CASE Stories

Odd, Even, and Square Numbers

There is almost no mention of odd or even numbers in the Ontario Mathematics Curriculum. Square numbers pop up beginning in grade 7, with perfect square numbers - those with a whole number root - arriving in grade 8. This is odd indeed as there are so many fascinating patterns and concepts to be found in odd, even and square numbers. These concepts are not limited to the number strand, in fact these numbers play an important role in measurement, geometry, patterning, algebra, data management and probability. If these numbers have such a broad reach, perhaps, they might be more important and interesting than their formal absence in the curriculum might suggest. We wondered if some of the youngest of our students, our grade ones and twos, could explore square numbers, find where odd and even numbers hide, and understand these concepts at a rudimentary level.

### The Mathematics

According to the Ontario Mathematics Curriculum, grade ones and twos need to be able to identify and describe two-dimensional shapes like squares. They also learn to decompose whole numbers, and while the grade ones deal primarily with repeating patterns, a growing shape pattern would not be too much of a stretch. Grade twos learn about growing and shrinking patterns, especially those dealing with addition and subtraction. This is all the fertile ground into which we wanted to plant the seeds of mathematical curiosity and surprise.

Through the sharing of the story My Full Moon is a Square, we initiated an exploration into square numbers. In the story, fireflies engage in precision formation flying, making ever bigger squares of light in the night sky: 4, 9, 16, 25 and on. The students made their own squares using bingo dabbers and grid chart paper. Four dabs made a square, nine dabs made a square, sixteen and on. As the students searched to make the next biggest squares, they discovered that not all numbers can be made into squares.

One student switched bingo dabber colour and made an L-shaped 7 to add to the 3 x 3 array that was already marked on our chart. This let us lead them into a startling discovery. “We added 7 to out square of 9 and it gave us the the next square of 16. I wonder what number we will have to add next?” The students quickly discovered that an L-shaped 9 was next. “Hm, 7 and 9 are both odd numbers. Can you find any other L-shaped odd numbers in the squares we have already made?” The students traced their fingers over the bingo dabbed spots of their squares finding L-shaped numbers in all of their squares. “I found 5 and 3. And there’s a 1 in the corner!”

“Are those all odd numbers?” we asked.

“They all seem to be hiding. Where do odd numbers hide?”

“In squares!!!”

To cement this idea in place, we had the students construct odd-numbered Ls out of linking cubes. These Ls fit pleasingly together like spoons forming squares. Square numbers can be decomposed into a series of odd numbers. You may recall from your own experience in middle school or high school math class that the sum of the first n odd numbers can be expressed as n x n. The physical proof for this now lay in the hands of six and seven year old children.

Next, we wondered aloud about where even numbers might hide. This set the students off on yet another round of discovery as they used linking cubes and bingo dabbers to explore the different possibilities of shapes made by a series of even numbers. Eventually, some of the students began discovering that the answer to this puzzle was already inside their squares full of odd numbers. Even numbers are just one less than odd numbers. By reducing each of their odd-numbered Ls by one, they got even-numbered Ls. Again, looking back at your own mathematics learning, you might see that they way the students found even numbers using manipulatives is the physical model of how the expression 2n+1 can produce odd numbers.

### The Students

The students doing this activity were completely immersed in discovering mathematics concepts through trying to solve the puzzles presented by a story and the teachers. By beginning with a good story, they were set down in the comfortable space of enjoying a good book. Like all good books, My Full Moon is a Square makes us think about more than just the words in the story, it leaves us with a puzzle in our minds that we have to solve. As the students solved one puzzle, they found that there was another puzzle posed by their solutions. As they went from puzzle to puzzle, making squares, finding where odd numbers hide, finding where even numbers hide, they were completely engaged in learning in a manner that exceeded even our expectations. In this case the mathematics learning did not run out signaling to the teachers that it might be a good time to end the activities, time ran out.

As a consolidation to their learning, the students were asked to take on a completely different role than learner. They were asked to prepare for presenting their new mathematical puzzles and surprises to their parents - they were to become the teachers. In order to help them visualize and practice for surprising their parents, the students created comics. These depicted what they thought the exchange might be between them and their parents - a creative way to prepare for a lesson… and to cement new concepts in the students minds.

### The Teachers

Discovery activities can seem messy to an observer on the outside. There are students going in all sorts of different directions mathematically, and there is a sea of talk going on between partners. In fact, these types of activities have to be even more highly structured than a regular teacher talks, students do type of lesson. The teachers have to start with an end goal in mind, then structure a series of puzzles or activities that lead to these goals, and that account for individual differences amongst the students. Within this structure has to be contingencies for other learning like when the students discovered something that led to them finding the odd numbers in squares.

This type of activity has a low floor and a high ceiling. The low floor is the making of squares by using bingo dabbers. The parts in between this low floor and the high ceiling are the series of odd and even numbers. Above that are the concepts of sums of series, area, arrays, even cubic numbers as we were to find out. Even though we had an end goal planned, we had contingencies for all of the students to exceed that learning and to go as far as they could.

### The Parents

The parents role in this activity was to be the student to their child’s teacher. This type of receptive role is not necessarily one that most parents are familiar with doing. The stereotypical role of homework helper or homework completion guarantor has been flipped over. It is the parents who have homework learning the mathematics their child has learned in school. The feedback that we have received from the parents was excellent. They were asked to give us feedback on what their child had shared with them and what they had learned.

• I learned a pattern with odd numbers 1, 3, 5, 7, 9 11…
• She learned about shapes, number, odds and evens.
• My son told me that odd numbers do not have pairs like the number 2 or 4 or 6 or 8 or 10.
• Maths has always been fun for her. From the current math learning activity, she shared with me about interesting shapes she found in the chocolate bar. She was able to find L-shapes in the chocolate bar that amused her. After this math activity she tries to find every shape in different things at home.
• It is good idea for me to explain math to my son, easy understand keeping interest.
• I learned that my son can differentiate between odd and even numbers and that he is learning about geometric shapes.

From the math-at-home activity I learnt that different shapes and patterns are all over around us. We often ignore the geometry and shapes present in everyday life. By exploring different shapes we can enhance the learning of children in the field of maths and polish their "problem solving skills" that play an important role in our lives.

This is the type of active role the parents like to take in their young child’s learning. They are learning together, discovering interesting and pleasing mathematics concepts as a team. As a bonus, the parents now have great answers to their question, “What did you learn in school today?” No more “nothings” or “I don’t knows.” Now they get a math surprise of their own.

### Implications for Practice

These activities do much more than just teach the students on the surface about mathematical concepts. Embedded in each activity is more mathematics that can be taken to the extremes of what the children are capable of learning. Even if the students do not go down these paths, they have been familiarized with this mathematical landscape so that they will ultimately be better prepared for learning in the higher grades. Consider how making squares and rectangles, and finding patterns of numbers within them prepares students for learning about area, even linear algebra. Being familiar with finding all of these patterns and linked concepts predisposes the children to look for these links when they are learning. Learning is an active role not a passive one.

Being prepared to do dramatically more mathematically than the grade level expectations is a different role for teachers to play than they might be used to fulfilling. Letting the students lead the teachers to different areas is the ultimate in differentiation. We almost always differentiate by ‘dumbing down’ or simplifying lessons so that there are multiple entry points for students who might encounter difficulties. In this instance, we create a low floor where everyone gets into the lesson together. Couched in a surprise, and embedded in complex mathematics, the activities let students rise up to their mathematical level and exceed it. This is differentiation in a different direction. We never fail to be surprised by how students who are not always the most successful in school are the ones who are very successful when learning with this approach. It is as if a hidden potential has been released in these students to learn complex mathematical concepts.

For other students, this allows them to learn at an even higher level than we would normally let them given grade level expectations. This approach of having a low floor and a high ceiling allows for differentiation at the other end of the spectrum as well. This has led us to wonder if there is really a limit to what six and seven years olds can learn given the chance.

### Conclusion

We wondered if some of the youngest of our students, our grade ones and twos, could explore square numbers, find where odd and even numbers hide, and understand these concepts at a rudimentary level. We were rewarded with a classroom full of curious, purposeful mathematical investigators who did not want to stop learning. The students not only understood these concepts at a rudimentary grade level, they exceeded these simple expectations.

Differentiation is supposed to allow for all to learn, but it is almost always designed for the students who have a more challenging time learning rather than differentiating for every single student in the classroom. Low floor high ceiling mathematics activities allow for each and every one of our students to be successful to the extremes of their abilities. They are designed for everyone to be successful and to become familiar with or even understand the sophisticated, complex mathematics they will encounter in future years. While we want to ensure that the students experience pleasing and especially surprising mathematics, that they can go home and share their surprise with their parents, we are almost always experiencing surprise ourselves at the amazing abilities all of our students show us when learning this way. While these surprises drive on the learning of the students, our teacher surprises drive us onward as well, improving our practice. All we need now is for our families to sit around the dinner table and ask us, “What did you teach today, Mom or Dad?” We certainly can’t answer, “Nothing or I don’t know,” any more.

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).