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Off The Beaten Math by The Pi Project - 5d ago

Corrinne and Kathleen are offering a workshop in Palo Alto, CA, on Aug 10.

If you’re interested, and have enjoyed the ideas in our blog, registration is here.

We’d love to meet anyone able to attend!

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Off The Beaten Math by The Pi Project - 1M ago

Corrinne and Kathleen are offering a workshop in Palo Alto, CA, on Aug 10.

If you’re interested, and have enjoyed the ideas in our blog, registration is here.

We’d love to meet anyone able to attend!

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Off The Beaten Math by The Pi Project - 3M ago

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?
What IS surprising
  • Measuring an angle that isn’t there…

  • Measuring the LENGTH instead of the angle:
  • Looking for AREA …
How to correct misunderstandings and build the concept of what an angle is:
  1. Spaghetti to the rescue, again. By holding down one of the angle legs, and moving the other, we can TALK through the opening and narrowing of an angle, COUNTING as the angle grows and shrinks. The counting helps students know what the size is – for example 60 not 120 degrees. It helps them internalize the idea of what an angle is; the gradual opening (like a fan) of the 2 angle legs.  Note – some students have to do this MULTIPLE, multiple, multiple times before it becomes intuitive. Whenever they’re confused, have spaghetti around!

2. If you have Geometer’s Sketchpad software, it functions similarly. Students can drag one leg of an angle to demonstrate different angle sizes and pairs. Students created videos similar to the one below.

3. Finally, give LOTS of opportunity for review. Here’s a game we played — we’re trying to offer a couple levels of challenge in each slide. CW angles

Game Rules: Half the class sits in a circle in

the middle of the room, at their desks.

The other half sit INSIDE that circle, where they

work in teams (7 teams for example, at right).

After every slide, the students on the inside rotate to the next table. This decreases competition and yet keeps the groups conspiratorially quiet and motivated. (You can keep score by team number, but it doesn’t mean a lot because of the rotation.) Smaller groups than 4 are even better.

Final note: The LEARNING going on in this unit should be the visual concept of what angles are — by size, by pairs, in triangles. Vocabulary should be a useful acquisition, but a secondary one. The goal is the solving of complex angle puzzles – an activity most children enjoy naturally. More on that next posting!

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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.

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.

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

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

And to decimal division:

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

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.

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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!)

Activity:
1. We pass out 2 or 3 spaghetti pieces per student, and papers with different multiplication problems written on the top (a different one for each student). Each problem has factors between 1.1 and 1.5, and has a grid beneath it:

The grid has bold marking around a 10 x 10 block.

We have the students break off a spaghetti piece as long as one side of the 10×10 grid. This is now ONE unit long. ONE SPAGHETTI UNIT, not one square unit, obviously. Then we ask them to break off another piece that is about 1/10 as long as the first. They glue these down on the left and top edges of the rectangle assigned on their sheet.

These are the SIDES of the rectangle. This is when all our work with blocks and area models through 4th and 5th grade pays off!

We have the students color in the WHOLE, the TENTHS and the HUNDREDTHS.

They are familiar with the appearance of tenths and hundredths from our games with decimal cards (see last post). They can count up the total value of the product; the area of the rectangle created.

From this point on, all our teachers can use the key word “Spaghetti Units” when they are referring to perimeter, length, or linear units in algebra. It gives us a common visual language that means something to students.

Follow up:  It is important to keep this at the visual level for a couple weeks. This conceptual introduction to decimal multiplication builds number sense and estimation skills. In a week or two, we will move on to multiplication with decimals LESS than one. Here’s the  word doc for these grids.

Here are some examples of student work:

Whenever students struggle with this (for example, seeing the SIX tenths on the 2 sides of model of 2.3 x 2.3 above) we simply ask them to lay out the BLOCKS for that problem, and it clears up. The concept is always best developed at the concrete level.

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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

For 2, 3 or 4 children.
Materials Needed: Deck of decimal cards.

To play: Shuffle the deck and lay face down on the table. Player #1 turns over the top card and lays it face up on the table. She looks at the card and says which card she hopes to draw in order to make 1 (For example: if she sees 3 tenths, she says “I hope I get 7 tenths!”). She draws a card. If she can make 1, she tells the other player which cards add up to 1 and takes those cards. If not, she lays her card face up to the table top. Suppose she draws 5 tenths — then she has to lay it on the table.
Player #2 then says which cards he would like to draw in order to make 1 using the cards already on the field. For example, since .3 and .5 are lying on the table, (all cards on the table are shared) he says “I hope I get 7 tenths, or five tenths, or 2 tenths! (considering he could add the .3 and the .5 to make .8, and he would need a .2).  He draws and then either makes 1 or, if not, lays his card on the field and play goes back to Player #1. Combinations of more than two cards are possible and desirable, since the player with the most cards at the end of the game wins.  We like this game because no one says “I hope I get POINT SEVEN”. They can see the 3 tenths in red, so they hope they get 7 tenths (of course, 70 hundredths would be fine, too!)

ALTERNATIVE GAME:  “WAR” Each student gets their own stack of decimal cards,  face down. At the same time, they both turn over one card. The largest decimal wins. Gradually one student’s pile gets bigger and bigger… it’s competitive, but it relies on luck, so students accept defeat when it happens!

ACTIVITY TWO – Number line

In 4 groups, students get 3 or 4 cards (mixed decimal and fraction cards).
The rules only allow one student at a time to tape up a card on their number line. Students take turns putting up one card, then they go to the second round, etc.  They cannot grab someone else’s card.
They usually start with their most obvious cards (1/2, 1/3…) and work from there. And yes, they make mistakes, but they’re thinking. Sometimes we point to 2 or 3 cards (if one of those 3 is wrong) and ask if everyone agrees.

ACTIVITY THREE – Make 21

In groups of 2 or 3, students play the traditional game of Blackjack, but their goal is to make 21 tenths. Suppose a child receives 2 cards from the dealer (we let the dealer play, too), let’s say 5 tenths and 75 hundredths. They usually say “12 and a half tenths” and they usually ask for another card. (“hit me”…:)   If they get 4 tenths (or 400 thousands), they have 16 and a half tenths, and they should hold, but usually they ask for another card and go over. You can play it online, too, at decimalsquares.com

We find we don’t even have to teach any “addition of decimals” rules. These games create a natural understanding of quantity.

Next post: Introducing Decimal Multiplication

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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”!

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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:

Problem #1 (all HW’s):    Fractions.  We’re still having them draw. We’ve had them draw addition, subtraction (with borrowing) and multiplication. We’re still withholding algorithms.  If we teach the algorithms too early, students discard their visual understanding and use the (more easily acquired) algorithm. The danger here, of course is that learning is superficial and temporary.  Fractions are a make or break concept in algebra and beyond, and the time spent drawing is worth it.

Problem #2:   2-digit Multiplication.  Again, we are keeping this visual, withholding the traditional algorithm.

Problem #3:  One-digit multiplication.  This is an optional problem composed of dot arrays to help students finish mastering their times tables. Students who do not yet know their times tables completely in 5th grade are probably not going to learn them with the traditional methods of flash cards and drill.

Problem #4:  Multiplication/Division by powers of 10.  This is a valuable way to build place value concepts while encouraging mental math skill development.

Challenge Problem:  Calculation in different bases. The logic behind this work is that it is good for the fastest students (those who have time to get to these challenge problems). They are forced to ignore traditional algorithms and focus on visualization of the concepts of multiplication and division.

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