Skip-Counting – The Threes & Sixes, Plus a Game

Today's post winds up our "From the Trenches" series by Colorado classroom teacher and former Yellin Center Learning Specialist Beth Guadagni. In Beth's prior posts, she explained how our brains learn math facts and how she uses songs to help her students -- all of whom have dyslexia -- learn the fours multiplication facts.

Our last post gave background and instructions for teaching multiplication facts for four and eight. Using different songs, Beth explains how the same techniques can be used to teach the math facts for threes and sixes.

Skip-Counting by Threes
“Three” is repeated three times to get the rhythm to work out. We add “and thirty-six” at the end in the same way people add “and many more” to the “Happy Birthday” song. Jazz hands, while optional, are highly recommended.

Row,                   row,                      row                      your                     boat,

Three,                 three,                   three,                   six,                        nine


Gently                down                    the                       stream,

Twelve,               fifteen,                   eighteen                        

Merrily,             merrily,                   merrily,                 merrily,

Twenty-one                 Twenty-four                   Twenty-seven


Life is but a dream.

Thirty   thirty-three

…and thirty-siiiiix!



Skip-Counting by Sixes

Happy                birthday                to                     you,

Six                      twelve                    eighteen          twenty-four


Happy                birthday                to        you,
thirty                  thirty-six               forty-two


Happy                birthday               dear           [name]

Forty-eight               and                 fifty-four


Happy                birthday                 to you!

Sixty                   sixty-six                 seventy-two!

Game: Domino Draw

Purpose:
To give students practice applying skip-counting sequences to real math problems.

Materials for the game:
A set of dominos, turned face-down or in a bag.

Procedure:
If you don’t plan to play long enough to go through a whole set of dominos, use a timer so that students play for a set amount of time. Be sure, once it goes off, that everyone has had the same number of turns.

There are two variations here.

1. To target the sequence students are learning:
On his turn, each player draws a domino at random. He adds the number of dots on the domino, then multiplies that number by the sequence you’ve been practicing. For example, if his domino had 11 dots on it and you were practicing the threes, he’d get a product of 33 and earn 33 points.

2. Once students have learned all the sequences, try this variation:
On her turn, each player draws two dominos at random. She adds the number of dots on each domino, then multiplies them together. For example, if one domino had four dots on it and the other had twelve, she’d get a product of 48 and earn 48 points.

Skip-Counting by Fours ... And a Multiplication Game

In her last post, Colorado teacher Beth Guadagni shared the concepts behind one of the methods she uses to  help her students (all of whom have dyslexia) learn number sequences. As part of her advice "from the trenches" we share the specific techniques that she uses in teaching them to work with fours, getting them ready to do multiplication and division. 

Purpose:
To help students memorize number sequences that will help them with multiplication and division.

Procedure:
First, make sure the students know the song. I like to start with the fours and use "Take Me Out to the Ballgame"  because the rhythm of the song fits perfectly with the number sequence, which is not always the case. Don’t try to teach the whole song-number connection at once! I usually like to start with the first five or six numbers, or the first two lines. In this case, the song breaks nicely after 28.

1. Write out the numbers from 4-28. Sing the first line and then ask the students to repeat it. Do this a few times.

2. Next, ask the students to close their eyes. Cover a number, then ask them to open their eyes. They should sing the line, filling in the missing number when they get to it. Do this a few times, covering different numbers each time.

3. Once they seem to be comfortable, add the next line of the song and repeat the procedure.

4. Cover or erase all of the numbers in your chosen lines and sing through them with the students once or twice.

5. Ask students to write the sequence as far as they’ve learned it.

For additional reinforcement, I like to hand out a sheet on which I’ve written the sequence they’ve learned a few times. Each time, it is missing more and more numbers. They have to complete the sequence by filling in the missing numbers, then cover the top part of the sheet when they’re ready to tackle the next, more challenging section. At the end of the sheet, they have to write the whole thing from memory. I also like to include a few multiplication and division problems from that family that they have to solve, using the memorized sequence to help them. They are not allowed to peek at the sequences from the top of the sheet unless they’re really stuck!

Teach the whole song, bit by bit, this way until you’ve taught the whole thing.

Take            me              out                    to the                     ballgame, 

Four          eight            twelve                sixteen                    twenty 

Take me out                to the crowd, 

Twenty-four               twenty-eight 

Buy me some                 peanuts and                      cracker-jacks, 

                             Thirty-two                      thirty-six                          forty 

I don’t care                           if                           I ever get back…. 

Forty-four                   and                            forty-eight

 

Once they’ve memorized the song, there’s one more part to add: students have to count on their fingers while they sing it. (I allow my too-cool high school students to keep their hands on their laps, as counting on hands that are held in the air feels babyish to some of them.) This is essential if they’re going to use the song to solve math problems.

Each time they say a number, they have to hold up another finger, so that by the time they get to, say, “thirty-six,” they are holding up nine fingers and will know that thirty-six is the answer to four times nine. This is trickier than it sounds. At first, when they have to both sing and track on their fingers most kids lose their place in the sequence. To practice this skill, we sing the song until I call out “Stop!” and students have to write down the last math fact they sang. For example, if I stopped them right after they said “twenty-four,” they should have six fingers extended and so they’d write “4 x 6 = 24.”

Why This Works:
Songs are incredibly powerful mnemonics. Most students seem to remember tunes easily, and this prompts them to recall the number that goes along with each change in tone and matches the number of syllables for that particular line. Students see the number sequence while they are hearing the sequence and the song, meaning that they store the information in several formats in their memories. Eventually, they count on their fingers while singing as well, adding a tactile element. Teaching the song in segments is quite important, too; as with any new skills, students must demonstrate mastery before they can tackle new material.

Once students have gained some comfort with skip-counting, you may want to introduce a game to help reinforce their skills. My kids like the card game "War".

Game: Multiplication "War" 

Purpose:
To give students practice applying skip-counting sequences to real math problems.

Materials for the game:
One or two decks of cards (Use two decks shuffled together if you’re playing with three or more students)
Procedure:

• Ace = 1
• Jack = 11
• Queen = 12
• King =0

Because War can take ages to wrap up, I often set a timer for around seven minutes while my students play, and the winner is the one with the most cards when their time is up.

There are two variations here.

1. To target the sequence students are learning:
Each player lays down a single card, face-up. They have to multiply the card by the sequence you’ve been practicing and say the product aloud. The player with the highest product keeps the cards from that round.

2. Once students have learned all the sequences, try this variation:
Each player lays down two cards at once. The player with the highest product gets to keep all the cards from that round.

If two players get the same product, they lay down two cards face down, then use a second pair to get the tie-breaking product.

Why This Works:
Students who aren’t focused don’t learn well, and games keep kids engaged in the learning task. This game is fast-paced enough to ensure that students have to use the skip-counting sequences they’ve learned many times during the allotted interval.

Strengthening Paired Associate Memory with Song

We are continuing our series of posts by Beth Guadagni, who shares the strategies she uses teaching her students with dyslexia in Colorado. 

Like many students, mine have struggled to learn their math facts. Automaticity with the multiplication tables is essential for math far beyond simply multiplying numbers; students use multiplication when working with fractions, doing long division, calculating area and volume, and in so many other applications that it seems rather silly to try to list them!

Perhaps most importantly: students need to have a sense of multiplication to determine whether a solution to a math problem makes sense. As Dr. Yellin will tell you, memorizing math facts involves a particular part of memory called paired-associate memory. Paired associate memory involves linking and storing two related data bits, retrieving one piece of information when presented with the other piece (eg., a sound with a symbol, or the number 28 when presented with 4x7).

Paired-associate memory is what we use when we learn someone’s name, remember that the color of the sky is called “blue,” pair the /ch/ sound with a "c" and an "h" together, etc. There’s no immediate context for these associations (although savvy students and educators can invent contexts to make information “make sense”); they just have to be memorized. Paired-associate memory is generally not a strength for dyslexic students, like mine, although people who don’t have dyslexia may struggle with this skill as well.

I learned skip-counting songs from a colleague and was amazed by the ease with which her fifth graders learned the number sequences. I was eager to try this concept in my class, but I was a bit apprehensive, too. Would my high school students be willing to sing strings of numbers to the tune of “Twinkle, Twinkle Little Star” and “The Wheels on the Bus”? The answer was a resounding “yes!” Although they were a little hesitant at first, my students were as pleased as I was that they could commit number sequences to memory with only a little practice. In fact (and this is true), one day one of my students, frowning darkly, exploded, “I’m really mad that I made it to eleventh grade before anyone taught me this!”

I’m going to spread the sequences over a few posts, which also is what one should do when teaching these songs. I’ll share a game for practicing math facts in each post, too. Learning the songs is important, but it’s not enough; one has to practice using the sequences to answer actual math facts, too. We'll present detailed instructions on how to use this technique in your classroom in our next post, but you can get a sense of how this process sounds from this YouTube video, posted by another teacher who used this technique. 

Rewriting Equalities to Build Numeracy - Part 2

In our last post, we wrote about the work of Dr. Barbara Dougherty, the Director of the Curriculum Research and Development Group at the University of Hawaii. Dr. Dougherty's work emphasizes the importance of of instilling a solid sense of numeracy (being able to reason with numbers and other mathematical concepts) and operations (recognition of the relationships among addition, subtraction, multiplication, and division) to provide students with a true understanding of math.

One way to do this is to teach students to "Rewrite Equalities", a process that begins with understanding the meaning of = .

How do we do this?

Step 1: Establish an understanding of =

= does not represent “the answer” (as most kids think it does). It means that whatever is on one side has the same value as whatever is on the other side.

This seems simple, but it’s a big intellectual leap for kids. One of the biggest revelations will probably be the insight that it’s possible to have multiple numbers on each side of the =.

Try modeling this with manipulatives and sticky notes. Start with this:

  

Then show students that you can also do this:

Or this:

Or even this:

In all of the examples above, there are still a total of seven pennies on each side of the =. Challenge them to build their own models.

Step 2: Entry-Level Equalities

When it’s time to move from manipulatives to numbers, the teacher should demonstrate how to rewrite an equality in a few different ways. Begin by writing and explaining a series like this:

3 + 5 = 8

8 = 3 + 5

8 = 4 + 4

7 + 1 = 8

Now, invite students to take over with their own ideas. How many can they come up with? 

The student who remembers the commutative property doubles the number of expressions s/he can write, but don’t tell them this! Wait for someone to figure it out.

In my classroom, after they’ve worked for a while, students pick their two favorite expressions to write on the board and explain to the class.

Questions to Push Thinking:

• Can you rewrite the equality using a different operator, like a subtraction sign?
• Can you rewrite the equality by putting two or more numbers on each side of the equal sign?
• Can you rewrite the expression using decimals or fractions? What about negative numbers?

Step 3: More Advanced Equalities: Focusing on Operations

Challenge students to rewrite an equality using only the numbers in the inequality. They may use any operators and any format they’d like.

So a simple expression like this:

4 + 6 = 10

could be rewritten like this:

10 – 6 = 4

A more complex expression, like this:

3 x 4 x 2 = 24

could be rewritten like this:

2÷24 = 4 x 3

Students will notice that more complicated expressions can be rewritten in more ways.

I’m my classroom, we’ve learned that there is tremendous power in simple activities and procedures. I hope other educators find this activity to be as valuable as we have!

Rewriting Equalities to Build Numeracy - Part 1

Recently, I was fortunate to attend a two-day seminar on teaching math to struggling learners in the middle and upper grades, led by Dr. Barbara Dougherty. Dr. Dougherty, the Director of the Curriculum Research and Development Group at the University of Hawaii, is an insightful and innovative educator with a passion for helping all students develop of deep appreciation for math. If I could sum up the most important lesson I learned during my two days with Dr. Dougherty, it would be with a statement she repeated many times over the course of the conference: Pay now, or pay later.

Dr. Dougherty was referring to the time it takes to build numeracy skills and operational sense. These two critical components of math education are often given a mere nod in the classroom, if not overlooked completely by teachers scrambling to cram as many procedures into students’ memories as possible before exam time. This rushed approach, while understandable, is a mistake, according to Dougherty. A solid sense of numeracy (being able to reason with numbers and other mathematical concepts) and operations (recognition of the relationships among addition, subtraction, multiplication, and division) underlies a deep understanding of math. Happily, these competencies are available even to students who struggle with numbers and even those with dyscalculia. But learners need time and the right platforms for exploring the relationships between numbers.

There are lots of ways to set the stage for this kind of inquiry, but one of the simplest is an exercise in which students rewrite equalities. This powerful strategy is deceptively simple: give students an equation and ask them rewrite it, keeping both sides equal. So,

6 + 4 = 10

might be rewritten as

10 = 6 + 4

9 + 1 = 10

15 – 5 = 10

5 x 2 = 10

7 + 3 = 20 – 10

etc.

My fifth graders have loved this activity, and even those who will swear they hate math cheer when I announce that we’ll be warming up for class with a new number sentence to rewrite. There are lots of reasons to try this in a classroom setting or with an individual student. 

Rewriting equalities:

• It's fun! – My students, without any guidance from me, treat these challenges like riddles, and they are tickled when they come up with novel solutions.  
• Builds operation sense – Crunching numbers with a real goal in mind makes practice relevant and motivating. 
• Builds numeracy – With practice, students develop a sense for how to point the magnitude of a quantity in the direction they want; for example, they discover that most of the time, multiplying by a positive number will make a quantity much larger than adding by the same number. 
• Accommodates a wide range of skill levels – More advanced students can use fractions, roots, exponents, etc., while those still building basic skills can gain comfort with simpler operations like subtraction or multiplication. 
• Sets the stage for algebraic thinking – Many teachers take for granted that students who have only ever seen a mathematical expression with a single number to the right of the =, structured like this:  5 + 3 + 1 = 9 will be able to transition to algebra, when problems may have multiple numbers and variables!) to the right of the =, like this: 5 + 3 = 9 – x.

A misunderstanding of = is often behind the confusion because students haven’t been taught to think flexibly about an equality. (Check out Part 2 of this post for examples of how to implement this process in the classroom and for more about the “equals” sign.)

Different Ways to Solve the Problem of Algebra

Professor Jon Star of the Harvard Graduate School of Education, a former math teacher, has been working to make algebra instruction more effective. We know that algebra is a tough subject for many students and have written in the past about ways to make algebra more readily understood by middle and high school math students.

Professor Star’s project, as described in Ed. magazine and online in the Graduate School Newsletter, has two aspects: first, he is working with teacher volunteers to show them that providing more than one way to solve a problem is more effective than insisting that problems be solved in one particular way. Second, he is having math teachers include more discussion in their classrooms. The idea is not to just focus on getting the correct answer, but to discuss how that answer was found or why someone took a particular approach to solving a problem. 

Working with colleagues, Star has created a set of curriculum materials designed for middle and high school students, which incorporates these approaches to teaching algebra. The materials are available at no cost for teachers to use in addition to their regular modes of instruction. And the U.S. Department of Education’s Institute of Education Sciences has included this approach in two new publications, a problem-solving guide for grades 4-8 and an algebra practice guide for middle and high school students.