Category: inquiry

When is Inquiry not “Best Practice”?

When is Inquiry not “Best Practice”?

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My co-teaching partner leaned across her desk and asked, “So at what point do we stop ‘inquiry into this’ and start teaching them something?”. I responded with nervous laughter because her question sounded like blasphemy. It was an absolutely valid question though and has lingered in my mind for ages, particularly since I am aware of Professor John Hattie’s meta-analysis on high vs. low impact influences on student learning. According to Hattie’s Visible Learning Rankings, inquiry-based learning has a low ranking on its impact scales. I do recognize that inquiry-based learning comes in a variety of flavors, which is well represented by Sarah Plews’ breakdown in her Concept-based Inquiry Mini-PD course:

So I have often felt skeptical of Hattie’s findings because I think generalizing inquiry-based learning is doing this approach a disservice. However, I have come to accept that some content lends itself better to inquiry than others, and some concepts must be developed in a more structured way. I think the reason why inquiry-based learning falls flat on Hattie’s Impact Scales is that many practitioners aren’t sure when and why we might use different approaches.

So, are there moments in our PYP classrooms when we need to put inquiry-based teaching methods on the back burner and provide explicit and direct instruction? Yes, yes there are! And that doesn’t mean we are tossing out all the other PYP practices. This is not a “either/or” situation–either we teach through inquiry OR we are NOT a PYP school. Having a strong PYP program means that practitioners know what content and skills would be best taught through a variety of methodologies.

Ok……so what content and skills might those be, you wonder? Well, here’s the thing, it’s not just the WHAT but it is also the WHO. We have to consider the needs of our students before we decide HOW best to teach them. So what students benefit the most from direct instruction? I think the guidance below is helpful:

Students who learn and think differently. Without explicit instruction, students who struggle with attention or working memory may not focus on the most crucial ideas in a lesson. With explicit instruction, you cue students in to the most essential information.

English language learners. When you use consistent and clear language in each step of instruction, English language learners (ELLs) aren’t overwhelmed with new language demands. Research has shown that explicit instruction can lead to achievement gains among ELLs.

From Understood.Org

When we consider our students with learning needs, it seems obvious that we might alleviate the cognitive load by providing direct instruction. Consider this-When students have a clear understanding of what they are learning about, they are not using their brain power for meaning-making; instead, they are using their working memory to process this information and store it for later use.  This actually helps them, later on, to engage in an inquiry because they have the knowledge to draw upon during a more open-ended lesson. It’s a support, not a deterrent to more inquiry-based approaches.

Explicit and Direct Instruction for the Key Conceptual Lens of Form and Function

One of the filters I use for deciding when we need to explicitly teach something is considering if this is going to be new knowledge to students. Typically it is more efficient to teach the key concepts of What is it? (form) and How does it Work? (function) through a more direct instruction approach when students are acquiring new understandings of concepts or skills. Students need to know the definitions of things and procedural knowledge. Let’s not leave them guessing or feeling confused.

Think about it- can we just send kids with a stack of books and have them generate theories about how language works and then teach themselves to read? No, that would be professional malpractice! We have to explain foundational concepts to students.

Image from https://www.evidencebasedteaching.org.au

Let’s think about the words “explicit” and “direct”, another synonym is clarity. You might also think of this as “scaffolding”, in which the teacher is explaining and demonstrating a concept or procedure, and then giving students the opportunity to have guided practice. Educational research shows that when we “show and tell” how something works, it has a greater impact on learning.

Students require more explicit instruction with some of the following content and skills:

  • phonics and decoding skills
  • research skills
  • collaboration skills
  • using mathematical tools and strategies
  • safety skills
  • spelling and grammar
  • understanding time and chronology

So what might this look like or sound like in a PYP classroom?

Sounds like: Today we will be unlocking our thinking through the concept of Form and answering the question, “What is a closed syllable?” . We will be successful at becoming knowledgeable of this syllable type when we can:

  • Define what a closed syllable is.
  • Identify what words contain closed syllables and which ones do not.
  • Read at least 5 words with closed syllables in them.

So you may be wondering, What is a closed syllable? A closed syllable is when a vowel is followed by (or closed in by) one or more consonants.  The vowel sound is ‘short’.  This pattern is often referred to as the CVC pattern (consonant – vowel – consonant), although a closed syllable does not always begin with a consonant. Let’s take a look at some examples……

Now it’s your turn, can you sort these 10 words into the categories of Closed syllables and Not Closed syllables? When you are done sorting, read your words aloud, do you hear the short vowel sound in the word? When you think you have sorted the words correctly, you can compare your answers with a partner…..

Okay, before we go today, I want you to whisper to a friend what a closed syllable is and write at least one new word that contains a closed syllable on this sticky note. You can put your sticky on our Definition Chart for closed syllables. And, if you find any other new words when you are reading, grab a sticky note and write it down with your name on it, and put it on our chart.

Looks like:

  • teacher-selected materials and examples
  • the teacher talks more than the students
  • the students are engaged in a specific task, with little agency over how it is done.
  • there is an explicit “right answer” based on attaining the learning goal.

A closing thought

Teaching is complex, especially in the PYP. Although I have made a stark generalization in what Key Concepts might demand a teacher-directed approach, I ask you to think carefully about your unique learners and consider how best they acquire the knowledge, concepts, and skills they need to be successful. There are moments when we are not actively soliciting curiosity, asking students to problem-solve, or generating theories in our PYP classrooms. And if you choose to develop a lesson that isn’t inquiry-based, it doesn’t mean that you are desecrating the PYP principles.

We want to emphasize that being ‘an inquiry teacher’ does not necessarily mean using an inquiry approach for EVERY lesson.

-Carla Marschall and Rachael French, from Concept-Based Inquiry in Action

So, let’s be thoughtful and skillful in our pedagogical decisions, taking a more teacher-centered approach and providing direct instruction when the moment calls for it. Perhaps if we are more mindful of the merit of the precision of explicit teaching, then our inquiry moments will be more profound.
Are We Asking “Beautiful Questions”?

Are We Asking “Beautiful Questions”?

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We are hard-wired to be curious. Have you ever been around a little baby before?  When a newborn begins to realize that they have a body and becomes fascinated with their hands, they study them intensely. They put them in their mouths, they linger on different textures, wanting to squeeze them to feel them oozing through their fingers.

We are born curious, our brains pattern-making machines, trying to make sense of our environment, both outer and inner. Our schools shouldn’t be a place where student questions go to die.  Schools should be a place where curiosity is nurtured and sustained.

visual questionsIn  The Book of Beautiful Questions: The Powerful Questions That Will Help You Decide, Create, Connect, and Lead,  Warren Berger pronounces “I am a questionologist.” I love that! When you look at the graphic that summarizes Berger’s book, you get a sense of possibility that deepening our inquiries can create through broad questioning techniques. The questions are not complicated, but the path they lead you on can branch into new avenues and creative opportunities. As educators, we should not only be modeling these broad-reaching questions but encouraging tangents of thought through open-ended questions.

A poem comes to mind which reminds me of the wonder and inspiration within the power of a question. Its words penetrate my soul and awaken the child within me, the one with a million “whys”.

Be patient toward all that is unsolved in your heart and try to love the questions themselves, like locked rooms and like books that are now written in a very foreign tongue. Do not now seek the answers, which cannot be given you because you would not be able to live them. And the point is, to live everything. Live the questions now. Perhaps you will then gradually, without noticing it, live along some distant day into the answer.

-Rainer Maria Rilke

When I consider the excitement of beginning a new unit of inquiry, despite its familiarity, a fresh set of questions always come to mind. Just like the students, I am there with them, embarking upon the inquiry, seeking new understandings. I want to “live the questions” that Rilke speaks of, knowing that curiosity is a way of being in the world, experiencing awe and elegance in the search for answers. It is more than a pedagogical approach, it is a way of being. 

the searchSo to develop our “questionology” is not only important for our classroom culture but it when you think of it, it generates well-being. To question is to shake hands with possibility, and possibility opens our focus, inviting new information into our awareness.  So this drive to wonder is what makes us  “inforvores”, and is a psychological need. In fact, science is beginning to show that if we are not organizing our classrooms in such a way that spark interest, we are literally deadening the brains of our students. I’d also like to add that our own teaching practice becomes joyless when life is all answers and no questions.

So let’s take a page from Berger’s playbook and start generating opportunities for curiosity by asking more “beautiful questions”. It’s a habit worth cultivating.

 

 

 
Designing for Humans: Thinking Beyond a Checklist for the Enhanced #PYP Planner

Designing for Humans: Thinking Beyond a Checklist for the Enhanced #PYP Planner

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This past year we trialed a new PYP planner, and the intentions were good with letting the students’ responses to our provocations direct and lead the unit, but we ended up having a planner that was so complex that it became cumbersome to actually fully complete. It was christened “The Big Book”, which should have clued us in that this was an exercise in paperwork. Clearly, it’s back to the drawing board.

So what are “The Basics” that have to be on the planner? As I see it, there need to be 12 components that are fundamental to the planning document:

  1. Transdisciplinary Theme
  2. Central Idea
  3. Lines of Inquiry
  4. Key Concepts
  5. Learner Profile
  6. Approaches to Learning (ATL)
  7. Questions
  8. Provocations/Engagement Activities
  9. Resources
  10. Assessment
  11. Action
  12. Reflection

As I began to wonder what is the “special sauce” that would move a planning document beyond “the basics” and make this planner “enhanced”, I decided that I needed to go back and listen to the webinar that addressed this aspect of the enhancement.

My big takeaways from the webinar were:

  • The document takes us through a PROCESS of CO-CONSTRUCTING learning.
  • It encourages COLLABORATION with staff.
  • It fosters REFLECTION.
  • It not only documents STUDENT AGENCY but reminds us that this is central to the learning. Teachers need to consider the WHO just as much, perhaps more so than the WHAT.
  • It influences the ROLE OF THE TEACHER and how they inspire ACTION in students to support SELF-MANAGEMENT skills.

While I considered the ideas shared, I was thinking “What would be the purpose of even re-designing the PYP planner?” I mean, they have given us a “refreshed” and updated example that we may use and other schools have already created other templates that could be integrated into our school. Truly, there is no immediate demand that schools HAVE to create their own planner.  But now schools have the liberty to design their own, yet it isn’t a mandate. So, if schools were to embark on creating their own, it would only be for the sole purpose of improving their collaborative planning at their school in an effort to increase student agency.

Agency is about listening.

Sonya terBorg

As I contemplate the benefit of redesigning the PYP planner, I wouldn’t dare create a copy and paste version of the templates shared. Not because they aren’t wonderful, but because they aren’t unique to the needs of my school.–which would be the purpose of even embarking on this journey. I 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 (7)In my past school’s pilot of the re-designed planner, it was a hard copy only. This worked well for our initial planning session, but on-going additions to the planner weren’t possible unless you were to have the hard-copy in your possession. And because it was a “big book” it took up a lot of space on one’s desk area, which became problematic since we had 6 Units of Inquiry plus 6 stand-alone Math planners. You might imagine the frustration of all those paperwork piles in one’s workspace, which created a disdain for planning since it meant that one teacher had this A3 sized booklet taking up a lot of real estate on their desk. This was an unintended consequence of going “retro” with our planning. I wouldn’t recommend this. So, with this in mind, if the planner isn’t digital, with equal-access available to all teachers, then it’s set up to fail. That’s like putting square wheels on a bike–it is taking us nowhere with collaboration.

With this in mind, I would utilize Design Thinking, focusing on human-centered design principles of really understanding what would be the needs of the users of this planning document. Also, since human-centered design considers the interaction along with the actual “product”, the experience is of vital importance. Here is the overview of the process:

designhc
Designed by Dalberg

Framing the Context: Understanding our Users and Their Problems

Human-Focused Design optimizes for human motivation in a system as opposed to optimizing for pure functional efficiency within the system. -Yu-Kai Chou-

What is the challenge: Let’s be honest, the main reason why teachers don’t appreciate using the PYP planner is that it seems like a time-consuming document that doesn’t seem to support their day-to-day planning of the unit of inquiry.

So how might we design a planner that is collaborative, compelling and ultimately results in better learning outcomes and increased student agency?

Hmm…..

In the first phase of design, Planning, we have to consider the audience for this document. Teachers, right? So, when we consider feasibility, we should ask ourselves what might be the biggest barrier that we will need to overcome in order for this document to work?

I’m rather practical so as a teacher, I would say TIME poses the biggest challenge to collaboration.

Thus, when we create this document we need to think about the amount of time it might take to fill out this document, especially since we might imagine that the initial planning will involve multiple teachers who represent a variety of subject areas. Trying to get all those educators in a room can seem like putting the planets in alignment. So, if we UNDERSTAND these teachers, then we must take into consideration that this document will most likely require at least 40 minutes of time to begin the planning process, with opportunities to plug into the document to give feedback and feedforward into the learning (at least another 30 minutes of individual or grade level teacher time). Lastly, there will need a final block of at least 40 minutes for teachers to get together to reflect on how students responded to this unit of inquiry. So, with that in mind, the document, from start to finish, needs to be completed in 3 planning periods; 2 of which will include multiple voices and perspectives in the room, and at least 1 planning period in which teachers or a grade level team get together to discuss how the unit is progressing and what direction it might need to take. So let’s just say, this collaborative document takes at least 2 1/2 hours to complete, give or take 1/2 hour.

Then, as we peel the layers of the onion, we know that the 2nd biggest challenge will be ensuring that this document is truly collaborative, with the opportunity for multiple voices to be present, particularly our subject area specialists, who often feel marginalized during planning.

Furthermore, this document must create a holistic process of learning about our students, so we can create learning opportunities for our students, in that we can examine what learning came from our students. It has to fuel conversation and inspiration among teachers to develop student-directed inquiries and motivate student-led action. Moreover, it should get teachers discussing how they can access the larger community, whether local or global, to tap into resources that expand the learning outside the 4 walls of the classroom.

Lastly, when teachers engage with this document, I would want them to feel excited and anticipating the best that could happen during this unit of inquiry. I wouldn’t want this to feel like “ticking a box” but instead designing learning that changes lives. (Because, truly, that is what we are doing, every day. How cool is our job, right?!)

Learning Phase: Perspective and Use by Teachers

I know that this planner has to contain the “Basics” but I’d think about the teachers first and not the “boxes” that it needs to tick. Already I’ve made some assumptions, such as identifying some barriers and challenges to using the planner. However, those are inferences and my own biased opinions. I have yet to tap into the perspectives of the teachers directly at my school, which might produce different ideas. I must put on my researcher hat and use some of the methods of Human-Centered Design to get an accurate picture of the challenge and its possible solutions.

empathymapdesignFrom a design point of view, I might start from one of the PYP planner templates shared, observing teachers “in the wild”, using the document during the collaborative planning process.  I would record reactions with the Empathy Map to evaluate their experience with the planner. Since I’m not just considering the physical experience with the document, I need to collate the responses of the emotional experience of the teachers when deciding how to help craft a new one. Remember, I’m not trying to devise a fancy planner, I want the planner to actually get teachers to have rich discussions that connect and extend the learning of students so that students can ultimately become self-motivated and feel a great urgency to take action. I’d need to be a fly on the wall, leaning in to listen and notice how planning is being “enhanced”.

Brainstorming Ideas

First of all, this is not me, alone, on my laptop or with a pad of paper and pen in hand, ready to sketch out ideas. It takes a team to cleave through the data and create mock-ups that will ultimately result in a prototype document. Every one of those template planners on shared on IB’s PYP resource page took a team of dedicated individuals to shape and mold the prototypes that we see today. And I use the word “prototype” very intentionally because no doubt these planners will evolve as those teams reflect on what works and what doesn’t work with its use. Just as our teachers have spent time reflecting and evaluating the “big book” planner that was created at my past school, all schools need to stand back and be critical of their work so that it can be refined and improved upon.

So when brainstorming ideas, it will require a group of diverse and interested educators who will not only ensure it contains “The Basics” of PYP principles but develops our teachers understanding of our student learning and improve collaboration among teachers. That’s a big ask. Needless to say, where we go from here is To Be Continued…….

If any brave and like-minded individuals want to share how their school is approaching this project, I’d be keen to hear more. Please post in the comments below so everyone can benefit from your learning and experimentation. 

“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”?

#PYP Déjà vu or Jamais Vu? Approaching Familiar Units of Inquiry in Unfamiliar Ways

#PYP Déjà vu or Jamais Vu? Approaching Familiar Units of Inquiry in Unfamiliar Ways

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Picking up the strand of LED lights, I felt overwhelmed at the Chinese Hardware Market, I had this disorienting feeling that I’ve been here before, discussing the color of lights in broken Mandarin. As I walked out with 2 meters of lights, I felt like I was in a dream world, realizing that this whole experience was a  déjà vu.

But having the luxury of teaching a unit of inquiry year after year creates the same experience.  You read over last year’s planner, reliving the experience and ready to proceed in the same way. Easy, right?  Then you can tick that off your To-Do list and move onto other things like setting up your classroom or having meetings. But this year, I can’t do that. I’ve promised myself to take myself and the students “where the streets have no name” and that means that I have to approach units of inquiry from a stance of jamais vu, selectively having amnesia about what provocations and activities we used in this unit.

So why on Earth would I toss aside all the thoughtful planning of the past? Because it’s the past. And we’ve grown professionally a whole year since our team originally designed that unit. Yes, we may be re-inventing the wheel a bit, but our experience and knowledge require us to develop more dynamic and empowering units of inquiry. We know more pedagogically. Moreover, we have a whole new group of students, with new interests and questions. We need to readjust our sails because we are going on a whole new adventure.

So when we examined our current Who We Are unit (Our choices and actions define who we become as a community), we decided to use “the end”, with a water-downed version of our summative task, a “learning fair”, to begin our current unit. It made sense that they needed more practice making learning choices so they could cultivate their self-identity and self-management skills. Now we can use this data to reflect and refine how we might use this jumping off point to have them become leaders in their own learning.

5_album_photo_image
Making choices helps us to appreciate how they see themselves as learners.

I think using the end as the beginning is an approach that we may use again in future units because it provides the context for all the skills and knowledge that we would have “front-loaded” on the students in past units. For example, last year we did several lessons on Kelso’s choices and How Full is Your Bucket?  before we gave them the agency to make learning choices. How silly, right? It’ll be so much better having the context of conflict as a provocation to really engage in deeper conversations. If we bring these resources into the unit, it would because the students needed it, not because we wanted it, because it was on LAST year’s planner.  In fact, coming from this angle has really helped us to see how capable and eager our students are to be in control of their learning. Maybe we don’t have to waste time on the previous year “staple activities”.

As we embark on another year of learning, I intend to embrace the jamais vu, putting old planning aside and coming at familiar units from unfamiliar approaches. And I wonder what insight the children we give me about how I can amplify learning and empower them. This is what I look forward to so much: I grow as they grow. How fun is that?

#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.

 

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