What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (24 page)

9. Never time children or encourage faster work. Don’t use flash cards or timed tests.
16
,
17
,
18
Scientists can now examine brain scans when people work on math, and these have shown that timed conditions create math anxiety. Math facts are held in the working memory part of the brain, and scientists have found that when people are stressed—adults or children—their working memory becomes blocked and math facts cannot be accessed. The emphasis on speed in US math classrooms is one of the reasons we have widespread underachievement and a nation of math-traumatized people.

10. When children answer questions and get them wrong, try and find the logic in their answers—as they have usually used some logical thinking. For example if your child multiplies 3 by 4 and gets 7, don’t say, “That’s wrong.” Instead say, “Oh, I see what you are thinking. You are using what you know about addition to add 3 and 4. When we multiply we have 4 groups of 3 . . .”

11. Give children mathematics puzzles. These have been shown to inspire children mathematically and are great for their mathematical development. Award-winning mathematician Sarah Flannery reported that her math ability and enthusiasm came not from school but from puzzles she was given to solve.
19

12. Play games, which are similarly helpful for children’s mathematical development. For young children any game with dice will help. Some board games I particularly like are:

• Place Value Safari

• Mancala

• Blokus

• Yahtzee

• Guess Who? (great for logical thinking)

• Mastermind

More games and puzzles are available on www.youcubed.org.

In addition to these twelve pieces of advice, which summarize many ideas in the book, I want to add a special message for teachers. Teachers hold students’ mathematical futures in their hands, and you should never underestimate the power of your words and actions to inspire or defeat students. Almost any successful person will tell you about a teacher, usually one teacher, who believed in them and changed everything for them. You can be that teacher for all of your students.

I have talked with and been sent evidence from thousands of teachers since I started sharing the evidence and messages in this book, and I know that you can implement the changes I have described and that they can transform everything for your students. Teachers can create beautiful mathematics environments for students regardless of their past experiences or the negative district or school regulations that are imposed on them. I know teachers who are given dry repetitive math curricula to use, but they make the curricula come alive with the changes they make. Any mathematics problem can be brought to life when it is made open, more visual, and creative and is infused with students’ own thinking. You can make these adaptations to problems, cut out unnecessary repetition, and invite students to go deeper into mathematical journeys
with you. I am the greatest supporter of teachers, and I am fully aware from my own years as a teacher how demanding and time-consuming good teaching is. I want teachers to have rich tasks—and we are putting as many as we can on YouCubed—but I also believe in teachers’ abilities to take any task and bring it to life for their students and to combine this with positive messages about students potential.

In addition to posting tasks and resources on YouCubed, I regularly post short, readable papers that summarize the research evidence we have on the ways mathematics should be taught, which you can give to administrators and parents who do not understand the need for change or who want to learn more. You are an expert on children’s mathematical development and learning, and you need to use your expertise and be prepared to be strong at times to help others know the best mathematical pathways for our children to follow.

In Conclusion

Mathematics is an engaging, creative, and accessible subject that enables people to be their most powerful selves, interacting with their environment in thoughtful and informed ways. Whether you are a parent, teacher, or other education leader, you can change children’s mathematics lives for the better with the knowledge we have of the brain and of powerful teaching and learning environments. We are stronger together in bringing about the changes that we need, and we should not waiver in that mission because all children deserve the very best mathematical future possible. Mathematics will play a central role in all children’s lives, and together we can enable them to apply powerful, quantitative, and creative thinking to the problems they meet in their work and lives. Together we can inspire children who in turn will go on to create a brighter future for their children, filled with the scientific, creative, and technological
discovery that mathematics enables. Let us move together from the mathematics trauma and dislike that has pervaded our society in recent years to a brighter mathematical future for all, charged with excitement, engagement and learning.

Viva la
Revolution
.

Solutions to the Mathematics Problems

Introduction

The Skateboard Problem

A skateboarder holds on to the merry-go-round pictured below. The platform of the merry-go-round has a 7-foot radius and makes a complete turn every 6 seconds. The skateboarder lets go at the 2 o’clock position in the picture, at which time she is 30 feet from the padded wall. How long will it take the skateboarder to hit the wall?

One Solution

The first step is to find out how far the skateboarder travels after letting go. In other words, we need to figure out the distance AB in the drawing below.

To do this we first need to work out what the angle θ is so we can use the properties of right triangles. To do this, draw the line
h
that passes through the center of the merry-go-round and meets the padded wall at a right angle. Since the skateboarder is at the “2-o’clock” position, which is 1/6 of the way around a clock, the angle α is 1/6 of 360°. So α = 60°. The angle β = 90°, since a tangent line of a circle always meets the radius at a right angle. Finally, α + β + θ = 180, since they are same-side interior angles between the parallel lines
h
and
l
. Therefore θ = 30°. This means that triangle ABC is a 30-60-90 triangle! Using the side relations of 30-60-90 triangles, we find:

and

So now we know how far the skateboarder travels. The next step is to figure out how fast she is traveling. The merry-go-round makes a complete turn every 6 seconds. In a complete turn, the skateboarder travels the entirety of the circumference, which is

C = 2π (7) ≈ 43.98 ft.

So the skateboarder is traveling at 43.98/6 ≈ 7.330 feet per second.

Since

(distance) = (rate) (time)

then

(time) = (distance)/(rate)

So the time it takes the skateboarder to reach the wall is

The Chessboard Problem

Solution

What makes this problem difficult is all the different sizes of squares on a chessboard, from the smallest 1 × 1 squares, to overlapping 2 × 2 squares, all the way up to the entire chessboard, which is itself an 8 × 8 square.

In situations like this, it is often helpful to be organized. One way to organize the problem is to count all the different sizes of squares separately. So, let’s start with the 1 × 1 squares. There are 8 rows and 8 columns on the board, so there are 64 of these. Next, let’s look for the 2 × 2 squares. These are more difficult, as they can overlap, as the two gray squares below do:

Even overlapping squares have distinct center points, though, so an easy way to keep track of the overlapping squares is to mark the center of each square with a dot. Here are some examples of squares with their center points marked:

Marking all the center points of the 2 × 2 squares, we get a grid of center points:

Notice this is a 7 × 7 grid of points. So there are 49 2 × 2 squares.

To keep track of the 3 × 3 squares, we can also mark the center points, as in the following few examples:

When we draw all of the center points for the 3 × 3 squares, we get a picture that looks like this:

This is a 6 × 6 grid of points, so there are 36 3 × 3 squares. Continue this process until you have counted the 4 × 4 squares, the 5 × 5 squares, and so on. The total number of squares is then

8
2
+ 7
2
+ 6
2
+ 5
2
+ 4
2
+ 3
2
+ 2
2
+ 1 = 204

This process of counting works for any size of chessboard. In general, for an
n
-by-
n
chessboard, the number of 1 × 1 squares is
n
2
. The number of 2 × 2 squares is (
n
- 1)
2
and so on. So the total number of squares is

n
2
+ (
n
– 1)
2
+ (
n
– 2)
2
+ . . . + 3
2
+ 2
2
+ 1

Chapter 3

The Railside Pattern Problem

Juan’s problem: “See if you can work out how the pattern is growing and the algebraic expression that represents it!”