Reasoning & Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

leadership oame

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Capture

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning, and to get our students really thinking), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.


Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).


A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

  • identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

triangle congruency

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!


A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Triangles 2

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

triangles 3

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)


Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

Graham Fletcher
  • Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
  • How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
  • If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
  • What does Day 1 look like when learning a new concept?
  • Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
  • Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
  • The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

Advertisements

Reasoning and Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

leadership oame

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Capture

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.


Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).


A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

  • identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

triangle congruency

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!


A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Triangles 2

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

triangles 3

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)


Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

Graham Fletcher

  • Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
  • How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
  • If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
  • What does Day 1 look like when learning a new concept?
  • Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
  • Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
  • The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

Making Math Visual

A few days ago I had the privilege of presenting at MAC2 to a group of teachers in Orillia on the topic of “Making Math Visual”.   If interested, here are some of my talking points for you to reflect on:a1

To get us started I shared an image created by Christopher Danielson and asked the group what they noticed:

a2

We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image.  After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture.

a3

I asked the group of teachers what mathematics they noticed here… then how they believed parents might have answered the question.  While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table…..  the group correctly guessed that many parents do not typically notice the mathematics around them.

So, my suggestion for the teachers in the room was to help change this:

a5

I then asked the group to do a simple task for us to learn from:

a6

After a few minutes of thinking, I shared some research of the different ways we use fractions:

When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about.

a13

Data taken from 1 school’s teachers:

a14

We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem:

a15

Followed by a quick story of when a student told me that the following statement is true (click here for the full story).

a17

So, why should we focus on making math visual?

a18


We then explored a statement that Jo Boaler shared in her Norms document:

a19

…and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually?

If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually.


I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description):

a20

Along with a few different responses that students had completed:

a21

Most of the students in the class had responded much like the image above.  Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2).

a22

In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals.

a23

We wrapped up with a few suggestions:

a25

a26

And finally some advice about what we DON’T mean when talking about making mathematics visual:

a27

a28

You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers.

a29


I want to leave you with a few reflective questions:

  • Can all mathematics concepts in elementary school be represented visually?
  • Why might a visual representation be helpful?
  • Are some representations more helpful than others?
  • How important is it that our students notice the mathematics around them?
  • How might a focus on visual representations help both us and our students deepen our understanding of the mathematics we are teaching/learning?

I’d love to continue the conversation.  Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).


If you are interested in all of the slides, you can take a look here

 

 

Developing Spatial Reasoning through Guided Play

For the last few months, a team of kindergarten teachers and myself have been working together to deepen our understanding of early years mathematics, spatial reasoning, and the importance of guided play as a vehicle to engage our students to think mathematically.  Below is a copy of our slideshow presentation we shared at OAME 2017, and some of the documents we have created over the past few months.

Spatial Reasoning - presentation

A quick synopsis of our work first:

1

While our research led us toward Doug Clement’s work about trajectories, and research about spatial reasoning and early mathematics, much of the tasks we actually did with students directly came from the book shown above (Taking Shape) which I can’t recommend enough – if you can, get yourself a copy!


2

We discussed the quote above to help us realize what actually underpins mathematics success.  More details about how the quote ends here.


3

4

We shared research showing just how important early mathematics is, and specifically what the kinds of instruction could / should look like to accomplish this learning.  Duncan et. al., is a widely quoted piece of research that has led many to realize that early math learning needs to be a focus in schools – even more so than early reading!


6

We played a few games that helped us stretch our spatial reasoning abilities.  The image above was part of our “See It, Build It, Check It” activity (found in Taking Shape).  Everyone saw the image for a minute, then was asked to build it once the image was removed.  What we noticed is just how difficult spatial tasks are for us!


 

5

After we had the opportunity to play for a bit, we dug back into the research about spatial reasoning and the jobs typically chosen based on (high school) spatial ability.  Hopefully you noticed something interesting on the graph above!

10

So, we know how important spatial reasoning is, but the 3 pieces above (taken from Paying Attention to Spatial Reasoning document) might help us realize how important a focus on spatial reasoning is for both our students and us.


7

In our time together, we learned a lot about the importance of observing our students as they were engaged in the learning.  From the initial choices they made, to how they overcame obstacles, to understanding the mental actions that were happening… Observing students in the moment is far more powerful than collecting correct answers!

8

See the link at the bottom of the page for our connections to Doug Clements’ work.


9

We also discussed the specific connections between the mathematics behaviours and the learning that happened beyond.  In our Kindergarten program document, our students’ expectations fall under 4 frames (see above) so we linked the learning we saw to the program document in a way that helps us see the depth and breadth of the kindergarten program (see document linked below).


We then ended our presentation with a synopsis of what we learned throughout our work together.  While the slideshow might be helpful here (I’d love for someone to comment on those slides at the end), the conversations that Sue, Kristi and Kristen had with others have shown me just how valuable it is to spend time learning together.  I couldn’t be prouder to be able to work with such reflective and dedicated teachers!


Our presentation:

Guided Play Slideshow

A few of our handouts:

4 Frames handout

Developmental Trajectories – Adapted from Doug Clements’ work

Spatial Language


As always, I’d love to hear about your thoughts or comments.  Leave a comment below or catch us on Twitter:

@MarkChubb3

@mrs_dt

@mrs_penlington

@kristinWillms