The Importance of Contexts and Visuals

My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing.  Here’s a quick story about one of those times, along with some considerations:

A while back Anne-Marie and I were preparing lunch for our three children.  It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup.  If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water.

Lipton soup

Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup.

nesting cups.JPG

Here is how the conversation went:

Anne-Marie:  How many of these (3/4 cups) do I need to make 4 cups?

Me:  I don’t know.  How many do you think?  (attempting to give her time to think)

Anne-Marie:  Well… I know two would make a cup and a half… so… 4 of these would make 3 cups…

Me: OK…

Anne-Marie:  So, 5 would make 3 and 3/4 cups.

Me:  Mmhmm….

Anne-Marie:  So, I’d need a quarter cup more?

Me:  So, how much of that should you fill?  (pointing to the 3/4 cup in her hand)

Anne-Marie:  A quarter of it?  No, wait… I want a quarter of a cup, not a quarter of this…

Me:  Ok…

Anne-Marie:  Should I fill it 1/3 of the way?

Me:  Why do you think 1/3?

Anne-Marie:  Because this is 3/4s, and I only need 1 of the quarters.

The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making.  My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm.  To many students, however, division of fractions is learned first through a set of procedures.

I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book Knowing and Teaching Mathematics.  In her book she shares what happened when she asked American and Chinese teachers these 2 problems:

liping ma1

Here were the results:

Liping ma2

Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:

  • How does the training of American and Chinese teachers differ?
  • Did both groups of teachers rely on the learning they had received as students, or learning they had received as teachers?
  • What does it mean to “Understand” division of fractions?  Computing correctly?  Beging able to visually represent what is gonig on when fractions are divided? Being able to know when we are being asked to divide?  Being able to create our own division of fraction problems?
  • What experiences do we need as teachers to understand this concept?  What experiences should we be providing our students?

Visual Representations

In order to understand division of fractions, I believe we need to understand what is actually going on.  To do this, visuals are a necessity!  A few examples of visual representations could include:

A number line:

three quarters NL

A volume model:


An area model:

Starting with a Context

Starting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday).  Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced.  I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn:

Design your own problem that others could use to launch a discussion of division of fractions.  Share your problem!  

A few things to reflect on:

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)


Strategies vs Models

Earlier this week Pam Harris wrote a thought-provoking article called “Strategies Versus Models: why this is important”. If you haven’t already read it, read it first, then come back to hear some additional thoughts…..

Many teachers around the world have started blogs about teaching, often to fulfill one or both of the following goals:

  • To share ideas/lessons with others that will inspire continued sharing of ideas/lessons; or
  • To share their reflections about how students learn and therefore what kinds of experiences we should be providing our students.

The first of these goals serves us well immediately (planning for tomorrow’s lesson or an idea to save for later) while the second goal helps us grow as reflective and knowledgeable educators (ideas that transcend lessons).  Pam’s post (which I really hope you’ve read by now) is obviously aiming for goal number two here.

Models vs Strategies

In her article, Pam has accurately described a common issue in math education – conflating models (visual representations) with strategies (methods used to figure out an answer).  Below I’ve included a caption of Cathy Fosnot’s landscape of multiplication/division.  The rectangles represent landmark strategies that students use (starting from the bottom you will find early strategies, to the top where you will find more sophisticated strategies).  Whereas the triangles represent models or representations that are used (notice models correspond to strategies nearest to them).

fosnot landscape - strategies vs models

In her post, Pam discusses 3 problems that arise when we do not fully understand the role models and stragegies:

  1. Students (and teachers) think that all strategies are equal. 
  2. Students are left thinking that there are an unlimited, vast number of “strategies” to solve a problem.
  3. Students get correct answers and are told to “do it a different way”.

I’d like to discuss how this all fits together…

Liping Ma discussed in her book Knowing and Teaching Elementary Mathematics four pieces that relate to a teacher having a Profound Understanding of Fundamental Mathematics (PUFM).  One of these features she called “Multiple Perspectives“, basically stating that PUFM teachers stress the idea that multiple solutions are possible, yet also stress the advantages and disadvantages of using certain methods in certain situations (hopefully you see the relationship between perspectives and strategies). She claimed that a PUFM teacher’s aim is to use multiple perspectives to help their students gain a flexible understanding of the content.

Many teachers have started down the path of understanding the importance of multiple perspectives.  For example, they provide problems that are open enough so students can answer them in different ways.  However, it is difficult for many teachers to both accept all strategies as valid, while also helping students see that some strategies are more mathematically sophisticated.

models and strategies2

As teachers, we need to continue to learn about how to use our students’ thinking so they can learn WITH and FROM each other.  However, this requires that we continue to better understand developmental trajectories (like Fosnot’s landscape shared above) which will help us avoid the issues Pam had discussed in her original post.

If we want to get better at helping our students know which strategies are more appropriate, then we need to learn more about developmental trajectories.

If we want teachers to know when it is appropriate to say, “can you do it a different way?” and when it is counter-productive, then we need to learn more about developmental trajectories.

If we want to know how to lead an effective lesson close, then we need to learn more about developmental trajectories.

If we want to know which visual representations we should be using in our lessons, then we need to learn more about developmental trajectories.

If we want to think deeper about which contexts are mathematically important, then we need to learn more about developmental trajectories.

If we want to continue to improve as mathematics teachers, then we need to learn more about developmental trajectories!

While I agree that it is essential that we get better at distinguishing between strategies and models, I think the best way to do this is to be immersed into the works of those who can help us learn more about how mathematics develops over time.  May I suggest taking a look at one of the following documents to help us discuss development:

Might I also suggest reading more on similar topics:

Zukei Puzzles

A little more than a year ago now, Sarah Carter shared a set of Japanese puzzles called Zukei Puzzles (see her original post here or access her puzzles here).  After having students try out the original package of 42 puzzles, and being really engaged in conversations about terms, definitions and properties of each of these shapes, I wanted to try to find more.  Having students ask, “what’s a trapezoid again?” (moving beyond the understanding of the traditional red pattern block to a more robust understanding of a trapezoid) or debate about whether a rectangle is a parallelogram and whether a parallelogram is a rectangle is a great way to experience Geometry.  However, after an exhaustive search on the internet resulting in no new puzzles, I decided to create my own samples.



Take a look at the following 3 links for your own copies of Zukie puzzles:

Copy of Sarah’s puzzles

Extension puzzles #1

Extension puzzles #2

Advanced Zukei Puzzles #3

I’d be happy to create more of these, but first I’d like to know what definitions might need more exploring with your students.  Any ideas would be greatly appreciated!

How to complete a Zukei puzzle:

Each puzzle is made up of several dots.  Some of these dots will be used as verticies of the shape named above the puzzle.  For example, the image below shows a trapezoid made of 4 of the dots.  The remaining dots are inconsequential to the puzzle, essentially they are used as distractors.


If you enjoyed these puzzles, I recommend taking a look at Skyscraper puzzles for you to try as well.