Number Talks Meet Fractions

We had our fourth meeting of our “Making Number Talks Matter” book club last night.  Our focus for this meeting was on Fractions.  We spoke a little bit about decimals and percents, but we spent most of our time looking at how we can support our students in developing conceptual understandings of fractions.

This blog post is meant to serve as a recap for those who were there, a fill-in for those who couldn’t make it, and a record for anyone else who is interested!

We started off our meeting with our usual conversation about how things are going in classrooms with number talks.  Some participants shared their ideas for how they are keeping track of student thinking.  One teacher has tested out incorporating our student self-assessment and another has been using her document camera instead of the whiteboard to record student thinking – she then has a record of strategies being used with student names attached to help inform her Number Talks planning.  It is so inspiring to hear about how excited students are about participating in Number Talks.  I hope you will all continue to carve out time in your class for them!

We then looked at a “Fractions on the Number Line” activity as a group.  We used a double number line for this activity.  We placed the benchmarks of 0, 1/2 and 1 on the top number line and then each participant had a number to place on the line.  First, we had teachers talk in groups to order the numbers at their table and then one-by-one the tables came up to put their numbers on the bottom number line, re-arranging as necessary to make it make sense.

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The benchmarks (once again a photo re-enactment)
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The Double Number Line (imagine the cards are hanging on the wall on two pieces of yarn)

From this activity, we moved on to talking about the BIG IDEAS for fractional thinking in the new Grade 4-7 curriculum.  We used this quote from the book as our jumping-off point:

…for success in high school, there is no avoiding fractions.

We talked about: what do our students struggle with in terms of fractional thinking? And: what do we want our students to understand about fractions?

Some thoughts that came up:

  • We want our students to understand that the size of the piece changes depending on the size of the whole.  It is possible to have a quarter that is bigger than a half if the two wholes are different.
  • We want our students to understand that fractional pieces have to be the same size but not necessarily the same shape.
  • We want our students to understand that fractions are numbers that exist on the number line.
  • We want to help our students make connections between their existing understanding of number and their understanding of fractions.

We then looked at the BIG IDEAS from the curriculum from Grades 3 – 9: where are our students coming from in primary, and where do we want them to go in secondary?  Now that all the fraction operations have been moved to Grade 8, we have the opportunity to solidify a conceptual understanding of fractions in elementary school so that students are prepared for fraction operations and linking of fractions to algebra in Grades 8 and 9.

From here, the teachers did another activity that connects a visual representation of a fraction to its place on the number line.  (Activity adapted from this blog –  printable download of the activity cards are available).  Teachers coloured in a section of the given square and determined what fractional part they coloured.  They then placed their fraction on the number line again.

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As a wrap-up, we briefly reviewed the other three types of Number Talks for fractions that are described in the book: More or less (give a fraction and have students defend whether it is more or less than a half); Closer to 0, Closer to 1/2 or Closer to 1 (give a fraction and have students decide which benchmark it is closer to), and Which is Greater (give two fractions and have students defend which one is greater).

Last but not least, we had a mini “make and take” – teachers took home yarn for a double number line and a package with coloured fraction, decimal and percent cards.  I will update this post with a link to the printable package as soon as I add some improper fractions and mixed numbers to it!  I will also have these packages at our final meeting for people who were unable to join us this week.

Here are a few useful links that we talked about in our meeting:

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‘fessing up – Number Talks gone awry

One of my professional goals this fall as a Math Liaison in my district is to spread the message of Number Talks far and wide in intermediate classrooms in my district.  Between the readings I have done (Making Number Talks Matter and Number Talks, blog posts, articles), the Pro-D workshops I have led (Number Talks book club, Number Talks and the Curricular Competencies, Intro to Number Talks), and the various Gr. 4-7 classrooms I have visited, I am starting to feel like a bit of an “expert” on the subject… and yet, Number Talks are still complicated and challenging.  I think that’s one of the things I like most about the Number Talks routine – it is simple enough to be accessible, but challenging enough to keep both students and teachers engaged.  So, today I thought I would share a blog post about a “failed” number talk that I have been pondering and what I learned from the experience.

The setup:

I was visiting a Grade 4 class – This was my third visit to this classroom this year, and I have done Number Talks with them on each visit.  We have done some dot talks and some addition number talks, and on this visit, we were going to be working on subtraction. Students in this particular class (and at this school in general) are very capable but tend to just do the traditional algorithm in their heads – this happens much more frequently here than at other schools that I visit (my theory is that it is related to high levels of parental involvement).

The Problems:

Rather than just one number talk, I brought a number string with me… My plan:

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I really thought that these (especially the first one) were going to be easy, but when we got going on the first question, I ended up with 4 different answers.  I have led a lot of number talks with multiple answers, and 99% of the time (100% of the time before this particular visit), the errors work themselves out nicely during the discussion of the problem.

So, for this particular problem, I got the following 4 answers:

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Take a minute and see if you can figure out where the errors come from.

I wish I had thought to take a photo of the board after we were finished (need to get better at documenting things for the blog!).  First, I had a few students who defended the correct answer of 6 with some good strategies – adding on, counting back, making 44 into 45 etc.  If I had taken a picture, you could note my nice use of number lines and whatever else I did to record student thinking…

But what I really want to discuss is the student who wanted to defend the answer of 14.

She said something like:

“I did the 5 minus the 4 to get 1 and the 4 minus the 0 to get 4.  The answer is 14.”

This is not really earth-shattering… probably the most common mistake made in subtraction by Grade 4 students.  Here is the interesting thing: this student had just listened to 3 or 4 of her peers defend (very clearly) their answer of 6 with very good strategies (and I’m quite sure she was listening).  This is the first time I have had a student sincerely defend a mistake after multiple other students have made their case for the right answer – she had no recognition that her answer might not be correct.  In hindsight, I can think of quite a few good ways to respond, but in the moment, I was caught off-guard.  I wish I could tell you that I referred her back to the original problem to see if her answer made sense… or that I asked a classmate to respond to her thinking… or that I asked her to explain why what she did made sense to her.  But… I just told her that you couldn’t flip the numbers around and subtract from bottom to top.  Sigh. Fail.

Even in the moment, I knew that my response was woefully inadequate… I could tell from the look on her face that I had done nothing to convince her.  I think she believed me that her answer was wrong, but she had gained no understanding to move her thinking forward.  Other students had not learned anything useful from our exchange.  And, possibly (hopefully not!), the experience has discouraged her from taking another risk to share her thinking.

Moving Forward

So, what have I taken away from this experience?  I went back to Making Number Talks Matter and reminded myself of some guiding principles…

  • Through our questions we seek to understand students’ thinking: It is not my role to be the judge of student answers, or even to correct mistakes.  It is my role to try to understand why students are thinking the way they are.  I need to focus my responses on questioning with the genuine desire to understand student thinking.
  • One of our most important goals is to help students develop social and mathematical agencyThis exchange would have been a great opportunity to encourage students to respond to each other.  By “explaining” the right answer, I removed the opportunity for students to be the thinkers and brought the responsibility for “correct mathematical knowledge” back to the teacher.  My new #1 goal for number talks: stop talking so much and LISTEN.
  • Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics: In hindsight, it is really interesting how uncomfortable I felt dealing with this mistake… as teachers, it is very hard to let go of our instincts to help our students through their struggles.  I am totally on board with the IDEA of stepping back and letting my students wrestle with mistakes, but in the moment, it is still a challenge to stop the “traditional teacher” who hides out in the back of my brain.

I am thankful that teaching is such an interesting job – regardless of how much experience we have, there is always more to learn.

I am going to better prepare for my number talks… I have been lazy about anticipating student responses. For our last workshop, we prepared a “cheat sheet” of phrases and sentence stems, and I have printed a copy to refer to.

I am giving myself some grace… making mistakes is the best way to learn, even for teachers!

And I found this lovely quote from Ruth Parker to help me remember why I am so excited about doing Number Talks in the first place:

I’ve come to believe that my job is not to teach my students to see what I see.  My job is to teach them to see.

So… who else wants to ‘fess up?  What surprises have you been faced with during a Number Talk?