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 ).

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Differentiated Instruction: comparing 2 subjects

I’ve been thinking a lot about how to meet the various needs of students in our classrooms lately.  If we think about it, we are REALLY good at differentiated instruction in subjects like writing, yet, we struggle to do differentiated instruction well in subjects like math.  Why is this???

In writing class, everyone seems to have an entry point.  The teacher puts a prompt up on the board and everyone writes.  Because the prompt is open, every student has something to write about, yet the writing of every student looks completely different. That is, the product, the process and/or the content differs for each student to some degree.

Teachers who are comfortable teaching students how to write know that they start with having students write something, then they provide feedback or other opportunities for them to improve upon.  From noticing what students do in their writing, teachers can either ask students to fix or improve upon pieces of their work, or they can ask the class to work on specific skills, or ask students to write something new the next day because of what they have learned.  Either way the teacher uses what they noticed from the writing sample and asks students to use what they learned and improve upon it!


In Math class, however, many teachers don’t take the same approach to learning.  Some tell every student exactly what to do, how to do it, and share exactly what the finished product should look like.  OR… in the name of differentiated instruction, some teachers split their class into different groups, those that are excelling, those that are on track, and those that need remediation.  To them, differentiated instruction is about ability grouping – giving everyone different things.  The two teachers’ thinking above are very different aren’t they!

Imagine these practices in writing class again.  Teacher 1 (everyone does the exact same thing, the exact same way) would show students how to write a journal (let’s say), explain about the topic sentence, state the number of sentences needed per paragraph, walk every student through every step.  The end products Teacher 1 would get, would be lifeless replications of the teacher’s thinking!  While this might build some competence, it would not be supporting young creative writers.

Teacher 2 (giving different things to different groups) on the other hand would split the class into 3 or 4 groups and give everyone a different prompt.  “Some of you aren’t ready for this journal writing topic!!!”  Students in the high group would be allowed to be creative… students in the middle group wouldn’t be expected to be creative, but would have to do most of what is expected… and those in the “fix-up” group would be told exactly what to do and how to do it.  While this strategy might seem like targeted instruction, sadly those who might need the most help would be missing out on many of the important pieces of developing writers – including allowing them to be engaged and interested in the creative processes.


Teacher 1 might be helpful for some in the class because they are telling specific things that might be helpful for some.

Teacher 2 might be helpful for some of the students in the class too… especially those that might feel like they are the top group.

But something tells me, that neither are allowing their students to reach their potential!!!


Think again to the writing teacher I described at the beginning.  They weren’t overly prescriptive at first, but became more focused after they knew more about their students.  They provided EVERYONE opportunity to be creative and do the SAME task!

In math, the most effective strategy for differentiating instruction, in my opinion, is using open problems.  When a task is open, it allows all students to access the material, and allows all students to share what they currently understand.  However, this isn’t enough.  We then need to have some students share their thinking in a lesson close (this can include the timely and descriptive feedback everyone in the group needs).  Building the knowledge together is how we learn.  This also means that future problems / tasks should be built on what was just learned.

ambuigity


We know that to differentiate instruction is to allow for differences in the products, content and/or processes of learning… However, I think what might differ between teacher’s ability to use differentiated instruction strategies is if they are Teacher-Centered… or Student-Centered!

Differentiated Instruction.jpg

When we are teacher-centered we believe that it is our job to tell which students should be working on which things or aim to control which strategies each student will be learning.  However, I’m not convinced that we would ever be able to accurately know which strategies students are ready for (and therefore which ones we wouldn’t want them to hear), nor am I convinced that giving students different things regularly is healthy for our students.

Determining how to place students in groups is an important decision.  Avoid continually grouping by ability.  This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.  Instead, consider using flexible grouping in which the size and makeup of small groups vary in a purposeful and strategic manner.  When coupled with the use of differentiation strategies, flexible grouping gives all students the chance to work successfully in groups. Van de Walle – Teaching Student Centered Mathematics

If we were to have students work on a problem in pairs, we need to be aware that grouping by ability as a regular practice can actually lead students to develop fixed mindsets – that is they start to recognize who is and who isn’t a math student.


Obviously there are times when some students need remediation, however, I think we are too quick to jump to remediation of skills instead of attempting to find ways to allow students to make sense of things in their own way followed by bringing the learning / thinking together to learn WITH and FROM each other.


To make these changes, however, I think we need to spend more time thinking about what a good problem or rich task should look like!  Maybe something for a future post?


As always, I want to leave you with a few reflective questions:

  • I chose to compare what differentiated instruction looks like in mathematics to what it looks like in writing class.  However, I often hear more comparisons between reading and mathematics.  Do you see learning mathematics as an expressive subject like writing or a receptive subject like reading?
  • Where do you find tasks / problems that offer all of your students both access and challenge (just like a good writing prompt)?  How do these offer opportunities for your students to vary their process, product and/or content?
  • Once we provide open problems for our students, how do you leverage the reasoning and representations from some in the room to help others learn and grow?
  • Math is very different than Literacy.  Reading and writing, for the most part, are skills, while mathematics is content heavy.  So how do you balance the need to continually learn new things with the need to continually make connections and build on previous understanding?
  • What barriers are there to viewing differentiated instruction like this?  How can we help as an online community?

For more on this topic I encourage you to read How do we meet the needs of so many unique students in a mixed-ability classroom?  or take a look at our Ontario Ministry’s vision for Differentiated Instruction in math: Differentiating Mathematics Instruction

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