Alternative Assessments: Multiple-Choice Tests

Have you ever given a multiple-choice test and wondered whether restricting your students’ creativity to a small number of available options truly assessed their comprehension of a concept? I have, but I didn’t toss the multiple-choices out. Call me an optimist, but I always try to find the benefit of something.

Multiple ChoiceI asked myself, “What could possibly be the benefit of giving a multiple-choice test?” The last time I created a multiple-choice test, I found it far more involved than creating other types of assessments. Once I typed out the question, I knew the correct answer and struggled to write three more false answers for a total of four possible answers.  So, I wondered if I could harness some of that critical thinking that went into creating the multiple-choice test. What if I could take a different approach to this type of test that required a higher level of critical thinking? I thought for a while about it and decided to create a complete multiple-choice test, but without the questions.

That’s right! I gave my students a multiple-choice test with four answer choices (with the correct answer marked) for each question, but without the questions. Using the answer choices provided, they had to figure out what the question could have been. The first time that I tried it, the students seemed extremely engaged in the different way of thinking. What I liked about it was that it really got them to critically think about the material before writing down a question.

What alternative assessments have you created to inspire critical thinking?


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).



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.


Classroom Management: Social Contracts

Social ContractAlong with involving students in the selection of topics to be covered, I would also involve them in developing behavior guidelines for the class. According to Marzano, Marzano, and Pickering (2009, p. 13), “Probably the most obvious aspect of effective classroom management involves the design and implementation of classroom rules and procedures.” In their book, Discipline with Dignity, Curwin, Mendler, and Mendler (2008, p. 68) agree with the importance of involving student in the development of the classroom rules and procedures, introducing their view of the classroom rules and procedures as a social contract, “The social contract is an agreement between teacher and students about the values, rules, and consequences for classroom behavior.” Using a social contract would not only increase student responsibility, but it would also affect their academic achievement.


Classroom Management: Student Involvement

Classroom management is essential in providing students with an environment that is conducive to learning. According to Marzano and Marzano (2003), “One of the classroom teacher’s most important jobs is managing the classroom effectively.” There are several factors that highly impact the effectiveness of how a classroom is managed. First, the teacher must utilize instructional strategies and curriculum design that foster student engagement and accommodate the students’ learning needs, “Effective instructional strategies and good classroom curriculum design are built on the foundation of effective classroom management” (Marzano, Marzano, & Pickering, 2009, p. 4). While the teacher should be knowledgeable of current pedagogical trends, it is important that the students understand the instructional strategies being used and participate in planning the curriculum to be covered. By involving students in the selection of the topics to be covered, the students are more intrinsically motivated to participate in the lessons. Likewise, the number of topics decreases, allowing for deeper exploration of each topic. According to Glasser’s concept of a quality curriculum, “Quality learning requires depth of understanding together with a good grasp of its usefulness. Learning a smaller number of topics very well is always preferable to covering many topics superficially,” (Charles, 2005, p. 78). In fact, fewer topics allow for greater clarity and specificity.



Workshop: Introduction to the Standards for Mathematical Practice

121diagramEarlier today, I had the opportunity to deliver a workshop on the Standards for Mathematical Practice (SMP) to a room of OST staff. The workshop was designed to:

  • Help them develop a basic understanding of the SMP,
  • Familiarize them with the language used in the SMP,
  • Discuss strategies for coaching their staff on the SMP.

After reviewing the workshop’s Agenda and introducing the Learning Goals, I presented them with a picture of a house and asked them, “What do we need to build a house?” Before moving to the next slide, I had the participants share their ideas of what was needed to build a house. I wrote their responses on the board and together we drew arrows between each idea, creating a sequence of events. Then, we moved through the next eight slides, comparing the information on the slides with the ideas that they shared. The order of the slides paralleled the SMP to provide the context for comparing the different steps of building a house with the SMP. I did this so that the participants focused on the process of building a house as a prelude to focusing on the process of solving problems. The emphasis here is on the process of solving than on the product of solving (i.e. the answer). Ultimately, I wanted the staff to witness the learning opportunities that could take place when the teacher focused more on the process of solving problems than on the product (i.e. the answer). I concluded this series of slides with a question that required the participants to consider what building a house has to do with mathematics. I used a general question to allow for greater discussion.

We transitioned to TED Talks video presented by Dan Meyer, in which he discussed the need to rethink how we engage students in mathematics. This led us into an introduction to the SMP. Two slides are provided for each SMP. On the first slide, the SMP is listed with a few bulleted focus items to consider when addressing the SMP. On the second slide, I provided two to three strategies to utilize when addressing the SMP. Instead of reading aloud the information on each slide for the participants, I engaged them in a discussion of what the SMP means to them and how they could address it in the classroom. We used the information provided on the slides to validate their ideas and consider alternatives.

Then, we explored three different problems that are samples provided by the SBAC. Two of the problems provided are third grade level problems and one of the problems is a sixth grade level problem. As we approached each problem, I modeled for the participants how their staff should be guiding their students’ thinking. Instead of disseminating information, I engaged the participants in:

  • Thinking about the problem on their own and writing down their ideas
  • Discussing the problem and their ideas with their partner
  • Conceptualizing the problem using different visual models
  • Sharing their solution and their reasoning with the rest of the class
  • Ask questions and critiquing the reasoning of others
  • Defending their solutions based on evidence provided in the problem

Finally, we concluded the workshop by brainstorming strategies for introducing the SMP to their staff and coaching them on addressing the SMP in their classrooms.

Slideshare link:

Instructional Strategies: Project-Based Learning

PBL 2Project-based learning (PBL) is another student-centered model of learning that emphasizes 21st century learning skills (i.e. communication, collaboration, creativity, and critical thinking). According to the West Virginia Department of Education, “When engaged in standards-focused Project Based Learning (PBL), students are working in teams to experience and explore relevant, real-world problems, questions, issues and challenges; then creating presentations and products to share what they have learned.” There are many ways to implement PBL, but they all begin with problem situation. From that situation, the students analyze the situation provided and investigate possible solutions. They gather more information through research and assess their findings. Together, they develop a solution and prepare a presentation to share with the class.

PBL is not unique to mathematics education. In fact, PBL can be used in many content areas. Using PBL in mathematics education allows students to explore mathematical concepts in a more engaging way than working through a series of problems. Project-based learning provides a means of engaging students through social interaction, while providing a way to formatively assess the students’ understanding of the material. This is particularly important in mathematics, where students easily grasp the calculations, but struggle with the language used to explain the concepts and read the problems given to them. By increasing the amount of time students spend communicating and collaborating with their peers, they are simultaneously developing their critical thinking skills and their mathematical literacy.

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).

Instructional Strategies: Direct (or Explicit) Instruction

The direct instruction (or explicit instruction) model is based on scaffolded learning, gradually releasing the responsibility of learning from the teacher to the student, “Explicit instruction models support practice to mastery, the modeling of skills, and the development of skill and procedural knowledge” (California Department of Education, 2013, pp. 17-18).This model is particularly beneficial in teaching procedural knowledge to students regarding mathematical concepts. Typically, direct instruction begins with the teacher modeling how to perform a particular task. The teacher may utilize the think aloud strategy at this point to model how the students should reason through the task. According to Walqui (2006, p. 170), “When introducing a new task or working format, it is indispensable that the learners be able to see or hear what a developing product looks like.” Then, the teacher engages the students in guided practice, eliciting student participation in performing the next task. At first, the teacher may ask students if the steps taken to perform the task are correct. The teacher may continue with similar tasks or calculation that are progressively more difficult, eliciting greater student participating in performing the task. Following the guided practice, the teacher assigns the students an assignment or activity to complete individually. During this time, the teacher should circulate around the classroom observing student work, asking guiding questions, and offering help when needed. Throughout the lesson, the teacher checks for understanding through a variety of formative assessment techniques. These include asking the students questions, having them respond with whiteboards, or circulating and listening to their conversations as they engage in a think-pair-share. At the end of the lesson and throughout the unit, the teacher tests the students using summative assessments (i.e. quizzes, mid-chapter tests, and chapter tests).

The direct instruction model is not explicitly unique to teaching mathematics. In fact, direct instruction is easily implemented as a model for teaching reading comprehension (Gersten & Carnine, 1986) and science (Adelson, 2004). The focus of the direct instruction model is on the method or procedure used. It is a teacher-centered method that specifically addresses procedural knowledge. The benefit of direct instruction is that students acquire the skills and procedural knowledge for effectively performing certain tasks, “Teacher-centered methods of instruction are often necessary to educate students on difficult material that requires multiple steps, and for procedures which are unlikely for students to discover on their own” (Cohen, 2008, p. 4). Because the direct instruction model is teacher-centered, it does not foster the development of reasoning and metacognitive awareness that is necessary for students to think mathematically.

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.

Integrating Technology: Visualization using Aurasma


It seems like there is always some new app or some new device that has come out on the market. Staying current with and integrating all this new technology is a challenge, but it builds credibility among your students and it makes learning more engaging. For example, all my students knew how to use their devices to read QR Codes, but none of them had ever used Aurasma.

AurasmaAurasma is similar to a QR Code Reader, but it allows the viewer to see a different image or a video when scanning a particular picture. So, for my class, we created interactive walls where students prepared pictures and video clips to play when someone scanned particular pictures or words on the walls. I never heard about this app until I attended a recent conference that I attended on transitioning to the Common Core. I participated in a workshop specifically on integrating technology with mathematics and learned all about this app.