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.

Things Come in Threes

Dan Meyer’s three acts are well known to math teachers. (For an explanation, see here. Excellent examples are here.) The tasks are engaging and get at the heart of mathematical modelling. According to Meyer, they follow a storyline structure:

  • Act One: Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.
  • Act Two: The protagonist/student overcomes obstacles, looks for resources, and develops new tools.
  • Act Three: Resolve the conflict and set up a sequel/extension.

Structuring math in threes led to these observations on teaching it:

Three Challenges

  1. Conceptual: Math is inherently abstract. It is disembodied ideas that become increasingly difficult to grasp as you advance through the grades. (Think formulas, functions, proofs etc.) Connecting concepts to the concrete can be a struggle.
  2. Weird: Strange symbols and jargon (not to mention the amount of content teachers need to cover to unpack and make sense of all the oddness.)
  3. Anxiety: The association of math with phobias and stress. For many students, math is terrifying, boring or meaningless. It can result in shutting down and feeling overwhelmed. Which leads us to the next trio…

Three Audiences

  1. Traumatized: Students who have had a terrible experience with math that has turned them off the subject. Or maybe they hit a wall along the way with long division or linear algebra. The ‘I’m just not a math person’ group.
  2. Perplexed: Students who see math as pointless. They are a bit lost, but compensate for it by working hard, following directions and overcoming failures. This group is often the silent majority in any class.
  3. The Naturals: Students who have a feel for math. It makes sense to them. It gives them satisfaction and they have innate talent. They’re not always the ones with the highest grades, but they usually go into careers in related fields.

Three Routes

  1. Illuminations: The eureka moments that help students come to love math after searching in the dark. Whether it’s through explanation, visuals or practice, these are the times that progress is made. Breakthroughs that allow them to keep going.
  2. Connections: Tying any subject to what we already know is effective and math is no different. Sports, music, science, movies, business and nature are all great sources for math inspiration. The traumatized and perplexed benefit and math class is more interesting.
  3. Conversations: Talking about math is underrated. Because it is a concise subject, the more we can grapple with concepts through discussion, the better. Providing opportunities for students to express and argue about ideas is important.

 

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Fermi and the Problem Solved

Word problems in math textbooks often give much of the information needed to solve them. There is no mystery. Students walk away from a math course with the only skill acquired being the ability to decode the textbook. They are just swapping numbers and plugging in different information. As a result, the so-called problems are no longer problems. They are routine and predictable. The problems are too scaffolded and the students realize that it’s an exercise in futility. An insult to their intelligence. While practice is indeed a fundamental part of math, when problems are variations of the same one, the motivation to complete them is lost. They don’t see the point of it all.

How can we make problems interesting and challenging?

Look to Enrico Fermi. The Italian physicist had a gift for making accurate estimates of seemingly unsolvable questions. Given little information, he was able to provide educated guesses that came very close to the actual answer. His most famous question, “how many piano tuners are in Chicago?” seems to make no sense, but through a series of questions, estimations and assumptions, he arrived at a reasonable answer. Legend says that Fermi calculated the power of an atomic explosion by looking at the distance his handkerchief travelled when he dropped it as the shockwave passed. He determined it within a factor of 2. For a discipline that is always looking for realistic applications, math class would do well to use Fermi problems. It doesn’t get more real-life than that!

(For more problems, see here)

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Using Fermi questions in our math classes will remove the pseudo-contexts of much of the problems. They are actually used in real life, unlike many of the questions from a math textbook. Companies use Fermi problems during job interviews as they offer a window into a person’s ability to think on their feet and their creativity. Scientists, economists and engineers use them in their work to get a ballpark idea of the feasibility of their projects. This power of estimation is a key aspect of mathematical thinking. Without it, all the math in the world won’t mean a thing.

Fermi problems emphasize the mathematical processes and help students practise estimation and reasonableness when solving problems. They strengthen number sense, dimensional analysis, and are important in developing a quantitative understanding of the world around us. They allow students to ask the right questions and break down complex problems into smaller, solvable ones. The problems don’t have a definite solution, providing room for interpretation and multiple approaches to problem-solving. The questions are not grade specific and can be used in a range of classes. The open-ended quality of Fermi problems is one of their strengths. 

Encourage intuition in math class through Fermi questions. Give students something meaningful to solve. Tired, old word problems should be a thing of the past.

Depth Over Speed

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Jo Boaler is a somewhat controversial figure in math education. The Stanford professor is in the ‘reform’ camp, arguing that new approaches to teaching math, that rely on a lot of group work, real-life examples and discovery should be emphasized over more traditional methods such as memorization, worked examples, repetition and the learning of key principles and facts. Back in 2012, she was accused by two academics (see here) of questionable research methods and inconsistent data in her Railside Report. For an excellent, in-depth post on the subject, see here. 

Despite the credibility storm that surrounded Boaler, she still has quite a large following. Her voice in math teaching is one of the loudest. Her opinions influence policy and make waves in education circles. When she suggested that memorizing times tables isn’t necessary for students to achieve success in math, it made headlines. It got people talking. It stirred the math pot. While many educators don’t agree with her philosophies, she continues to greatly influence the discussion on how to best teach math.

That said, like many things in education, it’s important to separate the politics and the egos from what works best. Education can sometimes suffer from too much self-righteousness. If Boaler can offer advice that will benefit math teachers, who really cares about the other noise. Leave the politics to the politicians. While you might not agree with everything Boaler says, she does offer valuable insights in her recent book Mathematical Mindsets. Sure, many are points that have been raised before, but they are worth repeating.

One of those ideas is depth over speed. The pressure to cover curriculum that many teachers feel leads to a rat race approach to math instruction. As a result, lessons are often a mile wide and an inch deep. Teachers get stressed out and students retain less as concepts are glossed over and enduring understanding is sacrificed. The train keeps moving down the track and if some get lost along the way, oh well.

This notion that mathematical skill is all about speed is just plain wrong. And yet, that’s the impression that most students have of math class. The best students are the fastest. Whoever can finish a problem the quickest must be the most capable. There is a beeline to the solution. We have been conditioned to look for easy answers, what Dan Meyer calls ‘impatience with irresolution’. To satisfy this dissonance, we rush through it to get it done. Maybe sitcoms are to blame, who knows. 

Boaler provides a telling example of her observation of a Chinese math class. With two of the top three PISA math scores, Shanghai and Hong Kong (along with Singapore) are the best in the world. It’s not even close. The assumption is that they use a lot of drill and kill instruction, where speed is valued, but the reality is much different. Students typically engage with no more than 3 questions per hour. Like mathematicians say, their work is done slowly and deeply. Justification and reasoning form the essence of math. And these take time.

We need to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. 

 

What Is Mathematical Thinking?

‘Mathematics is the music of reason’ – James Joseph Sylvester

Mathematical thinking is a lot more than just being able to do arithmetic or solve algebra problems. It is a whole way of looking at things, stripping them down to their essentials, whether it’s numerical, structural or logical and then analyzing the underlying patterns. Math is about patterns. When we are teaching a mathematical method, we are showing something that happens all the time, something that happens in general. Getting students to see these underlying structures, whether it’s in a math problem, in society, or in nature, is one of the reasons that studying mathematics is so worthwhile. It transforms math from drudgery to artistry.

As Steven Strogatz says, ‘Math is not just what we heard about in high school, the known and straightforward part of the subject. There are all sorts of interesting theoretical and applied problems out there.’ Helping students to see the beauty and wonder in math will go a long way in getting them to think more mathematically. Let them know that what makes mathematicians is the quality of their creativity and the quality of their technique. Combining these two is what leads to great mathematical thinking.

The problem lies in the fact that students usually have no idea about their strengths and weaknesses in these two areas. Schools emphasize the procedural side of math. There is the misconception that it only involves mechanical, robotic thinking. What pervades is a ‘we didn’t cover that’ mentality when they encounter a novel problem. 

Once you have identified a task or situation to explore, mathematical thinking involves these steps that are often done together and simultaneously:

  • break task down into components
  • identify similar tasks that may help
  • identify appropriate knowledge and skills
  • identify assumptions
  • select appropriate strategy
  • consider alternative approaches
  • look for a pattern or connection
  • generate examples

This all leads to formulating a conjecture (solution, generalization or relationship) that can then be tested for counter examples or special cases. Similar to the design process, if it doesn’t work, it’s time to iterate and try again. Proving the solution, making connections, considering limitations, and extending it further to ‘what if’ questions are additional aspects of mathematical thinking. Spending the time to think of and ask really good questions is at the heart of math.

John Mason’s questions and prompts provide a nice framework for ensuring and promoting students’ mathematical thinking. The verbs across the top of the chart highlight what is involved when we ask students to think about math. The specific examples are practical and useful.

 

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Open Questions

Differentiation in math is a relatively new idea. While it has become an integral part of any literacy program, it is scarcer in math. DI requires more planning and thinking through the logistics. Teachers may sometimes be reluctant to use it, because the majority of us experienced math as a subject in which everyone was on the same page. Deviating from the textbook was not encouraged. The class moved forward in lockstep. There was little room for varying the content, process or product.

Many of the questions we traditionally ask students call for a single number, figure, or mathematical object. These kinds of questions are closed because the expected answers are predetermined and specific. Conrad Wolfram estimates that they make up around 80% of math instruction! Students are sent the message that math is about simply finding the right answer. It is black or white. A static subject with no creativity that only relies on a series of endless computations. Motivation to learn math is lost. No wonder students tune out beyond a certain point.

Open questions, an approach popularized by Canadian math researcher Marian Small, offer an alternative. Creating a question that allows for multiple entry points gives all students the opportunity to find something meaningful and appropriate to contribute. They offer a variety of responses based on the students’ level of content knowledge and understanding. This gives students the confidence that is sometimes lacking in math class. Levels of anxiety are reduced as open questions close the various ZPD gaps among the students.

Math is seen as multi-faceted. The subject comes to life. 

The best open questions have a low floor and high ceiling. This effectively levels the playing field and gives more students a chance to engage in math. Open questions should focus on big ideas and curricular goals. They need just the right amount of ambiguity, ensuring that the question is broad enough to meet the needs of all students. Strategies for creating open questions:

  • turning around a question (think Jeopardy)
  • asking for similarities and differences
  • replacing a number with a blank
  • creating a sentence using certain concepts/definitions/numbers
  • using ‘soft’ vague words for flexibility

The first thing a mathematician has to do is pose an interesting question. These are virtually absent in most math classes today. Open questions are a way to change this.

The False Wars

‘In war, truth is the first casualty.’ – Aeschylus

The phrase ‘premature ultimates’ coined by the British literary theorist I.A. Richards describes those conversation stoppers that, ‘bring investigation to a dead end too suddenly.’ In education, this often takes the form of polarities, where there is a seemingly automatic taking of sides on educational issues based on whether one considers themselves to be progressive or traditional. Each side takes an ideological stand and the investigation grinds to a halt. No one wants to concede anything, dialogue breaks down and any thoughtful discussion is lost.

This false progressive vs. traditional dichotomy often strangely associated with politics (progressive = liberal, traditional = conservative) has played out in several educational ‘wars’, most notably in reading and math. In the reading wars, phonics was viewed as a right-wing suppression that deprives reading of its naturalness and destroys a love of literature in children, while whole language was attacked as a left-wing abandonment of the responsibility of teachers and parents to teach kids to read properly. The math wars, which started in the late 1980s, are still being fought, with the latest contention being over the standardized testing of times tables in the UK. Some wonder if these wars exist at all. Maybe it’s all just a front to sell textbooks, employ professors of education and provide a speaking tour for educational experts. The old Hegelian dialectic of problem-reaction-solution. To assume otherwise would be naive.

The wars of education, dividing issues into two camps, leaves little room for progress. Educators and policymakers are so busy getting dug in, they don’t stop to listen to the other side. This results in an echo chamber of ideologies which are self-perpetuating. It’s the same old, same old. Assumptions are not challenged and common sense is often ignored. Perhaps the solution to much of what ails education can be found in this: pragmatic changes in the structure of the ruling ideas.

Those involved often fail to find common ground and much like real wars, the fighting just continues on. Both are related to power, control and exploitation. Of the past 3,400 years, humans have been entirely at peace for only 268 of them, or just 8 percent. Of the past 150 years of modern education, educators have been at peace for very few of them.

It is time to end the education wars.

Developing Sequencing Skills

Sequencing is the process of putting events, objects and ideas in a logical order. It’s a skill that we use throughout the day, often without even knowing it. For adults, it is second nature. For many students, it is something they need to practice. It needs to be taught explicitly. It needs to be regarded as a thinking process that can be broadly applied across many areas. In school though, it is usually taught as simply the ordering of facts and information. This rote memorization reduces sequencing to lower order thinking.

Sequencing contributes to students’ ability to understand and develop their reading comprehension. It is a valuable tool for identifying the components of a story, examining the structure and retelling the story. Through an examination of sequencing in narrative texts, students also hone their writing skills. When applied to mathematics, it can be used in timelines. Just to refresh your memory, a timeline is a way of displaying a list of events in chronological order, sometimes described as a project artifact. It is typically a graphic design showing a long bar labelled with dates alongside itself and usually events are labelled on points where they would have happened (from Wikipedia).

Timeline Creators (click on logo)

Buzzmath – Engaging Practice

Buzzmath is a great 1-to-1 middle school math tutoring program that provides rich activities, instructional support and data management to track students’ progress. It was designed by a company from Montreal, so the units are aligned with Canadian Math standards. There are on-demand examples and solutions  for every practice activity. There is also a built-in calculator for students to use. https://www.buzzmath.com/
 
The students in my class have really enjoyed Buzzmath. The site uses a star & medal reward system that motivates them to build up their points and excel to the next level. It is a very engaging program that is easy-to-use and fun. It is a great way to assess their understanding. In our class, students spend about 3 periods per week working through the different units to reinforce the concepts they are learning. Many of the students have been using it at home as well. 

Foldify – Math in 3D

Foldify is an app that lets students create customized shapes to meet a huge variety of educational goals. Students can create 3D objects (from nets) that further understanding or demonstrate mastery with applications like these:

– identify/combine geometric shapes
– advance spatial recognition
– comprehend, calculate surface area and volume
– build number sense
– explore probability


Choose a template and customize it with multiple creative tools; you can even import your own photos to be printed on the foldables. Blank templates provide nets that form triangular pyramids, cubes, rectangular and hexagonal prisms, and spherical shapes, along with houses, vehicles, creatures and humanoids, hearts, milk cartons, and more. Your creators can see what their nets will look like as realized 3D shapes in real-time within a side panel on the screen.

Source: http://www.lessonplanet.com/professional-development/courses/ed-tech-foldify
Source: https://www.youtube.com/watch?v=vqCh8ILou_c