Two-Digit Addition – When Do I Show Them the “Real” Way?

Last week, I attended Carole Fullerton‘s parent presentation. She discussed strategies students have for adding two-digit numbers. Carole’s timing was great since I’ve been having similar discussions with teachers in recent weeks.

How many ways can you add 59 + 37?

The most common strategy that I see students use is to add the tens, add the ones, and then combine. Students working with ten frames naturally begin by grouping the 10’s, not the 1’s, together.

Students find other strategies. For example,

  • Add 1 to 59 to make 60. Take 1 away from 37 to make 36. 60 and 36 is 96. (make ten)
  • Add 1 to 59 to make 60. Add 3 to 37 to make 40. 60 and 40 is 100. Take the extra 4 away. (friendly numbers and compensation)
  • 30 more than 59 is 89. 7 more than 89 is 96. (add on)

These are the strategies I use to compute mentally. On paper, I fall back to the traditional right-to-left digit algorithm. It’s the result of performing thousands of such calculations in elementary school.

Students should be encouraged to write their mental math strategies down on paper. Some students will have to.

Teachers and parents appreciate these strategies. They make sense. Teachers and parents want mathematics to make sense to their kids. But at some point they always ask the question: “When should they learn the traditional/regular/real way?” They ask this because they are concerned their kids will not be prepared. “But do these strategies work for three-digit addition?” Yes.

“Relax. This will look familiar,” Carole joked. The same natural left-to-right strategy described above can be written vertically. We start with 50 and 30 is 80. Already, we know the sum is greater than 80.

Compare that with the traditional algorithm. We start with 9 and 7 is 16. We know the sum will have a 6 in the one’s place.

Which piece of information is more important? Carole made the point that accuracy is important. Always was, always will be. But it’s not where we should start. Strategies should be built on conceptual understanding. The emphasis of a left-to-right strategy is on number whereas the emphasis of a right-to-left strategy is on digits.

In her new blog, Amy Newman also writes about this. As well, she shares Carole’s key messages for parents helping children at home.

Teachers Make Excellent Pirates – Two Treasures from Blogs I Follow

If I had asked my Principles of Math 11 students “What can you tell me about linear inequalities?” I bet most would have said something about graphing a boundary line and shading one side of it. If I had asked “What does the shaded region represent?” I bet many would not have been able to answer correctly.

Similarly, if I had asked “What can you tell me about absolute value?” most would have said something about changing negatives to positives. Few would have been able to give a satisfactory explanation if I had asked “What does |2 – 7| = 5 mean?”

After reading John Scammell’s recent post on linear inequalities, I realized that I had it backwards. I’d begin by graphing the line. I’d explain that a line cuts the plane into two regions. Together, we’d determine which region to shade. I’d tell students that each point in this region is a solution.

Instead, John’s colleague begins by having students find x– and y-coordinates that satisfy the inequality. Then, each student plots these ordered pairs on a grid at the front. It becomes clear to students that each solution is a point in a half-plane and that a boundary line exists. I robbed my students of this discovery.

John’s scatterplot reminded me of something from Kate Nowak’s back catalogue. Kate’s activity involves having students guess the number of M&Ms in a container. Plotting the points (guess, distance from correct value) results in something like this:

Kate has students write equations and inequalities that model weather forecasting. “Today’s temperature will be more than 10 degrees off from the usual temperature” can be modelled using |T – 68| > 10. In a later post, she has students write an inequality that models this scenario:

She scaffolds this by asking students to explain why |t – 12| ≤ 1993 and |t – 12| ≥ 1993 are not good models and by having students write a sentence that begins with “The distance from…”.

Context is important. Not because it must answer “When am I ever going to use this?” but because it helps build conceptual understanding. I’m guessing Kate’s students can tell her more about absolute value than “it changes negatives to positives”.

By the way, the inequality for music that I find tolerable would be |t – 1985| ≥ 6. I graduated from high school in ’91 not ’85. I’m just not a fan of 80s music.

(The title of this post was “borrowed” from Nat Banting.)

“Under the M… the square root of 12”

On this blog, sometimes I share my thoughts about transforming math education. This is not one of those times.

Here, I’m using my blog as a digital filing cabinet.

One activity that my students enjoyed was MATHO (and its variations FACTO and TRIGO).

Have students select and place answers from the bottom of each column to fill up their MATHO cards. In some versions, I pulled prepared questions from a hat. In other versions, I translated answers to questions on the fly. For example, if I grabbed 2√3, I called out “Under the M… the square root of 12”. After a student shouts “MATHO!” ask potential winners to read aloud their numbers. (Remember to keep track of answers you have called.)

Nothing revolutionary here – just a fun way to review content.

Squares & Square Roots
Exponent Laws
Simplifying Radicals
Rational Exponents
Factoring Trinomials x^2+bx+c
Factoring Special Products
Trig Functions

By the way, if you are looking to read about changing things, please check out Sam Shah’s recent post, The Messiness of Trying Something New.

BCAMT Conference 2011

Here’s the list of puzzles & games from my session at the BCAMT Fall Conference.

Together, we brainstormed some ideas about connecting games to mathematics. I listed some of these ideas in an earlier post.

Also, I attended a couple of sessions in the morning…

“It’s the most interesting thing a graphing calculator can do and we don’t even have kids do it.” Dan Kamin was talking about creating a scatterplot to determine the linear, quadratic, or exponential regression equation. While lines/curves of best fit are no longer in the curriculum (regression functions were in Applications of Math 10 & 11), it is expected that students will solve problems by analyzing linear, quadratic, and exponential functions. This provides opportunities to have students use technology to answer questions such as:

  • Is the data best modeled with a line or with a curve?
  • What is the equation of the function that best models the data?
  • How does your best fit line/curve compare with the those used by experts?

As a person inhales and exhales, the volume of air in the lungs can be modeled with a periodic function. Dan asked his students to write the equation of the sinusoidal function. They couldn’t. It wasn’t a Ferris wheel. “No one had asked them to create something before.”

I’ve had difficulty making domain and range relevant to 15-year-olds. David Wees shared an interesting activity at his presentation. Have students draw a picture of an object using line segments. Then, for each line segment, have them write the equation and determine the domain and range. Finally, have them enter this information using GeoGebra and compare the result with the drawing.

A reminder… the 2012 BCAMT New Teachers’ Conference will be held on February 11 at Queen Elizabeth Secondary School in Surrey.

iPad as Mathematical Communication Tool – Part Deux

I have been learning about educational uses of the iPad. My daughter has been learning about repeating patterns at school (Grade 1). Also, she has been asking me to show her how to use iMotion. A win-win situation.

She built four patterns, taking a photo each time she added a piece. Then, she created a video which I dragged into iMovie. Finally, I recorded her as she talked about her patterns. The movie would be better if the audio were synced to the video, but I wanted to see what we could create in ten minutes. Here it is:

In primary classrooms, students could share their videos and have classmates describe or translate the patterns. Similarly, in high school mathematics classrooms, students could build functions and have classmates determine equations. See an interview of UC Berkeley Math Education Professor Dor Abrahamson for the inspiration behind this idea.

These student-created movies could be used by classroom teachers to assess what students are able to do. There are nine mathematics learning outcomes in the BC Kindergarten IRP. One addresses Patterns:

B1 demonstrate an understanding of repeating patterns (two or three elements) by
– identifying
– reproducing
– extending
– creating
patterns, using manipulatives, sounds, and actions [C, CN, PS, V]

In Grade 1, two small changes are made to B1 and a second PLO is added:

B2 translate repeating patterns from one representation to another [C, R, V]

What judgements could you make about my daughter’s performance in relation to the prescribed learning outcomes? A rhetorical question – I’m not expecting or even wanting a reply.

My daughter also told me that sometimes shape and size can be used to describe patterns (e.g., “circle, circle, square, circle, circle, square” or “small, big, small, big”). Our movie doesn’t demonstrate this knowledge. This speaks to the importance of having conversations with our students – from Kintergarten to Calculus.

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

“Me and Math … are barely on speaking terms.”

“Me and Math were good buds until high school. Then we started to drift and now we are barely on speaking terms.”

“I am naturally drawn to stories. The romance of a and b just doesn’t resonate with me like that of and j (Romeo & Juliet).”

This week, we met with a group of student teachers beginning their first teaching practicum. They were asked to reflect on their experiences learning math. Most of the responses were negative. I listed two of the more creative ones above.

Similarly, whenever we meet with a group of experienced teachers (and/or administrators) and announce that we will begin by solving a math problem, there is always a noticeable groan. You can feel the stress level rise in the room. And remember – this is always a group of educators who have voluntarily signed up to learn about teaching mathematics.

Two themes came out of the student teachers’ comments. First, for many there was a specific grade when math stopped making sense. Often this was grade 8, but grades 7, 11, and 12 were also mentioned. Poor teaching and/or an inability to remember which rule to apply were given as reasons for this. Second, many didn’t see themselves as “math people”.

From this, there are two things to consider. First, it is important that non-specialists be self-aware and not transfer this fear to their students. Second, it is important that we (secondary math teachers) provide our students with opportunities to make sense of mathematics concretely. An emphasis on the procedural may have worked for us math geeks (I might argue this later), but it didn’t work for the majority.

A problem-based approach addresses the above. Students (and teachers!) build conceptual understanding while having fun.

Aside from getting the chance to teach students in many different classrooms, the most rewarding part of my job is hearing teachers say “I used to hate teaching math, but now I love it”. This can’t help but be a great thing for Surrey students.

Multiple Multiples

“Can you show me another way?”

Multiple representations show students there is more than one correct way to do the math. This is an important message in itself.

Multiple representations also allow students to learn new mathematical concepts and procedures.

For example, division can be thought of as sharing or grouping.

8 ÷ 2 = 4 can be thought of as:

  • I have 8 items. I share them equally between 2 people. Each person gets 4 items.
  • I have 8 items. I put them in groups of 2. I can make 4 groups.

I prefer the adjective flexible over multiple. Adaptability of, not number of, is what is important.

To learn how to divide integers and fractions, students must be able to visualize both representations.

For example, -8 ÷ 2 can be thought of as sharing equally between 2 groups. Each group contains four negative counters. However, -8 cannot be put in groups of +2.

Alternatively, having a negative number of groups does not make sense. However, -8 can be put into groups of -2.

Think about why 6 ÷ ½ is 12 (without simply applying the invert and multiply rule). Having a fraction for the number of groups doesn’t make sense. However, students can explore how many halves there are in 6 using pattern blocks or number lines.

3 × 2 is more than simply 3 groups of 2.  An understanding of an area model of multiplication helps students to learn two-digit multiplication.

An understanding of this model will help students make connections between multiplying binomials and multiplying two-digit numbers.

As a secondary department head pointed out a meeting last year, teaching how to multiply binomials may be easier than teaching how to multiply two-digit numbers – in algebra, there isn’t the added complication of place value.

Now if only we would stop using the term FOIL…

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.