My philosophy of teaching mathematics and STEM subjects to students with exceptionalities is that I believe it is not a passive activity. “A passive learner will only see the object, not the relationships” (Van De Walle, 2014 p. 23). It requires active engagement from teacher and student. I subscribe to the research put forth by Jo Boaler of Stanford University. Most people are not born mathematicians, it takes practice to train our brains to think using a growth mindset. Using math applications and games students are able to see patterns and connections. Students’ often express mathematics understandings differently it is important to embrace this.
I believe in multiple representations of mathematical ideas (Boaler, 2010). First, students learn in different ways. Second, knowing multiple representations and methods makes it easier for students to problem solve. Arming them with different ways to solve a problem makes them less likely to give up and more likely to succeed in solving it. Using multiple assessment tools allows a teacher to evaluate students much more accurately such as interviews, portfolios, quizzes, writing assignments and asking learners to write and solve their own problems which leads to higher level thinking skills.
“Visual representations bring research-based options, tools, and alternatives to bear in meeting the instructional challenge of mathematics education” (Steedly, Dragoo, Arafeh & Luke, 2008 p. 8). It’s important to give students options to solving problems such as the lattice method of multiplication or the area model. The area model with its “rows and columns” automatically organizes multiplication information of equal groups.
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It is a “visual representation of the commutative and distributive properties” (Van de Walle, 2014 p. 184). It illustrates the same information as the standard algorithm but in a much easier to understand visual method. This strategy will assist teachers to support students with exceptionalities to engage in learning and successfully completing problems using visual representations. Students confidence will increase if they feel successful, thus engagement will increase.
There are two strategies from the National Council of Teachers of Mathematics (NTCM) excerpt titled What all kids can and will learn in Math Scope and Sequence that support my philosophy of teaching mathematics. These are understanding are number sense and, properties and operations and numbers and operations. Both strategies also coincide with Chapter two in Van De Walle’s Teaching Student-Centered Mathematics where teaching mathematics through problem solving is discussed. “The world is full of order and pattern” (Van De Walle, 2014 p. 7). It can also help support learners experiencing “math anxiety.” Students need to use visuals and other tools to include manipulatives and be presented with a multi step, three-phase lesson format e.g., “I do, We do, We do, You do”.
The two approaches teaching number sense and numbers and operations support my understanding of effective math instruction because they keep kids engaged. They are the goals and objectives I’ve identified for student success. Students can learn to be critical thinkers and problem solvers. Young learners can learn to be math lovers, confident in their abilities, love problem solving, communicate mathematics and, reason mathematically. Additionally, in Van de Walle’s chapter two, he argues that a good way to teach mathematics is through problem solving because it means that students are “actively learning” while working on problems not just applying math after it has already been learned.
Tools are to be used in a plethora of ways. Teachers may be creating a disservice to the students by not encouraging them to use the ‘manipulatives’ in other ways. Van de Walle made a point of reiterating the idea of the use of manipulatives in a classroom. While its true that just because teachers use them does not mean their students are gaining a full understanding of the concept. It is possible for teachers to “misuse manipulatives“ often while trying to get students to follow what they are doing with them and use the same ones they are. If students truly understand concepts, they should not need to follow along or use manipulatives in that way. It’s curious to think of the harm manipulatives could cause in math classroom if not used properly.
As NCTM points out for first grade “identify write and construct place value of ones and tens” is part of numbers and operations. It is important to have the manipulatives readily available for demonstration but not necessarily expect the class to follow one’s every step with them. When it comes down to it they need to also understand the standard algorithms. Manipulatives can be a valuable tool if used properly. For example, base 10 blocks and ten rods serve a valuable purpose for early foundations in developing whole number and place value concepts discussed in Van de Walle’s book, chapter ten, Developing Base-Ten Concepts.
Developing Students should be able to work with equal groups of objects to gain foundations for multiplication by make sense of problems and persevere in solving them, constructing viable arguments, critique reasoning and use appropriate tools strategically. While students will model repeated addition to write number sentences, a good teacher supports learners with changes made with this strategy(s) to meet the learning needs of their learner. For example, when children play the popular math game, “Repeated Addition Coverall” with a partner, they roll the dice. The number they roll is the number that will make a repeated addition sentence using that number. Children color in the squares for the “total” amount that they would have. This activity coincides with the learning task and allows the teacher to see what misconceptions students might have surrounding the topic of repeated addition or confusion that exists on what to do or how to perform repeated addition. It will better inform their teaching.
In a math class, formative assessment should be used frequently. It can be executed in the form of question strategies. For instance, if teachers will walk around the classroom as the students complete the learning activity, they can monitor and discuss one to one their completion of the repeated addition activity and alleviate possible math anxiety. Approaching students one to one can alleviate the feeling of being singled out in a math class and feeling as if they are on the spot. Assessment that supports instruction is the most valuable for classroom teachers. This can be done through anecdotal notes, checklists, problem based tasks and teacher observation (Van de Walle, 2014). Teacher observation can evaluate understanding on the spot.
Questioning strategies may be used one to one, in small group activities, or whole group. It encourages learners to ask and answer higher cognitive questions. For example, “why” and/or “how.” This also helps teachers determine the level and extent of the learners’ understanding in a much more fluid and natural way that will encourage math conversations using math vocabulary. According to Van de Walle in chapter three, Assessing for Learning, sharing samples of students’ work (both strong and weak responses) support all students ability to understand and generate more in-depth responses (p.40).
