From Patterns to Algebra: The Genius of Dr. Beatty and Dr. Bruce

From Patterns to Algebra, the second that I was given the chance to explore, play, and investigate, stands at the forefront of an amazing patterning and algebra resource.

This blog is essentially a love letter to Dr. Cathy Bruce and Dr. Ruth Beatty — and how they seamlessly weave play, patterns, and rich opportunities for students and teachers to immerse themselves in the joy of patterning (Hey!  I said I was going to write the blog, so here I am):

I’ll let the introduction of the book speak for itself (as seen on page 1, and in the pdf sample you can access here):

Mathematics has been called “the science of patterns” (Steen, 1988). Young children enjoy working with patterns, and older students enjoy discovering and manipulating patterns. In fact, it is human nature to nd patterns in our everyday experiences. Some educators and mathematicians would go so far as to say that patterning is the foundation of mathematics (Lee, 1996; Mason, 1996)

Let’s start with how the first lesson revolves around playing a game:  Guess My Rule.  Right away this game is A) super fun, and inviting to students — all of students really loved playing it, and B) gets students to make the relationship between term number and the output — and how the two interact.

My students loved this game so much that this happened:

(also cool is how this student used Google Draw to create the table, screenshot it, and add it to Kahoot).

My students willingly played this game — and often.  I added some elements by using cards- whatever you flipped up became the pattern rule — and putting the black line master in page protectors so my students could play when they were finishing other work – or for recess (yes, a lot of my students like to stay in and play math for recess).

Playing Guess My Rule was fantastic, because, when we got to creating and looking at patterns, we already had some understanding of the relationship between input and output.

This is another reason why this resource is so great — the patterns are visual.  Everything students do with this resource is creating and analyzing visual patterns; making students see how accessible and tangible patterns can be.  Students can manipulate patterns, and can play with them – and feel like anything they create can easily be added to, changed, or redirected without losing the visual representation.  Students were able to make connections quickly – and talk about the patterns we were examining.  Students could also identify the pattern quickly, move to graphing linear patterns, and determining the nth term.

Below are some of the awesome ways we used the resource — and really, we’ve only just touched on it.  Dr. Beatty and Dr. Bruce have created a place where non-intimidating patterning play can happen; and deep connections can be made.

See the images below – -some have captions so you can see a little more about what we did as a class.

Students used Ontario Mathies Coloured Tiles to play with, too.

Students created patterns, and went around and wrote down the pattern rule. This lead to great conversation. Is there pattern reflective of their pattern rule?

I laid out patterns with constants for students to guess.

Students talked about the pattern, then checked the pattern rule to see if they were correct.

 

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The Four Step Problem Solving Makeover

There is a lot to be said about the problem solving model that sits at the front of the Ontario Math Curriculum.  It’s common place to hear and see teachers discussing and using a model, or a similar model, in their classrooms.  It seems for some, it works, or, at least, it gives a framework for teachers to assist students to work through a math problem.

However, it never fully sat with me quite comfortably.  First of all, I will make many non-friends writing this, but, it created zombie-like problem solvers.  I witnessed many students who would be messy, take risks, and think during their math problem solving.  The four step problem solving model was introduced, and suddenly, I found these creative math thinkers felt like they had to follow a path, or a formula, to solve a problem.  They weren’t thinking as richly and deeply, and, in fact, were rushing to get through the steps rather than taking their time, and really questioning their math choices.

The Polya Four Step Problem Solving, as seen on page 13 of the Ontario Math Curriculum.

I have re-looked, re-examined, and re-thought of this model many a time.  In fact, it has been the focus of math workshops and district meetings I have attended, and I not once could say that following this model in a linear manner made any of my students any better at math.

Don’t get me wrong:  I see it’s purpose.  I have found it most helpful with students who struggle to organize their mathematical thinking, and I present it more of a way of thinking, or a framework, they might want to consider if they find they can’t focus on the math, and are getting jumbled in the process of their mathematics.  I never ‘forced’ a student to use it, as I always felt that solving a math problem – or any real problem in life, really – isn’t linear.  Many times we might jump to carrying out what we think is right, find it didn’t work, then go back to realize we probably didn’t quite understand.

I recently revisited the front matter of the the Ontario Curriculum to really investigate the process expectations, as I was thinking to myself:

An overview of the 7 Mathematical Processes as highlighted in the Ontario Math Curriculum (page 12)

Don’t we go through processes of thinking while we are solving mathematics tasks, and also taking steps to do so?  Could the two not be intertwined?

But, then I saw this:

1945?!?

A problem solving model hasn’t been re-investigated since 1945?

So now my head is really spinning.  I wanted to know a few things:

1. What do students really do and think while they are solving a complex mathematical problem?  Do they naturally go through ‘steps’?
2. What processes do they consider, or act through, when solving a math problem?
3. How do we bring a problem solving model back to the actual mathematics?

In my mind, I started to visualize what I had noticed my students were doing during math.  Over the months, I have been able to listen to many math conversations, and sat in on many math problem solving sessions with my students.

Right away, I wanted to depict how mathematics is a flow; people move in and out of their thinking process depending on what they are working on.

I feel that this is what my students do in the classroom while they are working on math.  More than that, this is what I would like my students to consider as they solve a math problem.

I decided to take it one step further:  I asked them to consider their thinking while they were working on some math.

I created a Google Form, and asked students to reflect what they were thinking before, during, and after their math task.  Here are some results.

Before:

You can see the top two for before were noticing and wondering the math information, and choosing a tool.  This was great — this means they were actively processing the problem, and selecting a tool (which is also one of the aforementioned process expectations).

During the problem, students were fixing mistakes and changing their ideas.  This means students were actively seeking math connections, and finding and fixing their misunderstandings.  Using a tool is still high on their choice, which means that tool must help them represent their thinking.

Students did reflect at the end, as the Polya Problem Solving Model states, which in the re-vamped model is embedded in the questioning themselves area — students, even at the end, are ensuring their mathematical ideas are complete.

I was really excited to see my student responses, and I am going to continue to check in with them as they work through math.

Feedback, please!

What do you think of my re-vamped Mathematical Thinking Flow Chart (because I am not sure what else to call it)?

Would you use this in your classroom?  Do you see your students doing this?

Walking the Fault Line: The Anxiety of Straddling Slow Math with Preparing for Standardized Testing

Earthquake Regions

I went to UBC for my undergrad, an experience that changed my life for many reasons.

The biggest shift was the immense beauty of ocean meeting mountains.  Never had I experienced this before, and to be thrown in and living in this beauty was something I never really became totally comfortable with.  It was almost always to good for me.

I grew up in Southwestern Ontario, where the Canadian Shield grounded me in safety and security in terms of natural disaster worries.  Suddenly I had to learn and be prepared for tsunamis and earthquakes.  It was strange and different for me, and although I loved and appreciated the beauty of B.C., I was always constantly thinking about the next possible earthquake.  Would I be in B.C. when the “big one” hit?  What would my reaction be?  Was there really any ‘safe’ place?

Perhaps silly, I had to mentally find a balance between appreciating where I, at the time, lived, and not constantly worrying – that small nagging voice in the back of my mind – of an earthquake of epic proportions.

The Fault Line in Teaching

At this point in my teaching career, I am having this giant existential-like questioning about why and how I am teaching.  In my mind, I am constantly not ‘doing it right’.  There is always something better I could present to my students, always a question I could have asked, and always a connection that was not made.

It’s like I am back living in B.C.:  I love teaching.  I enjoy the beauty and energy of teaching.  But I am standing on a fault line where that huge earthquake could come at any moment.  I am never truly comfortable.

Tectonic Math Plate Movement

In Mathematics, I feel that this is especially true for me.  As I learn more about Mathematics, as I do my best to bare my misunderstandings and misconceptions, as my personal philosophy of Math is developing (math is play!), I know that my ideals in education are shifting.  There is movement below the surface, and I just don’t know where to take this movement, and how to channel it.

What I mean is this:  I am a bigger picture teacher.  I teach in order for students to find meaning for themselves; to develop a sense of self in this overbearing world.  I want my students to be ok with not having the right answer; to not have to be perfect.  I want math for them to be interwoven in their lives, to be problem solving — to be an action they take with pride and honesty.

The shift is happening as I stand on the fault line of giving my students the bigger picture experience, and keeping my instruction focused and channeled.

Can really doing the math create successful EQAO results?

There.

I asked the question.

Here is where I am at, and what I mean by that question:  My students will be writing EQAO this year.  I am not about to discuss EQAO in itself, I am going to discuss how the act of performing EQAO is generally approached.

Here is an example.

Teachers engage students in using EQAO released questions (or EQAO like questions – focusing on application and thinking) to guide their instruction in a variety of ways, for a variety of reasons.  For example, to see if their students are able to understand the problem, if they can apply a workable strategy to solving the problem (was the student able to identify that this problem required more than one mathematical ‘step’ in order to be successful?  If not, what is our next step as educators?).  Are students able to reason and work their way through multiple choice, and have strategies if they are not sure of their choices?

Yet, my question is always this:  Will I be able to let students go deeply into all the math content so that they are ready for EQAO?

This is where I stand.  On this fault line of giving my students a rich, immersive math experience (and, I am still working and learning this!), and wanting them to feel like they can apply their learning in a testing setting.

Richter Scale

I tried something different this year.  Rather than make EQAO something students would “have to do, so we had better be prepared”, I had them examine the math in an EQAO released question.  I had them determine which tasks and problems in the past we had worked on and make connections.

This was me trying to bridge the fault line.

An example of this would be proportionality, and unit rate.  I started by asking students what was the best deal in a morning activity through grocery store photos I had taken (nothing new here), and to my surprise, students self taught unit rate — through no instruction.  Obviously students used other strategies (adding/multiplying the number of items until they came to the total cost, or estimating a reasonable cost of what one would be, then multiplying until they came to a close total), was one strategy, but many figured out the price of one by dividing number of items/weight and cost.

I gave them time to talk and share strategies, and discover if this was always true.  We played with the idea of proportion and rate, and compared it to ratio.  We made connections between ratio and rate — and how results can be effected by ratio.

But the nagging fault line struck me again:  Would students connect this new, and, believe me, we could still go a lot deeper into proportionality, learning to an EQAO question?

For a Board/Ministry Initiative I am involved in (Renewed Math Strategy), we were asked to give students a unit rate (well, a problem that can be solved through unit rate) problem, and at first my students had no idea how to go about solving it.

Even though they were disappointed at first, they begged me to go over the problem.  They wanted me to walk them through how it could be solved.  They wanted to know.  This struck me — if anything, I had created a classroom where, sure, they didn’t know what to do, but they had a desire to know.  They didn’t get wallowed in defeat.  They understood the yet — they weren’t sure how to go about solving it…yet.

Weeks later, after some exploration into unit rate, I decided I would have them own this problem.  I would let them figure it out – and on their own.  They would try the problem again, and they would self evaluate.  I gave them the codes; I had them lead the conversation, I had them determine and compare their previous work to their current understanding.  I had them discuss and identify what math they would need to use and apply in order to be successful.

I admit I did this rather randomly.  My purpose was for them to own the math.  I wanted to walk on that fault line.  I wanted to prove to myself that them as a team experiencing and feeling and playing with proportionality would transfer to a place where they would have to explain and justify in a testing manner.

Students are conferencing and working together to figure out the math they needed to use and apply. They themselves shared strategies.

 

Amazingly, the dialogue was rich.  My students were saying things like, “Oh! I see how when we found the best deal with the store prices, we had found the unit rate – we can do this here!”

“Oh, I see how you and I came up with different strategies, and I see where I went wrong – I forgot the last step of subtracting.  I see now how that makes the total cost make more sense.”

Before and after my students investigated and played with the bigger ideas of unit rate.

My students proved something to me — that really immersing in the content of mathematics, even though it is slow, and I feel a major pressure to go fast and cover more, is important.

My fault line fear — the earthquake of pressure to make sure I cover it all, and well enough that they can do well on EQAO, is still nagging me, but I can breathe a little more knowing that I am at least giving my students positive math experiences.

 

 

Math Self Talk: How can Collaborative Math Talk turn into Math Self Talk?

If you are a student in my classroom, every morning when you walk in, you are greeted with math talk and thinking right away. I greet you by name at the door, ask how you are, and then you move to the whiteboards or collaborative tables. There might be images for you to notice and wonder on, there might be a problem I want you and your team to tackle. There could be math tools laying around, and tasks for you to think through and explore. I have prompts for you to start the conversation.

Either way, before the bell, before O Canada, math is happening.

I purposely don’t interact right away. This is my time to check in with each student, quickly do attendance, and let the math conversation flow. Let’s be honest, the first thing you and I do at a PD session is catch up — talk about what happened since the last time we talked, and the kids do it, too. And it’s ok.

Because guess what? The math comes. Just wait. Watch for it. Listen for it.

It’s so nerdy to say, but it is my favourite part of the day. I come in to work to hear what they will do with what I put in front of them. I am excited to learn from them. To get a small glimpse into their beautiful brains.

Once attendance is done, I sneak around to the groups who are working. I listen in to the conversations. This is when I decide if I need to intervene or not.

I don’t tell them they are right or wrong. I don’t tell them how to solve, or where to take their thinking, I ask questions.

My goal through math talk is this: That students become aware of what they are thinking while they are doing math. They can do math self talk. I want my students to be good self math talkers.

Why?

It’s pretty simple.

I feel like there is still a shame when someone has to think and process math. Math is still awarded for speed and accuracy. Worse, when a math mistake does happen, many people flounder, shrug, and then give up.

I at least want my students to have the perseverance to reason through their mistake. I want them to live in a world where we are patient math thinkers. This will only happen if I teach and value patient math thinking. I want them to value if they fall off of a math track, they get back on. Even better — if they encourage and assist someone who is struggling by asking them to share their thinking, and help process through it. I want to create the next generation where it is socially acceptable to ask for help with math.

So, this is why I spend so much time talking math. I want math talk to be promoted, valued, and second nature.

To give you a feel for my room, here is an example of a math conversation I was honoured to overhear recently:

I took an EQAO MC and had students attack it without numbers first. I then added in the numbers later.

Student 1: OK — well, the books are in a box, in a container — so the amount of books is going to get higher. We are going to have to multiply at some time.

Student 2: I see that — but we need to find the cost of one book. We are given the total cost. What would be reasonable here?

Student 1: Books are kinda expensive — I think around 10 bucks? Should we play around with that?

Student 2: Hmmm…the total cost is $2592; I think a lot lower? Maybe books are cheaper in huge quantities like that?

Student 1: Could we try a cost then keep multiplying up? There are 12 in box — what if we started to 12? Would that get us the cost of a box?

Student 2: I don’t know if that would work every time, I think we need to divide somewhere too…

Wow. So much going on here. Student 1 and 2 are working off of each other’s ideas, questioning, and giving context to the problem, and playing with the idea of reasonableness.

I was thrilled!

But…

You see, here is where I am at. My students are getting pretty good at talking through their math with each other. I hear them correcting their thinking, changing their minds and their trajectories, without blame or negativity.

The more they do this, I assumed, the more it would become an innate independent practice. I assumed that since they were doing it with each other, they would be doing it in their own minds when given an independent task.

But it doesn’t seem so.

So, yes — I want math to be a social and collaborative language that everyone is fluent in. It seems like we are heading that way.

But now my goal is shifting. I want that math talk to turn into self talk that happens as students are working through the math task. I want them to imagine all of the questions and visuals and ideas in their minds as they solve math.

To start to move toward this, I have started consolidating our math talks with conversations like this:

• How will the math talk you just particpated in help when you are thinking on your own?
• What math words/visuals/strategies were easy to hold in your head?
• How was working through this problem going to help you through future problems?
• What did someone else say that stuck with you, made sense to you, or you want to try?

The fluidity of self and peer talk is so important — it’s where thinking becomes learning, and learning turns into new thinking.

40 Cantaloupes and Train Speeds: Why Math Can Be More Than Just a Problem to Solve

I have a dirty little secret: As a kid, until I became a teacher, really, I did not like math. Nor was I particularly good at it. I oscillated between Cs and Ds pretty much my entire school career.

However, when that dusty math textbook was opened, I found solace in those wordy math problems. You know, those ones about trains and speeds and stations and so and so buying 50 apples and comparing them to someone else who bought 30 apples.

They made no real logical sense (I mean, really? Why are we buying so many apples?), but, I relished in the language of them because I could do that: I could read and understand the senseless problem. Although I was always happy when one of these text book math stories came into view, I realize now, that, even though they were soothing — the pacifism of safe problems — it did nothing to increase or challenge my notions of math. Nor did they help me retain any of my math knowledge. I plugged in some math calculation, and crossed my fingers I was right. Then I never thought of it again. Book closed. Bell rang. On to something else.

I am not the only one with this experience, and interestingly, I find that as a teacher, we weigh so much on the problems we choose to give students.

We ask ourselves:

• Is it engaging?
• Is it logical?
• Can all students access it?
• Can it be solved in different ways across different strands?
• Will it make students think?
• Is it a higher order problem requiring students to apply their math understanding, not just regurgitate it?

These are all good questions.

If you are like me, you are always on the hunt for interesting and engaging math problems. There is a pressure to make sure your tasks are open, and are accessible to all students. There is a pressure to make sure that your problems push student thinking.

But, if you’re like me, then this is also not enough. Not only do you want these awesome engaging and open tasks, Marian Small in Canada is your go to for this, generally, but you want tasks that invite students to use specific math tools or manipulatives. (In 2018, why is “manipulatives” still something that receives a red you-spelled-it-wrong underline in so many computer programs, BTW?)

So, then you start to tweak the task, or, continue hunting, so that your students can use tools to represent and communicate their thinking.

After this, you go through the motions of a three part math lesson. You do the problem, students seem engaged, and you consolidate. You might have them Gallery Walk, walk through their thinking, and pull out highlights.

I find teachers can get ‘caught’ in this cycle. And, don’t get me wrong, for the most part, this is not a bad cycle to be in. Kids play with math within the problems, they are exposed to using math manipulatives, and highlighting their thinking.

But I have come to realize a few things: The problem is important, yes, but it’s not all that math class is and can be. Like those wordy senseless problems I’m sure we’ve all tackled, they may have soothed us; directed us to insert correct numbers at correct times, but did they challenge math stereotypes?

So, yes — Keep finding and using those awesome math problems, but, consider the other portions of math class that complete the image of what math is — a beautiful language that many do not get to experience.

Talk Math:

Sure, we do the think-pair-share thing, but it’s actually a process to really talk math; to get your students truly invested in sharing their conjectures, arguing differences, and seeing math presented to them in a different way — and not from teachers, but from other students.

In order to talk confidently about math, we have to make mistakes in math, and know where we made the mistake, and how to talk to it. It also helps if we can illustrate and/or represent our thinking in a clear way. The visual will help other students see the process the student was mathematically engaged in, and be able to compare, and talk to it.

These students are fully engaged in talking about math.

Wondering: Do we value math conversation?

Make Math Visual:

That’s right, noticing and wondering in our everyday lives begins to open the doors for students to make connections in math. It helps them see how we can dissect, appreciate, and understand math and how it interacts with the world. Find art, images, go for walks — and talk about the math. Let students tell you what they see, and honour their thinking.

Student thinking around a racing image. Students began to wonder about decimals and placement of the runners.

Wondering: Do we value how students perceive how math moves in their world?

Play Math:

Math games are great; so are logic puzzles. But, to me, it can also mean just messing around with a concept, an idea, a mathematical wondering. One of those open problems might ignite that math play. “Just try it!” “How would you test or explore that?”. This is where mathematical tools might get used, where students might want to use various technologies to support their investigations.

 

Tiling turtles at the math play table invite students to create and admire their tessellations. A more guided approach, on the right, has students use specific pattern blocks to make relationships between their creations, and mathematical content being studied.

Wondering: Do we value the learning from playing with math, even in unstructured settings?

Math is a Verb:

So do the math. Let it be an action rather than a statement. How can we get students to be in the math, rather than looking at it through window-like glass, from the outside? How can we encourage them to dive in, and tell them it’s ok if they need to come up for air?

Do we value the verbs in the Math Curriculum? Investigate, explore, student created algorithms? What does this look like in a classroom?

So, please — keep finding those open problems, and keep having your students solve problems. But the math may not come alive for all through a problem. For some, the problem may be too safe; a series of steps to complete. So let students experience math in different ways: Exploring images, talking about how one mathematical idea led to another, or how they researched a mathematician.

Let math spill into the crevasses of your classroom. Let students see how math inhabits the corners, leak into their thoughts, and be a part of their understandings.

MISA Grant Part One: Guided Intervention in Mathematics and use of Immediate Feedback

Note One:  I Applied for Me

Like many educators out there, I am always pushing myself, and questioning myself and my teachings.  I am inherently in love with learning; and naturally seek out research and PD.  As mentioned previously in many of my blogs, I was especially inspired by @MatthewOldridge — especially when during my Math Specialist course I really became interested in what purposeful mathematical play could look like (sidenote:  Still don’t have a full answer to that one yet), and this made me want to look into a TLLP.

But, at some point, the MISA Grant crossed my path, and, to be honest, I wasn’t ready for a TLLP (for many reasons), but I wanted to push myself, and do my own research.  This seemed the best fit for me – it was a smaller grant, true – but it would let me try out my own processes of classroom research.

Note Two:  Focus

Though I am still currently working through how math play can work, I wanted my MISA grant to be slightly more focused.  I decided to hone in on guided practice in mathematics, specifically, was there a way to work with intervention students through the use of interactive math technology, and immediate feedback?

How could I implement explicit math with these students, who needed to feel, experience, and immerse in the math, and, hopefully, be able to communicate and share that math within various contexts and problems?

How would that look for me?  For the students?  How would I know if mathematical understanding occurred?

My grant was granted.  Now I am going to do my best to blog about the journey.

Note Three:  I Started Small; and I am Still “Starting”

First, I say “I”, but this is a team effort between myself and the awesome @draperconnects.  We decided we would make flexible groups based on the strands we were focusing on at the time, and take the time to organize our groups based on who needed more specific math content.

I started doing research as this time — and looking at how others have given immediate math feedback to students as they were working.  See below for what I found in my relatively brief search.

Since my class was already used to the Google Classroom, it only made sense to use the comment features to give feedback.  Now what programs would best help students illustrate their math thinking, so I could pin point and assist in their content, and their representations and communications?

We tried Explain Everything, which I could see and watch my student thinking, and then I can give immediate feedback.  I need to play more with this app, or any app/Chrome Extension that will let me see how students are thinking and manipulating math.

but how will this get to the math?  What is more important in this grant – content, or giving students problem solving strategies so they can apply their math knowledge and understanding to a variety of problems?  They are both one in the same; I would imagine.  Both skills that need time to develop and act.

Note Four:  I am Still Thinking and Considering…

..and letting the ideas percolate and simmer in my mind about how I can make this a more focused and explicit experience.  I want the students involved; those who move through my guided math sessions, to feel empowered about their math learning and their math experiences.  I want them to feel confident to show their math reasoning, even in situations where they may be uncomfortable.

Any suggestions?  How are others working through guided math, that helps students reason and represent, communicate and consider, their math thinking?
What tools have you used to give immediate feedback to their math, so they can be reviewing their own math thinking, and become more aware of their mental process when doing math?  I would love to hear from you!

Links:

1. RTI and Mathematics:  IES PRACTICE GUIDE – RESPONSE TO INTERVENTION IN MATHEMATICS.  https://rti4success.org/sites/default/files/rti_and_mathematics_webinar_presentation.pdf

2.  Best Practice for RTI: Small Group Instruction for Students Making Minimal Progress (Tier 3). Reading focused, but applicable to math education.

http://www.readingrockets.org/article/best-practice-rti-small-group-instruction-students-making-minimal-progress-tier-3

3.  Book excerpt – RTI is a Verb.  Tom Hierck, Chris Weber.

https://us.corwin.com/en-us/nam/rti-is-a-verb/book241454#preview

 

 

Math Play in the Junior Classroom: Some General Observations

Before I start to document, I want to admit that I have not done as much around math play as I have wanted.  When I started to realize this, months ago, I felt guilty and upset with myself.  I had been so motivated and driven to really explore what math play could look like in a junior classroom.  Not only that, but after I decided what it could look like, I was eager to move my thinking into:

• What are students truly learning and creating through math play?
• What makes effective math play?
• How can play develop math dialogue and thinking within math conversations and tasks?

Upon further reflection, I realized that I should not feel guilty or upset.  Letting the math play just be — play and mathematics — is an even better place to start.

Just about this time, this tweet from the awesome @Mathgarden (with mention of other amazing math leaders on Twitter), which seemed to give me new focus.  Let the play and the math be enough.

Student Play:

I had a lot of ideas for what student math play could look like, but realized the new school year was fast approaching, and I knew I needed something more concrete to start.  Thanks to Sarah VanDerWerf and her idea of a math table, I started there.  I immediately created a space for students to explore and play with math.

I randomly began to leave out math tools for students to play with.  No direction.

At first, students hesitated.  I don’t have a quantitative reason why they hesitated, but I can assume that perhaps they did not have much experience with free math exploration.

After time, students began to play.  I noticed they would go to the table as soon as they came in.  It was a great way to start the day for many of my students.  I also observed that if one student went over, it was only a matter of seconds before a small group would begin and dive into the play.

I started with select pattern blocks and small mirrors.

I love the use of height by this student.

 

I used magnetic shapes next — oddly, I did not take any photos.  This is where my students started becoming comfortable with math play — their confidence and play began to expand.  I knew I had a winner when students were asking me to stay in for recess to continue their exploration.

I then discovered the great @Trianglemancsd and his amazing wooden creations.  My students adored — and continue to adore — our tiny turtles and our spiralling pentagons.

Turtle Work:

At first, my students had no idea what to do with the turtles.  They sat alone and untouched for about a day.  A few of my bound and determined students finally dove in.

My first student written tweet was the result:

Here are some other tessellations my students played with:

It interesting that at first, my students hesitated to work with all of the turtles.

Spiralling Pentagons:

Next, I pulled out the pentagons.  I thought students would go right for it, but there was still hesitation.  Again, once one started started to develop some designs, more followed suit.

What I Have Learned Through Student Play:

• That play is different for each student.  I can confidently say every student has manipulated these toys at least once throughout the year;
• Students have more persistence with problem solving.  Again, I can’t say there is a direct correlation, but since I have put out the play table from the ‘get go’ of the year, I can say that students ‘attack’ problems in math with a little more ease and confidence;
• Collaboration happens naturally, and even if a student starts solo, another student is always entering and at least asking questions about the designs on the table (and, I admit, there might not be the most math-specific talk here, but there is definitely “What are you going to do next?”, “What about these that you left here?”, “Can I try something?”)

What I Need To Do Next:

…and these are crucial for the happiness of math to continue:

• More time for play.  Students need more time, I need to provide more time for them to play, and give them more things to try and play with (how can I add numbers and number sense into my play table?);
• Silently Observe:  I need to watch more, and take notice.  I need to step back and listen to the conversations around the math toys;
• I need to start having students reflect on their play.  This can happen a number of ways:  Have reflection cards (or a Google Form ready for when students have something to say about their play), or talk to another student, or bring the conjectures and ideas they came up with into math tasks (“Remember when you were playing at the math table?  What did you do there that might connect here?”);
• More creative tools for students to explore and work with;
• Me asking more questions.  To start, as and after they play;
  • What are you thinking as you create?
  • What do you notice as you create (would be powerful to ask another student this question about someone else’s creation)?
  • What is your plan now?
  • What will this mean for future math play?

In short, I need to spend more time on math and play.  I need to be more cogniscent of my students and their play, and that it is being fostered as much as possible.  Math is play, after all!

Suggestions?  Please comment, or reply on Twitter!  I would love to hear your thoughts.  Thank you!

How student and teacher learning interconnect

Before I start, I should say that this is a really rough reflective piece for me — and may not be super clear to you, dear reader.

The other night I was honoured to discuss my learnings and experiences teaching math at a Mathematics Part 1 AQ course.  I was worried I would embark exactly what I don’t want in my classroom:  That I am a keeper of knowledge, and I should be spreading it.

Instead, I wanted to go in and discuss how I have learned, which has made my students learn, and how student learning has made me learn.  I wanted to come from a angle that I am constantly and still learning, and here are some of the mathematical things I have tried.  Some worked, many did not.  But I am still researching, learning, and trying.

I talked to these lovely teachers about how they know they have learned, really learned, something.  We padlet-ed our thinking out.  Then I asked if they could tell me how they know their students have learned.  Interestingly, it wasn’t so obvious.  It brought up great discussion about how we can ask students how they know they have learned something, and odds are, we’d see some similarities.

In the curriculum, there are metacognitive expectations.  These are the challenging expectations that we feel come after a unit or test — what did you learn?  How?  But now I am thinking how I learn.  It’s not linear at all — I learn in bumps and starts, ebbs and flows.  So I wonder, can I not ask my students as they are learning to start considering how they know they are learning, and if not, what has to happen to help them learn.  Wouldn’t having them stop and pause and reflect as they are learning be powerful?  Do we give students enough time to really reflect on how they learn, and when they know they have learned (and this will be different for everyone).

This should be the same for teachers.  We should be constantly going through how our students learn, and how we are taking that knowledge and applying it to our planning and moving forward for our students to continue learning.

We as teachers, and adults, should realize that learning is not neat — it’s messy.  Full of buzz, and, sometimes, frustrating.

So, tomorrow, I am going to ask my students – How do you know you have learned something?  Really understood and learned something?  I know for a fact those answers will help me learn about them, and how they learn, and therefore, how I will teach.  I want to know more about the reciprocity of student and teacher learning.

 

What if Learning Goals were Learning Questions?

It was early and silent in my classroom.  I like to get in early (freakishly early, from my husband’s point of view) and putter.  As I putter, I look around my room, I reflect and synthesize what’s going on in my room.  I feel my brain expand in a long sigh; as all of my general worries and thoughts as a teacher stretch.

I was standing at my chart stand, one of my brand new markers in hand, curriculum open, and ready to finally consolidate our learning goal and success criteria for math.

I am not sure about anyone else, but I don’t give my learning goal or success criteria up front.  I do assessment for learning, I let students explore and investigate (probably the most common verbs in the Ontario Math Curriculum), and I listen to what students are saying first.

As they work, are they making connections between concepts?  Are they seeing how their estimates and thinking connect to the bigger problem I handed them?  Once I have heard the mathematical language I want, once students start talking in conjectures, then I pull out the expectations.

I write them, and we put them in student language.  We reflect, and we determine what it takes to be successful under that expectation and learning goal.  We keep re-visiting, and we change and morph this living track of our learning as we explore and learn more.

Anyways, back to me staring at my chart stand — I was recently doing this with Data Management, and following the EDUGains Scope and Sequence.  The idea of choosing the most appropriate graph for data found was the big idea.

And, instead of writing it in a “we will learn…” my gut instinct took over, and I wrote “How does organizing data effectively influence the reader?  What do I have to include in my graph or chart to ensure my reader understands my data clearly?”

Then, our success criteria would be the answers we had discovered.

• Line graphs are better for data over time, as readers can see dramatic changes in data with ease
• I want to make sure that my scale is representative of my data, and my reader can follow easily
• Double bar graphs make comparing two similar data sets easily, and make the reader be able to find similarities and differences to make generalizations

Now, sure — this success criteria is massaged by me, but is generally the student ideas that have come up through our work.  But, really — doesn’t that question flow, make sense, to what we are learning in math?

But I panicked, erased it, and went back to a non-question. Because that would be wrong.

Now that I am moving into measurement, review of perimeter, area, and length with standard units, the same thing jumps to me.  I want students to understand the relationship between perimeter and area, and how this learning will lead them to area of triangles and parallelograms later.  So, the question, “What are the relationships between perimeter and area?  How does one relate to the other?”  seems powerful to me (Now, that’s not a perfect question by any means, but you get my gist).

Thoughts?

*February 11, 2018 — Update*

I have been working on creating learning questions, and, interestingly, other teachers at my school are trying it, too (cool ,right?).

So, these are very, very, very rough – but I have been using learning questions for students to connect their learning and experiences through the Ontario math expectations.

I have tried different layouts, and I am not sure on one that is the ‘best’, or one that is working any more or less than stating a learning goal rather than asking.  I do know that, once it has been established, it’s great to ask my students to reflect on the learning goal.

I also want to note, that these are not pretty, perfect, or even close to what they could be yet, they all come from my students.  My students voice is spread all through here.

This is currently what we are working on, so very early stages.  I do not like I wrote success criteria, since, really, it’s students reflecting on the questions with the interpreting data experiences we have had.

One effective task that we do is to write some of the curriculum words in our own words – so students have their own understanding of terms.

I tried to ask 3 questions and see how this layout worked — now I wish I had connected them all at the end.

I am going to keep trying, and see if I can model the stages we go through more visually.  For example, our first attempts at answering our learning questions, and then adding (in perhaps a different colour?) as we learn and add more.  I need to do a final consolidation at some point, to make all of these questions interconnect.

How could me asking the learning question for a final assessment piece work?

Estimating is justifying

Estimating to me means lots of visuals, lots of dialogue, and, generally, everyone feels safe estimating.  Estimation is a great way to start the year, and get the math talk flowing.  Also, it leads to the process of justifying.  Why and where did your thinking come from?  What do you consider a too high or too low estimate?  I can also throw in modelling open number lines, standard notation, familiarity and comfort with benchmark numbers (because generally kids naturally group in 5s or 10s), and – hey!  Even a little fraction talk (what if I had half?  Hmmm…the stadium isn’t full, I’d say there is about three one fourths present.  Does that affect your estimate?).

The power of estimation, and playing and thinking about estimation, is very powerful.  I find I can get a lot of information about the depth and perception of a students mathematical concept of number, and relationship between numbers, through estimation tasks.

I am going to highlight a few of the estimation tasks I did (and, honestly — they are not super amazing; they are just great for seeing student thinking).

I started out the year with one of my favourite tasks.  The classic Fermi Problem “How long would it take to sing happy birthday to everyone?” and right away the questions start.  “Wait, Mrs. Allum — who is everyone?”  “What do you mean ‘how long’?”  And I shrug, and pretend to go do something (OK, not quite, but I say, “well, what do you think?” and there is a moment of panic.  I remind them they have a group to help them, and sure enough, they are asking me for timers, and telling me that ‘everyone’ will be the members of this class).  And, as we know, once they start talking and thinking, they start coming up with estimates, or even better, questions for more information that would require their estimates to be more precise (my favourite this year was when I overheard one of my students say, “Hey!  Won’t it be different lengths depending on the lengths of people’s names?  Shouldn’t we factor that?”).

Then, I do the classic candy jar estimations.  Because, really — who doesn’t love candy?

 

This gave me a lot of information — how much of this estimation was just sheer guessing?  What strategies were students actually using?

We did a few classic estimate 180s, but I still wanted to get the justification of where their estimates were coming from.

So, I had a bit of a brainstorm. Rather than students estimating, I gave them 2 estimates to choose from, and then justify why they thought it was the better estimate.

We started with one estimate where students chose a side of the room.  See these Estimation slides for the bacon slide Estimation slides (thanks estimate180!), where I had students choose the amount, and then try to convince the other side to join theirs. We watched the third act video to see which side of the room was correct. Now students were beginning to understand they had to convince and justify their estimates to convince others, and to ensure their estimates were reasonable.

I then used the whiteboards, more photos, and a speed dating feel to the next task.  Students rotated around the room and looked at the photos, and justified their estimate (these photos are also in the aforementioned estimation slide folder).

 

The next morning, I had my students do an estimate match as they walked in the door.  One student got an image, and another student got a number.  Find your most reasonable partner.  They had to then share and justify with each other why they were the best match.  Some students even went on to discuss why it would be possible for more than one response.

Estimation Photo Match

I then went back to the estimation boards.  I wanted students to determine what they thought were the best justifications of the estimations.  I had them stick sticky notes on what they deemed the best estimates.  We decided as a class these would include:

• A clear strategy that someone could follow and understand by reading it
• using information in the photo to assist (size, shape, relation to another item)
• they would consider the ‘unknown’ factors (i.e., if they couldn’t see all of the items, they would consider the amount based on what they could see).

I told the class we would look for clusters of sticky notes.  I took photos of the boards, and we discussed about why there were clusters.  These were the justifications that were the most clear and made the most sense.

 

So, I went back to the candy jars.  This time, students had to estimate, and write their justification.  I was curious to see if I would get richer responses — hoping for students to tell me they were grouping, considering and comparing size, and even using number lines.

Then I decided that students would decide who had the better justifications and estimates.  I typed up all of the sticky notes, and then gave them to groups anonymously.  Students decided and determined who won based on best justifications and estimates. They decided the criteria would be the same as earlier: A logical, clearly explained (there words:  I should be able to understand your thinking from your reading as if we were talking) justification.

I have definitely heard a difference in students not only estimating and talking about number, but sharing how they know.  They are beginning to share their reasoning strategies and ideas, and I feel this time spent estimating will really help when we get deeper into computation.  It will increase their comfort with number, and reasoning through a problem.