Math Talk

Traditionally, math class was a quiet place. Kids alone in rows busy calculating with limited back and forth. A competition to see who could find the correct answer the fastest. The idea that discussion was a necessary tool for deepening and consolidating understanding was a foreign concept, given that much of mathematics was conveyed as symbols and numbers. There was a right and a wrong answer – so what was there really to talk about? Yet the kind of communication we want students to engage in is so much more than simply answering questions or reciting procedures. Of course these are a part of any math class, but they shouldn’t comprise most instructional time, as they often do.

Talking about math is not something that comes naturally to kids. 

There needs to be a shift from focussing on finding the answer to discussing the problem. When this happens there is a collective easing and the pressure is off of students who are reluctant to share their ideas for fear of getting it wrong. The potential embarrassment is not worth the risk. More than any other subject, math creates this anxiety among the less confident. As a result, executive functions such as working memory and regulating behaviour suffer and math proficiency is not fully developed (see research here). To alleviate this stress, teachers can redirect attention back to the problem. We’re in this together to find a solution.

With enough practice, these four simple questions will lead to profound math talk:

  1. Why?
  2. How do you know?
  3. Can you prove that?
  4. Can someone else disprove what’s been said?

Effective approaches to encourage math talk are Gallery Walks, Math Congress and Bansho (see here)In order to make students more comfortable sharing their mathematical thinking, the following strategies and sentence starters are a great way to scaffold dialogues. It is in these moments that some of the best consolidation of learning happens. Sometimes we don’t know what we truly know until we give it a voice.  

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Characteristics of math communication to look for (see rubric here):

  • precise – relevant choice of method that has accurate calculations
  • clear – logical organization that is easy to follow and requires little inferencing
  • cohesive – reasoned argument held together through explanations, diagrams etc.
  • elaborate – justification of ideas and strategies with sufficient detail
  • appropriate – proper use of mathematical terminology, symbolic notations etc.

Metaphors for Class Discussions

Class discussions are one of the best strategies for encouraging students to take a more active role in their learning. They allow students to find their voice. They create a space for the sharing of ideas and an opportunity to engage in debate. They also provide valuable feedback for teachers.

The problem is that the standard ‘review type’ questions that ask students to recall and give back information follow a familiar pattern. The teacher questions, a student responds and the teacher evaluates. It is sometimes known as IRE (initiate, respond, evaluate). There is a time and place for this type of questioning, especially when teachers want to activate prior knowledge, unfortunately though, it is the default style of discourse in many classrooms. Unless teachers make a deliberate attempt not to always use this style, it will prevail and limit the richness of discussions.

The IRE pattern of interaction is best described as resembling a ping pong match, with a back and forth between the teacher and a single student. Doug Lemov refers to it as playing catch, or static questioning. Much of the class is left out of the interaction, and as a result, they usually tune out. The teacher is the one getting most of the practice, with students not having to carry much of the load. It also focuses primarily on memory as the main cognitive function, offering a limited effect on the development of thinking and reinforcing the idea that learning is memorizing.

Instead of the ping pong style, a more useful metaphor for effective discussions is basketball. The ball, representing the question, is passed around and ideas are bounced off one another. There is a greater involvement from the whole class as each student has an opportunity to contribute to the discussion. This approach forces students to prove, justify, or defend their answers and those of their classmates. Students are held more accountable for contributing to the discussion, leading to a more engaging environment and livelier discussions.

The ice-cream cone is another metaphor for class discussions, with each scoop resting on the foundation of the one preceding it. By building on the comment that comes before it, the discussion becomes deeper, with ideas being connected together. There is a sense of cohesion. It also requires students to really listen to their peers and contribute only that which moves the discussion forward. It brings a focus to the discourse and a sense of purpose, as there is no place for random thoughts or tangents.

To get the most out of class discussions, it’s important to move away from the traditional game of ping pong and play more basketball, build more ice-cream cones.

What Happened to Free Play?

Growing up in the 1980s, my friends and I had two educations, one at school and one after-school. We played in mixed-age neighbourhood groups almost every day, often until dark. Weekends and summer gave us time to explore, time to daydream, and time to immerse ourselves in hobbies, all without adult supervision. We were independent, we were given so much freedom. It was a ‘golden age’ of free play.

But today, it’s impossible to imagine giving kids that. Everywhere you look, we have reduced children’s freedom to play on their own. The streets are a ghost town, the parks and playgrounds are closely monitored by parents. Playgrounds Without Parents, a recent post by Royan Lee and this article from The Guardian, speak to this problem.

Many parents micromanage their children’s free time, worried that without a diverse skill set in extra-curricular activities, their kids will lose that competitive edge. Childhood has been reduced to a time of resume-building with no time to be wasted on ‘frivolous’ play. In the process, we’ve replaced ‘pickup’ games with adult-directed sports leagues, hobbies with adult-directed classes, and fun with work. Performance has taken the place of play. Children are no longer given opportunities to play and explore in their own chosen ways. Free play doesn’t count for anything, because it’s just play – there’s no place for it on the college or university application.

When parents are asked why their kids don’t play outside anymore, they often cite safety concerns. But the reality is, cases of crime, abduction or abuse by strangers are greatly exaggerated by the media. News reports have manufactured this fear, manipulating parents into thinking that unsupervised play is dangerous. As studies have shown, the actual rate of such cases is low and has been declining in recent years.

While parents’ decisions and the influence of the media have had an impact on kids’ loss of freedom, Audrey Watters, in her recent post, Raising (and Educating) ‘Free Range Kids’, wonders how schools have shaped this reduction in play. Watters raises important questions that are worth discussing:

  • What role do schools play – not simply in monitoring students but in discouraging (or perhaps encouraging) students’ physical freedom?
  • Do we let students roam – physically and intellectually – at school? Why or why not?
  • How does rhetoric about children “in danger” – online and offline – shape how we treat them at school?

This loss of freedom for kids is something I want to come back to and explore in more detail in later posts. It is too large an issue to be covered all at once.

I want to leave you with a talk by Peter Gray on the decline of play. Gray is a developmental psychologist who wrote a book that should be required reading for all parents and teachers. As Lenore Skenazy says, “If you’ve ever wondered why your curious kid is turning into a sullen slug at school, Peter Gray’s Free to Learn has the answer. He also has the antidote.” Gray’s views will change the way you think about your students’ learning and development.

What’s the Point?

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What’s in it for me?  Why should I bother learning this?

When students enter the classroom, these questions are on their mind, and are usually unspoken. They linger in the background, and they will subtly undermine our efforts to make lessons enjoyable if students are unable to see the relevance of what we teach them. The degree to which students vocalize these questions depends on the subject. Just ask a math teacher. Making the effort to address these questions, answering the why before we approach the how to, is time well spent.

Teachers can discuss the ‘what’s in it for me’ (WIIFM) questions as a faculty, department or grade team, brainstorming for a particular subject or topic. What is in it for the students? How will they benefit from learning what we are trying to teach them? By focusing on the benefits, students will hopefully see the point of what they are learning. Like a good salesperson, we highlight the benefits not the features. To differentiate between the two, try putting a ‘so what?’ at the end of your explanation. If you fail to answer the so what?, try to rethink the WIIFM. Here are three examples:

“Arithmetic stirs up him who is by nature sleepy and dull and makes him quick to learn and shrewd.” (Plato on math)

“Studying English literature at school was my first step towards mental freedom and independence.” (Ian McEwan on English)

“The farther backward you can look, the farther forward you are likely to see.” (Winston Churchill on history)

Through marketing our subjects and putting together a list of WIIFMs to display in classrooms, teachers can build intrinsic motivation. Allow the students to come up with a list of their own. Their answers might surprise you. In discovering their own WIIFMs, students who previously seemed incapable of learning and unmotivated and apathetic towards school, can find what motivates them. A light is switched on in their heads, giving a better understanding of what drives them. The road ahead is a little clearer.

The answers will encourage a greater fascination with the subject, a sense of its relevance to life and the world, a feeling of accomplishment in mastering it, and a calling to it. Ideally, they will spark thinking and prompt students to start asking some of life’s big WIIFM questions: Where do they want to go? Who do they want to be? How is what they’re doing in school going to get them there?

And in the end, isn’t that the point.

Painting of the Voice

Writing is hard.

Finding your voice is even harder. For many students, it is the most difficult aspect of writing – one that all writers struggle with and search for. It requires looking deeply at who we are. A turning inward. Unfortunately being introspective is not an easy thing to do in a society that is always enticing us to look outwards. The external takes precedence over the internal. It can be tough to hear one’s voice amid all the noise of modern life. So many distractions, so little time.

Like most things, the key to becoming a better writer and finding a voice is practice. There is no substitute for hard work. Simply put, to improve at writing we have to write. Often. Every day if possible. It is a craft that needs to be honed. One that takes discipline. As Stephen King says in On Writing, “If you want to be a writer, you must do two things above all others: read a lot and write a lot.” Simple and good advice for young writers. Setting aside time each day for students to write is a great way to develop their skill and find their voice. Give students the freedom to write about what matters most to them.

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Unfortunately in schools, a lot of what students are asked to write about on a daily basis is nonsense. The writing tasks are dry and disconnected. They serve no real purpose. And they don’t offer real opportunities for students to develop their voice. The following example illustrates this problem: write a letter to a scientist explaining what you know about osmosis. Who would ever write like this? Why give students writing assignments like this? Prompts like this create an apathetic attitude towards writing. They remove the joy. No wonder students often dislike writing.

Many writing prompts also lack relevancy. They are busy work. And often the only person reading students’ writing is the teacher. When students are given an authentic audience, motivation then comes from the desire to share with others, rather than merely to complete an assignment. Blog writing sites like KidBlog provide students with a platform to share their writing with the world. Blogging makes their writing significant. Figment and Teen Ink offer students a popular forum to publish their writing and have their voice heard.

If we want students to value writing, we need to give them meaningful and important things to write about. They must be able to see and hear themselves in their writing.

We need to give them a real voice.