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.

Cuisenaire Rods for Intermediate Math

One of the larger goals of math in the intermediate grades is to learn about proportional reasoning (multiplicative thinking) as applied to fractions, percentages, proportions, ratios and patterning. This provides a solid foundation from which higher levels of mathematics can be explored, including geometry, linear and quadratic functions, and data management.

In the primary and junior grades, among other mathematical constructs, students work with decomposing larger numbers into smaller quantities, primarily as addends, and eventually working towards factors as used in multiplication.

Cuisenaire rods are a useful tool in math, allowing students to easily compose and decompose small quantities while showing the proportional relationships. For example,



Each block can represent an even proportion of units. For example, if we call the white rod "1" and the red rod "2", then it follows that light green is "3" and so on up to 10.

We can combine rods to create larger numbers, such as 12 by using red and orange, 20 by using 2 oranges etc.

We can also assign a larger value than "1" as our base (white) unit. If we assign it as "5", then we can count by 5's up to 50 without the need to add additional rods.

Using these relationships, intermediate students can use these rods to explore the concepts of proportion, ratios, fractions, and the distributive property.

Factoring and the Distributive Property With Cuisenaire Rods

Using the principles listed above, students can use the rods to factor a larger number. For example,
in the following photo, 12 is factored in a variety of ways:

 In the dark green row, we see 2 rods of 6 or 2 x 6. Likewise, in the red row, we see the reversed version where there are 6 rods of 2, or 6 x 2. Through the use of many examples, students can determine through exploration the commutative property of multiplication, and also the relationship between multiplication and division.

Factor trees can also be built in this way, by further reducing the larger factors. Eventually, each component of each row can be factored down to the prime number components.

If we decided to use a base unit other than one, we could compare similar numbers based on proportional relationships. For example, if we chose "3" as our base for the above example, the top row would be worth 36. The second row would be 2 rods of 18. The third row would be 3 rods of 12, and so on. We could also build the 36 by using more orange and red rods and keeping the base unit as 1.

In the following picture, "9" has been divided into various rows in a somewhat different way:

Moving down from the top, the second row shows 3 rods of 3, or 3 x 3. In the third row down, we see something new. Suddenly there are two different colours in the row. Can we write a multiplication expression for this row?
Similarly, the 4th row also has two different colours.
In the 5th row, however, we see a pattern emerge. There are 3 instances of red-white, or 2, 1 in this row. How might we express this mathematically?

This is an example of the distributive principle of multiplication. How else might we use cuisenaire rods to show this?

So far we've been looking at rows of rods to compare relative values. How might we move into an extra dimension where we can compare both rows and columns?

Eventually, algebra tiles will help with these visualizations, however, the challenge of combining grids to visualize proportions allows for concrete abstraction that will eventually lead into linear and quadratic applications.