Mathematical Mindsets Book Study Questions

Mathematical Mindsets Book Study

Well, September has hit, and the overwhelming craziness has started!  I hope to post more regularly here (do I say that after every holiday?).  I have 3 full days under my belt with my new class and already some thoughts to share…

But for today – I have had several requests to share the discussion questions that we used for our mathematical mindsets book study last year.  So – here they are!  I loved this book so much and hope they are useful to others who are doing a similar book study – it was such a powerful learning experience for all of our teachers who participated.

Mathematical Mindsets Book Study Chapter 1Mathematical Mindsets Book Study Chapter 2Mathematical Mindsets Book Study Chapter 3Mathematical Mindsets Book Study Chapter 4Mathematical Mindsets Book Study Chapter 5Mathematical Mindsets Book Study Chapter 6Mathematical Mindsets Book Study Chapter 7Mathematical Mindsets Book Study Chapter 8Mathematical Mindsets Book Study Chapter 9

Along with rich discussions, our teachers “played” with open-ended math tasks as part of our book club sessions.  We experimented with pentominoes, tangrams, Desmos, Kenken puzzles, Steve Wyborney’s Tiled Area Questions, and Zukei puzzles.  Teachers that participated loved the opportunity to DO some math as well as talk about it – if you are running a book club, I would strongly encourage you to set aside some time to showcase some rich tasks with your teachers.

I have just registered to go and see Jo Boaler in Vancouver in February (thanks to the BCAMT for bringing her to BC!) and am so excited to hear her in person!

Read more about last year’s book study here.

 

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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.