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.

 

Mathematical Mindsets Book Club

For the second year in a row, I had intended to participate in the MTBoS blog challenge, and for the second year in a row, I managed to post exactly zero times in January.  So… it’s February 1st and I am going to re-commit to reflecting on and cataloging some (hopefully) interesting math-related happenings.  January has been a busy month for workshops and classroom visits, so I have lots on my mind to write about… now to set aside the time to get it on the screen which seems to be the more complicated thing for me.

Yesterday was the start of our new Math book club (we hosted Making Number Talks Matter in the fall).  This time around, we are reading Mathematical Mindsets by Jo Boaler and our district’s entire Math Enhancement Team (our full-time Numeracy teacher and 5 of us part-time Math Liaisons) is participating, so we opened the book club up to teachers from Grades 3-10.  We have 21 teachers signed up and we will meet monthly until the end of April.  We have adjusted things slightly based on the feedback we received from our Number Talks follow-up survey and to better accommodate our secondary teachers.

This book is quite different in focus from the Number Talks book, so I have been really pondering what our sessions should look like.  We have only set aside an hour for the meetings, and I really want to make the most of that time, and also to make sure that teachers find the meetings valuable and worthwhile.  The Number Talks book was very practical and hands-on, and we had lots to talk about as teachers began implementing Number Talks in their classrooms.  This book is a little more theoretical and part of our goal is to shift the mindsets of our teachers as well as impact the way that they are approaching their math instruction.

I decided that we should start with a math task (given that this is a math book club and we are all math teachers).  I am a frequent reader of Dan Meyer’s blog and have been very interested in his posts on recreational math and becoming a better math teacher through DOING more math.  I think we (especially as elementary generalist teachers) don’t DO enough math just for fun, and it is hard to get kids excited about doing math if we don’t ourselves believe that doing math is fun.  (After all, how much buy-in would we get if we tried to get kids excited about reading and then admitted that we NEVER read ourselves… there are so many interesting double-standards around literacy and numeracy instruction).

Anyways – we started with these Zukei geometry puzzles that I have been itching to try since I saw them on Twitter in the fall.  I wanted to make sure that I chose a task that would be accessible and relevant for my elementary teachers, but also interesting for our secondary teachers and I think these puzzles did the trick nicely.  We had a nice hush over the room and some interested chatter – I was definitely hooked – they are challenging in a nice way and sparked some interesting table conversation about precise definitions for geometric shapes.  I realize that it has been a long time since I have thought about the exact definition for a rhombus…

After taking about 10 minutes for people to get settled and work on the puzzles, we dove into discussions.  This book has so many interesting and thought-provoking ideas, it was really hard to narrow down the discussion questions.  I was aiming for 5 and I ended up with 8.  Some of these questions are adapted from the Mathematical Mindsets #mathbookchat that was happening on Twitter in the fall, and others are just things that really resonated with me as I read through the Chapters. mathematical-mindsets-questionsI had intended to leave some time at the end for a whole-group debrief, but the discussions were going well at all the tables and I am a terrible timekeeper (definitely one of my biggest weaknesses as a teacher and a facilitator), so we ran out of time.  My table had a really great mix of expertise – Grade 4, 5, 7 and 8 teachers – and our discussion was so engaging and thoughtful.  It was a really fun experience.

At the end of the meeting, teachers left with two resources that we had printed off from the YouCubed website: Classroom Norms and the Building a Mathematical Mindset Community card.  For “homework” we asked participants to read Chapters 4 and 5 and to try some kind of activity in their classroom on growth mindset or mistakes or brain science and math learning or…

A few of my take-aways from this week:

  • I am really loving the book study structure for offering professional learning – I love that I get to be a learner/facilitator right alongside our participating teachers and I love that I can offer resources to teachers who participate in these groups.
  • I am continually grateful that I have the opportunity to work with individuals and groups of teachers.  Teachers are such creative, thoughtful and passionate people, and it is exciting to be with a group of people who are excited about improving their practice.
  • My office is a mess – I really need to figure out a system for organizing and re-using leftover handouts and activities from workshops…

I am really looking forward to diving into task creation across the grades with our teachers in late February!  Are there any other math/instructional coaches out there who lead book studies with their teachers?

 

 

More Clothesline Math

I’m having a bit of a hard time shutting work down for the Christmas break, so I thought I would see if I could make up these other primary clothesline cards that I have been pondering.

They are a bit addictive… as I make more cards, I keep thinking about more cards I could make…

This set has benchmarks of 0, 5, 10, 15, 20, and then uses ten frames and dominoes.  I added the numbers from 11-20 with the ten frames and then mixed them up to make addition cards with the ten frames.  I did all the make tens and the doubles and some random other combinations.  The dominoes have all the doubles and all the make tens and then some near doubles and some other random combinations…

Does anyone want to try them out?  I have no printer at home, so can’t test with my own kiddos and we are on holidays until January now (woohoo!).  If you have a chance to try them, I’d love to hear how it goes!  Feedback and suggestions welcome 🙂

See my original post about primary clothesline cards here.

Download the original primary clothesline cards here.

Download the new card set here.

 

 

 

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.

img_1757
The benchmarks (once again a photo re-enactment)
img_1758
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:

‘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:

img_1745

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:

img_1748
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?