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

Running naked again. This time, with scissors.

In an earlier post, I shared a poster that I created using photos of odd numbers taken by local photographer mag3737. He shared this pictorial representation of the Pythagorean Theorem with me. Very cool – both the image itself and the online sharing.

I created another poster from mag3737’s photos – a pictorial representation of exponential growth.

The rate at which exponential functions grow can be a difficult concept to visualize. Starting at 2^4 = 16, the area is doubling but the height is not. By doing this, I’m not sure if I accomplished my goal of illustrating exponential growth (although I did manage to have the numbers fit on the page).

I suggest taking scissors to this poster. The eight columns of eight 64’s that represent 2^6 stacked on top of each other reach a height of over five feet. This is a powerful (and perhaps surprising?) image of exponential growth.

Instead, because of necessary scaling, students often see something like this:

What other mathematical concepts could be represented using these photos of numbers?

PDF’s of the posters: sum of consecutive odd numbers & powers of 2

A pictorial representation that will have you running naked through the streets

The sum of the first consecutive odd numbers is a square number.

Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.

Some of us (okay, me) may have presented something like this:

1     +     3     +     5 + … + (2n – 1)
(2n – 1) + … + 5     +     3     +     1

The sum of each column is 2n. We have n columns. The total is then × 2= 2n². We added the sum twice so 2n² ÷ 2 = n².

Can you see what perfect squares have to do with odd numbers? Me neither.

Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.

Inspired by this pictorial representation, I created this poster below.

 ¹ Lockhart might say it’s not the fact that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the idea of chopping the square into these nested shapes.

Never let them see you smile.

At least ’til November.

Anyone else remember being given this advice by veteran educators at the start of your teaching career? The thinking here was that it would prove too difficult to get students back on track once you loosened the reins. If you must, loosen up at the end of the semester. I could never pull this off. My true self, or at least my true teaching self, would make a special guest appearance by the end of the first class.

I often struggled with planning for the first day of classes. I’m just not able to lecture students for 75 minutes about consequences of unexcused absences, procedures for handing in homework, and lists of food & drink items that are acceptable to have in the classroom. Imagine sitting through this four times on Day 1. Welcome back!

“And one more thing… here’s a review worksheet that covers everything you should know from Math 9. See me or a counsellor if you’re having difficulties with it.”

I was also uncomfortable with the let’s-get-to-know-all-about-each-other approach. No “Find someone who…” searches for me.

When students left my classroom for the first time, I wanted them to believe that

  1. We were going to get to know each other as people, and
  2. We were going to do this while learning mathematics.

Here’s a PMa 10 1st Day Jigsaw activity that, although not perfect, attempts to convey this message.

I cut the squares and placed them in envelopes. In small groups, students pieced the puzzle back together so that questions and answers shared a common edge. An answer key is not provided, but the jigsaw puzzle part of the activity does provide students with some feedback.

These are not rich problems – they are review questions of important concepts & procedures from Math 9. However, I did listen to some interesting conversations. For example, in many groups, there were debates about which power (3^-2, -3^2, or (-3)^2) was equal to -9. One student said he remembered that a negative means flip (his words, not mine) and matched 3^-2 with 1/9. His group members asked him to explain why this works.

Please let me know what you think of this activity. Also, do you have a Day 1 lesson to share?

As a new school year begins, are you looking for posters to decorate your classroom? Learn how to create a gigantic math poster of your own.