What Makes Math Interesting Anyway?

Many teachers are comfortable allowing their students to read for pleasure at school and encourage reading at home for pleasure too. Writing is often seen as a creative activity. Our society appreciates Literacy as having both creative and purposeful aspects. Yet mathematics as a source of enjoyment or creativity is often not considered by many. 

I want you to reflect on your own thinking here. How important do you see creativity in mathematics? What does creativity in mathematics even mean to you?

Marian Small might explain the notion of creativity in mathematics best. Take a look:

Marian Small – Creativity and Mathematically Interesting Problems from Professional Learning Supports on Vimeo.

Type 1 and Type 2 Questions

Several years ago, Marian Small tried to help us as math teachers see what it means to think and be creative in mathematics by sharing 2 different ways for our students to experience the same content. She called them “type 1” and “type 2” questions.

Type 1 problems typically ask students to give us the answer.  There might be several different strategies used… There might be many steps or parts to the problem.  Pretty much every Textbook problem would fit under Type 1.  Every standardized test question would fit here.  Many “problem solving” type questions might fit here too.

Type 2 problems are a little tricky to define here. They aren’t necessarily more difficult, they don’t need a context, nor do they need to have more steps.  A Type 2 problem asks students to get to relationships about the concepts involved.  Essentially, Type 2 problems are about asking something where students could have plenty of possible answers (open ended). Again, here is Marian Small describing some examples:

Examples of Type 1 & 2 Questions

Notice that a type 2 problem is more than just open, it encourages you to keep thinking and try other possibilities!  The constraints are part of what makes this a “type 2” problem! The creativity and interest comes from trying to reach your goal!

Where do you look for “Type 2” Problems?

If you haven’t seen it before, the website called OpenMiddle.com is a great source of Type 2 problems.  Each involve students being creative to solve a potential problem AND start to notice mathematical relationships. 

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Remember, mathematically interesting problems (Type 2 problems) are interesting because of the mathematical connections, the relationships involved, the deepening of learning that occurs, not just a fancy context.

Questions to Reflect on:

• When do you include creativity in your math class? All the time? Daily? Toward the beginning of a unit? The end? What does this say about your program? (See A Few Simple Beliefs)
• If you find it difficult to create these types of questions, where do you look? Marian Small is a great start, but there are many places!
• How might “Type 2” problems like these offer your students practice for the skills they have been learning? (See purposeful practice)
• What is the current balance of q]Type 1 and Type 2 problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
• How might problems like these help you meet the varied needs within a mixed ability classroom?
• If students start to understand how to solve type 2 problems, would you consider asking your students to make up their own problems? (Ideas for making your own problems here).
• How do these problems help your students build their mathematical intuitions? (See ideas here)
• Would you want students to work alone, in pairs, in groups? Why?
• If you have struggled with developing rich discussions in your class, how might these types of problems help you bring a need for discussions? How might this change class conversations afterward?
• How will you consolidate the learning afterward? (See Never Skip the Closing of the Lesson)
• As the teacher, what will you be doing when students are being creative? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

I’d love to continue the conversation about creativity in mathematics.  Leave a comment here or on Twitter @MarkChubb3

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Reasoning & Proving

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

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:

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.

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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).

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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:

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!

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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:

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:

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:)

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Final Thoughts

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

• 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 ).

Reasoning and Proving

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

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:

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:

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:

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:

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:

• 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 ).