 CASE Stories     # I WISH WE HAD MORE HOMEWORK LIKE THIS - Grades 1-8

“I wish we had more homework like this,” a grade three parent.

Making ten is a simple activity. Just add two numbers and you are done. There is nothing really special about this activity, right?

Or is there?

This set of math activities started off simply, but quickly and naturally went into mathematical areas the curriculum never anticipated grade three and four students could go. A simple exercise of finding missing numbers led to a wonderful exploration of pattern, graphing, integers, parabolas and hyperbolas. When the activities went home, the parents wanted more of this kind of homework the students gave them. Algebra is cool.

We wondered how much our students would understand when engaging with these beautiful, complex concepts. We were familiar with low floor high ceiling mathematics activities. Making ten is a low floor activity, so we wanted to find out just how high would the students comfortably rise.

The Mathematics

Algebra is cool. Many people might not know this secret yet, but we do now. In the primary and junior grades, pattern and algebra are treated separately. There are expectations for pattern like identifying, describing, extending, or creating growing and shrinking patterns. Separated from these are the algebra expectations like determining a missing number in a number sentence. It is not until grade six that the missing numbers are represented in expressions by letters, and not until grade seven that pattern and algebra are tentatively put back together in trying to find a “general term of a linear growing pattern.” No wonder algebra’s coolness is a secret. Starting simply with ___ + ___ = 10, we had the students roll a die, write the result in the first blank then calculate the second blank. They rolled until they exhausted all possibilities. Simple. Grade ones regularly find pairs of numbers to make ten. Grade twos find missing numbers. Next we showed them how to take their results, turn them into coordinates (___ , ___), then plot them rather easily on an x-y graph. They lined up! How could something that was randomly generated create a pattern like this?

A series of activities with different challenges all related to ten led the students to various levels of the mathematics curriculum and let even the youngest of students work with complex concepts that they would not see for many years to come if they were to just follow the curriculum. Taking these complex and interesting concepts from further along the linear path of mathematics is a really interesting way to create different big ideas for mathematics instruction.

1.We asked the students to try ___ + ___ = 8 instead of 10. The line for this is below the first line. The students are experiencing the function of the constant in a linear function.

2.We asked the students if the line formed by their ordered pairs stopped at the x- and y-axes. What might the ordered pairs be if the line continued? Looking at the 11 on the x-axis and tracing a finger down below the axis, the students looked for a minute, thought about it and almost all of them quickly came to the conclusion that the other number had to be “take away one.” You mean with a minus sign before the number? Where have you see numbers like that? In winter?

3.We substituted a die marked from 7 - 12. When a student rolled a 10, they were easily able to calculate the other number as zero. When the first 11 was rolled, there was a momentary silence followed by a chorus of, “Minus one! 11 minus 1 is ten.” They were then asked how that might be plotted on the x-y graph. 11 we knew how to plot, but - 1? Where might that go? The students naturally pointed to the line beneath the x-axis.

They were asked about minus one. Is that a number? When do we use that type of number? The students quickly mentioned temperature. They were very familiar with numbers like -28° as that is the wind chill temperature that triggers an indoor recess. Grade 3s and 4s have no trouble understanding the concept of negative integers.

4.We asked the students to find all of the pairs of numbers that would satisfy ___ + ___ < 10. They quickly found out that there were a lot more pairs than their first equation. One student looked at a teacher with an eyebrow cocked, “You know if we use negative integers, we’ll never be able to write them all.” Really? Why? “Because there’s an infinity of pairs!”  5.We asked them to see if they could change the equation to make the ___ + ___ = 10 line move or change. The students suggested all sorts of different ideas like ___ - ___ = 10; or (2 x ___) + ___ = 10; or even ___ x ___ = 10 (this generates a hyperbola). The students easily played with these ideas, getting introduced to the ideas of slope and hyperbolas. For the hyperbola, we changed the equation to ___ x ___ = 12 to as it has more natural outcomes than ten. When trying this in a grade 8 class, one student wanted to square one of the blanks and got a parabola. Another wanted to try squaring each blank and add them to make ten. They were very surprised to get a circle which they had been studying in measurement and geometry.  The Students

Throughout these activities, the students found that they could do complex mathematics that they had never done before. Their role was to discover complex concepts, and to keep trying new ideas, exploring the possibilities afforded by this simple idea of making ten. They were encouraged to show and share what they had found with each other and with the teachers. The students were also asked to communicate their learning with their parents to demonstrate all that is possible when making ten with their parents. This was rehearsed through such means as role playing, students taking the parts of child and parent to practice delivering the mathematical ideas they had learned. Another way they practiced for sharing was by making comic strips of what they imagined would happen when they shared at home.

At all times, the students were actively engaged in making meaning about mathematics or they were reflecting on their learning to figure out the best way to share their new knowledge with their parents.

The Teachers

The teachers’ role in these activities was to act as a guide, a questioner, and a listener. Bringing the knowledge of the students forward through a series of guided discoveries and leading questions, the teachers were able to let their students discover and work with ever increasingly complex mathematical concepts. Through listening carefully and critically to the students’ mathematical talk, they were able to guide the students on their journey to discover ever more complex ways to make ten.

The interactions between teachers and students were, perhaps, the most important part of the experience for both. By actively, and critically listening to the students, the teachers were able to choose the next mathematical steps for the students. By sharing their discoveries and wonderings with the teachers , the students were able to confirm their understandings, and formulate questions for their wonderings.

The Parents

The parents played a critical role in the students’ learning. The students had to explain their new understandings to them and to do some of the activities with their parent. It is one thing to understand, and to communicate that on an assessment or a task, it is quite another to explain and to teach an adult. The parents were quite surprised at the amount of mathematics their children had learned by ‘simply’ making ten.

An interesting story with math ideas. So surprised at how easily a younger kid is explaining about the plots on x & y axis. Wow!

I learned that the introduction to function can be done with generating numbers to be inserted into a simple equation (math sentence). The graphical results can be attractive to a child, thus sustaining interest in learning further.

She told me she learned 2 new words. One is parallel, and the other is writing an ordered pair such as (1,9). She showed me how to plot a number pair on the grid. She also identified the horizontal axis and the vertical axis. She also talked about the patterns in the number pairs.

I wish we had more homework like this!

Implications for Practice

When teaching about missing numbers in algebra, it is possible to take a very simple mathematical path, and to supply practice worksheets to have the students drill the idea of finding missing numbers. This method will work on the skill of working backwards to find numbers, but it will utterly fail to connect students to the reasons why missing numbers are important to know and understand. Lost will be the opportunity to explore constants, variables, lines, curves, x- and y- intercepts, circles, coordinates, negative integers, the four quadrants of the Cartesian plane and more. Young students are able to engage with these big ideas, these complex concepts at their own level with appropriately designed activities. By engaging with these concepts early, they are able to better understand the place of the mathematics that they are learning. Doing so in the context of a series of activities that enable students to discover complex ideas, was fun and filled with surprises for the students. It was a memorable experience. It was worth remembering.

Conclusion

When eight year olds comfortably state that they have no worries making ten by adding a twelve and a minus two, you have to wonder if we have not been consistently underestimating the mathematical abilities of these young students. When the introduction to coordinates and plotting points on the Cartesian plane takes a one minute demonstration followed by the students completely taking over, you begin to suspect that the mathematics curriculum is not too hard at all, it might just be too simple and straightforward - lacking in depth, beauty and wonder. When young students link pattern to straight and curved lines and to equations, you know that children are a lot smarter than we have thought. Algebra is cool, and the children know it. The Math Performance Festival is funded by the Imperial Oil Foundation, the Fields Institute, Research Western, the Faculty of Education at UWO, and the Canadian Mathematical Society. A project by George Gadanidis (UWO), Marcelo Borba (UNESP, Brazil), Susan Gerofsky (UBC), and Rick Jardine (UWO). 