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)

a) A student who is still struggling with algorithms, number sense and speed-based work. Her only alternative is to consistently depend on drawing, which means she got this problem right.

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b)  A student in the middle group, who might have rejected drawing too early, moving on to traditional algorithms (easy to pick up from parents, tutors… what we affectionately call “street-math”!).  She saw that algorithms wouldn’t get her far on this problem, but hadn’t mastered the concept well enough to use a visual approach, so got it wrong. It’ll all come together next year in 6th grade, when much of this year’s concept-building is transferred to the abstract.

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c) A normally top-scoring student, who skips drawing whenever possible (algorithms work, after all, and they’re efficient!) . She tried algorithms, tried drawing… she got close, but no cigar!

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Our Grade Estimate for Our Year of Fractions:  B+

Most (85%? 90%?) have developed a good conceptual understanding of fractions. (Why is it never 100%?)  What we’d do differently next year….?  We’re not sure. We’re not voting for spending even more time on fractions. We’ve been at them all year!  Maybe just wait and see how this all gels next year in 6th grade? Any suggestions?

II.  Multiplication with Area Models (decimals in this case)

We’re still having students draw area models as much as possible, too. Most of them seem to prefer the visual clarity of the area model over the traditional algorithm, so a majority are getting their answers correct. Here’s the most recent outcome:

a-1) A student who is still struggling with algorithms, number sense and speed-based work. Her solution is to (1) get base-10 blocks and build it, (2) count up the answer, and (3) draw what she built on the quiz. Brava!

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a-2) A similar student tried to skip the building step, and go straight to the drawing. Her area model understanding is not yet strong – she drew an area model that is not a rectangle 🙁 This is fairly easy to remediate…. just one session alone, with blocks  cleared it up (hopefully).

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b) A student in the middle group is just moving from the concrete/pictorial to the more abstract. She gets the 4-quadrant model; she knows that 2 times 3 tenths is 6 tenths. Her mistake came in the bottom-left quadrant, when 3 groups of 5 tenths became 3 groups of 5 hundredths. Again, this is easy to fix with blocks. It is also fixable with language. If  I say 3 groups of five tenths, I won’t write .15.   She’s almost there; her quiz corrections showed this realization around the usefulness of language.

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c) A  top-scoring student is able to do this multiplication at a fairly abstract level. She can pretty much do it mentally now.

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Our Grade Estimate for Our Year of Multiplication:  A-

We’re approaching 100% mastery of the area model as the year closes. The test will come in September (will they remember??) – we’re fairly confident that the visual mastery of the area model will stay with students, just because visual memory is so much stronger than abstract memory.

III.  Division ( including decimals)

Half our class is still struggling with division. Maybe it’s just that hard… Those who are willing to build or draw their solution have a good chance of getting to the answer, but of course it’s time-consuming:

a)  Concrete/pictorial

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This student tried to solve by drawing when she really should still be using blocks. We’re not even sure  exactly what her drawing represents. We’ll have to help her go back to whole numbers and blocks.

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b) Most of the middle group have learned to visualize the blocks, or use number sense to break 165 into number bonds of 120 and 42. Since their understanding is built on the concrete use of blocks, it doesn’t matter to them that they’re breaking up 165 hundredths, not 165 wholes. To us, this is almost as abstract as long division, and more transferable to mental math.

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c) Algorithmic solutions. The first student below has mastered long division, the second is trying to use it without enough place value understanding.

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This student is apparently using mental/short division in her first step. Nice!

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Our Grade Estimate for Our Year of Division:  B

We’re pleased with our progress at the concrete/pictorial level. Most students can do division if they resort to circles and blocks. We do wish, though, that more were already able to transfer to chunking (into number bonds),  or long division, or mental math. Maybe we’ll start earlier with next year’s 5th grade.

AHHH… We know the wise adage: THERE IS NO PERFECT.  It’s easy to say and harder to live with, though. Teachers always hope for miracles — 100% success, mastery for every child.  Since that’s unlikely to happen often, let’s set our goals around growth. Growth for each student, self-knowledge for each student, self-acceptance of “how I learn, what are my strengths, what tricks help me learn, how do I recognize my strengths and build on them?” Let’s set this as our #1 continued goal for next year’s teaching.

“Once more unto the breach, dear friends, once more!”





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: Continue reading Problem-solving beats vocabulary

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. Continue reading The Case for Withholding Algorithms (For a While!)

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  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.


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.