How many do you see?(Part 1)

A few days ago I had the opportunity to work with a grade 2 teacher as her class was learning about Geometry.  The students started the class with a rich activity comparing and sorting a variety of standard and non-standard shapes, followed by a great discussion about several properties they had noticed.

Shortly after, students started working on following the page as independent work. Take a look:

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Take a minute to try to figure out what you think the answers might be.  Scroll up and pick one of the less obvious shapes and count how many you see.

This isn’t one of those Facebook “can you find all the hidden shapes” tasks, it’s meant to be a straightforward activity for grade 2 students. However, I’m not sure what the actual answers are here.  So, I need some help…  I’d love if you could:

  • Pick one shape (or more if you’re adventurous)
  • Think about what you believe the teacher’s edition would say
  • Count how many you see
  • Share the 3 points above as a comment here or on Twitter

I’m hoping in my next post that we can discuss more than just this worksheet and make some generalizations for any grade and any topic. 

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Building our Students’ Mathematical Intuition

I’ve been asked to share my OAME 2017 presentation on Mathematical Intuitions by a few of my participants.  Instead of just sharing the slides, I thought I would add a bit of the conversations we had, and the purposes behind a few of my slides.  Here is a brief explanation of the 75 minutes we shared together:


I started with an image of the OAME 2017 official graphic and asked everyone what mathematics they saw in the photo:

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I was impressed that many of us noticed various things from numbers, to sizes of fonts, to shapes and other geometric features, to measurement concepts to patterns…

I decided to start with an image so I could listen to everyone’s ideas (the group could have simply noticed the numbers visible on the page, or the triangles, but thankfully the group noticed a lot more!).


I then shared a few stories where students have entered into a problem where they have attempted to do a bunch of procedures or calculations without ever doing any thinking, either before or after, to make sure they are making sense of things.

intuition3intuition4You can read the full stories on these 2 slides here and here.

The bandana problem above is a really interesting one for me because it shows just how likely our previous learning can actually get in the way of students who are attempting to make sense of things.  Most students who learned about how to convert in previous years in a procedural way have difficulty realizing that 1 meter squared is actually 10,000 cm squared!

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In an attempt to explain the kinds of mental actions we actually want our students to use when learning and doing mathematics I showed an image shared by Tracy Zager (from her new book Becoming the Math Teacher You Wish You’d Had).  We discussed just how interrelated Logic and Intuition are.  Students who are using their intuition start by making sense of things.  They start by making choices or estimates, which are often based on their previous experiences, and use logic to continue to refine and think through what makes sense.  This process, while often not even realized by those who are confident with their mathematics, is one I believe we need to foster and bring to the forefront of our discussions.


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I then shared the puzzle above with the group and asked them to find the value of the question mark.  Most did exactly what I assumed they would do… but none did what the following student did:

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Most teachers aimed to find the value of each image (which isn’t as easy as it looks for many elementary teachers), but the student above didn’t.  They instead realized that all of the shapes if you add them up in any direction would equal 94.  This student had never been given a problem like this, so didn’t have any preconceived notions about how to solve it.  They instead, thought about what makes sense.


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So, how DO we help our students use their intuition?  Here are a few ideas I shared:

  1. Contemplate then Calculate routine (See David Wees for more about this here or here, or purchase Routines for Reasoning by Grace, Amy and Susan)

intuition11intuition12The two images above show visual representations (thank you Andrew Gael and Fawn Nguyen for your images) where I asked everyone to attempt to think before they did any calculations.  I used Andrew’s picture of the dominoes and asked “will the two sides balance… don’t do any calculations though”.  For Fawn’s Visual Pattern, I asked the group to explain what the 10th image would LOOK LIKE (before I wanted them to figure out how many of each shape would be there, and then find a rule for the nth term).


We shared a few estimation strategies:

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and a few “Notice and Wonder” ideas:

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However, while I love each of the strategies discussed here (Contemplate then Calculate, Estimation routines, Notice and Wonder) I’m not sure that doing a routine like these, then going about the actual learning of the day is going to be effective!

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Instead, we need to make sure that noticing things, estimating, thinking happen all the time.  These need to be a part of every new piece of learning, not just fun or neat warm-ups!


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Building our students’ intuition means that we need to provide opportunities for them them to think and make sense of things, and have plenty of opportunities for them to discuss their thinking!

If our goal is for students to think mathematically, and use their logic and intuition regularly, we need to operate by a few simple beliefs:

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I ended the presentation with a final thought:intuition22

Here is a copy of the presentation if you are interested:

Building your Students’ Mathematical Intuitions

I’d suggest you scroll down to slide 49 and play the quick video of one of my students doing a spatial reasoning puzzle.  It’s one of my favourites because it illustrates visually the thinking processes used when a student is using both their intuition and logic.


To me, there seems to be so much more I need to learn about how to help my students who seem to struggle in math class use their intuition.  Hopefully this conversation is just the beginning of us learning more about the topic!

A few questions I want to leave you with:

  • What routines do you have in place that help your students make sense of things, use their intuitions and develop mathematical reasoning?
  • Do your students use their intuition in other situations as well (or just during these routines)?
  • How can you start to build in opportunities for your students to use their intuition as a regular part of how your class is structured?
  • What does it look like when our students who are struggling attempt to use their intuition?  How can we help all of our students develop and use these process regularly?

Special thanks to Tracy Zager’s new book for the inspiration for the presentation.


As always, I would love to continue the conversation here or on Twitter

Aiming for Mastery?

The other day, a good friend of mine shared this picture with me asking me what my thoughts were:

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On first glance, my thoughts were mixed.  On one hand, I really like that students are receiving messages about mistakes being part of the process and that there is more than one opportunity for students to show that they understand something.

However, there is something about this chart that seems missing to me…  If you look at the title, it says “How to Learn Math” and the first step is “learn a new skill”.  Hmmm…… am I missing something here?  If the bulletin board is all about how we learn math, the first step can’t be “learn a new skill”.  For me, I’m curious about HOW the students learn their math?  While this is probably the most important piece (at least to me), it seems to be completely brushed aside here.

I want you to take a look at the chart below.  Which teaching approach do you think is implied with this bulletin board?  What do you notice in the “Goals” row?  What do you notice in the “Roles” row?  What do you notice in the “Process” row?

Teaching Approaches - New

If “Mastery” is the goal, and most of the time is spent on practicing and being quizzed on “skills” then I assume that the process of learning isn’t even considered here.  Students are likely passively listening and watching someone else show them how… and the process of learning will likely be drills and practice activities where students aim at getting everything right.


I’d like to offer another view…

Learning is an active process.  To learn math means to be actively involved in this process:

  • It requires us to think and reason…
  • To pose problems and make conjectures…
  • To use manipulatives and visuals to represent our thinking…
  • To communicate in a variety of ways to others our thinking and our questions…
  • To solve new problems using what we already know
  • To listen to others’ solutions and consider how their solutions are similar or different than our own…
  • To reflect on our learning and make connections between concepts…

It’s this process of learning that is often neglected, and often brushed aside.


 

While I do understand that there are many skills in math, seeing mathematics as a bunch of isolated skills that need to be mastered is probably neither a positive way to experience mathematics, nor will it help us get through all of the material our standards expects.  What is needed are deeper connections, not isolating each individual piece… more time spent on reasoning, not memorizing each skill… richer tasks and learning opportunities, not more quizzes…

“One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world”
TIPS4RM: Developing Mathematical Literacy, 2005

So, while I don’t think putting up a bulletin board (or not) is really going to do much, I really do hope we are spending more of our time thinking about HOW our students learn and WHAT our goals are for our students (see the chart above again).

I still wonder what it means to be good at math?  I wrote about my questions here, but I am still looking for ways to show others how important mathematical reasoning is for students to develop.  Skills without reasoning won’t get you very far.  Maybe more about this another time…