The Case for Withholding Algorithms (For a While!)

In 1998, Tom Carpenter and his colleagues documented grades 1–3 students’ use of invented strategies and standard algorithms. The vast majority of students in the study used some invented strategies. The researchers found that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations than those who learned standard algorithms earlier.

We have seen it over and over: a child is tutored early in memorized arithmetic procedures by a well-meaning parent.

Decimal Multiplication – a Visual Approach

Our 8th grade teachers tell us that of the most confusing ideas for students is the difference between linear units (the sides of a rectangle) and quadratic units (the area of a rectangle). Students routinely confuse x,  x-squared, and x-cubed, without realizing what each represents.

This concept SHOULD reach back to a conceptual development in 4th and 5th grade: the idea of area versus perimeter units.

Our solution: We use spaghetti to illustrate the linear units on the sides of a rectangle. (Actually, if available, linguine works better because it’s flatter!)

Teaching Decimals with Decimal Squares

The most visual, most fun way to learn decimals is with decimal squares.

We use decimal squares we bought from https://decimalsquares.com  several years ago.

They show decimals in tenths, hundredths and thousandths, and the equivalencies are obvious. (see below)

The website has fun interactive games, too, but you need to download the Shockwave app to play them.

And Quizzes? How are we doing?

Word doc:   Quiz 3

1. Fractions.

Yay – we’re getting this!  Multiplication plus subtraction with borrowing. Example 3 below is one of the few with a mistake, and its last step is correct, it’s the first step that’s not.

THIS  problem, however, lead us into the pit of confusion.  Only the last example below is correct, but the others came very close. It tells us what we still need to work on.

2. (Level 2) Multiplication and (Level 3) Different Bases

Multiplication is going well, too. Our students did a lot of area modeling in 4th grade. Exploding Dots helped many students visibly see what it means to borrow and carry within the place value chart.

About 40% of the students tried the Challenge Questions. These challenge questions were useful in clarifying place value relationships for our fastest students who otherwise would say, “just tell me the steps”!

Homework: Goals and Structure

Recent Homework assignments:  (mostly available as Word documents)

Solution Keys:  HW_keys#1to#8

What are our main goals in assigning homework?

1. Provide thoughtful practice, not drill. Hopefully in an independent setting, without help, so we can see how the learning is going.
2. To allow for long periods of time (weeks) to repeat the visual representation of important topics. We only do one problem or two per topic. Fractions, multiplication, division, decimals. Here we withhold the algorithm for weeks, until the concept is internalized.

A few notes on our HW problems:

Experimenting with 3-act Activities

We’ve heard so much about 3-Act activities. Here’s this week’s experiment, with our verdict at the end.

Day 1…

Does \$1 million fit in a briefcase?

Show this GIF:

Ask… “Would \$1 million fit into a normal briefcase, and if so, could an average adult carry it? Assume the bills are all 100s.

When students asked for more info, we looked up the following  (on the internet):

Part 2 of “What Does Remediation Look Like?” Michael’s silence. Plus Exploding Dots.

In our last blog, we asked, “How do we humans react to confusion and difficulty?”

• Amelia claims grumpily “I don’t get the multiply and divide by 10 thing.” She tends to get grumpy when confused.
• Michael withdraws into silence. “Shall we get the blocks for this, Michael?” Silence. “Which part is confusing?” Silence. Sigh.
• Tomas minimizes his struggles. “I’m fine now. I was just confused on yesterday’s quiz. I’m good now.”
• Nicole writes notes to us on her quiz. “I need more instruction in this concept. It makes no sense at all.”

What Does Remediation Look Like? Aka “Bring Your Knitting”

How do we humans react to confusion and difficulty?

• Amelia claims grumpily “I don’t get the multiply and divide by 10 thing.” She tends to get grumpy when confused.
• Tomas minimizes his struggles. “I’m fine now. I was just confused on yesterday’s quiz. I’m good now.”
• Nicole writes notes to us on her quiz. “I need more instruction in this concept. It makes no sense at all.”
• Michael withdraws into silence. “Shall we get the blocks for this, Michael?” Silence. “Which part is confusing?” Silence. Sigh.

Welcome to what might be the hardest job in the world to do well.

Looking at Quizzes- Part 2 … How are we doing?

Quiz 2, Trimester 2, as a Word Document:

Question #1: Fractions.  Everyone was able to solve this problem correctly!  This is huge. Fractions matter.

Looking at Quizzes- Part 1 … How are we doing?

Quiz #1 as a Word Document:

Question 1, Fractions.  It’s Payoff Time!

We’ve been having students draw  fractions only for weeks now. Weeks! We are gradually increasing the difficulty of the questions, to cover addition, subtraction and multiplication of fractions. We assign about 1 problem a day.  And… drumroll…90% got this problem right now. It involves multiplication and borrowing! Neither of which we’ve taught. We’ve just given students the tools to make sense of fractions.