Rethinking the Order of Operations (i.e. PEMDAS)

I was talking to my students the other day about the Order of Operations (i.e. PEMDAS). I quickly reviewed the Order of Operations as we worked through several algebraic expressions. Then, I asked my students why we followed the Order of Operations. They said, “It’s just what we’re supposed to follow.” I decided to investigate this further and I asked them, “But why does the Order of Operations follow the order that it does?” They didn’t know.

Why do we follow the order that makes up the Order of Operations? I asked several other people and got the same response — their teacher told them it’s what they were supposed to follow. Welcome to the era of education that focused primarily on the product. Why study the reason behind the Order of Operations, when teaching students that they’re supposed to follow it works? That is like asking, “Why teach an archer about trajectory physics, when you could just teach the archer to hit a specific target?”

So, I entertained the challenge of understanding the Order of Operations further with my students. I wrote PEMDAS vertically on my marker board and analyzed the directionality of each operation.

  • Parentheses only refers to embedded operations, so we left that aside.
  • Exponents were essentially a form of multiplication, since  x^2 is equivalent to  x \cdot x.
  • Multiplication and Division are usually computed together from left to right. I stopped my students at this point to consider the two different operations. Could multiplication be computed bidirectionally? Yes. What about division? If we think of division as multiplication, then  \frac x 2  could be written as  x \cdot \frac 1 2  (i.e. multiplying x by the multiplicative inverse of 2).
  • Addition and Subtraction are much like Multiplication and Division. While addition could be computed bidirectionally, subtraction won’t allow for it. Fortunately, we are able to write subtraction as the addition of an additive inverse. For example,  5-3  could be written as  5 + (-3).

Therefore, instead of PEMDAS, we have PMA. This simplifies things a little, but it doesn’t answer the question why we follow the order of PEMDAS (even if it’s written as PMA).

PEMDAS

 

Let’s visualize a situation where PEMDAS would be needed. In the above picture, we have four stacks of squares with nine circles in each square, a single square with nine circles in it, and four circles by themselves. If we had to put this in an expression, we could write it as  4(3^2) + 3^2 + 4. If we had to add up all of the blue circles, how would we do it? Would you want to count all the circles one-by-one? This may be a little time-consuming. What if we computed each of the squares as nine circles (from  3^2 = 9). Then, if one square equals nine circles, four squares must equal 36 circles. Thus, 36 circles + 9 circles + 4 circles. Interestingly, we just followed PEMDAS. Wait! I don’t remember bringing PEMDAS into this. It seems that the reason for PEMDAS is right in front of us.

Think about it.

Four squares of nine circles + one square of nine circles + four circles

4 ( 9 circles) + 9 circles + 4 circles

36 circles + 9 circles + 4 circles

Basically, we had to convert the original expression into an expression that allowed for addition to happen with quantities of similar terms. There was no way to add a square with four circles. Instead, we had to convert the square to nine circles to allow for addition to be possible. Ultimately, PEMDAS follows PMA, ranking the operations in priority from the embedded operation (P), to multiplication (M), and then to addition (A). If addition is the final operation performed, then all quantities must be similar in order to be added. Likewise, if multiplication is the final operation performed, then all quantities must be similar in order to be multiplied.

In summary, we follow the Order of Operations to allow for multiplication and addition with quantities of similar terms.

 

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Instructional Strategies: Collaborative Group Work

It has been a tradition in many mathematics classrooms to follow the direct instruction model, but with the emphasis of the Common Core State Standards on developing students’ mathematical fluency, using a method that requires students to develop their thinking individually and then collaboratively (in that order) is consequential. This is particularly the reason why integrating formative assessments into the sequence of lesson planning is essential to teaching mathematical fluency. This strategy reverses the gradual release of responsibility by having the teacher give the students a task to complete in class or as homework prior to the lesson, in the form of a pre-assessment. The reasoning for this is to provide the teacher with an understanding of the students’ use of mathematics and any areas of opportunity. On the day of the lesson, the teacher passes back the students’ work with feedback or questions provided on each student’s work, encouraging the students to engage in self-reflection, “When we use assessment for formative purposes, students should receive growth-producing feedback and have the opportunity to make adjustments to their work based on that feedback,” (Rutherford, 2009, p. 139). This allows the student to consider their reasoning for using the mathematics that they did. Then, the teacher divides the class into small groups and has them work collaboratively on developing a joint solution. After students have developed a joint solution, the teacher holds a whole-class discussion. Prior to holding the discussion, the teacher should circulate around the classroom and note the types of approaches that each group is using to solve the task. The teacher can use these notes to help guide the discussion and explore the methods used.  If time is available, the teacher could pass out several sample responses to each group, allowing them to analyze the different approaches and compare them to their own approach. Then, the teacher could hold a whole-class discussion on the different approaches used.

TeamworkTeaching mathematical fluency is not only about teaching content, but about teaching students how to use that content, how to think about that content, and how to apply that content to the real world. Collaborating with others offers an opportunity for students not only to interact with their peers, allowing them to develop their mathematical fluency through listening and speaking with other students, but also to engage the content at a deeper level, “The fact that students are actively exchanging, debating and negotiating ideas within their groups increases students’ interest in learning. Importantly, by engaging in discussion and taking responsibility for their learning, students are encouraged to become critical thinkers” (Dooly, 2008, p. 2). As students interact and engage their peers in this model, they become active agents of their own learning and experience learning at a whole new level, “Educational experiences that are active, social, contextual, engaging, and student-owned lead to deeper learning” (Cornell University, 2013). Because this strategy offers a student-centered approach that builds on collaboration, it is generally applicable to all content areas. However, in terms of mathematics, it offers students a unique approach to mathematical fluency by improving how they think about mathematics. Based on the nature of this strategy, it provides for an excellent means of formative assessment, “Formative assessments…first allows students to demonstrate their prior understandings and abilities in employing the mathematical practices, and then involves students in resolving their own difficulties and misconceptions through structured discussion” (Mathematics Assessment Resource Service, 2013).

Rethinking the Use of Real World Examples in Mathematics

While connecting mathematics to the real world may seem like it would be a simple task, it ends up being a real challenge. I always tried to find some application to the real world that the students could use to understand the mathematics better, but it was never that easy. I realized that my struggle was not in finding the real world examples, but in what I was looking for. I usually imagined career-based real world examples. Then, if students asked why they had to learn something in mathematics, I could refer to the type of career that would need that type of mathematical background. Later, I considered re-thinking the types of examples I wanted to use. Instead of career-based real world examples, I explored real world examples that students experience day-to-day. This subtle change made it easier to think of examples. After a while, my students were even thinking of their own examples. I realized that the examples I originally presented them were far removed from anything that they could really relate to. By selecting examples that they could relate to, they were more open to working through the problem.

Approaching Mathematics in a Less Than Mathematical Way

I recently had the opportunity to teach a lesson to a group of 6th grade students on writing algebraic expression involving addition and subtraction. The group consisted of 18 students, of which 10 were classified as English language learners two were classified as students with special needs. I started the lesson by introducing myself and the objective of the lesson that we would be addressing throughout the lesson. Then, I moved on to the pre-assessment by posting the five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. The students wrote all five of the expressions on their papers and began translating. I circulated around the room and noted a few observations. Some students worked through each expression sequentially, while others provided at least one phrase for each expression before adding more phrases. Interestingly, the first phrase that every student wrote for addition expressions included the word “plus” and for subtraction expressions included the word “minus.”

Algebra WritingAfter administering the pre-assessment, I modeled how to represent algebraic expressions using visual models. One of the models that I wanted to focus on was the bar diagram. I used the bar diagram to differentiate between the operations of addition and subtraction and to illustrate the commutative property of addition. Since there were 18 students, I divided them into six groups of three students each and gave each group a set of algebraic expressions to represent using a bar diagram. I circulated around the groups and engaged them with questions to further their thinking. After five minutes passed, I brought the class together and had the groups share their solutions with the rest of the class.  This provided an excellent opportunity to record their responses using the document camera and LCD projector. I had the students help me organize the responses to see the order of complexity of each response. For example, for the expression “n + 5,” we categorized “n plus 5” as a simple statement and “n increased by 5” as a complex statement. Then, we looked at the phrase “5 more than n” and contrasted it with the first two based on the order of the factors. I had each group brainstorm other ways of translating the expressions into verbal phrases. We did the same for “10 – m” and highlighted the impact of commutativity on writing verbal phrases. This led into our discussion on multi-step algebraic expressions and the syntax necessary to translate these expressions into verbal phrases.

I administered the post-assessment by posting a set of five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. I changed the factors from the expressions used during the pre-assessment and gave the students several more minutes to complete this task before concluding the lesson. Compared to the pre-assessment, the students took almost double the amount of time to consider verbal phrases to represent each algebraic expression with during the post-assessment, signaling the students increased level of understanding. I also observed an increase in the number of grammatically written phrases used to represent the expressions given. Unlike the syntactic errors that the students made on the pre-assessment, they rearranged the factors in the correct syntactic order.

After reflecting on the pre- and post-assessments that I administered, I would consider changing the post-assessment. For the post-assessment, I would have the students organize a list of verbal phrases and explain their reasoning for organizing the phrases the way they did. Instead of repeating the same style of assessment as the pre-assessment, this type of assessment would require the students to apply their critical thinking skills as they consider the application, the syntax, and the logical implication of each verbal phrase. This could lead into a nice introduction for the following lesson by having the students work with a partner, reflecting on each other’s responses and critiquing their partner’s reasoning.

Throughout this whole experience, I really appreciated the students’ level of engagement. The students were eager to participate in class discussions and provide responses that facilitated the critical thinking process. They were also open to the concept of analyzing mathematics through language. As I circulated around the classroom, I heard groups engaged in collaborative discussions, using their communication skills to create solutions that made sense to everyone in the group. They shared ideas and built consensus as they brainstormed different verbal phrases to represent the algebraic expressions given. It was great to see so many students receptive to approaching mathematics in a less than mathematical way.

Solving Multistep Percent Problems (Sample Lesson Plan)

Lesson Plan - Multistep Percent Problems (Thumbnail)In designing this lesson plan, I wanted to approach the concept of multi-step percent problems in the context of problem situations. According to the Common Core State Standards Initiative (2012), “Mathematically proficient students make sense of quantities and their relationships in problem situations.” This means that for students to be mathematically proficient, they need to be able to decontextualize and contextualize quantitative problems. By teaching students to decontextualize problem situations and represent them symbolically, students will develop their quantitative reasoning skills. The lesson explores ways of decontextualizing problem situations that relate to percentages (i.e. sales on merchandise). Students are then shown how to use different operations to compare sale prices during the Teacher Presentation and practice decontextualizing problem situations in collaborate multi-ability groups during Class Activities.

For English Learner students, the process of decontextualizing problem situations requires a certain level of English language proficiency that may be beyond their proficiency level. In order to facilitate their English language development while lowering their anxiety, I integrated the collaborative model into the Class Activities portion of the lesson. The collaborative model is beneficial not only for English Learner students, but for all students. As students interact and engage their peers in this model, they become active agents of their own learning and experience learning at a whole new level, “Educational experiences that are active, social, contextual, engaging, and student-owned lead to deeper learning” (Cornell University, 2013). For English Learner students and especially for Jonathan, the profiled English Learner student, collaborating offers an opportunity to interact with other students, allowing them to develop their English language proficiency through listening and speaking with other students. While developing their English language proficiency, English Learner students will also be able to engage in a model of learning that develops their critical thinking skills, “The fact that students are actively exchanging, debating and negotiating ideas within their groups increases students’ interest in learning. Importantly, by engaging in discussion and taking responsibility for their learning, students are encouraged to become critical thinkers” (Dooly, 2008, p. 2). In order to facilitate the active exchange of ideas within these collaborative groups, I integrated the use of posters, strategically placed around the room, with a list of discussion and sentence starters. John Larmer (2013) suggests the use of discussion and sentence starters as a way of helping English Learner students acquire 21st century learning skills and develop their English language proficiency, “Help English learners access the project by thinking carefully about the language functions called for in specific 21st century learning activities; place them in supportive teams; provide models such as sentence starters and graphic organizers to help with collaborative discussions.”

At the beginning of the lesson, I engaged the students in brainstorming as a way of activating background knowledge of the word “sales.” Exposing English Learner students to new concepts through vocabulary correlations facilitates their learning and their English language development. By activating background knowledge, the students will be able to make connections to the learning that are meaningful and personally engaging. It also helps to engage the students in activating background knowledge to ensure that students are approaching the new concept with a basic foundational understanding of the concepts necessary for learning the new concept. Using a Circle Map to record the students’ responses help graphically organize the responses for the students, especially the English Learner students, to visualize their relationships to the word. For Jonathan, the profiled English Leaner, the Circle Map will facilitate his acquisition of the meaning of the word “sales” and the relationships of the students’ responses, especially considering his interest in creating art. Individuals with spatial intelligence enjoy creating art and easily learn through visual presentations. Giles, Pitre, and Womack suggest that teachers utilize visual presentations to accommodate the learning needs of students with spatial intelligence, “Teachers can foster this intelligence by utilizing charts, graphs, diagrams, graphic organizers, videotapes, color, art activities, doodling, microscopes and computer graphics software.” I have also integrated PowerPoint in the Introduction and the Teacher Presentation as another way of visually presenting the concepts, example percent problems, and the steps for solving each problem.

During the Introduction, I emphasized the time (or extended time) necessary to wait on students to provide their responses to questions. It is crucial to allow students enough time to think about a question and develop a response that they are confident in sharing with the class. For English Learner students, sharing with the class may already be a high-anxiety situation. There is no need to exacerbate the situation by requiring speedy responses. In order to lower the students’ anxiety, I emphasized implementing a longer wait time. This not only provides the English Learner students and Jonathan, the profiled English Learner student, with extra time to process content, but also to linguistically plan their response.

The lesson incorporates a number of strategies, supporting the 1/3 Plus Model. The students are intentionally placed in collaborative multi-ability groups to allow the top third the opportunity to lead the collaborative discussion and provide support to his/her peers. During the Introduction, the top third may offer more advanced vocabulary when they think of the word “sales.” The middle third of the students will learn because they have a basic understanding of the concepts underlying percent problems and will be able to participate in the collaborative discussions. During discussions, they may respond on a voluntary basis or they may wait to hear another student respond first. By giving students extra time to respond after asking them a question, the middle third (and bottom third) will be more encouraged to participate in class discussions. The bottom third will be first engaged by the activation of background knowledge. This will facilitate their learning and help them make connections to any new vocabulary or concepts. Using graphic organizers, like the Circle Map, will help the bottom third visualize vocabulary and their relationship to the concept being taught. Including PowerPoint as a means of visually highlighting key aspects of translating percent problems and providing images for students to relate to facilitates the learning process. By collaborating in multi-ability group, the bottom third are able to work with other students, developing their mathematical and English language proficiency through observational learning and social contexts. The questions asked at the end of the lesson are progressively leveled according to Bloom’s taxonomy to encourage higher order thinking of all students. Having students participate in a think-pair-share to engage their learning and have them share their responses helps the bottom third and middle third take the time to reflect on their learning and hear other students’ responses. It helps the top third reflect and consider further investigations while sharing their ideas with other students.


PowerPoint Presentation Used with Lesson

Applying Bloom’s Revised Taxonomy to Mathematics

A few months ago, I designed a training on applying Bloom’s Taxonomy to create higher order thinking questions in mathematics.

I showed two figures: (1) A right triangle and (2) an equilateral triangle.

220px-Rtriangle

Right Triangle

triangleEquilateral

Equilateral Triangle

The following were questions that I developed to help the instructors think through the various levels of Bloom’s Revised Taxonomy in regards to mathematics.

  • Knowledge – What can you tell me about the first triangle?
    • The students provide any information they know about the mathematical concept.
  • Comprehension – What makes the first triangle a right triangle?
    • The students use the information they already know about triangles to rightly identify a specific triangle.
  • Application – Based on what you know about right triangles, why is the second triangle not a right triangle?
    • The students apply the information they already know about triangle to differentiating one triangle from another based on their characteristics.
  • Analysis – How is the first triangle similar to a rectangle?
    • The students compare the characteristics of a right triangle with those of a rectangle.
  • Evaluation – How would you prove that all right triangles fit in a circle, with each vertex (or corner) of the triangle touching the circle?
    • The students extend their understanding of triangles by proving a well-known theorem of geometry (see Thale’s Theorem)
  • Synthesis – How could you use the right triangle to design our next engineering project?
    • The students integrate information they know and understand about the right triangle into designing a new project.

Any comments/suggestions?