3.+Supposition+Develops

= Superior Greenstone District School Board =

Student Work Study Initiative
= Learning Supposition for 2012-2013 = = Written by Kathleen Schram = = = **Supposition Develops**

When the political situation impeded on traveling I was able to spend a lot of time directly working with students in School A. As I worked with the grade 3 and 4's to see what types of strategies they were using, I learned a lot. That their favorite seemed to be that of counting on and splitting, and that when asked what it might look like on a number line, they had difficulty.



Students were working with 2-digit addition and had completed an activity where they measured

each others heights in cubes and were working through problems combining and comparing

heights. Students were able to share their strategies and articulate their thinking. I modeled what

it was they were sharing, and as is evident from the picture, they were able to use strategies such

as splitting, adding on, and breaking apart to help them arrive at their answers.



When asked what the problem might look like on a number line, they had difficulty, and even through modeling it became evident that they saw adding more as 'putting together' and did not see how it was also 'putting more on'.

We spent some time building this understanding through the use of the vertical number line that was generated with their cubes and how it related to the horizontal number line. Time was spent with them developing games and problem solving strategies. The broken hundred chart seemed to pose problems....especially when counting through a period. With practice in making sense of the number line - they were then able to see how counting on/adding could be used to help them in a subtracting situation - but that the true meaning was represented differently. In the beginning students recognized that in a subtraction problem where the start was unknown, that you could just add the numbers and end up with the correct answer.

They also know that all problems had a "mystery" to solve and began to use "mystery boxes" in their problem solving to help them to represent what the problem was truly asking them to do.

Student work evolved as they practiced representing various situations on the number line.

Working through a situation like this where they knew immediately that the 'mystery box' was 2 because they know that 2 and 8 made 10, was difficult to model and make sense of on the number line.

It took time and students generating and sharing number stories of their own to make sense of how the number line could represent different situations.

Number talks evolved to be me saying things like "68 plus a number is 123, where would the mystery box be and what would it look like on a number line"....

Students would then complete independently and then take a look at each others to see how the number story they represented was the same and/or different.

Some who were fluent with their numbers where even able to solve the 'mystery' quickly.



Independent practice after showed that some students were able to use this model to make sense of why adding worked in a 'start unknown' situation and could model the equation correctly with a 'mystery box' as well as represent it correctly on an open number line and then use their previous way of thinking to check their answers. Some could still come up with the correct answer, but struggled to represent the situation correctly on the number line. Or take the time to prove why different models/strategies were resulting in different answers.

=**What the Research was Saying**=

It was during this time that I began to take a look more at what the research was saying.

In Small Steps, Big Changes, Chris Confer & Marco Ramirez outline 8 Essential Practices for Transforming Schools Through Mathematics. Each phase of the change process is brought to life through the stories and perspectives of teachers, coaches, and principals. r //“We have learned that teachers, coaches, and principals who intertwine their roles and together research math instruction realize what is possible for children to achieve.”// //“We know that success is possible when educators search for that gold coin in a different place.”// T heir Chapter 7: Patterns in Instruction talks about Pattern 2 – Keep the Math Visible Pg. 114- 115 //Most of us are visual learners. When students can “see the math,” they better understand the relationships that are being discussed. “Now I see!” is what students often say, and it can literally be the case.// // Before teaching a lesson, consider which visual models will most likely illustrate mathematical relationships. During lessons, consistently connect equations to that visual model. //** Teachers who explicitly and consistently make those connections **// improve the likelihood that struggling students or English language learners will comprehend what their classmates are saying and be able to participate in class discussions. Successful students who make those connections for themselves – and develop the habit of doing so – keep in mind the meaning of symbols. // //Visual models do not in and of themselves communicate their meaning to students.// ** Number lines can be especially elusive to children unless they have regular opportunities to create number lines, to use them to tell stories, to represent relationships on them…and to talk about what they see //.//**

The article Teaching Our Children to Communicate: The Importance of Symbolic Language also resonated as it states that: //The symbols used to communicate important information within the human world are neither simple nor easy to understand.// //“Why is it that young children are innately good problem solvers, that they can solve complex problems and yet after a couple of years in school, they seem to be worse off than when they come to school? In other words, what are we doing to our baby birds that seem to make them less well able to communicate with the flock?// //We know that they have not lost their innate capacities for thinking and reasoning. In fact, as they are growing and developing, they should be better able to use their growing capabilities to thrive and communicate even more effectively…but they don’t. So why don’t they?// //I suspect the answer lies hidden within the nature of the symbol systems we use and the ways in which we teach them to our young children. Our symbol systems, be they numeric or alphabetic, are very abstract.//


 * ???Wonderings??? Are teachers explicitly and consistently making connections between visual models and multiple representations to help our children learn how to communicate effectively using the language of mathematics??? **

In Arthur Hyde's, Comprehending Math: Adapting Reading Strategies to teach Mathematics, K-6, his Chapter on Visualization He notes on Pg. 88 how: //“Figure 3.16 shows six major ways of representing the mental images in our heads. To build conceptual understanding to create multiple representations of a situation, and to be able to flexibly move back and forth across them, is critical. Teachers should ask students, ‘What does each representation reveal that the others don’t, and what does each obscure?’// //Our goal for mathematics teaching must be real conceptual understanding, and that means that at least some of the time, if not most of the time, students must work on complex, real-world problems, building mathematical models.// //Models are mental maps, representations of relationships. They are ideas, constructs, schemata that have been generalized across a number of contexts.”//

On pg. 87 he notes that: //“Of course, I feel strongly about the representational strategies being used in problem solving. Once students hav a good ‘feel’ for the problem from visualizing the situation, they should use one of the other representational strategies to work on it. In other words, they should try to represent the problem in a way that will help them to either understand it better, to understand it in another way, or to lead them to a good solution path......When students create and share multiple representations of the same problem or situation, they are continuing to keep their thinking alive. Multiple representations also may provide deeper, more elaborate understandings of the underlying mathematics, and fresh, new insights into the problem.”//


 * ???Wonderings???Do we do enough modeling and relating visual representations??? **

This lead to the initial supposition of: When visual representations are paired with effective questions a deeper understanding of mathematical concepts are attained. It is important for multiple visual representations to be used so that students can make connections and see the mathematical relationships.

= Continue on to 'The Work Continues'  =