In the primary and junior grades, patterning and algebra are kept separate. Patterning and patterning rules are learned in one sub-strand. Equalities, expressions and equations are learned in the other sub-strand. Why is this? It is not until grades seven through nine that students learn that these equations and those patterns are one and the same and that they can be represented graphically and algebraically and that their function is controlled by constants, variables and slopes.
We wondered if it were possible for some of the youngest students to understand the relationships between number patterns, expressions, equations and their graphical representation. Through the careful use of manipulatives, simple graphs and a puzzling story to wrap them up in, would grade twos and threes be able to understand linear algebra and analytic geometry?
In grade two students begin the move away from attribute patterns and towards numerical patterns. In grade seven, students begin to understand that numerical patterns can be expressed as a function and that the resulting values can be graphed. By grade nine, students are able to decompose a linear pattern and through investigation determine its slope and its equation.
By using linking cubes strategically arranged to represent variables and constants within a pattern, and by making numerical charts and simple graphs representing these ideas, we thought we could bridge some of the gap between the simplicity of patterning in grade two and the more robust idea that patterns are a function in grade seven as well as an equation with a slope in grade nine.
The students were asked to assume two different, but complimentary roles during these activity. While listening to segments of the story Math Trains they were active listeners, listening for context and for clues to the puzzles presented in the story. When we paused the story to solve pattern puzzles, the students were asked to be active meaning makers. Normally this is a role that is taken on while reading, looking for themes, big ideas, messages, but in this case they were also helping to make meaning of the mathematics as well. While they all listened as individuals, the students made mathematical meaning in pairs and small groups, talking things through, building models, graphing patterns, and making inferences about the story and the mathematics.
Our role as teachers was to facilitate the learning of these deeper concepts, we acted first as story tellers and puzzle posers, setting up the students to jump into the mathematics together and to find meanings and concepts. During this time, we moved from group to group asking what the students were doing and for explanations of what they were finding. At no time did we ‘give away’ answers or directly deliver content. We listened, complimented student thinking, or perhaps asked a leading question.
After all the student groups had made their mathematical discoveries, we helped summarize everything the students had brought to the discussion, and helped them to come up with terms for the mathematical phenomena discovered. Some of this was achieved by using prompts and charts from the story Math Trains, as well as feedback from the students.
The biggest challenge, we thought, would be bringing the students to an understanding of slope. Looking at an example of the two patterns we had been working with, we asked which one seemed steeper. The students quickly came to the conclusion that the graph on the right was steeper. We then challenged them to make their own pattern which was even steeper still than this pattern. They were asked to use cubes as well as chart paper and bingo dabbers to make their patterns. To our surprise the students did come to the conclusion that something about the variable part being bigger made the graph steeper and that increasing the constant did not really make a steeper graph. They had the beginning idea about the concept of slope.
Parents were invited to take part in these activities at home with their children acting as guides. Copies of the stories were sent home to be shared, and students were prepared in class to deliver a math surprise to their parents. “Look, Mom and Dad. I can do high school math!” The parents were asked to provide feedback about what their children had shared and what they personally had learned.
What the students had shared about their math learning:
“They learned that math can be taught by using graphs. When shown in a graph format, the students can comprehend and grab the concept faster.”
“She shared patterning and described its flow. She explained how it can apply in various situations.”
“My child shared an easy way to learn math patterning and adding.”
“My son is really into the story and tries to solve the problems as the little piggy did.”
What the parents learned:
“Representing the data in different ways (charts, graphs) can bring out more observations.”
“I learned the table and the bar graph relates to the algebraic functions.”
“I learned that using graphs can assist the child with solving math equations.”
“There are several ways to collect math data and also present it to make conclusions about patterns, rate of change and quantities.”
“Using stories can not only simplify students in developing math skills but make it fun.”
“I learned about the interesting steeper bar graphs.”
Using this feedback from the students and parents, we picked and chose a wide sample of what they had written and turned them into lyrics for a song which we made into a video to share on the internet at the Mathematics Performance Festival. This video documents our shared learning and wraps it up in a song and video so that we can always go back and see what we have done.
Implications for Practice
The students demonstrated for us that by using good, flexible models - linking cubes, graphs made with bingo dabbers - they can infer and learn sophisticated mathematical concepts at their own level. There has been a concerted push for many years to include concrete manipulative models of mathematics in student learning before proceeding to the abstract level of using numbers. We believe that there is another step in between these two that also needs to be taken, a representational component. This is a visual component that is built by students, in this case by making graphs using colourful bingo dabbers representing the same number patterns as the linking cube patterns they had made. As well as including the representational between the concrete and abstract, we also believe that wrapping up the mathematical puzzles that were presented in a story helped to make the experience more enjoyable and memorable. The surprises in the narrative were mathematical surprises for the students. The extent of their learning of sophisticated mathematical concepts was our surprise.
From simple ideas come sophisticated mathematics. Building with blocks, strategically using two colours to represent constants and variables is simple. The concepts themselves are sophisticated and integral to real mathematics that these young mathematicians will learn in more detail later. Using bingo dabbers and grid chart paper to simplify the creation of graphs that represent number patterns and functions is simple and easy for the students to do. Inferring that there is a function in a linear equation that controls how steep a graph can be is deep mathematics embedded in analytic geometry that eventually winds up in the study of calculus. Our youngest students readily grasped these sophisticated concepts, and will be prepared and eager to learn more about them when they get into grades seven to nine.
In recent years, the scores for mathematics in Ontario on the yearly EQAO tests have been declining. One possibility is that too much is being asked of the students. Another possibility that perhaps we have revealed here is that we need to allow and encourage students to learn more. Instead of a curriculum which fragments mathematical learning by separating patterning from algebra in the early years could be made whole again by taking on big mathematical ideas from linear algebra and analytic geometry.