## 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.”

## 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.

## 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:

## 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”!

## 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.

## 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!)

## 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.

## Teaching Angles

We are surprised every year (re-surprised?) at the most common mistakes fifth graders make around angles.

## What’s NOT surprising:

• The difficulty of choosing WHICH of the 2 numbers you encounter on the protractor… Is is 60º or 120º?

## Problem-solving beats vocabulary

Sorry we’ve been incommunicado for a while! In the last month, we’ve had one week of outdoor education (camping), one week of spring break(yay!), one week of standardized testing(no comment) and a week of school-wide theme-based learning(fun).

The trick is not to stress about curriculum 🙂 All those other things matter! If we are measuring LEARNING as our main goal, then each of those (except the standardized testing – yuck!) involves lots of learning.

So this week, we had to start by spending a couple days reviewing the unit we started last month – Angles. Here’s what we believe:

## Nearing the end… How are we doing? Self-assessment time for us.

It’s time to look at recent quizzes and evaluate the progress we’ve made this year. Why is it we teachers remember the slips, the failures, the lessons that didn’t work? We’ll try to be honest with ourselves here, and evaluate our outcomes so far for the year. We’re evaluating the three concepts we feel are VITAL, non-negotiable skills for 5th/6th grade.

## I. Fractions

We’re still having students draw their fractions as much as possible. Some students lean towards not wanting to draw – to use their algorithmic shortcuts instead. Which is fine! We do approve of algorithms — they’re excellent shortcuts based on centuries of refinement.  Except that we want BOTH! Algorithmic and conceptual knowledge.    So we’ve switched from computation-style problems to word problems in Level 1 on quizzes. Here’s the outcome:  (we decided to use all female pronouns today)