“This is Easy”: How Simple Language Can Discourage Students

During our class discussion this week about equity and access, I was particularly interested in the idea of setting specified group norms for the classroom. In looking more into creating norms in the classroom, I found an article from NCTM’s journal Teaching Children Mathematics titled “‘This is easy’: The little phrase that causes big problems.” This article discusses how the comment “this is easy” from a student in the classroom immediately discourages other students who may have been struggling with the problem. To combat this, the teacher and students discussed what the students meant when they said the phrase. The two most common reasons were that students either saw the problem as something familiar they had already encountered or that they had an idea of how to solve the problem. The teacher encouraged students to use more precise language to explain more about how they viewed the problems. After this change, students began to see how everyone has varying abilities and that different tasks may be more difficult for other people. Students also began to acknowledge how saying the phrase “this is easy” affected others and began to encourage other students who were struggling.

While the situation described in the article takes place in a second-grade classroom, these ideas can still be applied to secondary classrooms. As we discussed in class, it is important for students of all ages to all have an opportunity to attempt a problem. Similar to the “no hands, just minds” technique, ridding classrooms of the phrase “this is easy” creates an environment where students don’t feel as much pressure from their other classmates or that they can just sit back and rely on the “smart” students to get the answer first. I believe that this idea would work well paired with collaborative learning. After a discussion with students about how easiness is relative for everyone, they will be more likely to help other students in their group who are struggling with the problem.  

I was surprised by this article and the ideas that were discussed. It seems like such a simple and obvious idea, but it had never occurred to me how damaging this phrase can be. In my mind, when math students are claiming that something is easy, it is most likely because they’re excited to understand a new concept or problem and they don’t necessarily intend the phrase to be harmful. Regardless, it is important to discuss with students how the language they use can discourage other students. Additionally, it’s important that teachers be aware of what they deem “easy” in front of their students. When I first began tutoring, I found myself saying “Oh! This is easy” to students when they asked a question, when instead, what I really meant was “I understand why you’re stuck” or “I understand what the question is asking and know how to help you.” Now looking back, I see how that was disheartening to the students I was helping. From now on, I’ll focus on getting both myself and students to use more specific language to express what their opinions are on approaching a problem.

Link to the article referenced:



Classroom Layouts to Inspire Cognitive Demand

This week we continued to discuss the same portion of Principles to Actions that we read last week which focused on the varying levels of cognitive demand in the classroom, as well as a little bit of reasoning and sense making. A few of the specific types of problems that we have discussed in class that have a higher cognitive demand for students are Shift Problems, and Three-Act Problems. Shift problems are aimed at creating a sequence of tasks from one specific task in a textbook, replacing many of the traditional homework problems in a textbook with these sequences. Three-Act problems are the brainchild of Dan Meyer, and are mathematical “stories” in which the teacher introduces the conflict of the story within the First Act, the protagonist overcomes obstacles in the Second Act, and the conflict is resolved and a sequel/extension is set up in the Third Act. In Meyer’s explanation of the Three-Act problems, he discusses that historically, teachers have seen their role in the Second Act. However, with the tide of mathematical classroom layout changing to flipped classrooms, and tools such as Khan Academy becoming more and more useful, the mathematics teacher’s role is changing too. It is seeping into the First and Third Acts as well. These two types of problems are new ways of looking at mathematical tasks. Similar to the ticket annulus problem that was performed in class, these types of problems provoke a higher cognitive demand from students– they have to have a clear understanding of the mathematics behind the problem in order to accurately develop a solution. It asks them to make connections between previous topics, use deductive reasoning, and more as they strive to find a solution to a problem.

While reading, particularly about the flipped classroom in Dan Meyer’s blog, I began to think about my own experience in both high school and college. Though I never had a flipped classroom, I have thought about both the advantages and disadvantages of a flipped classroom. The collaboration that allows students to problem solve and find solutions to any given problem is harder to find in a flipped classroom. Furthermore, students may not get as deep of an understanding of the material as if they were in an inquiry-based classroom. However, talking to one particular teacher that does have a flipped classroom, there are advantages to flipping when compared to a traditional classroom. This teacher asks students to watch the videos while they are in class. They then have an opportunity to go more at their own pace, and ask questions as needed. I believe the distinction of the advantages of a flipped classroom does have to made in that is advantageous compared to the traditional classroom, but perhaps not when compared to the classroom that we are striving to create that inspires reasoning and sense making, with tasks that have high cognitive demand. In the future, I would hope to have a blend of sorts where some tasks are embedded into technology that allows students to first work through a problem by themselves.

Although not a Three-Act Problem, I was able to find a problem from NCTM called “Thunder and Lightning” based on how two people in different locations hear the sound of thunder at the same time. In this problem, students combine their knowledge of science and mathematics to find both where the lightning struck, where the people that heard the strikes were located, and to find where three different people would be. To do so, the students use the Geometer’s Sketchpad, and properties of triangles and circles. The activity sets the stage for looking at equidistance, prompting students to discover properties of the perpendicular bisector. There is a worksheet associated with the activity, but this can easily be changed so that students are working collaboratively. A teacher could distribute different scenarios to each small group of students, and ask that students present their findings for their particular solution to the rest of the class at the end of the period. There are various modifications that a teacher could make to this activity as they see fit to change the layout from a worksheet to a more collaborative activity.

Cognitively Demanding Tasks

This past week, we read about the different levels of cognitive demand that can be implemented in a mathematics classroom. Typically, in a traditional mathematics classroom it appears that there is a low level of cognitive demand. For example, the ticket and annulus problem that we did in class was a higher level cognitive demand activity but when we were given an opportunity to look at the original questions presented in the textbook about the problem, the questions led the students to the answer rather than asking them to problem solve their way through the question– wondering what information they would need to know in order to answer the question. Although the strategy of leading students to the answer is psychologically sound as it can be seen as scaffolding, especially if students are still in the zone of proximal development where they would need help to solve the problem, it does not contribute to developing essential problem solving skills in students.

As discussed in Principles to Actions, there are four different levels of cognitive demand for students:

  1. Lower-level demands that include memorization
  2. Lower-level demands that contain procedures without connections
  3. Higher-level demands that contain procedures with connections
  4. Higher-level demands that include doing mathematics

Many of our lecture-based mathematics classrooms only, at the most, employ higher-level cognitive demands where the students are executing procedures while making connections between them. However, even more common are students who simply memorize the formulas but later forget their purpose and do not truly understand them, such as making the connection between proportions and the area/arclength of sectors of circles. This reading, and discussions in class prompted me to begin to think about how frequently teachers employ tasks that are cognitively demanding at a higher-level where students have a full understanding of the mathematics that they are doing.

From this, I was able to find a dissertation entitled “Teacher challenges in implementing cognitively demanding tasks in the mathematics and science classrooms” (Monarrez). This dissertation studied the professional development opportunities available to teachers as they attempt to implement cognitively demanding tasks in the mathematics and science classrooms. As a basis for the study, Principles and Standards for School Mathematics (NCTM, 2000) was cited, as well as the TIMSS Video Study that was discussed in class, where a majority of United States mathematics teachers only required students to regurgitate formulas rather than engaging them in cognitively demanding tasks. In discussing the challenges that teachers face, the study found that many teachers cited the students, content knowledge, and external factors as their main challenges when creating cognitively demanding tasks rather than the mathematical task itself (Monarrez, p. 128). Previously, cognitive demand had been discussed as part of the mathematical task, which is why it is significant that teachers discussed other outside factors as roleplayers in the challenges in construction of the tasks themselves (Monarrez, p. 128). This information gave me an opportunity to reflect on what I will need to consider as a future teacher when creating/implementing cognitively demanding tasks for my students. In any given classroom, students of various levels of content knowledge will be given the same task. This is where differentiation of mathematical tasks can be employed, and a vital tactic in ensuring that students are able to get the most out of the tasks assigned to them. Using the information given in the dissertation, I believe that pre-testing students would be useful, and tracking their performance and levels of understanding in previous tasks when creating lessons. This also displays how imperative it is that a teacher gets to know his/her students so that they are able to be aware of how to best help certain students. In a kindergarten classroom where I volunteer, the teacher frequently takes notes over what each student is struggling on (i.e. sounds of letters) so that she can better help them in the future. Practices such as this would be helpful in combating the challenges that arise when planning and implementing truly cognitively demanding tasks for students. In the future, I can use the information found both in Principles to Actions, and the dissertation to better plan cognitively demanding tasks, and to remember the importance of them in the mathematics classroom.
Monarrez, A. M. (2017). Teacher challenges in implementing cognitively demanding tasks in the mathematics and science classrooms (Order No. 10278789). Available from ProQuest Dissertations & Theses Global. (1924679389). Retrieved from http://excelsior.sdstate.edu/login?url=https://search-proquest-com.excelsior.sdstate.edu/docview/1924679389?accountid=28594

Using Learner-Generated Examples to Increase Understanding

Learner-Generated Example (LGE): Examples of concepts that are created by students

In the majority of mathematics classrooms, teachers use what is called lower-order thinking to teach their students. Some verbs that are in Bloom’s Taxonomy, which shows what’s higher- or lower-order thinking, that are a part of lower-order thinking are words like calculate, solve, apply, complete, produce, and manipulate. These are all words that teachers use after teaching a lesson and giving students a large amount of problems and tell them to do finish them. This approach is fine. Practice does help. However, if we want our students to have a deeper understanding through the use of higher-order thinking, we should use activities that use verbs like create, compose, invent, and formulate. One way to promote this higher-order thinking is by having students create their own examples and interpretations of concepts, called Learner-Generated Examples (LGE).

I learned about Learner-Generated Examples from an article in NCTM’s Mathematics Teacher titled “Learning about Functions Through Learner-Generated Examples” by Martha O. Dinkelman Laurie O. Cavey. The article describes how they used a multiple-step process of formal formative assessment using LGE. They started by teaching about functions. After this step, they had students list everything they knew about functions. They worked up to having students give an example of a function in four different formats. They later came back to this method after seeing what they needed to teach and teaching it to the students. This is a great use of LGE, because students may not be able to put into words what they know about functions, but having them give examples of functions shows what their preference is. The teachers learned that while students could have written out the equation, drawn graphs, or written what the function did in words, students usually just wrote a series of numbers for x-values and y-values. This showed that most students thought of functions and x- and y-values, which meant that the teachers had to work on helping them understand all the forms of functions.

In our 7-12 Math Methods class and our Curriculum and Instruction for Middle/Secondary School class we’ve learned about the value of having multiple correct answers. If there’s different correct answers, that means that there’s multiple ways to come up with those answers, which provides an opportunity for students to struggle and collaborate. There is certainly multiple correct answers when students have to come up with their own examples. For an example of an example, if students have to give examples of how a difference in slope in a linear equation changes the appearance of the line on a graph, there’s an infinite number of potential answers. They also have to identify what number in an equation symbolizes the slope and make the connection between the equation and the graph.

Another useful part of LGE is that it can be used for many different concepts. Students can create their own graphs. They can create their own equations. They can create tables or dimensions of objects or shapes. They can also use technology to create these examples. Students could draw a random line and try to figure out the equation of it. This is a great way to use Desmos. Students can experiment with entering equations and seeing what shapes they can create.

The way that I would use this in my classroom is as a formative assessment to check and see what students have learned and what I need to work on. I like the idea that the article had about asking students to provide multiple examples for a single function. This can show what they’ve learned and what they need to work on. It can help me as a teacher to figure out what my weaknesses are in a lesson. Most importantly, I think it deepens students understanding of a concept by having them create an example that fits the main idea.

Link to NCTM article:

Learning About Functions Through Learner-Generated Examples

Coherence in the Mathematics Classroom

This past week has seen a shift in the focus of our class from incorporating reasoning and sense making into the classroom to learning about coherence in mathematics. Coherence is defined as artfully piecing together segments, creating tensions and dilemmas, building toward a conclusion, and creating links day-to-day in the mathematics classroom. If implemented deliberately in a classroom, students should leave the classroom with a deeper understanding of why within mathematics. In turn, this would aid them in their pursuit of mathematical understanding. An article I found was from NCTM called “Curriculum Vision and Coherence: Adapting Curriculum to Focus on Authentic Mathematics” (link: http://www.nctm.org/Publications/mathematics-teacher/2009/Vol103/Issue1/Contemporary-Curriculum-Issues_-Curriculum-Vision-and-Coherence_-Adapting-Curriculum-to-Focus-on-Authentic-Mathematics/) that discussed how in a standards-based curriculum, the textbook that a teacher is supposed to follow may not connect the topics day-to-day so students are not able to see the flow of the mathematics. This prompted me to reflect on the activities that we completed in class this week, my own experience in high school mathematics, and what I hope to accomplish for my future students in my own classroom. The connection that was made between proportions, arclength of a circle, and area of a sector of a circle was new to me. While it makes sense, teachers had never before attempted to show that to me so that I would get a deeper understanding of the material. Similar to the geometry textbook that is mentioned in the Mathematics Teacher article mentioned above, I had been taught in a way that required me to memorize formulas but had not made connections, or attempted true coherence in my high school career. The study done concerning making connections in the mathematics classroom that was discussed during our class time reveals that my personal experience was not unique in the United States. In this way, I hope to change that trend for my future students. Whether it is simply showing a simple geometric proof to help them understand a geometric theorem, incorporating cross-disciplinary mathematics to show how the proportion relates to arclength and area of sectors, or giving students a visual demonstration of the mean, there is so much that can be done as a teacher to make coherence a focus in the mathematics classroom.

The teacher in the Mathematics Teacher article began to have a “vision for [his] curriculum” in which he began to change how the theorems were taught. While they were listed in the textbook, he started to make activities where the students would explore the theorems and make their own conjectures so that it was not “random text in the green box” for them. Although teachers may see textbooks as a constraint, the teacher (Matt) took advantage of the layout of the textbook while changing the way that he presented the information to his students. This would take time and planning on behalf of the teacher, but it is an efficient way to make connections within the theorems of the mathematics classroom. The same idea can be applied to the standards that a curriculum is built around. For example, my chosen portfolio item for the Reasoning and Sense Making portion of our class follows a Common Core standard about proving the Pythagorean Theorem. However, the standard states that the Pythagorean Theorem should be proven using similar triangles, but the proof is approached in a different way. Although it does not then fully follow the standard, it draws connections between scaling, different theorems about triangles, and properties of kites and parallelograms. It asks students to make connections between what they have already learned– coherence. Changing the presentation of the Common Core standard is another way that teachers can begin to create coherence in the classroom. Creating lessons that make the connections that textbooks do not is the primary way in which teachers can begin to create coherence in their mathematics classrooms. Thus, I began to search for activities that promote coherence in the classroom.

When searching for coherent mathematics lessons I found Activities with Rigor and Coherence (ARCs) from NCTM. These are activities that NCTM has chosen, similar to the Reasoning and Sense Making Activities, that they believe evoke rigor and coherence in the math classroom. While many of this short list of activities are aimed at elementary to middle school, there is one called Barbie Bungees Again (link: http://www.nctm.org/Classroom-Resources/ARCs/Barbie-Bungees-Again/). Although I never took AP Statistics in high school, I know that the teacher did this activity each year with her AP Statistics class, though on a higher level. This activity focuses on asking students to collect and analyze data to predict the longest bungee cord that Barbie can safely use. It is a three day lesson that first introduces the activity to students where they collect data, then find the line of best fit for the data that they found, and finally make a prediction about the longest length of the bungee cord that Barbie can safely use. The Common Core standards that it follows are listed beside the activity, and it is clear that students will be making connections between the collection of data and the basic algebra that they have learned. This lesson asks students to get involved with the collection of data, put it together, and evaluate it rather than simply evaluating data. In this way, Barbie Bungees Again would be an example of a mathematics lesson that promotes coherence in the mathematics classroom.

Math Tiles: The Greatest Thing I Learned this Week

Imagine learning how to do a mathematical concept that never made any sense and  was completely memorized by you, and then FINALLY LEARNING WHY AND HOW TO GET IT! This has been the highlight of my week: learning how to complete beyond just the concept and why it is called completing the square. I leaned it through a magical tool called, math tiles.

The math tiles were tiles of three different sizes: a large square, a small square, and a rectangle with the length of the large square and the width of the small square. The large square represented x^2, the rectangle represented x, and the small square represented 1. The tiles could then be combined to represent factored and unfactored forms of equations like this:Image result for math tiles

This tile arrangement shows (-x+2)(-x+3) = x^2-5x+6. I am only assuming this equations by taking red as negatives and tan as positive.

How this method involved completing the square was we physically had to figure out how to complete the square with tiles. Right now the image above is a rectangle, but to turn it into a square we could add a 1 tile to the left column and then the square would be complete as (-x+3)^2 = x^2-6x+9. Learning how to do this finally made me realize why we consider it completing the square and helps me to visualize and remember that the formula for it is (b/2)^2.

This method is so amazing to me, because I would have never thought to represent equations this way. Which is exactly the point of the tiles, they are supposed to represent equations and completing the square in a different way for students who may not understand the “formula” for doing it. Having this in my toolbox, I am now more equipped to show students a different way to distribute and factor equations and completing the square.

I definitely want to use this in my classroom someday, because I think that it helped create a deeper understanding of completing the square. One way I would modify it is before I give students the tiles I would assign them to think of a way to visualize represent it and see what they could come up with. Then later give them the tiles, and see how close they were of if they came up with something similar.

Learning an in-depth way to visualize and realize a pretty basic algebraic concept made me wonder if there were other ways to use the algebra tiles to solve for math problems. Upon my search of the internet I came across this lesson on NCTM that has an online interactive apparatus that has you use tiles to solve for different equations. Link:  https://illuminations.nctm.org/activity.aspx?id=3482

It is an online way of solving similar problems to what we did and class, and also has other applications. It allows a way for students to save and print their answers, that way teachers can check the students work. I really like the online aspect, because many schools now are trying to find ways to integrate technology so it is a free lesson that requires less storage and cost of the tiles. I think that this is a really good alternative to having the tiles so that students could also have access to it outside of class.