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.

Some children do fine with those traditional algorithms. These are linear thinkers with a facility for memorization. (We would argue that even the best memorizers would benefit from a more investigative, explorative stage of learning, coupled with complex problem-solving).

For about a third of our students, however, this approach does not work. These students retain the procedure for a few days or weeks, and subsequently confuse it with the next algorithms taught. They fall behind, need frequent re-teaching, and lose confidence in their own conceptual abilities. In an earlier century, they would have ended up in agricultural or factory jobs. Today they run the risk of ending up in a long-term underclass, with poor career choices. (See this  Article for data.)

Alternatively, if students spend as long as they need to at the concrete/pictorial level, they can then comprehend and internalize the resulting algorithm. They create their own meaning behind that algorithm. They are able to apply it to new, more complex situations and they have an internalized, often visual conceptual understanding to fall back on if they get confused later on. Perhaps most importantly, the acquisition of algorithms through exploration is more fun than drill, more respectful of students’ intelligence than direct teaching, and more likely to instill confidence in learners.

We DO believe in the usefulness of algorithms. They are quick, efficient and expedient. In fact, one of the final goals of mathematics teaching is to help students master its widely accepted abstract representations. We just believe that they are most useful when they are understood.

So… which 5th grade concepts are worth spending that much time on? Our answer: Division, Fractions and Multiplication. Let’s look at division.

TEACHING DIVISION

  1. CONCRETE

We start with concrete representation; mostly base-10 blocks, but we also play a game called the remainder game. It uses a planting tray from a nursery and plastic spheres, but blocks would work, too.

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2. PICTORIAL

Since our students had numerous fraction investigations in 4th grade, we move quickly to pictorial division, but with the blocks available for the few who need them.

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For most students, this transfers quickly to semi-mental math:

 

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3. ABSTRACT.    Finally, all this work transfers to the long-division algorithm. IMG_1970

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And to decimal division:

 

The child at right still needs drawings, the child below is doing most of the division mentally.

 

 

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This problem is only visible as 36 tenths if it is first illustrated with blocks or drawings. 36 tenths divided among 6 groups is 6 tenths each. The rest becomes 12 hundredths, also divided by 6. Voila — division with less pain, less drill, and more number sense.

 

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!) Continue reading Decimal Multiplication – a Visual Approach

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.

ACTIVITY ONE – The “I HOPE I GET…” Game

Continue reading Teaching Decimals with Decimal Squares

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.

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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.

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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”!

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Homework: Goals and Structure

Recent Homework assignments:  (mostly available as Word documents)

HW#1       HW#2         HW#3     HW#4     HW#5      HW#6   HW#7   HW#8

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: Continue reading Homework: Goals and Structure

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:

Episode 15 Money GIF - Find & Share on GIPHY

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):

Continue reading Experimenting with 3-act Activities

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.”

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

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.

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

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