Forensic Fridays

Fantastic professional development leads to fantastic ideas. Last spring at Thames Valley STEAM conference,  I was inspired by Jen Brown's talk about Mistakes Monday, wherein students are given a problem with an incorrect solution, and are asked to work in groups to prove why this solution and the reasoning behind it are wrong.

This summer, at the CEMC Math Teacher's Conference, Michael Jacob's talk, Mind the Gaps, was also very enlightening. He showed us incorrect answers, and had us work out how students came about those answers to determine the misconceptions behind those answers. It was enlightening to take the time to discern where seemingly random answers had a basis in logical, if faulty, reasoning.

The idea that it is OK and expected to make mistakes, and that doing so can be beneficial in the learning process, is reflected in the growth mindset work of Carol Dweck and the mathematical mindset work of Jo Boaler. When students never or rarely make mistakes, it may also mean that they are not being appropriately challenged.

And I thought, why not combine these ideas, and create "Forensic Fridays".

Using one or more of the following sources: incorrect student work or correct student work that uses a unique approach (either one that will work consistently, or one in which the answer is coincidentally the same but the strategy is faulty), old EQAO test examples, and CEMC math contests, I search for problems that match misconceptions associated with underlying concepts of the math we are doing in class.

On Friday, students work in pairs or small groups on the question I have posted. They must determine several things:
- is the answer correct?
- how did the person go about solving this?
- what was their train of thought for each step?
- what (if anything) is faulty about their reasoning?
- how can I prove this is correct/incorrect?

Then they must solve the problem correctly in a way that shows their strategy. As students get more comfortable, we will begin to discuss what constitutes a mathematical proof and how to use this to support their thinking.

As a class, we can create a flow chart to demonstrate how to go about these tasks together, in order to break it down for those who struggle with "just knowing" that something is right or wrong, as well as for those who might need extra support.

Since I do not currently have my own class, I have not had the opportunity to try this out. I'd love to hear your feedback if you have done something like this in your classroom.

Prioritizing Purpose

How many times do we here the words, "When will I ever use this?" in math class? In the media? Every couple of years, there is an editorial in a major newspaper questioning the relevance of teaching students algebra, and questions about whether or not it is necessary for all students to learn.

Math teachers often lament hearing their students ask this very question. We can reply with general, well-meaning answers including statements that it helps build problem solving skills, good work habits, the ability to follow procedures, and the ability to think logically. I've seen lists of careers that depend on higher math (most of which will be sadly outdated by the time students graduate), or a chart of expected income based on the level of math successfully completed. But few actually answer these questions specifically, head-on. I often find myself wondering why this questions persists, and also, why so many of us dread it and skirt the issue.

When I took math in high school, the purpose of most of what we worked on was a complete mystery. However, I did not feel free to ask questions such as "How do we know this?", "How is this used?" and "Who discovered this, and why?", for fear of being considered rude and disrespectful. The answers would have meant a great deal to me as a student. It might have given me reason to finish the pages of textbook-based problems assigned each night on a more regular basis. I also wonder if this lack of connection might not lie at the root of our society's math phobia problems. When learning happens in isolation without the benefit of connecting to relevant applications, it becomes by default a mystery, much like an untranslated ancient language.

Luckily, the Rosetta Stone for math exists through simple internet searching. Math appreciation through learning about how, where, why and by whom the concepts were developed and learning about abstract and concrete connections of concepts and applications are all readily available.

Math teaching has changed somewhat since then, but there is still a tendency toward abstraction without explanation and connection as students move into more complex math, and this, I believe, is one area in which math instruction can and must improve. While it is true that there is a great deal of material to "cover" in the higher grades, what purpose is there to this if students fail to see the relevance and drop or fail out of it, or simply go through the motions of applying algorithms without seeing the purpose and beauty within?

There are countless YouTube resources about the history and demonstration of various math concepts and applications. There are also wonderful apps and online manipulatives such as Gizmos and the line-graph intuition app by Sal Khan that help students visualize and make those connections. As teachers, we need to use these tools not only in the lower grades, but through middle school and beyond. For quirky takes on concepts, there is ViHart.

But why not ask the students to work across the curriculum and discover those answers for themselves? They could create a historical video, a comic strip, a simple paper, a song/parody, booklet, interpretive dance (like the trigonometric functions dances and circle dances that already exist), etc. around the history and use of a given mathematician or concept. Yes, this will take time from "covering the material" in class, but taking a period or two to do this just might improve student engagement and investment over the long term.

Resources:
Videos:
It's OK To Be Smart
Standup Maths (Matt Parker)
Numberphile
TedEd (also search this channel for "math" for a more specific playlist)
ViHart

Representation:
Hidden Figures
The ADA Project (also see this NPR article)
Multicultural Mathematics

General Math History:
History of Maths
The Story of Maths

Apps and other resources:
Gizmos
National Library of Virtual Manipulatives
YouCubed
Solve Me Mobile Puzzles

Here We Go Again...

Every couple of years we see a huge public outcry about how we need to "go back to basics". Sometimes this is in response to test scores being "below average". Here I will (once again) weigh in on this.

Because most people have spent years in classrooms, they feel like they are qualified to weigh in on educational policy. They appeal to politicians who are not likely to have a background in education or psychology. However, the fact that Canadians are concerned about and value education is something that we can definitely be proud about.

Several news agencies have recently published articles calling for a "back to basics" approach to mathematical education. But what exactly are "the basics"? Are we speaking of numeracy, or simply a fluency with basic number facts?

Many opinion pieces cite methods such as Dewey's constructivism used in the current approaches. What is actually being taught in teacher education is inquiry learning, which shares some features with constructivism, but is not entirely the same thing.

An inquiry model does not require a student to "construct" their understanding of a concept and then leave it at that. The basic lesson has three parts: a minds-on section in which a problem that builds upon prior knowledge is introduced and students are asked to think about how they would go about solving it; an action section in which students collaborate and share their ideas, applying them to a new problem or problem set that extends the concept, with the same concept; and then a consolidation phase in which students share their work. Various approaches used by the students and introduced by the teacher as needed are compared and evaluated for clarity, consistency and efficiency. This phase is where the students consolidate their learning. Students are often asked to complete a new problem or problems using the concept as an "exit ticket" to show their understanding. The teacher uses these to determine the next steps needed for the class, as well as individual students, in order to further their learning.

In subsequent lessons, students are also asked to apply their mathematical understanding in various hands-on ways, which might include projects built in maker spaces, coding, or geometric art.

Taking a lesson to look at mistakes every now and then is also common. Students are asked to look at a teacher-chosen problem and solution, and demonstrate why the reasoning used is not correct. The ideas are that in learning from mistakes, students realize that making mistakes along the way is part of the process, and it also encourages them to work on their own mathematical reasoning skills and means of communicating their mathematical thinking.

What is missing from this approach? Memorization of an algorithm and repeated practice. Memorization of an algorithm provided by the teacher, with detailed steps on how to complete the algorithm, is what many adults equate to math instruction. It is what is familiar to them, since many learned it this way. However, simply knowing the times tables and how to do long division alone do not make a person numerate, any more than knowing the alphabet and phonetic sounds makes someone literate. Maybe you can sound out a simple word, but to gain meaning from the text requires comprehension skills. This is also true of math.


It is true that memorization of times tables helps with the quick completion of worksheets in higher grades. Computation abilities are still important. Even though we have tools everywhere that can complete this with greater speed and efficiency than people can, being able to process these smaller steps with ease and fluency frees up working memory needed to manipulate more complex problems. However, we do have computational tools (calculators, electronic devices, computers), so placing our priorities on those computational skills alone is not beneficial and does a disservice to our students. We need students who are able to apply those concepts, program the computers, choose a strategy, solve problems, make connections, find patterns and apply and extend those patterns, plan and strategize. We need to prioritize higher-order thinking skills that allow us to move beyond basic computation. Students need to develop a sense of number, quantity, additive and multiplicative reasoning, proportional reasoning, patterning, balance, spatial reasoning, estimation skills and so on.

To remain stuck at memorization of number facts and algorithms alone is simply not enough.

Practice is one area that in my opinion could use more balance. We have gone from reams of worksheets, usually all of a single problem type that does not require reflective thought, to the use of 1-3 problems in a day to illustrate a concept. Somewhere in the middle is a place where students have a chance to work on problems that reinforce a concept while being required think critically and strategize, not only with the algorithm of the day, or by matching a pre-determined vocabulary list with a given operation, but in visualizing and manipulating the information given until they make sense of it, then applying an appropriate strategy and computation for solving it. Students need to also be encouraged to search for and find the answers to the age old question, "(when) will we ever use this?". If they don't see a purpose in it, how can we expect them to find the motivation to struggle through a problem or concept? The purpose must be clear.

Another recent push in education is the concept of developing a Growth Mindset, as described by the work of Carol Dweck, and elaborated upon by Jo Boaler. The ideas here are that students need to be open to learning, and accept that there will be some struggle when they are truly learning, but that they are capable of working through this struggle to gain competency. This is especially important in math, since there are many myths that abound about people having a "math brain" or not having one, which is simply not how brains work. While we'd never shrug off being illiterate, common phrases and ideas such as "I'm not a math person" and "you must be so smart to understand math" show how our society reflects an idea that numeracy is out of reach for many people. If students are to learn math, they need to first believe that they can learn it, and the adults around them need to also believe they can.

The Latest False Dichotomy in Education

Years ago when I was a student teacher, there was a battle of sorts being waged in the language arts curriculum: the phonics vs. whole-word approaches to teaching reading. On one hand, students were asked to sound-out all words they did not know, and on the other, students were asked to memorize thousands of words. Reality: fluent readers use a variety of techniques including but not limited to the ones espoused by such methods. When some people tried to call this approach "whole language" it created confusion as many missed the difference between "whole word" where individual words are memorized out of context with 'whole language" in which context, along with phonics and sight words is an important component.

Many moons later, we see a similar battle being waged in the teaching of mathematics. 

There are the traditionalists who value "sage on the stage" and "drill and kill" methods in which students memorize algorithms and focus on answer-driven tests. Students become walking calculators, and weaker students are often left without the conceptual understanding to allow them to know when and how to apply these algorithms to solve everyday problems. Mnemonics such as "Yours is not to reason why, just invert and multiply", "FOIL" (which only works for up to two terms), and what I've only recently discovered, the "butterfly method" for multiplying fractions are examples of how conceptual understanding is replaced with memory tricks in order to gain a correct answer on a standardized test.

Then there are the constructivists who believe that students must create their own knowledge set through experimentation. They offer an overwhelming range of options for students to explore, but often neglect the final stages of consolidation and review, as well as time for practice with additional problems. Critics argue that since it took centuries to develop the fields of mathematics, expecting students to "reinvent the wheel" is a waste of time for everyone. Such teaching can also be time-consuming, and students who are struggling can become overwhelmed and confused with the large variety of methods to solve a given problem.

Again, just as in the reading example, the polar extremes reflect a false dichotomy when it comes to learning.

Since there seems to be a reluctance for educators, policy makers and the general public to consult the literature, examine what we know about cognitive development and read the studies, there becomes a tendency to grasp onto the methods one is familiar with and hold these as sacred. In many cases in North America, this means that the traditionalist methods are held in higher regard than the constructivist methods. Looking to other countries that tend to do well in mathematics, there are some interesting cultural differences that appear in the approach to teaching and evaluation. One example can be seen in this video with Phil Daro http://vimeo.com/30924981.

In the middle, is student inquiry (again, the name is often used as a substitute for pure constructivism, which causes confusion) in which students are encouraged to try out problems using whatever means they can, discuss the various methods that worked and didn't, share and yes, memorize the methods that work consistently, and connect these methods and patterns to problems they encounter in everyday life. There is structure to the lesson, but there is also a place for students to work with problems on their own terms, experiment and make connections with prior learning. Lessons are scaffolded so that they build on concepts already mastered. Consolidation happens with the whole class and is reviewed again at the start of the next lesson. Students still memorize times tables and formulas, but they also understand where these come from and what is happening with them. They can use a matrix to show multiplication and can tell you why ax + by + c= 0 is a different way of showing y=mx +b, and how various different values of "m" will change the slope of a line when graphed. They can relate this to situations in their everyday life. They know what to do when confronted with 

(2x + y)(3x +2y -z)

 because they have learned the underlying pattern of how this works, rather than just a convenient but limited mnemonic. The understand that BEDMAS is an mnemonic shortcut that helps them use the distributive property, and that the multiplication/division are interchangeable as are addition/subtraction.

Without context, understanding of the underlying pattern, and sufficient understanding to apply the concept widely, a math student's abilities are no more useful than a calculator, and are likely much slower at that. We need people who can not only calculate, but choose the appropriate algorithm and problem-solve in a variety of situations. We need people who understand how to program the algorithms in the first place. Math is not a religion to be taken on faith; it is a science that stands up to scrutiny. We would do well to remember this as we approach the subject in the classroom.

Students learn by doing and thinking, by struggling through problems. When this is connected to their everyday experiences, it becomes meaningful. If we can recognize this in other subject areas, then why not in math?

The Value of N

I've been thinking a lot lately about the way math is visually presented to young students. Looking at equations and the meaning of equal signs, as written about in this recent post, I also began to wonder about the use of answer blanks, such as can be found in questions like this one:

5-3=___

There are reasons to write it like this, most obviously to give the student a place to put the "answer" or, better put, to complete the equation. However, following the equal sign with a blank might also be the cause of some confusion when students reach introductory algebra. They may have learned to associate the = sign, and/or the blank with "this is where the answer goes" rather than understand that this is an equation in which the value of each side must balance.

Consider the following way of representing the same question:

5-3= n

which could be followed with:

n = ___

or simply the words, "What does n equal?" or perhaps better still, "What is the value of n?".
(I chose the letter "n" because it can stand for the word "number", but any letter would work as well)

While there is still that problem of the equal sign followed by the blank, the way the first part is represented manages to help convey some information that is missing in the first example, such as "what are we looking for?" and "how can we represent the unknown number that will balance the equation?".

Maybe a picture would help get the idea across better (please forgive my crude drawings!):


Of course, if you have a balance and unit weights handy, you could always use those to help solve the problem.

I wonder if presenting simple arithmetic with a variable rather than a blank from the outset would help students better understand the concept of equation and equality better, and also predispose them to accept variable notation when it becomes more crucial in algebra.

If you choose to use this idea with your students, I'd be very grateful if you would let me know how it goes.

What is 4?

I'm currently reading The Glass Wall: Why Mathematics Can Seem Difficult by Frank Smith.

Early on, he describes how number is not the same as quantity. He uses 4 as an initial example, but then moves on to use a large number, somewhere over 7 hundred million, to demonstrate that the number is valid whether you have an associated quantity of something that it represents or not. Number is number. However we develop number sense, he argues, it occurs separate from natural language which tends to be ambiguous where math, buy its nature is not ambiguous (at least not to those who understand it!).

So, what in fact is 4? How do we truly understand the concept of 4 (or any other number)? Smith argues that a number can only be put into context when it is compared with other numbers. He sees mathematics as a separate existence than the rest of the world.

I'm not quite sure if this rings true for me or not, but it is an interesting thought to explore.

This also made me wonder what certain young mathematically inclined students might think about it, which led to the following idea for math enrichment.

Lesson Plan Idea: What is 4?

Students are challenged to brainstorm how they would explain the concept of 4 to people who had no numeracy (aliens, young children, etc.). They then move together into groups and share their ideas. The group chooses several to share with the class. One person (either the teacher or another student) plays the role of the learner while the students attempt to explain the concept. The learner should do their best to avoid using any previous mathematical knowledge and base their "understanding" purely on the information given by the students.

Class discussion should include:

•  counting--number/object connection; meaning of each number name
•  quantity--did they use concrete items (manipulatives) to demonstrate their ideas, and how successful might this be in getting the idea of number across without ambiguity (were the shape of items, colour, function or other characteristics confused with sense of number)
•  use of geometry and/or other drawings or models
•  other ways of relating the concept
•  would their system of explanation work for very large numbers, fractions, decimals, negative integers, zero, etc., and if not, how could they adapt or change it so that it will
•  was this difficult, and if so, why do the students think it was

The activity is open-ended and intended to provide deeper insights into the complexity of seemingly simple mathematical concepts.

Other related lessons might include:

• working with different number systems
•  working with different bases 
• writing computer programs to solve very basic mathematical problems (using a machine-based language or something without built-in mathematical algorithms that the students can access)
• a study of the historical use of "zero" and what it really means in mathematics (hint: it has a more complex meaning than simply "nothing")

For more math activities, see the Lemonade Math Page

On the Concept of Balancing Equations

So often in the early grades kids become accustomed to seeing problems written out as below:

3 + 4 = ________

When the answer blank appears in different places in the equation, such as on the left-hand side, it can help, as can the creative use of manipulatives to represent the symmetry of equations. However, the connection between the equal sign and the demand for an answer may continue to confuse some students. Some students learn to think of the = sign as meaning "insert answer here" rather than as the fulcrum of the equation.

What do I mean by this?

Consider the term "balancing an equation".
If you envision an equation as a balance scale, you can put the = sign at the centre, or fulcrum of the scale. In this way, the equation is balanced when both sides are equal to each other. There is a symmetry in the weight on each side.

This can be used to demonstrate the mathematical meaning of the equal sign in a hands-on concrete way.

For students who have difficulty with the concept, consider having them use the balance with weight manipulatives. x might be the name of the 1 gram weights, y the two gram weights etc. Let them play around.

What if they put 2x on one side and y on the other? When the balance is level, the sides are equal.
What if you put 2y on one side and x on the other? When they balance is tilted one way or the other, the sides are unequal. Instead of an equation, you have an inequality ≠.

You can take this a step further if your scale is the kind that has an arrow on the fulcrum. Label the point of balance with an equal sign =, and the space on either side with an inequality sign ≠.

Encourage students to write the equations as they work with the balance to solve problems and also to make predictions of what expressions will be equations and which will be inequalities, as well as determine what is needed to turn an inequality into a balanced equation.

More of my math activities can be found here.

Why I Love Khan

Khan Academy is one of the best known free educational sites on the internet. Here you can view hundreds of short videos on a wide range of topics, work on sample problems and chart your own progress all for free.

I know that the lecture format isn't the most ideal way to learn, but there are some excellent reasons why this format is popular. Here are the reasons I love Khan:

1. The videos are short, well sorted and well explained. This allows anyone to quickly look up a topic and clarify points easily.

2. Khan has a friendly, relaxed manner that puts you at ease. He doesn't re-film his errors, just corrects himself as he goes, showing students that it's OK to make mistakes and that you just need to double-check and keep going.

3. You can access these anywhere, replay them as often as you want and can learn at your own pace in privacy. Chances are good that if you still don't understand a topic, someone will have posted the question you have in the comments. The community here is strong, and the questions tend to be answered thoroughly. The explanations are clear and broken down into logical steps. He makes no assumptions about the connections the viewer will make automatically--everything is explained. When you understand a part, it is very easy to skip ahead as needed.

4. It is available free of cost to anyone who has an online connection. The videos do not require high speed connections to run.

5. Sal Khan is an excellent role model. He took a huge risk in order to develop this site, and has put much of his own time and money into it without resorting to charging fees or hosting ads to pay for it. As an owner of a website, I can tell you it is not easy to avoid the need to allow advertising since site hosting does not come free. When you consider the hundreds of hours spent making and organizing the videos and website, the immense size of this task is truly overwhelming.

6. You can progress from basic operations right through to graduate level math through this site. Although it is always best to use a variety of resources for your studies, the list of topics covered within the math section at least is quite extensive and thorough.

7. You can access all parts of the site without having to provide any personal information. The only thing you miss out on if you don't provide info is a record of your progress through the videos and lessons.

8. The colour coding helps keep things visually organized.

9. Unlike textbooks, it provides both visual and auditory explanations.

10. The short length of the videos makes them more manageable and allows for natural breaks between sections and concepts.

11. They are accessible to people who struggle with the concepts. They are non-threatening and can be watched and completed in privacy, which makes them more attractive to students who need a little more or a little less time to master a concept and would prefer to move at their own pace without being put on the spot to answer questions in public.

12. Vi Hart has partnered with Sal Khan for some videos and if you are familiar with her work, you will immediately understand that this is a very good thing!

There are other free educational courses available online, including many open courseware options, but the short video format fills a need not found in many other places. Both are excellent opportunities to extend your learning of specific topics without having to travel or commit large amounts of time or money.

Thank you Sal, and all the others who provide us with valuable and accessible educational resources.