Principles of Math Videos

This summer, Marc and I made several videos designed to help parents understand what mathematics their children are learning. As well, we wanted to give parents a feel for how their children are learning in their math classes. We added Mathematics 6 & 7 videos to the previous summer’s 8 & 9 collection. The work of producing videos for Foundations of Mathematics and Pre-calculus 10 is well underway; I expect to add two more videos–Solving Systems of Linear Equations Graphically & Algebraically–this week. Although intended for parents, we believe that this series could be a helpful resource for teachers, especially those having to teach in an online or blended environment due to COVID-19.

In this post, I’ll share some of the principles that guided us when creating the videos. This’ll be a peek behind the curtain of interest more to educators than to parents.

Make it visual.

Math is visual. Videos are visual. So math videos should be visual. It is disappointing how often makers of digital content fail to take full advantage of visual aspects available to them. The animation of symbolic representations–line-by-line equation solving or drawing little arrows to show the distributive property–should not be the extent to which content is presented visually. It’s ballsy to sell this to educators as visual if that’s all you’ve got. By “make it visual,” I mean include images and animations that help viewers make sense of the mathematics at hand or the context in which the mathematics is situated. For example, we show that 2:3 is equivalent to 8:12 by repeatedly extending a black-red-red-red-black pattern of beads; we don’t describe two candles with different heights and different rates at which they burn, we show it–so long as we can figure out how to do it in Keynote.

Mathematics 6: Ratios & Percents (1/2)
Foundations of Mathematics & Pre-calculus 10: Systems of Linear Equations (1/2)

In our videos, we make use of virtual manipulatives–or virtual virtual manipulatives?–like pattern blocks, colour tiles, counters, multi-link cubes, base ten blocks, algebra tiles, tangram-like puzzles, Solo cups and paper clips, etc. We employ other pictorial representations such as hundred charts, decimal/percent grids, number lines, double number lines, factor trees, factor rainbows, tables of equivalent ratios, graphs, etc.

Make it conceptual.

I once watched a short video to fix an issue with my dishwasher. I carefully followed the directions, pausing the video at each step along the way. But no one is ever going to mistake me for a handyman! I don’t really understand how dishwashers work. I couldn’t connect the problem to any knowledge of the machine’s mechanical or electrical systems. It didn’t matter; I set out that morning to make one small repair, not become an appliance repair technician. If the solution shared online didn’t work, I was hooped. And even though I was successful, my procedure for fixing my dishwasher was useless for fixing my washing machine, let alone a different make and model of dishwasher. This skill–long-forgotten, by the way–didn’t transfer from one household appliance to another.

But it didn’t matter. I set out that morning to make one small repair, not become an appliance repair technician. Mathematics is different. The emphasis in math class must be on sense-making, not answer-getting. The same should be true of math videos. In our videos, we attempt to always address the why.

For example, we answer “Why is a negative divided by a negative a positive?” by revisiting what it means to divide whole numbers and then applying these two fundamental meanings to dividing integers. Later, a thermometer example reinforces dividing as measuring.

Conceptual understanding means seeing mathematics as a coherent whole rather than isolated procedures. Digital content can support students in developing conceptual understanding by creating opportunities for them to connect models and representations. For example, we ask learners to connect multiplying binomials to what they already know about multiplying two-digit numbers (i.e., an area model, partial products, the distributive property).

Of course, procedural fluency is important. Effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency. However, procedural fluency depends and builds on a foundation of conceptual understanding.

For example, percents are presented as fanatical comparisons to 100. No part-whole-percent triangles or is-over-of-equals-percent-over-100 cross-products here. Relating percents to decimals does not appear until grade 7. So, in our Mathematics 6 video, the emphasis is on equivalency and benchmark percents (i.e., 50%, 25%, 75%; 10%, 20%, … , 90%).

Procedural fluency includes the ability to apply procedures flexibly. Throughout each video, multiple strategies are discussed. For example, see the strategies–and representations–used in this proportional pizza problem.

To bridge conceptual understanding and procedural fluency, we try to build on learners’ own mathematical ideas. For example, before the elimination method for solving systems of linear equations is introduced, viewers are first encouraged to solve a puzzle using their intuition. Then, they are presented with a pictorial representation of a solution to a problem. It’s at this time that algebraic symbols and notation appear:

(Note: Each row and column is a sum. None of that fries times Coke nonsense!)
Foundations of Mathematics & Pre-calculus 10: Systems of Linear Equations (2/2)

This process plays out when solving equations in one variable (boxes of doughnuts; algebra tiles) in Math 7 and when solving systems using substitution (scale puzzle; two types of tickets problem) in Math 10.

Of course, we can build on viewers’ own ideas only to the extent to which they engage with and participate in the learning experiences that we design. This segues into our last principle…

Make it inviting.

I think that this is the right adjective. To me, inviting goes beyond accessible.

One way in which we make an effort to invite parents to “do the math” is to use open questions. Sometimes, this means open-ended. For example, Show me one-quarter in as many ways as you can, What could the numbers be? and What comparisons can you make? (and later What is being compared in the ratio 1:2?) are open-ended; they allow for many correct answers and signal that a range of responses are valued.

Other times, we used open middle problems. An open middle problem may have one correct answer but multiple ways of getting it. For example, there is an arrangement–or two–of number tiles that maximizes the expression below. The animated placement of the number tiles is meant to model one strategy and includes me making missteps and backtracking as I went along.

(Here’s a number tile factoring task from Math 10 that has both an open end and an open middle.)

In addition to openness, we try to hold off on introducing formal symbols and notation early. For example, Two numbers add to 12. What could they be? comes before Solve the system x + y = 12 & y = 2x. (See also Burgers, Fries, and Cokes and Tees and Hoodies above.)

Sometimes, when a task is not a soft place to start, we may still present it up front and then return to it later, after we’ve built up some knowledge. In this way, we hope to “make it inviting” by piquing the curiosity of viewers. For example, asking parents to pick two numbers that differ by two and multiply them is accessible whereas asking them to explain why this product is one less than the square of the number between them is not. A similar approach was taken with a gas vs. electric vehicle application of linear systems; it serves as a hook in the video’s introduction.

* * ** *** ***** ********

We believe that learners of mathematics should be active participants rather than passive spectators. There’s a tension between this belief and video. In our videos, we put a lot of trust in parents pressing pause when prompted. It’s in these moments that they “do” math, that they play, notice and wonder, solve problems, visualize, look for patterns, make conjectures, generalize, reason, explain, connect ideas, take risks, etc. We were limited by the medium–or our tech skills within this medium.

In a mathematics classroom–be it face-to-face or remote–this tension can be resolved. And this is one reason why we’re just as, if not more, excited about teachers using these videos. At the moments when we ask viewers to pause, students could be placed in visibly random groups or breakout rooms. Teachers are not limited by our prompts–or these moments. They can observe and adapt to what’s happening with their learners in the moment and ask How else might you have solved the problem/represented your thinking? What does this remind you of? How are these the same? How are they different? What would happen if… ? etc. There’s no need to “fake it” coming out of a pause as we had to do (e.g., “You might have noticed that…”).

These videos were intended to capture the big ideas or enduring understandings or key concepts of a topic–a whole chapter or unit. Although each clocks in at about twenty minutes, it would be inappropriate to have students experience an entire video in one sitting. Instead, a task or two clipped from a downloaded video could make up one day’s learning experience.

If you find this video series helpful, we’d love to hear from you. Drop a comment, question, or complaint in the comments.

Grade 3/4 Fraction Action

Recently, I was invited into three Grade 3/4 classrooms to introduce fractions.

Cuisenaire rods give children hands-on ways to explore the meaning of fractions. After students built their towers, flowers, and robots, I asked, “If the orange rod is the whole, which rod is one half?” Students explained their thinking: “two yellows make an orange.” I emphasized, or rather, students emphasized that the two parts must be equal.

yellow orange

I asked students to find as many pairs as they could that showed one half. I let ’em go and they built and recorded the following:

one half

Once more, with one third:

one third

As children shared their pairs, we discussed the big ideas:

  • the denominator tells how many equal parts make the whole (e.g., two purple rods make one brown rod, three light green rods make one blue rod)
  • the same fraction can describe different pairs of quantities (e.g., one half can be represented using five different pairs, one third can be represented using three different pairs)
  • the same quantity can be used to represent different fractions (e.g., white is one half of red and one third of light green, red is one half of purple and one third of dark green, etc.)

Something interesting and outside the lesson plan happened in each of these three classrooms.

Some students described each pair of rods using equivalent fractions (e.g., 1/2, 2/4, 4/8):

equivalent fractions

I asked the “we’re done” students to represent their own fractions using pairs of rods and determine each other’s mystery fraction. Many students chose fractions like 2/5 or 3/4, not simply unit fractions:

two fifths three quarters

After students shared the three pairs of rods for one third, I asked if anyone found any more. “I did,” said one student, unexpectedly. Check this out:

four twelfths

I asked her why she chose to combine an orange rod and a red rod to make the whole. She explained that twelve can be divided into three equal parts. Without prompting, the rest of the class starting building these:

five fifteenths six eighteenths

adapted from The Super Source

Marriage Problem

Last week, we wrapped up our winter sessions with over 50 elementary school math teams. Part of these sessions are devoted to having teachers work together to solve problems. Having teachers “do the math” helps brings meaning to important topics in mathematics education. We gave the following problem, from Van de Walle:

In a particular small town, 2/3 of the men are married to 3/5 of the women. What fraction of the entire population are married?

This is a challenging problem, but only because traditional algorithms get in the way of sense-making methods. The gut reaction is to do something with common denominators. Time after time, with each group, primary and intermediate. Through questioning, the mistake can be recognized.

“In this context, what does the 15 over here represent?” [points to 10/15]
“The total number of men.”
“And over here?” [points to 9/15]
“The total number of wom–OOOOOh…”

Sometimes, it takes longer to reach an ‘OOOOOh’:

“What does the 10 represent?”
“The number of married men.”
“And the 9?”
“The number of married wom–OOOOOh…”

Once teachers realize that having 10 men married to 9 women is somewhat problematic, most model the problem using colour tiles. Two out of three men being married becomes four out of six and six out of nine. Three out of five women being married is equivalent to six out of ten. Six pairs of husbands and wives can be formed. We have 12 out of 19 people being married.

marriage (concretely)Others think logically to solve the problem. The number of husbands must equal the number of wives. The number of husbands and wives are represented by the numerators.  Therefore, the numerators must be made equal. With all due respect to Dr. Math, it just makes sense.

marriage (pictorially)The use of manipulatives to construct meaning continues to be a focus of teachers involved in the numeracy project, both for themselves and for their students. Long before I became involved in this project, my fellow Numeracy Helping Teachers (Marc Garneau, Selina Millar, Sandra Ball, and Shelagh Lim) worked tirelessly to set a climate in which teachers and students felt comfortable using a variety of manipulatives.

At these sessions, we present teachers with problems, not practice. It’s a pleasure to work with such an amazing group of educators so willing to explore, take risks, and persevere. But as much fun as these sessions with teachers have been, I’m looking forward to the real fun: problem-solving with their students.

Quadratic Patterns

Having students write an equation that describes a pattern involving toothpicks, pattern blocks, or colour tiles is nothing new. However, students (teachers?) often focus on patterns in the table of values rather than properties of the pattern itself. Visualizing the pattern can help students write the equation. For some, this approach may be new.

For example, consider the following pattern:

In each figure, students may see a rectangle with two squares attached, one above and one below. That rectangle has a width of n and a length of n + 2. The expression is n(n + 2) + 2.

Some students may see the pattern in a different way. But what about the students who don’t see anything? For them, some scaffolding is necessary. Note the scaffolding in the pattern below.

Students may see one red square, two green rectangles, and two blue tiles in each figure. That is, they see n^2 + 2n + 2. The use of colour is intended to be helpful. Of course, some students may ignore this hint. I’m cool with that. They may see a large square with one tile attached, or (n + 1)^2 + 1.

Again, look for the scaffolding in the pattern below.

Students may see a rectangle with a number of tiles being removed, as suggested by the dotted lines. That rectangle has a width of n + 1 and a length of n + 2. The number of tiles being removed is equal to the figure number. Alternatively, students may visualize  2(n + 1) + n^2.

Did you notice that each of the expressions above are equivalent? They must be. Each of the three patterns begin with 5, 10, and 17 tiles. Each pattern/expression tells the same story, but in a different way.

My goal was to design three parallel tasks. Have students choose one of the three representations… just don’t tell them they’re the same.

My three-part lesson plan:

Marc and I created two more sets of patterns. All three:

For more, please see Fawn Nguyen’s Pattern Posters.

A Deconstructed Learning Outcome: Sum of Its Parts

Maybe I’ve seen one too many deconstructed Caesar salad or peanut butter and jam sandwich on TV. Or maybe I’ve heard “This workbook covers the curriculum” one too many times¹.
 
Whatever my reason, I wanted to take a closer look at a learning outcome from the WNCP Math 8 curriculum document:
 
It is expected that students will demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially, and symbolically [C, CN, ME, PS]
 
“It is expected that students will”
It’s about students’ learning. Worked examples on the whiteboard or in a textbook may be evidence of the teacher’s or publisher’s learning.
 
“demonstrate an understanding of”
Not will be able to. Students need to make sense of mathematics. Justifications and explanations are required for answers and methods.
 
“multiplying and dividing positive fractions and mixed numbers”
This is a topic. Curriculum is more than a collection of these.
 
“concretely, pictorially, and symbolically”
No longer just suggested, the use of concrete materials (i.e., manipulatives) is prescribed² as is having students draw to represent their thinking (diagrams not decorations).
  
[C, CN, ME, PS]
From K to 12, seven processes are to be integrated within the learning of mathematics. The ‘C’, for example, means that students should be provided with opportunities to communicate their learning– to write about and discuss mathematical ideas.
 
¹ To my US reader(s)– in my province, curriculum is different than recommended learning resource (i.e., the textbook). In theory, the textbook is not the course. In practice…
 
² For many teachers, this is probably the biggest change to the curriculum. Earlier this year, I created the posters below. My intent was to generate conversations among teachers, not to teach the concept. Plus, I got to be artsy-fartsy. Enjoy.
 

CPS Poster Algebra Tiles
CPS Poster Counters
CPS Poster Pattern Blocks
CPS Poster Toothpicks

Fool me once, shame on… shame on you. Fool me… you can’t get fooled again.

Lately I’ve been enjoying Veritasium’s videos on misconceptions about science. From the Veritasium YouTube channel:

If you hold views that are consistent with the majority of the population, does that make you stupid? I don’t think so. Science has uncovered a lot of counterintuitive things about the universe, so it’s unsurprising that non-scientists hold beliefs inconsistent with science. But when we teach, we must take into account what the learners know, including their incorrect knowledge. That is the reason a lot of Veritasium videos start with the misconceptions.

I’ve been thinking about students’ misconceptions about mathematics. What math concepts are counterintuitive? How might starting with the misconception play out in the math classroom? Probability probably provides the most potential, from a pedagogical point of view. (Do robot graders give high marks for alliteration?) The classic Monty Hall problem or birthday problem are just two examples of this. Exponential growth can also be counterintuitive – see Chris Lusto’s alternative to the doubling penny problem.

One common misconception students have is that (a + b)^2 is equal to a^2 + b^2. In my classroom, I’d start with this misconception then have students substitute values before exploring this with algebra tiles. Not exactly Why does the Earth spin? type stuff. Still, addressing this misconception right off the bat provided us with a problem to solve – if (a + b)^2 is not equal to a^2 + b^2, then what is it equal to and why?

Recently, I was fascinated by Dan Meyer’s Coke v. Sprite question because my gut reaction was wrong. Twice. Please watch Dan’s act one video now. I’ll wait.

What fraction must you drink to balance the Coke can on edge?

My guess was that there was more Sprite in the Sprite glass than there was Coke in the Coke glass. After all, I reasoned, the Coke that was added to the Sprite also contained a small amount of Sprite.

When I did the calculations, I was surprised to learn that the amount of Sprite in the Sprite glass and the amount of Coke in the Coke glass were the same:

  • Assume the original amount of each is 100 mL.
  • Assume 10 mL of Sprite is transferred to the Coke.
  • 10 mL of pop is transferred back to the Sprite. Stirring means 10/110, or 1/11, of this is Sprite. 100/110, or 10/11, of this is Coke.
  • The amount of Sprite in the Sprite glass is now 90 mL + (1/11)*10 mL = 90 10/11 mL.
  • The amount of Coke in the Coke glass is now 100 mL – (10/11)*10 mL = 90 10/11 mL.

Before watching Dan’s act 3 video, my colleague Shelagh Lim and I modelled this with colour tiles:

  • Start with 12 green tiles on the left and 12 red tiles on the right.
  • Move 4 green tiles to the right. Now, 4/16, or 1/4, of the tiles on the left are green. 12/16, or 3/4, are red.
  • 4 tiles are moved back to the left. To simulate the effect of stirring, 1 of these 4 are green. 3 of these 4 are red.
  • The number of green tiles on the left is now 8 + 1 = 9.
  • The number of red tiles on the right is now 12 – 3 = 9.

Shelagh asked, “What if you don’t move back 1 green and 3 red? What if you close your eyes and take out 4 random tiles?” In other words, does stirring matter? I argued it did. “Something something proportions,” I said.

Mind. Blown.

I want students to experience this feeling of enjoyment at being led astray by their intuition. But, more importantly, students must also experience the feeling of enjoyment that comes from following their intuition and being correct. The former is not possible without the latter; to be amused by failure, there needs to be an expectation of success.

A Linear Functions Lesson Across the Grades

How many people can sit at 100 (or n) triangular tables? Square tables? Hexagonal tables? What if you join the tables so that one side of the next table touches one side of the previous table?

I appreciate this problem for a few reasons:

  1. I can present it in grades 4 through 10. In grade 4, students write a recursive relationship (e.g., for joined hexagonal tables, start at 2 and add 4 each time). In grade 6, students write a functional relationship (e.g., 4n + 2). In grade 8, students graph a linear relation (e.g., y = 4x + 2). In grade 10, students interpret the slope and y-intercept (e.g., each added table provides 4 additional seats, there are 2 additional seats at the ends of the table). When I teach and discuss this lesson at different grade levels within a school, I think a common activity helps teachers connect the big ideas across the grades.
  2. I can easily adapt and extend the task. When I have taught this lesson in grade 6 (see three-part lesson plan), most students can write an expression for joined square or hexagonal tables. Some students may choose to solve a simpler problem and write an expression for joined triangular tables. Other students can be challenged to write an expression for tables with any number of sides. All students can participate in the class discussion.
  3. The use of pattern blocks can help students gain a deeper understanding. Most students were able to make sense of the 4 in 4n + 2. Each time a table is added to an end, 4 seats are added. (Two seats are lost when tables are joined.) When one student showed how he added tables to the middle rather than an end, this helped his classmates make sense of the 2 in 4n + 2. There are two more tables at the ends. Pattern blocks allow students to make sense of the expression beyond “add 2 to make the numbers in the table of values work”.

This problem appears in several resources including The Super Source.

Math Manipulative of the Month – Pattern Blocks

MMM September 2011 Pattern Blocks (colour printer, double-sided)

Last year, a group of Surrey teachers suggested having a “Math Manipulative of the Month” at their school. Instantly, I thought this was a great idea. After this conversation, I created the brochure above. My hope is that this series of brochures can be used to generate conversations between teachers (and students, of course!).

Before trying the problems, I would ask teachers to get to know each MMM and list all they know about them. For example,

  1. “Two reds cover 1 yellow”, “Three triangles make 1 trapezoid”, etc.
  2. “All sides are the same length, except the base of the red trapezoid. It’s twice as long.”
  3. “The orange square and tan rhombus do not cover the other tiles.”

The symmetry problem ended up on the cutting room floor. Here it is: Pattern Blocks Symmetry.

Also, please see how the question “How many ways can you make 360 degrees?” becomes a problem-based lesson in Grade 6. Here’s the three-part lesson plan: Angles (format from Van de Walle).

I attempted to have a balance of primary and intermediate problems. How can each problem be adapted for the grade level that you teach?

Next month… Base Ten Blocks.