatial Reasoning. While you probably haven’t read this, I think it is really important to read through the whole article. Really… it will explain a lot better than I can.
If our students come into our math classes with an intuitive understanding of the spatial relationships behind concepts, shouldn’t we provide experiences that allows our students to access this form of reasoning BEFORE we start with formal rules, procedures, notations…?
For instance, how might a simple graph like this bring about a conversation of subtraction?
How might a number line be used so our students can visually see how subtraction works? And how subtraction is related to addition?
It is blurry, but the 132-94 was moved to show 138-100. How might this visual help us with future problems?
Or, if we know Mr. Stadel’s height is 193cm. How might that help us think about estimating Mrs. Stadel’s height?
How did you think about this? What spatial cues did you use? Can you see how others might see this as a subtraction or addition problem?
Spatial reasoning isn’t just for subtraction though. If we wanted to know how many chocolates were here what might students do that are just starting to learn about multiplication?
How many different ways can you think of solving this question?
An example that really illustrates the need to focus on Spatial Reasoning:
At the end of the year, the grade 7 students throughout our school board were given a year-end test. One of the questions our students did poorest on was this problem:
While the vast majority of students got this problem wrong because of issues with converting units, several students instead took a path that allowed them to make sense of the problem. Below is a diagram we saw a few students draw that represented what was going on in the problem quite nicely. What did they do here? What do you notice?
I notice dimensions that make sense. I notice a piece of cloth being cut up, much like what we would actually do if this were a real-life problem. I notice the student thinking about the space required to cut the bandanas.
So I am left wondering why did only a few students attempt to make a drawing? Why did so many make errors with converting between units? Why did so many students make errors that were not even close to an acceptable answer? What experiences did they have… and didn’t they have that led them to attempt this problem the way they did?
While many teachers might tell students that there are 100 x 100 cm in a square meter, and expect students to understand and remember… or show conversion charts… or offer a page of conversion questions… I think we might be missing a big piece of the learning process. We haven’t tapped into our students’ spatial reasoning at all!
After seeing so many questions like this attempted by students in such procedural ways, I think next year we will start to think more about the reasoning our students already come into our classes with. And continue to think about how we can support the development of our students Spatial Reasoning is in helping them make sense of things!
The National Research Council describes the current situation as a “major blind spot”in education and maintains that, without explicit attention to spatial thinking, the concepts, tools and processes that underpin it “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7).