Fractions: Thinking with Curricular Competencies

For our district’s Pro-D day, we (our 2 Intermediate Math Liaisons) decided to focus on fractions for Grades 4-7.  We have both tutored students in Grades 8-10 and know that fractions are a huge problem for many students – and are a big reason that students struggle in Math in high school.  We wanted to convince intermediate teachers to spend more time with fraction concepts and to share some ideas of how to address fractions conceptually.

In BC’s redesigned curriculum, a lot of the heavy lifting when it comes to fractional understanding is done in Grades 4-7.  They are formally introduced in Grade 3, and by Grade 8, students need to be able to work with fractions (operations with fractions).  As intermediate teachers, it is really our responsibility to help our students develop a deep understanding of fractions in all their complexity.

For today, I just wanted to share some images that we developed to walk teachers through our curriculum’s progression of fractions and focus in on some of the BIG IDEAS that students should be developing in these grades.

A HUGE thank-you to Graham Fletcher whose fraction progression video inspired us to think of the progression of fractions in our own curriculum and whose video inspired our images as well.  (His video is way more spectacular than our images, so go watch it now if you haven’t already).

early-fraction-ideas

At this stage (Grades 3-4 in BC), we want our students to understand:

  • Fractions are built of equal-sized pieces
  • We can partition shapes in different ways
  • Two fractions are the same if the pieces are the same size – even if they are a different shape!

fraction-models

At this stage (Gr. 3-4 and beyond in BC), we want our students to have practice with developing understanding of all three of these models and we want them to have the opportunity to use various manipulatives in exploring them.

comparing-and-ordering

At this stage (Grade 4 and beyond in BC), we want to help students use various number sense strategies to compare and order fractions.  These four strategies are: common numerators, common denominators, benchmarking and missing piece strategies.  We also want our students to recognize that the size of the whole must stay the same in order for us to compare.  For example, ½ can be smaller than ¼ if we are comparing ½ of a apple to ¼ of a watermelon.  

equivalent-fractions

At this stage (Gr. 5 and beyond in BC), students working with different representations and manipulatives will notice that different fractions “line up” and are actually the same size, but they have different “names.”  We want to encourage our students to see and make note of patterns in the numerator and denominator.

fractions-decimals-percents

At this stage (Gr. 6-7 and beyond in the BC curriculum), we look explicitly at improper fractions and mixed numbers as well as decimals and percentages.  Students can use manipulatives to explore what fractions look like when they have pieces that make up more than one whole.  Students will extend their understanding of fractions along the number line.  We want to help our students make connections between fractions, decimals and percentages and to think about how these concepts are related. 


 

As teachers, we can be so immersed in our own grade that we sometimes lose sight of the bigger picture – where our students are coming from and where they need to be several years down the road.  Thinking about the progression of concepts can help us to avoid relying on “tricks” and focus on helping our students develop the conceptual foundation that they need to be successful in the long term.

These pictures/ideas were a small portion of our recent workshop – hopefully I will be able to circle back around to this topic of fractions again in the near future and share some activities that we recommend for using BC’s curricular competencies to help develop fractional understanding.

 

Investigations with Number Talks

Our last *sniff, sniff* book club meeting was held this week.  We are so appreciative of the teachers who signed up to participate with us.  We are so grateful for the rich and thoughtful conversations and collaborative trouble-shooting that went along with our study of Making Number Talks Matter.  We really believe that professional learning is so much better with colleagues and that setting aside time for professional learning is great for our students, but also helps us (as teachers) stay excited about our work on a day-to-day basis.

This week, we looked at Chapter 9 – Investigations.

We started our meeting with our usual check-in about how Number Talks have been going in the classroom.  It sounds like most people have established a good routine with Number Talks.  Some folks are taking a short break and shifting focus but planning to come back to Number Talks in the New Year.   We did some trouble-shooting discussion about how to deal with students who offer silly answers or make up strange answers that don’t relate to the question posed.  We talked about using phrases that help the student connect their answer with the question (can you explain to me where in the question you got the numbers that you are using in your strategy?).  We also talked about moving on from a student who is having trouble explaining his/her thinking clearly with a statement like: I’m having a hard time understanding your explanation and I would like us both to think about it some more – can I check in with you about your strategy after the number talk is over?

Next, we talked about our BIG IDEAS for the day:

investigations-big-ideas

And outlined how to do an investigation:

investigation-procedure

We then delved into exploring the multiplication strategy of halving and doubling using the general procedure for an investigation.  We started with the question 8 x 26 to try to elicit the strategy of doubling and halving.  Once we looked at all the suggested strategies, we focused in on doubling and halving and talked about the big question of “Does it always work?”  The group split up into partnerships to explore this question – we provided graph paper, colour tiles, rulers, paper and scissors and then circulated to try to see how the investigation went.

It was interesting to note that it is really difficult to be a skeptic in Math – the strategy might make sense to us, but actually thinking about what it takes to PROVE that it works requires much more depth to our thinking.  Many groups got started by discussing WHEN it would be good to use this strategy (ie. what circumstances/numbers make it an efficient strategy).  Some groups explored odd vs. even numbers, some explored big and small numbers, some tried to delve into fractions to see if it worked there.  Some groups worked with the colour tiles to make arrays and others used the graph paper to show the strategy visually.

Then, we wrapped up with a discussion – different groups shared their approaches and it was interesting to note how varied the ideas were.  We looked briefly at the questions offered in the book to guide small group work for this investigation:

  • Will it only work with even numbers?
  • What would happen if, instead of halving, you took a third of one factor?
  • Can you represent this strategy visually/geometrically?
  • What generalizations can you make?
  • Would this work for division?

We had hoped to have time to also have teachers choose another investigation: either the same difference strategy for subtraction or the halving-halving strategy for division, but we ran out of time.  I think participating in an investigation was really valuable.  I enjoyed seeing how much mathematical thinking (curricular competencies) is involved in this type of activity.  Of note – mathematical investigations take time, and it is worth setting aside some time to do activities like this in class.

Once again, a HUGE thank-you to our teachers who participated!  We will be running some more after school PD for intermediate math teachers in the New Year – likely a Mathematical Mindsets book study at some point – so keep an eye on your email in January for information on how to join in.

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.

fraction-clothesline

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:

Number Talks Book Club

This fall, we (the two intermediate math liaisons in my district) have been planning a book study for the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker. Our 18 participants teach Grades 4-7 and come from 16 different schools across our district (our district has 31 elementary schools).  We will be meeting every second Tuesday until the beginning of December to work our way through the book.  We will be talking about number talks, strategies for mental math and doing some planning and practicing of the Number Talks routine. Participation in this book club is totally voluntary and we know how difficult it is as teachers to squeeze in after-school commitments and still have everything ready in the classroom – our book club meetings run from 3:30 – 4:30 and we have committed to getting everyone out of there on time.

Yesterday was our first meeting – we had a few people who had to miss the first meeting because of parent-teacher interviews and staff meetings, so I will do my best to recap our discussion and learning!

We had a few goals for our first meeting –

  • Understand why we do number talks
  • Identify the procedures and setup necessary to get Number Talks started
  • Discuss the underlying values that the routine of Number Talks is based on
  • Create a plan for doing a dot talk in the classroom before the next meeting

We first showed the teachers a clip from the DVD that comes with Sherry Parrish’s Number Talks book.  We asked the teachers to ignore the mathematical strategies (for now) and just to focus on the routine – what is the teacher doing? What are the students doing?  What logistics do you notice?  (View this YouTube video from 44:50 to 51:30 – this isn’t the exact same clip we watched, but close enough).

Afterwards, we asked each group of teachers to fill in a chart with their ideas from watching the video and from doing the pre-reading.  Here are the finished ideas:

img_1702img_1703img_1704 

We then had teachers do a “gallery walk” to look at all the ideas.

Next, we had intended to pull up the new BC Curriculum website to show all the places that Number Talks fit in both the content elaborations and in the Curricular Competencies from Grades 4-7, but the website was down (*deep breath*), so we skipped this portion. We are planning to do a full workshop on our district’s next ProD day on Number Talks and the Curricular Competencies, so we will have a chance to dive into this further on that day (if you are from SD57 and reading this, you can register on PD Reg for this session!).

From here, we provided groups with some discussion questions from the chapters they read and gave them a few minutes to talk/discuss and plan:

  1. What strikes you as most useful/valuable/exciting about the Number Talks routine?
  2. What parts of the routine are of concern? What do you think will be most difficult for you as the teacher/facilitator?
  3. What norms and structures do you need to have in place to be successful with Number Talks?
  4. What Guiding Principles (from Chapter 3) resonate with you?
  5. Which ones make you feel uncomfortable/concerned?

We provided groups with a blank template to record some guiding principles/norms for Number Talks that they thought they might like to use in their classrooms.  There was so much good discussion during this portion of our meeting – I feel so lucky that I get to facilitate and work with groups of teachers on things like this – what a thoughtful group of people! During these conversations, teachers discussed the importance of “wait time” and how difficult that can be, they talked about the difficulties of facilitating if we ourselves are unsure about some of the strategies (hopefully we can clear some of these feelings up in future meetings), they talked about the importance of students listening to one another and how this routine can connect across many math content areas.

Finally, I did a demonstration “dot talk” so that teachers could see what a dot talk would look like in action.  I used the same dot pattern from the Chapter explanation and showed teachers briefly how I set up a number talk to get started.

For our next meeting in two weeks, we have asked teachers to try a dot talk (or several) in their classroom, and read Prelude to the Operations, Chapter 4 and Chapter 6 – we are going to dive into addition and subtraction number talks next.

Some resources:

These are the Guiding Principles that I use when I start Number Talks in a classroom – they are adapted from Making Number Talks Matter.

Here is a planning page that Dorianna and I made for a workshop we did last year on Number Talks (also adapted from Making Number Talks Matter).

I borrowed and adapted several ideas for this session from this blog – I am so thankful for teachers who share their ideas and work so graciously online!

This is a great summary of Number Talks if anyone is looking for more information.

Reading, reading and more reading

I had good intentions to blog this summer, but rest and family time ended up taking priority.  It’s October already!?!  So I am trying (again) to be committed to this blogging thing.

One thing I did manage to do this summer was some reading.  Professional reading is a bit of a wormhole.  One book leads to the next, which leads to the next and I always seem to have about 5 waiting for me to get to them.  This is what I managed to read this summer:

Essentialism: the Disciplined Pursuit of Less by Greg McKeown
This is the best productivity book I have ever encountered.  It is minimalism, but for your time instead of for your stuff.  I loved everything about this book, but a few things really resonated with me.

  1. It is impossible to do it all, so set some selective criteria that help to outline what you really want to accomplish and then STICK TO IT!
  2. Create a buffer by adding 50% to your estimate of how long it will take to accomplish things.  I am a chronic under-estimator of how much time things will take and often have to pull things together at the last minute.  I’m sure I would experience more EASE in my life if I consciously added in a buffer.
  3. Set aside professional time to think and read – this is really hard to do as a teacher – there are so many demands on our time.  But some of the most creative insights and solutions to problems come when I give my mind time and space to think.  I want to be intentional about adding this kind of time to my workweek this year.

The Innovator’s Mindset by George Couros
This book is related so closely to the shift our province is currently making in our curriculum – it is so much more important to teach our students HOW to think and learn rather than worrying about WHAT they are learning.  There is currently a MOOC going on as a book study with this book that I was trying to keep up with, but… I am about 2 weeks behind (see the comment about the buffer above).  Luckily, the Live chats are being archived, so I can follow at my own pace. (#IMMOOC if you are interested).

What’s Math Got to Do With It by Jo Boaler
I applied to read this book and write a review for the BC Association of Math Teachers book club series.  Jo Boaler’s books are so inspiring and her YouCubed website is full of great resources.  You can read my full review of the book here when it gets posted.

Classroom Chef by John Stevens and Matt Vaudry
I enjoyed the creativity of the lesson ideas and tips around crafting lessons.  I think many teachers feel anxious about straying too far from “predictable” in Math class and this book reminds us that there are rewards for doing so.  I actually (for the first time ever) thought it might be fun to teach a Grade 8 or 9 class.  Luckily, that feeling has passed quickly 🙂

Teach Like a Pirate by Dave Burgess
I am a couple of years behind the bandwagon on this one, but I have had it signed out of the district library a few times and have never made time to get through it.  I am glad I finally read it – there are so many great ideas for making lessons interesting and motivating for students.  This book was a good reminder of why I became a teacher in the first place.  A very motivating read!

And… that brings me to the present…

img_1692-1
I need to learn how to take non-blurry pictures with my phone… if I wasn’t so lazy, I would re-take this one.

Recently, I have been very intrigued by the idea of thinking routines and instructional routines that support deep thinking.  Making Thinking Visible caught my eye on Amazon and after ordering it, I am seeing references to it everywhere – a good sign.

I started Mathematical Mindsets in the spring and got about halfway through it – I really wanted to read it slowly and carefully because there is so much to think about.  I am looking forward to digging back in this fall.  This is the book I am considering for my next teacher book club – depending on how successful our Number Talks book club is this fall (more on this next week!).

So, that’s my professional reading life over the last while… does anyone have any good suggestions for what to read next?

 

 

 

 

 

 

Hindsight

Math Poster Books and Co (1)

I stopped in our locally owned bookstore the other day and came across this sign.  Of course, I instantly noticed the “anything but math” comment and had to look more closely.  There are just so many interesting things about these responses…

For the past 6 weeks, I have been working as a curriculum coach in my district (a new position designed to help support teachers in transitioning to our province’s new curriculum), and we have been talking a lot about what we really want for our kids when they leave school.  We want them to be good people, and we want them to have the skills they need to be successful.  The content part is less important.  I am excited that our curriculum is starting to make this shift as well, and making it easier for teachers and students to focus on what is really important.

I think it’s interesting that pretty much everything (with the exception of “fresh avacado”) falls under interpersonal skills or real-world skills.  And of course, it’s also interesting that so many responses are math/money-related.  I am happy that financial literacy has been included in the curriculum starting right at Kindergarten.

So, what do you wish you had learned in school?