Category: Math
Creating Machines or Mathematicians? How Might We Use the Learner Profile as a Math Planning guide in the #PYP

Creating Machines or Mathematicians? How Might We Use the Learner Profile as a Math Planning guide in the #PYP

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How would you finish this sentence? Math is…..

  • practical, a part of every day life
  • happening all around me.
  • fascinating
  • a language
  • a mindset
  • an opportunity to build relationships
  • a form of creative expression

These are just a few of the ideas that pop into my head as I reflect on what Math means to me. But I wouldn’t say that I felt that way all my life. There was a time when I wouldn’t answered it as boring or hard. It really wasn’t until I studied Calculus that I realized that the journey to getting an answer was actually where “math” happened. The solution wasn’t as rewarding as the struggle. And puzzling over a challenge can be fun. 

But no one has to wait until high school or college to experience joy when doing math. I feel strongly that we have an obligation to use our PYP framework to intentionally develop mathematcians. Lately I have been thinking, reading and reflecting on how to support New-to-the-PYP teachers in shifting their practices. More than I care to admit, I’ve heard these fledgling PYP teachers retort, “It’s okay to ‘do the PYP’ for Unit of Inquiry time but for literacy and math? Nah?!”

I fight to keep a straight face when I hear them say things like that, while inside my heart goes

The PYP isn’t some jargon-filled, philosophical mumbo-jumbo, it’s designed on best practices. I know that the framework is a lot to take in for newbies, and, as a PYP coordinator, I must be patient. They are learners. And I am a learner too….and sometimes I am learning how to get teachers to not only “drink the kool-aide” before they can serve it to others. 

But I digress…..

Examining the research and approaches to rich math learning experiences, it’s obvious to me that our PYP standards and practices are grounded in not only what is joyful but what is powerful in math learning. Take a look at this chart–doesn’t it just scream our Approaches to Learning?! 

From the book, Everything You Need for Mathematics Coaching: Tools, Plans, and a Process That Works for Any Instructional Leader, Grades K–12

I love how this chart clearly articulates what competant mathematics do and how teachers can create the culture and opportunites for engaging in learning to solve problems through math. For teachers who struggle without a textbook or scripted curriculum, transforming their practice takes a lot of support and compassion. I think we all can acknowledge that most teachers who fear to stray off the pacing guide or curriculum resource is really just trying to do their best to ensure that students get the knowledge and skills they need to succeed. They are not trying to be defiant or stubborn, they are just don’t feel competant enough in their own decision-making abilities to support learners. To call them robots or machines because they can’t teach without a script would be cruel. Chances are they never had an authentic experience in which they embodied the spirt and curiousity of a true mathematician. Whether you are in leadership or just a peer, we have a duty to encourage them to take baby steps and take risks. 

The Learner Profile isnt just for the Students

Since we are in the throes of our IB Review Cycle, I’ve been reflecting on the new standards and practices. In particular, I’ve been thinking about the Learning standard practice:

Approaches to teaching 4: Teachers promote effective relationships and purposeful collaboration to create a positive and dynamic learning community. (0403-04)

Approaches to teaching 4.1: Teachers collaborate to ensure a holistic and coherent learning experience for students in accordance with programme documentation. (0403-04-0100)

Approaches to teaching 4.2: Students collaborate with teachers and peers to plan, demonstrate, and assess their own learning. (0403-04-0200)

The “PYP” isn’t something we do during our Unit of Inquiry time, it’s how we approach EVERY aspect of learning content to ensure a holistic and coherent learning experience for students. Moreover, it’s not something that the students do. EVERYONE DOES it!! Teachers promote….a positive and dynamic learning community. To think that teachers work outside of the jargon is to miss the point. We provide students with an everyday example of living the Learner Profile. But do we use it when we are reflecting on our planning?

As a PYP practicioner, do you ever ask yourself…..

How am I using the Approaches to Learning (Atls) to do math?

Who are the students becoming as I create opportunities to develop the Learner Profile in the context of solving problems using math?

Those questions need to be asked on a daily basis, as an individual teacher and within our teaching teams. When we live and breathe the values and philosophy, it’s easy to communicate it to other members in our learning community. 

Since I am working on supporting teachers who still learning how to shift their mindset and approach to designing learning experiences through the lens of the PYP, I’ve been thinking about how I might try to kill 2 birds with one stone:

#1: Elevate the implementation of our IB Standards and Practices

#2: Use the Learner Profile as a filter/checklist as we plan.

After a lesson, we might reflect, how did I create…

Risk-taking today?

Open-mindedness today?

Thinkers today?

Communicators today? 

etc…

Circling back to the “baby steps” a novice PYP practicioner might take could include embracing one or more of our Learner Profiles as we approach planning math, whether it is a stand-alone unit or transdisciplanary.  For example, perhaps they want to set a goal and become more of an INQUIRER in their math practices. I can help them then paint a picture and start to describe what kind of evidence they might see, hear or feel in the classroom environment to demonstrate that they are achieving this professional goal. Moreover, when I come into the classroom and see the teacher honoring the kids questions about math on a Wonder Wall, I need to acknowledge and provide accolades for their effort to shift their practice. Change isn’t easy and becoming a competant PYP teacher requires intention and a desire to be a creative educator. And, at the end of the day, I want them to experience the joy of engaging in math, not as a machine, but as a real mathematician–even if they have to live vicariously through the students.

I’m curious, can you think of other ways might we help support the development of PYP teachers through explicitly developing who we are as educators through naming and noticing the Learner Profile?  Please share in the comments below! 

When Numbers Divide and then Multiply

When Numbers Divide and then Multiply

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When I think about Who I Am as an educator at this moment, you could say that I’m 40% teacher, 40% student, 20% teacher-leader, but I am always 100% parent. Everything I think and care about in education is definitely shaded by my perspective as a parent and my hope for my daughter’s future. In fact, my love for her is the fuel which creates an urgency for changes in education and can blind my decision-making.

Although I am not a proponent of homework for young children, I do spend an evening a week “doing math” with my 3rd-grade daughter because, during our transition to Laos, her academics have dipped. We usually sit together to play Math For Love games but after her MAP test, we’ve been doing some lessons on Khan Academy to supplement her classroom learning. My husband and I have been trying to investigate other more “fun” options for self-directed learning since she is getting older and desiring independence. Since my daughter just recently stopped counting on her fingers, we’ve decided to explore the math website Reflex Math that was recommended by a colleague:

Full of games that students love, Reflex takes students at every level and helps them quickly gain math fact fluency and confidence. And educators and parents love the powerful reporting that allows them to monitor progress and celebrate success.

Sounds like its worth a try, right? We felt that if she could become more proficient in her math facts, she’d feel more assured when engaged in math. So for the last few days, she’s been “playing the game”, and the report we got made us gasp:

Screen Shot 2019-04-11 at 4.51.27 AM

So then we asked her to do this game daily, while she found fun in its novelty, excited to do it independently, eventually, it became a drag on her motivation for math learning. Last night, when I came in and asked her why she was staring at the screen, it brought her to tears. Perhaps it wasn’t one of my “good parenting” moments when I asked her that, but I was afraid that she was just allowing time to elapse until the “store” opened, and she could take her avatar shopping-something that a lot of kids might do to”play the game”.  I really felt awful that my words stung her heart. When I inquired further behind her emotion, she told me that she didn’t have enough “think” time to solve the unknown facts. Oh man, I really had misunderstood her blank stare!

And I, like many parents, had fallen into the trap of thinking that fast=fluent. Instead of creating confidence, I had crushed her esteem. Darn it! As an educator, I know better, so why didn’t I do better?

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After dinner, I decided to give her a little quiz and any math facts that she needed more than 5 seconds to solve, went on a flash card. On the flash card, along with the answer, we’d create a strategy to help her remember it better. We only focused on addition and subtraction.

Now all of a sudden, what had become a drudgery of math practice had suddenly become strangely exciting. Whenever she could quickly give me an answer, she started dancing around the room and laughing. And the ones she got “wrong” (aka, not quick)?-well, when she shared her strategy for solving it, she had solid mental math strategies such as using derived facts. I started giving her harder ones like 17+17 and, as she exclaimed “34”, I wanted to get up and dance around the room with her. Suddenly a “bad parent moment” had turned into a “good teaching moment”, for both her and I.

So what did I learn? Technology isn’t a teacher. I am. Conflict can help develop a deeper understanding of one another. Time may be relative, but conceptual thinking is not.

And most importantly, when I think about homework/home learning–it’s never the worksheet or activity that improves the performance of a student, instead, it is the parent relationship that builds understanding through compassionate attention and love of learning–it’s the US, not the IT.

 

“The Standards” Aren’t a Race: The Importance of Assessment in Getting to a Finish Line

“The Standards” Aren’t a Race: The Importance of Assessment in Getting to a Finish Line

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I didn’t enjoy Math until I was in high school. Trigonometry was the first time that I remember gazing up in amazement and wonder. Sin and Cosine. Identities, theorems, and proofs. Parabolas and Ellipses.  It suddenly became interesting even if it was hard. I loved using the nifty functions on the calculator as well. But why did it take me so long to appreciate the beauty of math? I wonder where and who I might be if I had learned less about standard algorithms and more about number concepts and reasoning at an earlier age.

I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide. ……. to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is all about?

From A Mathematician’s Lament by Paul Lockart-

For me, if I can invoke wonder and surprise, then the beauty of communicating in numbers becomes self-evident and a student’s heart awakens to the joy of an interesting problem or question. Creating this experience is a passion of mine. After spending a week with Lana Fleiszig, it’s hard NOT to be more inspired to create a love of math in our classroom. Her enthusiasm is contagious, and her advice about inquiry is clear–know your destination, but don’t worry about how you get there. Don’t be afraid to throw students into the “pit of learning” and allow them the experience of confusion. As I have come to appreciate her point of view, I recognize that when students climb out of their “pit”, that’s where beauty lies.

So here we are, in another stand-alone unit, which might be considered the “place value” unit, which is not typically the most exciting math concept. It’s a ho-hum inquiry into base-10 blocks in how we express large numbers and use it to develop strategies for addition and subtraction. But what if we threw them into the learning pit and took our time to really develop number sense. How might we approach our planning and execution of the unit if this wasn’t a race to tick off a curriculum math standard?

The Standalone

Let me break down the basics of the unit for you:

Central Idea: Numbers tell us How Many and How Much

  • The amount of a number determines its position in a numeral.
  • How we know when to regroup.
  • How grouping numbers into parts can help us find solutions

(All lines of inquiry and Central Idea from conceptual understanding in the PYP Math scope and sequence and subsequent learning outcomes in  Phase 2)

Knowledge and Understandings, aka, “The Standards”

I’m going to cross-reference 2 commonly used national curriculum, Australian and American Common Core, because our team needed clarity into exactly WHERE our destination needs to be in this unit of inquiry:

Australian:

Count collections to 100 by partitioning numbers using place value (ACMNA014 – Scootle )
  • understanding partitioning of numbers and the importance of grouping in tens
  • understanding two-digit numbers as comprised of tens and ones/units
Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015 – Scootle )
  • developing a range of mental strategies for addition and subtraction problems

The Common Core:

Understand place value.

CCSS.MATH.CONTENT.1.NBT.B.2
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
CCSS.MATH.CONTENT.1.NBT.B.2.A
10 can be thought of as a bundle of ten ones — called a “ten.”
CCSS.MATH.CONTENT.1.NBT.B.2.B
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
CCSS.MATH.CONTENT.1.NBT.B.2.C
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
CCSS.MATH.CONTENT.1.NBT.B.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Use place value understanding and properties of operations to add and subtract.

CCSS.MATH.CONTENT.1.NBT.C.4
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
CCSS.MATH.CONTENT.1.NBT.C.5
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
CCSS.MATH.CONTENT.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

 

Planning the Unit

If you “peel” back these standards, what (math) concepts and skills seem evident to you? What are the “big ideas” that students need to walk away with?

  • Collection or Group
  • Place Value
  • Position
  • Partitioning: composition and decomposition
  • Reasoning

Since I teach 1st grade, we would be exploring the key concept of Form and Function, mainly, throughout this unit. But we would also look at the Connection between using groups of 10s and developing mental strategies for problem-solving in which we can Change addends/subtrahends around to make friendly numbers. Students would also need to consider the Perspective of other mathematicians in our class when it came to solving a problem in different ways.

With this in mind, we looked at these standards and identified 5 main guiding questions  that will be the basis of our inquiry and the purpose of every provocation that we create:

  1. How does the place value system work?
  2. How does the position of a digit in a number affect its value?
  3. In what way can numbers be composed and decomposed?
  4. In what ways can items be grouped to make exchanges?
  5. How can we use place value patterns for computation?

Provocations to Explore and Reveal Math Thinking

Once we had clarity around the big ideas in our unit and created our guiding questions, it became easy to start planning provocations.  Using a guide like this one, Task Identification Tool_Identifying High-Quality Tasks (1),  from the work of John J. SanGiovanni in his book series on how to Mine the Gap for Mathematical Understanding really helps teams like ours to create a high ceiling, low-threshold activity for inquiry-based maths.

We knew from a previous provocation, (14 or 41–the position of a numeral doesn’t matter. Agree or Disagree. Prove it.), that students still were developing an understanding of what a written number means. We needed to further explore it. So we began with place value.

Guiding Question #1: How does the place value system work?

We decided to launch the unit with an emphasis on language since we noticed that a lot of students were mixing up their teen numbers when explaining their ideas. So we started with Teen vs. Ty, is there a pattern or a rule about these numbers?

  1. Sixteen and Sixty, What do you notice about these numbers?
  2. Seventeen and Seventy? How are they different, how are they the same?
  3. What do you think “teen” means? What do you think “ty” means?

We then began exploring expanded notation with showing the tens in a number. Students were introduced to how expanded notation is related to the place value mat, which can be represented as:

43=40 + 3 or 4 tens and 3 ones. 

The students played a partner game called “guess my number” in which they had to express a number in tens and ones and have the student create it with base-10 blocks and numerals.  They did really well. We thought we were smashing it and ready to move on to using it for addition and subtraction.

But how could we be sure they “got it”? ……….

Assessment

We decided to assess if they got the idea of base-ten and how we use the place value mat as a structure to show the parts of numbers. We used this SeeSaw prompt to assess if they truly understood:

How we know when to regroup – Using a collection of objects – how do you find out how many items you have?

We decided to use unifix cubes because the “tens” weren’t prepackaged, sort of speaking, as they are with base-10 blocks. In this assessment, we had them grab a handful of unifix cubes and organize them on the place value mat, explaining to us what number they thought they had. What we observed stopped us dead in our tracks and ask what misconceptions do we see? Here is an example of a common surprising result:

As you can see, this student didn’t connect the quantity he had in their collection at all. These students would need some additional support with connecting the amount of a number to how it is written and presented.  We felt we needed to go deeper into how we “bundle” tens to count things efficiently. In fact, we felt we needed to do an inquiry into 10, so they could appreciate how this is the basis of base-10.

Back to the Starting Line?

We are in Week 4 of this unit, and we are going back to the starting line. Based on our observations, it seems that the students don’t quite have the idea of ten yet, and, we have a group of students who just need to work on skip counting by tens. It would be easy to move ahead and push through so we can tick off our standards, but we’d rather spend more time immersed in context and play that develops their number sense than to push them along. We understand our future impact. Moving ahead hoping that they “get it” later on would seem like a disservice, as they’d lose the interest and motivation to do more complicated mathematics and have half-baked conceptual understandings.

Since have a free flow of student groups, in which children choose what Must Dos and May Dos they want to participate in. However, ideally, we have 3 primary activities that we want the students to work through in small teacher groups throughout our math learning time:

The Big Idea of our teacher-directed groups: 10 can be thought of as a bundle of ten ones — called a “ten.”

  • The Base-10 Bank

Students will pick a numeral and build numbers using “ones” which they can exchange for tens. As partners, one person will be the “bank”, which the other partner can trade in their ones for 10. No place value mats, only the base-10 blocks.

  • Race to 100

Using dice, a hundred’s chart and a place value mat, students have to roll and add their way to 100. As they roll their way up to 100, they have to build the new number, using the place value mat to show how the quantity that is ever-increasing, as well as giving a context for exchanging units.

Making Bundles: In this activity, students are given a collection of objects and they have to bundle them up into tens, so that they have an appreciation of the value of a number.

Additional Games and activities that they can do independently, when not working with a teacher. The May-Dos:

Traffic light (Partner Game): One partner comes up with a “mystery” number and, using a place value mat, has to try to guess what digit is in what position.

Big 4 (Independent or Small Group): In this game, we use a hundred chart to try to get to the biggest number in just 4 moves. A child rolls a die and moves that many spaces, moving in any direction, forwards, backward, diagonally, upwards or downwards. This game gives them practice at thinking about number patterns as they move around the hundreds chart.

Ready or Not?

After all that exploration, we hope that these games will prepare them for the following formative assessment:

4+4 = 44. Agree or Disagree? Show how you know. (This actually is inspired by a misconception that we observed) Students can use 10-frames, the Hundreds Chart, Math Racks or Base-10 blocks to provide evidence of their reasoning. (We determined that these sorts of materials would help them to “see” patterns and make connections, rather than loose parts alone)

If they can articulate and demonstrate a firm understanding of place value in this provocation, then we feel that we can move into applying our understanding of using the base-10 for addition and subtraction, examining the guiding question:

How can we use place value patterns for computation?

This is the ultimate reason why place value is such a critical understanding after all. However, it is the journey into number sense that makes this a beautiful experience. We are not quick to move them onto pencil and paper. We want them to experience numbers and segue them into contextual situations.

The Summative

We are still in process with determining the actual prompt, but we feel that we need to give them choices with the task. Choosing a task that shows how they apply grouping strategies to solve addition and subtraction problems will ultimately be our goal.

For our low-level readers, we will give them an oral word problem and then hand them a collection of objects that need to be counted. We want them to observe if they create groups of tens to determine the number. No place value mats offered, but they can request one. For our stronger readers, we will give them a word problem, and, again, offer them concrete materials, but other tools to solve the problem are upon request.

At the end of this task, we can identify the skills and understandings they have acquired. Although we have “mapped out” where we think this unit will go, we can be flexible and stop to address misconceptions along the way. Will they arrive and “meet the standard”? That is entirely up to us, and how effectively we observe, challenge and question our students’ thinking as they playfully and joyfully experience numbers. At the end of the day, that goal–to appreciate and be fascinated with numbers--that is the true destination of math inquiry.

 

Into the “Pit” or upon the “Clouds”: Kensho and Satori Moments in the Development of Number Concept

Into the “Pit” or upon the “Clouds”: Kensho and Satori Moments in the Development of Number Concept

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It’s Sunday morning and as I soap up greasy dishes, I hear Susan Engel say on the Heinemann Podcast: 

One of the things that I think that our schools have unwittingly done is ignored all the processes that kids use at home and try to replace those with a set of formal procedures that aren’t always as effective…. But it’s a shame because while we are busy trying to sort of force these somewhat formal kinds of learning beacause we think they are more “efficent” or “high powered”, we waste a lot of the natural learning skills that students have. And often a lot of the natural teaching skills that grown-ups have.

Huh, I think I know what she is talking about. Whether we are teaching a genre or the scientific process, teachers are constantly “telling” kids what to pay attention to and to think about. When I start examining my current practice and reflecting on Who I am as a teacher, I have come to see my role as a provocateur and coach. I am always considering who is REALLY doing the learning in our classrooms?–is it me, or is it the students?

egg
I think of this quote often, reminding myself that if I  tell students, then I’m “breaking their egg” and killed the opportunity for their learning.

So I am constantly asking myself that question because I know that “the person who does the work, does the learning“. But when I say “work”, I mean thinking, and there are so many of these micro-moments in our classroom in which I have a chance to tell kids what to do or to ask them what they think they should do to approach a situation or problem.  Sometimes these moments of learning are Kensho, growth through pain, and other times it is Satori, growth through inspiration. I first encountered this term when I read this blog and Kensho immediately reminded me of our teacher-term, the learning pit. You need determination and resilience to get out of that pit and your reward is Kensho. However, we rarely talk about it’s opposite, Satori. Up until this morning, I didn’t think we had a name for Satori in education. It Kensho is the “learning pit” than Satori must be up in the “clouds”, having a clear view and understanding. But Susan Engel articulated best in the podcast:

There are certain kinds of development that children undergo that are internal and very complex and they don’t happen bit by bit. They happen in what seem to be moments of great transformation of the whole system. ……

At that point, I stopped and turned toward my device. I recognized exactly what she was talking about it. I observed it the other day. My ears perked up some more as I moved closer to listen:

When children are little, their idea of number is very tied up with the appearance of things. So, this is a famous example from Jean-Pierre, a line of 10 pebbles to them is a different quantity than a circle of 10 pebbles, because lines and circles look so different.

The idea that it’s 10, whether it’s a circle or it’s straight, is not accessible to them. At a certain point, virtually every typically developing child, no matter where they’re growing up, acquires this sense that the absolute number of something stays the same no matter what it looks like. Whether it’s a heap or a straight line or a circle, that may sound like a tiny discovery, but it’s the beginning of a whole new way of experiencing the abstract characteristics of the number world.

You can’t teach that through a series of lessons. That’s an internal, qualitative transformation that children go through. Once they’ve gone through that, there are all kinds of specific things that you can teach them about the nature of counting and number and quantity.

Yes! I totally know what she is explaining. I was a witness to it. And perhaps, when you reflect on these Zen philosophical terms as development milestones, you may make a connection to your own classroom learning.

Here’s a snapshot from a recent example in our Grade 1.

Some context

There’s a math coach that I love, Christina Tondevold. She always says that “number sense isn’t taught, it’s caught”.  I’m always thinking to myself, how can I get them to “catch” it. This past week, we did just that using the Visible Thinking Routine, Claim, Support, Question making the claim:

The order of the numbers don’t matter–12 or 21, it’s the same number.

The students took a stand, literally, in the corners next to the words and image for Agree or Disagree, with  I Don’t Know, in the middle. This was great formative data! Then we provided the students with a variety of “math tools” to Support or prove their thinking is correct. They had to “build” the numbers and show us that they were actually different. It was neat how the students who stood in the I Don’t Know and Disagree areas were developing an understanding of what a written number truly “looks like”. We didn’t jump in and save them at any point, but some of them were experiencing Kensho. It was painful because they didn’t know how to organize their tiles or counters or shapes or beads in such a way that they could “see” the difference between the 2 numbers. Meanwhile, the students who chose the unifix cubes were experiencing Satori- and it became very obvious to them that these were different numbers. 

In our next lesson, we introduced the ten frames as a tool to help them organize their thinking and develop a sense of pattern when it comes to number concept. We did the same two numbers: 12 and 21, and they could work this time with a partner. Oh man, was there a lot of great discussions that came out as they talked about how the numbers looked visually different. The concept of Base 10 started to emerge. As observers, documenting their thinking, it was exciting to see the connections they were making. But the best part was yet to come.

We then brought in the Question part of the thinking routine. We asked them “if the order of 1 and 2 matters to 12 and 21, then what other numbers matter?” They told us:

“13 and 31, 14 and 41, 24 and 42, 46 and 64, 19 and 91, 103 and 310.”

A Hot Mess of Learning

Once unleashed, the kids grouped up and flocked to resources. There was a buzz. Giving students choices allowed the opportunity to choose whether they wanted to stay with smaller numbers or shoot for the BIG numbers even if they had no idea how they might construct a number past 100. They could use any math tool they wanted: cubes, blocks, 10 frames, Base 10 blocks, number lines, counters, peg boards–anything they wanted. Those choices, of itself, really provided some great data.

Here is an example of one of the groups who went with lower numbers:

But the ones who went for the BIG numbers, were the most interesting to watch because they were Kensho. Most of them grabbed unifix cubes, thinking that the same strategy they used before with 12 and 21 would work with 103 and 310. big numbersOh man, they persisted, they tried, but it took a lot of questioning and patience on our part to help guide them out of the pain that their learning was experiencing. Only one group naturally gravitated toward the Base-10 blocks, and when they realized how the units worked, it was a moment of Satori. They moved on from 103 and 310 quickly; they tried other numbers and invented new combinations. And interestingly enough, those groups, at no point, looked over to the ones engaged in the struggle to suggest that they might try another math tool. It was as if they knew that when one is in Kensho, best to leave them alone to make meaning on their own.

And there we were, in the midst of this math inquiry, and we felt like exhausted sherpas but satisfied that we were able to let them choose their own path of learning and made it to their “summit”.

As I consider how the role of the teacher is evolving in education, I think it is recognizing these moments of pain and insight in learning, and guiding them towards the next understanding in their learning progression. I absolutely agree with Susan Engel that when we see children fumbling around, we should be asking if they are within reach, developmentally, to even acquire the knowledge of skill that we are working on. For me, inquiry-based learning is the BEST way in which we can observe, engage assess our learners to truly discover their perceptions and capabilities. It is through capturing the student conversations and ideas that emerge as they give birth to a new understanding that is the most exciting to watch and inspires me in our planning of provocations that lead to their next steps.

How about you?

 

 

Math in the #PYP: Can you really “kill 2 birds” with one planner?

Math in the #PYP: Can you really “kill 2 birds” with one planner?

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I’ve been doing a little light reading and exploring the new PYP: From principles into practice digital resource in the PYP resource center. This led me to nose around the Programme standards and practices documentation to see if anything had dramatically changed. I was surprised at how much it had changed in wording, not just swapping section letters for numbers but how some of the ideas have shifted to articulate the “enhancement” of the programme.  Here’s something that stood out to me:

(2014)Standard C3: Teaching and learning

Teaching and learning reflects IB philosophy.

1. Teaching and learning aligns with the requirements of the programme(s). PYP requirements

a. The school ensures that students experience coherence in their learning supported by the five essential elements of the programme regardless of which teacher has responsibility for them at any point in time.

 

(2018) Learning (04)  Standard: Coherent curriculum (0401)

Learning in IB World Schools is based on a coherent curriculum.

Practices: The school plans and implements a coherent curriculum that organizes learning and teaching within and across the years of its IB programme(s). (0401-01)

This led me to question and scan through the standards and practices documentation to examine how “stand alones” are being viewed in the enhancements. Since I wonder how they fit in with this idea of “coherency”, (which was not defined in the glossary of terms, oddly enough) they could be problematic as they might conflict with transdisciplinary learning.

And why do I think this?-because I’ve been struggling with trying to “cover” the math standalone along with the transdisciplinary maths. At schools in which TD (Transdisciplinary) Maths and SA (Stand Alone) Maths are taught simultaneously during a unit of inquiry,  I’m sure many of you PYP educators share my pain and are trying to “fit” it all in while not sacrificing the main UOI.

Oh, I can hear you–

Judy, but TD Maths is supposed to be embedded naturally into our UOIs. We shouldn’t know where one subject begins and where ends in transdisciplinary learning. 

But math is not a noun, it’s really a verb. And unless you write units of inquiry that create the context to do mathematics organically, it hardly lends itself to transdisciplinary learning. Perhaps it is for this reason why our school has created a whole Math Programme of Inquiry (POI) around the strands of Number and Pattern & Function. Christopher Frost wrote a brilliant blog post that articulated his school’s challenge with the PYP planning puzzle: mathematics so I can appreciate why our school has attempted to create a Math POI. However, because we only developed it within those strands, in my opinion, this has further complicated the challenge of integrating math into our units of inquiry.

For example, our last Math UOI  in 1st Grade was:

Patterns and sequences occur in everyday situations.
–Patterns can be found in numbers.
-Types of number patterns
-Patterns can be created and extended.

This was our conceptual rubric for this Unit of Inquiry:

Screen Shot 2018-10-28 at 9.52.48 AM

The lines of inquiry came from the learning outcomes (which we refer to as “learning territories” at our school) from the IB’s Math Scope and Sequence, under “constructing meaning” in Phase 2 in the Pattern & Function strand.  But then this stand-alone wasn’t enough, and we had to then create a TD math focus to go with our How We Express Ourselves unit:

Language can communicate a message and build relationships.
-Different forms of media;
-The way we choose to communicate;
-How we interpret and respond.

So there we were, as a team, staring at this central idea and wondering what would be a natural match, conceptually, with this unit. We could definitely DO data handling as a component of this unit, creating graphs and charts that reflect the 2nd and 3rd lines of inquiry. However, since we were stuck on the CONCEPT (rather than the skills), we ended up focusing on the word LANGUAGE and eventually wrote another conceptual rubric based upon the conceptual understanding (from the Math Scope and Sequence): Numbers are a Naming System (Phase 1, Number), using the learning phases from the Junior Assessment of Mathematics from New Zealand–a standardized assessment that we use across all grade levels.

Screen Shot 2018-10-28 at 10.08.44 AM

Although we felt that we “covered” the learning outcomes or “territories”, we definitely felt dissatisfied with how we approached planning and learning these of concepts. Recently, I read the Hechinger Report, OPINION: How one city got math right, something stuck out at me and made me reflect deeply on our process and purpose of math in the PYP.

The top countries in education have shown that going deeper and having more rigor in middle school are the keys to later success in advanced math. Compared to high-performing countries, American math curricula are a “mile wide and and inch deep.” Students who want to go far in mathematics need a deeper, more rigorous treatment of mathematics…..

Going for depth of understanding in the foundational years, and accelerating only when students have solid backgrounds and have identified their goals, has paid off. This is progress we can’t risk undoing by returning to the failed practices of tracking and early acceleration.

Here are the questions that surfaced after reading that article and reflecting on our context:

  1. Is having TD math and SA math taught during the same unit of inquiry really “best practice”? Are we creating a “mile wide and an inch deep”?
  2. Is focusing on conceptual understandings vs. skills really the best approach to transdisciplanary learning in math?
  3. Do broad conceptual understandings help or hinder the assessment of a math UOI?

Now I’d like to add one more question after reading the Standards and Practices……

4. How can we create coherency, not only by “covering” all the learning expectations for our grade, but create authentic math connections for transdisciplinary learning?

 

Where we are in place and time with Math in How the World Works.

Our new unit began this week. Originally our upcoming Number SA Central Idea was going to be:

Making connections between our experiences with number can help us to develop number sense.

As we were beginning to develop lines of inquiry for our “learning territories”, we decided that this central idea seemed hard to approach and written for the teacher, rather than the learner. (In my opinion, if students find Central Ideas to be goobly-gook, then how on Earth can they make meaningful connections?) We went back to the IB’s Math Scope and Sequence to provide clarity and direction to developing skills.

Will mathematics inform this unit? Do aspects of the transdisciplinary theme initially stand out as being mathematics related? Will mathematical knowledge, concepts and skills be needed to understand the central idea? Will mathematical knowledge, concepts and skills be needed to develop the lines of inquiry within the unit?

When we looked at those questions, our team nodded their heads in agreement–Yes, of course this is a TD Math unit–it’s a scientific thinking unit, for heaven’s sake–the best kind to connect with!

Thus we rewrote the Central Idea and created our lines of inquiry based upon what they might be “doing” with number, recognizing that other math strands might be employed in our How The World Works unit (Central idea: Understanding sound and light can transform experience), thus combining the “Stand Alone” with our “TD Math“. Here is the unit we created:

We collect information and make connections between our experience and numbers.
–use number words and numerals to represent real-life quantities.
-subtitize in real-life situations.
–understand that information about themselves and their surrounding can be collected and recorded
-understand the concept of chance in daily events.

To be honest, I’m not sure if this is the best approach either and I spent a good amount of time cross-referencing pacing calendars and scope and sequences from other national curricula. However, this not only would help us to “kill 2 birds” with one planner, but it also helps us lean towards creating math units that develop the context of discovering vs. “being told” when and how to do math. This is true inquiry, in my mind, whether it is through a SA or a TD Math lens of learning. But when you are trying to squeeze in teaching two maths (TD and SA) during a unit then there is the challenge of approaching problem solving as a rote skill instead of having enough time for students to make decisions based on their math understanding. Documenting and analyzing those student decisions require time in order to evaluate appropriately what our next steps might be and in order to guide them towards a deeper understanding and more flexible thinking. So stay tuned.

If any other schools have been fiddling around with integrating math into units, I’d love to hear some of your stories–indeed anyone reading this blog would!! So please share your approaches in the comments below.  It benefits all of us trying to put “Principles into Practice”.

 

 

The PYP Planner: A Shift in Our Approach to Planning Inquiry (#enhancedPYP )

The PYP Planner: A Shift in Our Approach to Planning Inquiry (#enhancedPYP )

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Quick Quiz: What is the first “box” in the PYP planner? Did these things come to mind?:

What is our purpose?   To inquire into the following:

  • Transdisciplinary theme:    
  • Central idea :  

summative assessment task(s):

What are the possible ways of assessing students’ understanding of the central idea? What evidence, including student-initiated actions, will we look for?

Now with the enhancements in the Primary Years Programme (PYP), we can redesign our planner which has to lead to an overhaul of our collaborative planning. If you notice in the Box #1, aside from clarifying our theme and central idea, we would start planning our summative. However, we haven’t done any assessment of student’s prior knowledge, and I often found that determining the summative assessment before we have even launched a unit of inquiry (UOI) has always led to more teacher direction in our units, as if we are nudging and, sometimes pushing the students toward our end goal–The Summative Task. Think about it, when we plan in this way, we are already dictating the terms of what we want the students to Know, Understand, and Do (aka: KUD) before we have even gotten them to SHOW US what they already know, understand and do. A bit presumptive of us, really, eh?

Needless to say, since the reigns are off, and schools get to design PYP planners in the Enhanced PYP, there’s been a shift in how we approach planning. And the new “Box 1” (figuratively) is about planning our provocation FIRST so we can let the students reveal to us what they know and lead the direction of the UOI, rather then us marching them towards the summative. It may seem trivial, but when you consider how AGENCY is the new core of our curriculum, we need to be approaching our units in different ways.

Let me provide a context, looking at our Math Stand Alone:

Patterns and sequences occur in everyday situations.
-Patterns can be found in numbers.
-Types of number patterns
-Patterns can be created and extended.

Key concepts: Connection, Form, Reflection

Related concepts: pattern, sequences, collections/groups

As a team, including our Math Leader of Learning (Olwen Millgate), we sat down and discussed the many different ways that we could plan a provocation around this central idea. At the end of the day, we determined that the most open-ended, the better, so that students could exercise as much creativity and skills as possible. We would just be the “researchers” in the classroom, observing and noting what the students came up with when given the challenge–Create as many patterns using one or more of the materials provided. 

As teacher researchers, we divided up the students so that we could take notes, making sure that all students were given the time and attention to “show what they know” about patterns. Here is the simple observational sheet that was created for this provocation: (Free to use)

We gave the students a variety of math tools to work with:

  1. counters
  2. ten frames
  3. unifix cubes
  4. Cuisenaire Rods
  5. beads
  6. pattern blocks
  7. peg boards
  8. tanagrams
  9. popsickle sticks
  10. white board and markers
  11. stampers and paper
  12. dominos
  13. magnetic letters and boards

As you can see, they had a plethora of options, and the students engaged freely, making their own choices and creations. Some students preferred to work by themselves while other collaborated–another aspect that we noted along with capturing their conversations. Here are just a few of those creations:

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There was a very loud and animated group at the Unifix cube station–which surprised us all by how excited they got about building “tall patterns”, with a lot of debate about whether they were just snapping cubes together or generating a true pattern. Although to outsiders, it may have felt chaotic, there were fantastic conjectures going on, which we saved a few examples to use for future provocations. (The Guiding Question: Is this a pattern? Why or Why Not?)

Later, our team met to discuss what we observed and we were able to sift through our documentation. We unpacked the provocation, and shared our photos and notes, describing some of the interactions that we had with them. Then we went to our curriculum and examined the phase the students might be achieving at in their conceptual understandings. Our central idea comes from the PYP Maths Scope and Sequence in Phase 1, so we needed to challenge it —is this the phase they are actually in or are we seeing evidence of Phase 2 understanding? We decided to stick with our central idea and lines of inquiry because we felt like we saw and heard evidence that most of our learners were on the tail end of this phase, applying their understanding of pattern.

After this conversation, we went on to determine what our next steps could be. Most of the patterns were ABABAB–do we encourage them to make ABCABCABC or other more sophisticated patterns? At the end of our deliberations, we decided that rushing them might create conceptual gaps so we wanted to stick with their ABABAB, but create a series of opportunities to look at how we could manipulate only 2 variables to generate a variety of patterns. What can we do with only 2 attributes?–This became the guiding question for our upcoming provocations.

So here we are, in Week 2 of this unit and we still haven’t nailed down our summative task. Gasp, right? But, on Friday, after this week’s follow up provocations, we can safely appreciate our learners, where they are and where we can take them during the remaining weeks of the unit. I find that thrilling. We will create our conceptual math rubric, using this generic one as our guide:

math standalone 2

Hopefully, you can see that we are thinking about planning not as boxes but phases in our inquiry. We are using this “tuning in” period to dictate the terms of our how we will ultimately assess students. And we are spending more time researching and planning our provocations so that they can make the children’s thinking visible and expose their understanding of the math concepts. I believe that as this approach to planning evolves, our attitudes toward our students also evolve when we are observing how they are competent and creative when expressing their ideas.

I’m wondering how other schools have begun to consider the impact of planning on agency and how it will look in the Enhanced PYP. The planner has always been a tool for us to shape our collaboration and thinking about how best to meet our students’ needs in the inquiry. I think it will be fun to see how schools begin to shift as they reflect deeply on the importance of it–it’s more than just an exercise in paperwork, it is an artifact of learning.

So what’s your “Box #1”?

#BuildMathMinds18: How Slow Thinking, Playing and Challenge Create Mathematicians

#BuildMathMinds18: How Slow Thinking, Playing and Challenge Create Mathematicians

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In the Build Math Minds Summit, Dan Finkel elaborated on this notion that “what books are to reading, is what play is to math.” And as he said this, my ears perked up, I leaned in and listened intently because this is all I’ve been thinking about for the last week as I begin to plan for next year’s inquiry maths with play as a pedagogical stance. He articulately beautifully how math thinking comes from asking questions, solving problems, playing and exploring.

So as I marinated in his words and ideas, I began binge learning all over again. Glutton for punishment?–I guess I am. But they say that when you teach others, you learn twice. So I want to share my takeaways from some of the presenters, Dan Finkel included, for the Math Minds Summit (which you should go watch right now if you read this post before August 6th, 2018). And because I know that the brain is more switched on when you present ideas as questions, my gleanings are represented in that way for this blog.

I hope it inspires you…..

During unstructured play, what kinds of questions can provoke analytical and divergent thinking?

How many? (number)

What kind? (classification)

How big? (measurement)

What if? (creativity and logic)

What makes games good to develop mathematical thinking?

  1. Anything with Dice
  2. 5 in a row
  3. Number sense cards (that show alternate variations of number patterns)
  4. Checkers
  5. Nim
  6. Anything with cards
  7. Anything that you can advance pieces on a board
  8. Games that involve making choices so that children develop strategy and thinking.

Provocations= Puzzles and Challenges

These can be concrete opportunities to explore estimation and making conjectures, but the heart of a mathematical provocation is that it must be intriguing to get the students curious and motivated to solve the problem. Consider if the provocation is going to…

  1. Allow all students to show their thinking and understanding in interesting ways.
  2. Invite conversation and collaboration among peers.
  3. Provide opportunities to assess what students know and can do mathematically.
  4. Have an ROI (return of investment) of time and resources–all the set up is worth it because of the cognitive demand and depth of learning that is going to come from this provocation.

(These 4 criteria were inspired from Jon Orr  and his work with starting a Math Fight)

These are some examples that I think were great examples:

“About” how many ketchup packets do you think can fit inside these containers?

estimate

Prompts that incite a variety of answers:

mathbefore bed

What language encourage matheI remember thinking that students should learn the way I taught; they should adjust to me. I could not have been more wrong. A great teacher adjust to the learner, not the other way around (3).pngmatical dispositions?

  • Demonstrate that wrong conjectures can be the jumping off point for refining our ideas with counterexamples which enrich our thinking and deepen our conceptual understandings.
  • Using descriptive and numerical language to highlight the math concepts  (He gave the simple example of saying “Get your 2 gloves” vs. “Get your gloves”.
  • Use language that shows that we, as adults, aren’t afraid of making mistakes, so they feel safe also.
  • Do NOT use words that suggest that you have to be “smart” or “fast” to do the math.
  • Likewise, do NOT give praise for being “fast” or “smart”.
  • Ask questions that provide challenge and make students take a position (conjectures):
    • Do you agree or disagree with this idea?
    • Why?
    • How do you know?
    • Say more about that?
    • In your own words, could you explain….?
    • Would you rather….. or would you rather……?
    • How might you represent your thinking?

What routines or thinking systems encourage mathematical conversations and develop conceptual understandings?

(Click to learn more on the links)

  • EVERYTHING WE KNOW ABOUT THIS routine: Present a problem or puzzle, asking them to….Write down, tell a neighbor, tell me EVERYTHING you know about this.
  • Number Talks: a simple problem shown that students try to solve mentally in a variety of ways.
  • Number String: a specifically structured string of number problems in which the numbers get progressively harder.
  • Counting Collections: The routine speaks for itself- students count set collections of objects. This develops a variety of counting strategies.
  • Claim, Support, Question: providing a claim (conjecture) that students have to provide evidence to support their claim. In order to deepen the conjecture, students can use counterexamples or ask questions that help develop a better math argument.
  • Two Truths and a Lie: students are presented with a math problem or graphic.  Students are instructed to create two truths and one lie about the math.  Then, students share their “truths” and “lie” and have other students decide which are the truths and which is the lie.
  • Which One Doesn’t Belong?: These are visual puzzles that have multiple answers. Click on the link to see a plethora of them. There’s also a book written by the same title.

Next week, when our 1st graders start piling into the classroom, I have an arsenal of ideas that I’ve gotten from this summit. (And it’s not even over!!) I really would invite you to check it out. I know, beyond a doubt, that our students are going to fall in love with math at an early age because they will engage in play, feel challenged at their level and construct meaning on their own timeline. I wish the joy of math for all children (and adults) out there. Don’t you?

 

#InquiryMaths: Planning for Play as a Stance for Math in the #PYP ?

#InquiryMaths: Planning for Play as a Stance for Math in the #PYP ?

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I’ve been binge learning through the online conference on The Pedagogy of Play. It’s been really inspiring for me. Last year, I felt like I was moving away from play-based learning and into more formally academic structures when I began teaching first grade. This has been a challenge for me because I miss the discoveries (theirs and mine!) and creativity that are natural by-products of a play-based approach. So as I embark on this school year, I have two questions that I am holding in my mind: How do I make math more fun and authentic? and How do I provide rich open-ended tasks that allow for multiple approaches with low threshold, high ceiling tasks?

These questions come from this quote from Jo Boaler, a math educator hero of mine.

Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels.
― Jo Boaler, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching

loris malaguzziAs I reflect on that research, I believe the answer to my questions is to play. Not just because it develops curiosity and self-expression, but it cultivates self-motivation and an appreciation for the pleasant surprises that our mistakes bring us in our learning process. Moreover, from Boaler’s academic point of view, “they become more deeply engaged and perform at higher levels”. Um…so why on Earth wouldn’t we connect play and math?

What is play?  Play is the ultimate What If question in my mind because it allows us to explore with possibilities. Most Primary Years Programme (PYP) Early Years educators feel that the word “play” is synonymous with the word “inquiry”. As teachers, we can be intentional about marrying the joy of learning through play with our learning outcomes. I don’t think we have to suck the fun out of everything to make it “learning”; in fact, I think it has to be injected back into the process, especially when I consider that real * (think Albert Einstein and Euclid and Leonardo Pisano aka Fibonacci) mathematicians are exceptionally creative and playful with their ideas. (*Actually, I think ALL of us are REAL mathematicians, but not all of us embrace and delight in this aspect of ourselves).

So then if I approach inquiry maths through the lens of play, I need to consider ….

What tools can we use for play?

  • Loose parts?
  • Technology?
  • Each other?
  • Math resources (traditional, like geometric shapes, Unifix cubes, hundreds chart etc.?)
  • Math resources (non-traditional materials that allow students to create. ie: a bridge)

What mathematical ideas can be developed and deepened through play?

I actually believe that most of the time, when we are authentically engaging in math decisions, we are not doing “number” and then “measurement” and then “data handling”–it’s not that discrete in real life and often time these strands are happening simultaneously and overlapping. Play expresses this transdisciplinary nature.

What language can I use to invite “playfulness” with math?

I think our teacher talk is actually a critical component of shaping our mathematical identities. Also, the enthusiasm I communicate, either through my speech or through non-verbal cues is something that I want to be mindful of. My favorite book that addresses this is still Mathematical Mindsets  but I also love the simplicity of Peter Johnson’s ideas on language and I recently read In Other Words: Phrases for Growth Mindset: A Teacher’s Guide to Empowering Students through Effective Praise and Feedback which had a lot of gems in there that can be applied to developing our language around math learning.  I’ve been ruminating over how I can embed more sophisticated math language in our classroom vernacular, especially with our English Language Learners (ELLs). I really want students to talk like mathematicians, explaining their algorithms and debating approaches to problem-solving in a way that is light and spirited as if we are having a cool conversation. I know that deepening my ability to express the “fun of math” is going to be an area of growth for me because I have been brainwashed into thinking (like many of us were) that math is serious and hard. I STILL have to unlearn this when working with older children.

How can I document their learning decisions so I can create more opportunities to engage, process and reinforce key concepts while also expanding their cognitive boundaries? Right now I am reading A Guide to Documenting Learning: Making Thinking Visible, Meaningful, Shareable, and Amplified by Silvia Rosenthal Tolisano and Janet A. Hale in the hopes of deepening my knowledge and finding answers to this complex question. I also find that this Math Mindsets Teaching Guide from YouCubed will be incredibly helpful in my professional learning journey.

So as I think about our first unit of inquiry in our stand-alone Programme of Inquiry (POI), I find this a wonderful opportunity to develop play as a stance to inquiry maths. Here’s the unit:

Central Idea: Exploring patterns and solving problems empowers us to think mathematically

An inquiry into how mathematicians . . .

1.Construct meaning based on their previous experiences and understandings
Make meaning from what they understand

2. Transfer meaning to connect and deepen their knowledge and understanding
Make connections to deepen their knowledge and understanding

3. Apply their understanding of mathematical concepts as well as mathematical skills and knowledge to real life situations
Use what they understand to solve problems

CONCEPTS – Connection Reflection
ATTITUDES – Independence Confidence
LEARNER PROFILE: Knowledgeable Communicator

 

I am considering what provocations would allow the students to “to show what they know”–which is really the essence of our first unit.

Before I do any provocations though, I have to survey and collect data. Nothing fancy, but I need to know their answers to the following questions and then analyze their answers to make informed choices on how we can create invitations to play in mathematics. Also, it helps me to assess the Key Concept of Reflection. aflThese are the open-ended statements that can help me understand where the students are now:

  1. Math is……
  2. Math makes me feel…..
  3. Math is fun when….
  4. I do math by…
  5. Math is everywhere (agree or disagree) because…..

Here is some of the brainstorming that I am considering for “provocations” to begin to shape our awareness in our daily lives and help create an authentic invitation to play. (By the way, this is my first thinking–I haven’t collaborated or researched with peers–so this is raw and rough ideas, happening in real time on this blog):

  • The ole’ suitcase: Place inside a seemingly odd collection of items from everyday life  that represent mathematical strands* like a pair of pants (measurement), a bottle of water (shape and space), a license plate (number and pattern), a bag of candy (data handling), a clock (number), a map (shape and space), some rocks or shells (data handling/number and pattern), some tape (measurement). Then have students pair up, select an item, and discuss the guiding questions. Record their thinking onto SeeSaw.

(*May I just say that I know that selecting those items and arbitrarily labeling them in particular strands is a bit comical because I know that the students will come up with more interesting ideas and connections than I ever will. But this is just an “accounting task” to ensure that, in my adult mind, I’ve covered all possible topics.)

The Guiding Question(s): If math is everywhere, then how are these things related to math? What math might someone have used to create these things?–What ideas were people thinking about when they made these items? (Key Concepts: Connection, Perspective)

The next day, we would need to share those survey results with the class so that students can start developing their identities as mathematicians. We’d probably come up with a display and have the students do a gallery walk and discuss what they noticed. Then I would set out these items and ask a follow-up question: If you were to sort these items, which things would you put together and why? (This is just to further identify the connections they’ve made)

Up until this point, I am just trying to kill two birds with one stone: plant a seed that math can be everywhere and collect data about their thinking. But now I have set up the opportunity to have purposeful math discussions through invitations to play.  Of course, the types of tools and learning situations that can be engaged through play will obviously vary based on the survey and the data collected from the provocation.

But I think we could set up a variety of “challenges” or authentic contexts that can be steeped in play-based situations.

Example: The Challenge: Your mission should you accept it……

  • Fill the cup: using a straw and this bowl of water, how might we fill the cup to the line?

Possible Tools: drinking straw, spoon, soap pump, timer, popsickle sticks, paper, pencils

Because I didn’t ask for a particular tool to be used, then this becomes a more open-ended task, allowing more choice and helps me to get data on the student’s thinking. This amps up the play quotient and math possibilities.

Possible teacher questions: What if you used a spoon (or straw, or soap dispenser, etc..), how might this change your results? How do you know that you have completed this challenge? How might you do this challenge faster? How do you think we could record your success?

This forward planning for a provocation and “play-storm” is really just the beginning. In less than 2 weeks, the doors will officially open and learning will officially commence for the 2018-2019. I couldn’t be more eager to approach this year’s learning with a dedication to play, taking their ideas and imaginings and connecting them to math learning that matters to them is going to be important and fun work. As I consider the possibilities with play, it gets me really excited. I hope, no matter what age we teach, educators see the value and need for play in developing mathematical thinking.

 

#InquiryMaths: Mathematical Thinking meets #Social Leadership using #Seesaw

#InquiryMaths: Mathematical Thinking meets #Social Leadership using #Seesaw

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Social media is viewed as “entertainment” by many. However, many people use social media accounts like Facebook and Twitter to find their news -2/3rd to be exact. As I ponder this, I have to recognize what a powerful influence this is over our culture.  Social media as entertainment may be what IS, however, it doesn’t have to be what WILL BE for our children’s future. I believe we can change that. I think we can educate students to view it as an avenue to have true intellectual discourse and human connection. In Social LEADia, Jennifer Casa-Todd suggests that digital citizenship should evolve into using the internet and social media to improve the lives, well-being, and circumstances of others and I don’t think we have to wait until students are old enough to have social media accounts to begin to develop this mindset. So we have begun to test out this idea during our current unit of inquiry:

Language communicates messages and builds relationships

  • Different forms of media (form)
  • The way we choose to communicate will affect relationships. (reflection)
  • How we can interpret and respond (causation)

During this unit, we have been using Seesaw as our social media ‘training wheels’ to explore what it means to consume content and respond to it by examining how we share our mathematical ideas through posts.  Our team had realized that students were posting different ideas of problem-solving and we wanted them to examine alternatives to their thinking. We could show the students these clips as a whole class and do number talks around them but we felt that allowing students the choice to select the ideas would help them gain independence.

So we started by explaining how you could find these great mathematical ideas in their journals.

Since we wanted the students to construct meaning, we didn’t tell them what they should post as comments, we just explained how you could show your response to listening to them. aidan'sThose first comments became the fodder for discussion–Were “hearts” and emojis really helpful for growing ourselves as learners? And they also talked about how we presented our learning online. One student expressed a chronic sentiment: “Sometimes I can’t hear them speaking. I think people should listen to themselves before they post. ” As a teacher, I loved this observation which really has improved their presentation skills overall. As a result, students have naturally begun to articulate how they really wanted to engage better online.

Through the Activities feature, students can peruse and select math ideas that they would like to view instead of just going to their friends’ journals. img_7460This has also helped to spread mathematical thinking around. I can see students nudge one another and say “Hey did you so-and-so’s idea? Go check it out!’

Aside from developing “friendly feedback and helpful comments“, we have been inquiring into how we can interpret and respond to these comments. My partner created a few “starters” for them to get them thinking about the need to be polite whether you agree or disagree with the person:

  • Thanks. I hadn’t thought of that.
  • Wow, that made me think that I can now…
  • Thanks but I disagree because….

Although we scatter these sentence starters around, it has been lovely to see them create their own messages, showing us that they have transferred the meaning and personalized it. img_7416Now we are at the stage in which we are encouraging and educating parents about how to make helpful comments and responses. It’s a bit hard to get them to “unlearn” some of the social media habits that we have as adults, so we get parent comments like “Love you boo-boo. Great work!” I hope that the students challenge their parents and ask them what they connected to in their post.

Since this is my first attempt at teaching younger students these skills, I am excited at how we can improve their communication skills through the use social media next time. However, I feel immensely proud of how serious they have taken their learning and their need to connect with their peer’s ideas. I feel confident that if we approach social media from this perspective, we can indeed shape and transform what social media can be like in the future.

I’m wondering how others have used Seesaw to develop these skills and what strategies they found successful. Please share in the comments below so we can all learn from each other. (:

 

#PYP: What is a Provocation?

#PYP: What is a Provocation?

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I love the International Baccalaureate but the jargon really can get you jumbled up, especially when you are new to the program. In the PYP, we use a lot of terminologies that others would just call “best practice”.  However, there is a word that pops up quite a lot: provocation.

Now someone might call it the “hook”, something that draws student’s attention into a lesson. But when I say “hook”, I don’t mean an attention grabber like a joke or cute anecdote or a routine of some sort that gets students on task. No, that’s not a provocation!   A provocation is a thoughtfully constructed activity to get students excited and engaged, but a really powerful provocation creates cognitive dissonance that throws kids into the Learning Pit (of inquiry).  Students should be examining their beliefs and ideas as a result of the provocation.

Here is a list of questions that were shared by Chad Walsh which can help filter activities and perhaps refine them in order to transform them into provocations:

  • Is the provocation likely to leave a lasting impression?
  • Is there a degree of complexity?
  • Might the provocation invite debate?
  • Might the provocation begin a conversation?
  • Might the provocation extend thinking?
  • Might the provocation reveal prior knowledge?
  • Is the provocation likely to uncover misconceptions?
  • Does the provocation transfer the ‘energy’ in the room from the teacher to the students?
  • Does the provocation have multiple entry points?
  • Can the provocation be revisited throughout the unit?
  • Might the provocation lead learners into a zone of confusion and discomfort?
  • Does the provocation relate to real life/their world?
  • Is the provocation inconspicuous and a little mysterious?
  • Might the provocation lead learners to broader concepts that tend to carry more relevance and universalitMight the provocation be best during the inquiry, rather than at the beginning?
  • Does this provocation elicit feelings?

That is a very extensive list, isn’t it?

Well, let me share a  few examples of provocations:

How We Organize Itself, The Central Idea: Governments make decisions that impact the broader community.

Students come to class that morning and are treated according to the government system that is being highlighted. (Example, Totalitarian) This goes on for a week and each day students have to reflect on what it was like to be a citizen of this type of government.

Where We Are In Place and Time, The Central Idea: Personal histories help us to reflect on who we are and where we’ve come from.

The “mystery box” (which I think originated from the work of Kath Murdoch): inside a box (or a suitcase, in this example) there is a bunch of seemingly unrelated items that students have to guess what the unit might be about. This is a “tuning in” activity. And since this is a central idea about personal histories, it might include a family photo, an old toy, some cultural artifacts or relics of things we enjoy doing, a clock, a map.

Math Stand Alone, The Central Idea: Mathematical problems can be solved in a variety of ways 

The  “sealed solution“: there are 5 envelopes that have the sum of two numbers “sealed” inside them. Students have to use the digits 0-9 only once to create those sums. What could be the sums inside?

Hopefully, this is helping you to discern what a provocation might be. Even if you are an experienced PYP teacher, reflecting and refining our provocations is something that is critical to developing our student’s learning and sparking curiosity.  A well-designed provocation will not only make it to the family dinner table conversation that night but will have a longer shelf life in a child’s mind and ultimately develops important conceptual understandings.

What have been some of your favorite provocations? What questions or engagements have led to deeper learning? Please share in the comments below so we can all benefit from your experience! (Thanks!)

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