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SECTION 1: DEVELOPING MATHEMATIANS
What We’re Aiming For:
In Prairie Spirit School Division, we believe in developing students who think like mathematicians. The belief is embedded in one of the strategic objectives on the PSSD Strategic Plan Framework for 2023-26; it states that our goal is to strengthen the foundational numeracy skills of PSSD learners so that they are competent, confident, accurate, flexible, and efficient when dealing with mathematical concepts. These last three characteristics are also echoed on the Saskatchewan provincial holistic rubrics, which were developed to identify desired characteristics for grades 1-9.
The Saskatchewan Mathematics curricula also directs how student mathematicians should be educated. On page 8 of the current (2008) Mathematics 8 curriculum there is a section titled, “Mathematics as a Human Endeavour.” In this section it notes, in part, that in order for students to see the value of mathematics as a human endeavour, they need to have experiences where they experience varying perspectives and approaches to math; value reflection and sharing as a part of developing understanding; recognize errors as part of the learning process; take risks, self-assess and set goals related to math learning; and have opportunities for enjoyment, curiosity and perseverance. This is about creating environments where students welcome productive struggle, embrace the errors that come while risk taking, and work collaboratively to develop understanding.
Further specific examples from the Saskatchewan Mathematics curriculum are provided later in this document, but the main theme is that it is vitally important to build a culture that supports the learning of mathematics in every classroom. Therefore, the aim of this document is to highlight some of the factors to consider when designing a classroom with a positive math culture and suggest some structures that can aid in this task.
Developing mathematicians
To help students reason and understand, we would do well to focus on the following:
Help students develop mental models (schema)
Work through the continuum of concrete to abstract representation
Be aware of how language impacts conceptual understanding
Pay attention to differentiation
Each of these points is expanded on below.
Help Students Develop Mental Models (Schema)
Students can understand any number of mathematical concepts once they have developed mental models for those concepts. But for that to happen, we need strong mental models of simple concepts upon which to build more abstract concepts. Stated another way, in order for students to do well with abstract math, they must understand the simple concepts. This begins with ensuring that students develop a very solid grasp of number sense. The front matter of the Saskatchewan Mathematics curriculum highlights this when it states, “in order for students to become flexible and confident in their calculations abilities, and to transfer those abilities to more abstract contexts, students must first develop a strong understanding of numbers in general.” (p.8) Therefore, many of the activities that we do with students in their early years should be focused on helping them develop really strong number sense.
In addition, students develop deeper understanding of concepts when they discover these concepts through working on tasks. Whenever possible, we want to avoid explicitly giving students the content that they can construct themselves based off their previous mathematical knowledge. The SK Math curriculum states that this content that students can construct themselves are typically things such as procedures, strategies, rules, and problem solving strategies. When an understanding about math can be grasped as a result of how math is logically structured, we want to give students the opportunity to construct this knowledge for themselves. There are most definitely parts of the mathematics curriculum that teachers must cover explicitly so that the students can understand the concepts and ultimately have success. The things that students need explicitly taught (i.e. the parts they cannot discover on their own) include the customs of math – specific symbols for operations, math terms, and conventions regarding recording symbols. For a great example of the difference between covering and discovering, see the discussion about linear relations on p. 15 of the Saskatchewan Grade 8 Mathematics curriculum.
Concrete to Abstract
Further, how we ask students to represent mathematical concepts can develop their mathematical understanding. There is a continuum of ways to represent math concepts that runs from concrete to abstract, and students will understand math concepts more deeply when they can represent their ideas and strategies in a variety of ways – and make connections between those different representations.
For example, if a student is learning the concept of “three items” we would want them to connect the idea of having three poker chips with having three circles drawn on a piece of paper. And we would further want them to connect that understanding to the numeral “3,” the word “three” and also to the oral sound that you just produce in your head when you just read the word “three.”
Student will also develop stronger understanding when they have the opportunity to compare their representations to others that the find in their classroom, including those done by other students (p. 17). As well, the act of creating and understanding models helps learners to develop deep and meaningful understanding of math concepts. When students can switch between different models of a mathematical concept, it helps develop flexibility, which will lead to greater fluency with mathematical concepts.
When teaching students to move from concrete to more abstract representations, note that this is not necessarily a linear process. There are times we may need to bounce back and forth between different types of representations in order for students to solidify a deep understanding of a concept. For example, by the time we teach students about multiplying fractions we will likely have already learned about what a fraction is and how to add and subtract them. We may be tempted to skip over the concrete representation of how to multiply fractions, but we want to allow for both concrete and abstract opportunities for students to learn this – the concrete representation might be what the student needs in order to solidify their understanding.
One final idea to consider when thinking about the concrete to abstract continue and how we can help students develop understanding: it’s useful to look for local connections to build upon. For example, if the students come from a rural background, examples from that context can help them build upon the concepts that they already understand and have a mental model developed. The example used in a textbook or worksheet might not resonate with students, and in some cases can be more of a barrier to understanding then an aid. Be sure to use examples where students have a solid understanding to further the understanding of the mathematical concept in question.
Importance of Language: Conceptual Understanding
Mathematics, like any discipline, has language that can be highly specific. In order for students to be successful, teachers want to be sure to help students translate between math language and their common language – while at the same time develop a conceptual understanding of what the math language represents. To do this, teachers will want to start with language that students already understand and have a mental model for. For example, if we are learning about the Pythagorean theorem, it can be much more helpful to introduce the topic to students by talking about angles (a concept they have a mental model for), and then looking for patterns with right triangles, and then help them communicate what they are seeing. In this case, students might notice that the side opposite the 90o angle is always the longest side, at which point we can introduce the idea of a hypotenuse.
As another example, when we think about the mathematical notation for fractions, students might start by identifying there is a “top” part and a “bottom” part of a fraction. We want to build off that in at least two ways. First, we’d want them to attach the term “numerator” and “denominator” to their understanding of “top” and “bottom,” respectively. But we don’t want to stop there. The second understanding we want them to have is that the numerator represents part of something, while the denominator represents all the parts necessary to make a whole. Language can help us understand the current mental models that students have, and can help us build upon these mental models to develop deeper mathematical understanding.
Importance of Language: Questioning
Questioning is critical to keep encourage student thinking, which is foundational to developing deeper understanding. The effective use of questioning can guide students’ investigations, critical thinking, problem solving, and reflection. The following questions are examples of questions that move students to deeper understanding:
“When would you want to add two numbers less than 100?”
“How do you know you have an answer?”
“Will this work with every number? Every similar situation?”
“How does your representation compare to that of your partner?” a
Questions should be open ended (there may be multiple correct answers or multiple correct approaches to get an answer), help students make sense of the math being studied, and encourage student make connections. The key is to ask questions that cause thinking and reflecting. Getting students to justify their answer is an effective strategy to do this. Teachers will want to plan out some questions in advance that have these attributes. (SK Math Curriculum, p. 22)
Importance of Differentiation
Different students start at different places of understanding, so it can be helpful to use tasks that have a low floor and a high ceiling. These types of tasks give a place for everyone to get started, and an opportunity for students to increase the complexity of their thinking and challenge themselves at a higher level. The goal is for everyone to have the opportunity to have success at the end of the lesson/period/unit, etc., and develop a deeper understanding of the concept being studied. These are often problems that have more than one answer, instead of only having one right answer. For example, instead of asking like this:
Betty walks into the store and buys a can of bean for $2.25, some bread for $3.29, and some an apple for $0.58. How much money will she have left over if she hands the cashier a $10 bill? (There’s only one right answer here!)
Ask a question like this:
Betty walks into a store and buys three items. But it’s been a while since she used a calculator, so instead of adding the totals together, she multiplies them. When Betty gets to the cashier, she says, “I owe, $5.88, correct?” The cashier adds the items together (correctly), and says, “that’s right!”
Can you find the values of the three items? (There are different ways to answer this.)
The second question allows people to come up with different solutions, using different strategies. Want to see how students can solve this? Check out: https://nrich.maths.org/shoppingbasket
Head back to the Math Culture page.
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