Video #4: Lesson on Variables

Purpose for the Activity

The purpose of these videos is for education students to be able to see how an experienced math teacher would go about teaching a math lesson in the fourth grade. There are other videos that one can access through the website and the goal of these videos is for students to see the variation in instructional strategies and content across the grade levels. The teacher in the film used many helpful strategies that are worth emulating. I quickly notices the progression of the lesson and ways she allowed the students to practice with the material and experiment and reason through problems as well as the very thought provoking questions that she asked the students.
The students in the film were working on learning variables, in compliance with the state standards and NCTM guidelines. In a group with her peers the teacher addressed the issue that the students were familiar with using letters and symbols to represent numbers but were struggling to understand the concept of variables. The teacher knew this information because she is very cognoscente of her students abilities and their prior knowledge, this is a key element of the lesson and frequently becomes apparent as you see her ask different questions of different students and the degree to which she probes students to elicit a desired response. For some students this lesson is very challenging and when they are on the right track she offers praise. For the more advanced students she requires that they explain their answer more fully and even worded her questions in such a way that it made the students explain the reasoning and steps in their thought process.
I think it was really interesting to see the degree in which the teacher was able to connect the mathematic content with other subject matters. Obviously, the lesson involved a lot of letters and words, as that was the basis of the variable machine. However, she was also able to incorporate vocabulary when she asked the students about their prior knowledge with the word ‘variable’. She connected their prior knowledge with their social studies project, studying the weather in Belize and Guyana. Both cities have very constant temperatures which she contrasted with the variable weather in their home town. The teacher also made connections between student’s names and made the class pet part of the lesson. I really commend the way that she made this material attainable for the students by using so many elements that they were comfortable with.



Questions:

#1 Describe the primary task in this lesson and identify the mathematical skills and concepts that this task is designed to develop.

This lesson was about variables. The students needed to learn the vocabulary word itself as well as understand what a variable is, how it can be altered and how altering the value of the variable can affect the outcome of the problem. In order to teach the lesson the teacher had the students make a “variable machine” which could be used to assign the twenty six letters in the alphabet different variables. Initially the teacher provided examples and then allowed the students to use their own names to practice the concept. By the end of the lesson the students were assigned the word “bear” and attempted to find the lowest and highest possible score for the word. The students were really required to use a lot of trial and error initially but as the lesson progressed they began to recognize the patterns that arose when working with the variable machine.


#2 Identify an example of when the teacher responds to students by offering clarification, by explaining, by questioning, and by letting the student struggle.

At one point in the lesson all of the groups but one had identified the lowest possible score for the word ‘bear.” Instead of allowing the group to move on or allowing them to struggle and potentially become frustrated, a problem that may be been exacerbated by the presence of the cameras and newcomers in the classroom, the teacher asked a group who had gotten the answer to give some strategies to the group that was having trouble. This was a way of asking the groups to explain how they reached their answer that also is part of her classroom climate, an environment where the students help each other.

#3 Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.

An aspect of teaching that I have really come to value is the debriefing part of a lesson. Many teachers overlook this essential component of teaching. The teacher wrapped up the lesson by drawing attention to the main goals she wanted to accomplish. She wanted the students to have a firm grasp of the word ‘variable,’ she wanted them to recognize that the value of a variable can change, she wanted them to understand that changing the value of the variable effect the outcome and she wanted them to realize the patterns that arose when working with the variable machine. The teacher accomplished these objectives by asking very strategic questions. She asked the students to explain what a variable was and asked them would happen if you changed the value of the letter in a word. As a teacher she was able to informally assess if they understood the content and she boosted the student’s confidence in the material. One of the things she mentioned in her conversation with her fellow teachers was that some of the students needed to feel successful with the material, they needed the positive feedback and sense of accomplishment, likely because they had previously had negative experience with math and their lack of confidence was affecting their ability to absorb new information.

Overall Impression

Overall, I feel that the video was a very useful tool. However, having just studied the equity principal I was very aware of some of the teaching and conversational phrases that the teacher used in her instruction. When she used phrases like “big ticket item” and “hike your score” I was struck by the face that many ELL students, as well as some English speaking students may not be aware of the meaning of these particular phrases. Furthermore, when she asked students to contribute their ideas she would indicate that she wanted them to answer by saying “speak.” Personally, this is something I would not do, I found it disrespectful. Also her primary mode of instruction was verbal. In an attempt to reach visual learners, I might have also written or drawn more of the examples on the white board. However, I do feel that the teacher from the films was a very experienced mathematics teacher and her strategies and skills are defiantly something I would use in the classroom.

Reasoning and sense making. Reasoning and Proof Process Standards

W. Gary Martin & Lisa Kasmer
Reasoning and sense making

What is reasoning and sense making? Essentially, it is learning. Learning is an active process; data is imputed into the mental machine, processed and filed away with all the other data, according to its type. Students learn how to make sense of new data, new material by an active reasoning process. It is the process of breaking large pieces of information down into manageable pieces, reassembling the information, assessing the intended computation and working through the problem to reach an answer. Students need to challenge the question, they need to challenge the answer, and they need to challenge each other. It is essential that students learn how to talk about math. It is essential that students learn how to defend their ideas. Students are naturally curious and endless worksheets do not foster that curiosity. Students who are taught in this way are given the foundation for advanced problem solving and creativity, skills that are important not only in high school and college but in the world, as they enter the job force. A teacher can create this type of atmosphere by making students feel comfortable. Students have to feel that they can make mistakes. The teachers must learn the right questions to ask.

Consider a specific example: the teacher asked the student to work through a word problem on subtraction. The students were permitted to use any subtraction process they liked. One of the students decomposed the numbers in the problem before subtracting. The teacher and students asked questions about how and why the student worked through the problem and she, the student, was very comfortable and eloquent in defending her choice. This is wonderful example of a classroom where mathematical reasoning is used and questions are encouraged.

Martin, W. G. and Kasmer, L. (2010). Reasoning and sense making. Teaching children
mathematics 16(5), 284-291.

Reasoning and Proof Process Standard

This article focuses on four key concepts:

All students, from the pre-kindergarten classroom through senior year of high school should be able to:

1. Understand proof and reasoning methods as a ways of understanding, explaining and comprehending mathmatics.
2. Develop mathematical conjectures based on the problem given
3. Develop,analyze and represent mathematical reasoning
4. Understand when and how to utilize the various reasoning methods


Students need to be able talk about mathematics. From an early age children are able to justify and explain what they see in the world around them. One of the first things a child will recognize is a pattern and they can quickly learn to anticipate the next element in a pattern and explain why it is the correct choice. As students get older and enter school this is generally called reasoning or proof. When most students think of "proofs" they think about geometry; however, proofs and justification are a part of daily life and an essential component of mathematics and true mathematical understanding. As students grow older their level of reasoning will become more refined and sophisticated; however, the skills are quite the same.

Every time a teacher asks a student to justify his or her answer the teacher is asking for the reasoning, asking that the student demonstrate the proof or line of though that led the student to the answer. This is a important aspect of mathematics that can often become overlooked in worksheets and endless problems.
The second point revolves around conjecture. Conjecture, or guessing, is a part of mathematics, it is a part of learning. Students have to feel that they are able to guess, make mistakes, be wrong and learn from the experiences. Mathematics is a stressful subject for many students, due in part because they fear being wrong and this fear can stop them for interacting with the material. As defined in the process standard, students learn best when they are able to work cooperatively with their peers and reason through their mistakes.
As students learn how to go through the process of reasoning they need to also learn how to represent their work. In the elementary grades it may be most appropriate to have students draw and color their work. In the older grades, students should practice showing their work with sketches, numbers and even in paragraph form, not only as a two columned proof.
Finally, students need to be able to recognize when and how to utilize various reasoning methods. For instance, the nuances of an algebraic equation may vary considerably from a geometric problem. Essentially, as they progress through their schooling students should start to recognize how best to go about a problem and various strategies that they can use.

Is mathematics a universal language? by Tim Whiteford

This article dives into the realities of teaching in a diverse setting. The challenge is more than just communication but rather truly understanding the student’s prior knowledge regarding mathematics. Some students enter the classroom with a strong mathematics foundation; however, there are significant differences in the way that math is taught in other countries. Teachers struggle with the conceptual differences in the way math is taught and understood. Yet, other students may have virtually no prior knowledge of math at all. There are times when students have essentially the same mathematics background but have learned different procedural ways to solve problems. Whitford offers a wonderful example, Molita from Bosnia struggles because she has been taught the equal addition method of subtraction rather than the traditional American decomposition method, taught in schools today. I can strongly connect with Molita as when I was a child I was home-schooled by my mother who taught me the equal addition method. When I attended fourth grade my math teacher publically called me out on my work and penalized me on assignments because I did not know the “correct” way to subtract. Looking back on the experience, I can recall how embarrassed and defeated I felt. I never want my students to feel that way and I want to make myself aware of the cultural differences and sensitivities that children may have. As teachers we must learn about the various procedural methods in order to make ourselves aware of the multitude of ways that mathematics can be taught. I think that one of the key ways that these differences will be understood is by talking about mathematics and engaging our students in conversation.
The next hurdle in teaching mathematics to a highly diverse student body is the difference in the way that people think about mathematics. Teachers fail to realize how important the way they speak is. Subtle variations in the way we word questions can have a profound impact on how a student perceives the question. Students who have limited English proficiency can be perceived on having very poor math skills simply because they lack the ability to communicate their ideas. Teachers must seek out the proper support, be it an interpreter or an aid, in order to understand the students meaning. Teachers must also be sensitive to the challenges that the English language pose to an ELL student. Even the number system can be challenging to a student who isn’t fluent in English. Students may also struggle to understand the units of measurement, which vary considerably from culture to culture.


In conclusion, teachers have to invest the time to understand a student’s background, their prior knowledge and their comfort level with the English language, the thinking process, the units of measurement and the method of instruction. The bottom line is that teachers have to believe that all students can learn mathematics. They have to understand what support a student will require in order to understand the material. Furthermore, teachers should embrace the opportunity to learn about the cultural variation in how mathematics is taught and understood.


Whiteford, T. (2010). Is mathematics a universal language? Teaching children mathematics 16(5), 276-283.

The Equity Principal

The core idea of the equity principal of mathematics is that all students are entitled to mathematic education which is challenging, meaningful and purposeful. Teachers do not have the right to choose which students will be successful and unsuccessful in mathematics; rather, it is their responsibility to make every effort to see that each and every child is able to be taught and able to learn mathematics. Sometimes this means that special accommodations must be made to reach the specific needs of the child. One of the key dispositions that all teachers must have is high expectations for all their students, free of discrimination of all types. Technology is a tool that can be used to reach students who would otherwise struggle. New technologies can help overcome language barriers and assist students with learning disabilities. Furthermore, in the teaching profession, equity requires the support of the school administration. Teachers must be supplied with the technology, funds, time and personal support to make this concept a reality.

One of the notes that really resonated with me was the final comment that the author made to the teachers personal biases. It is so disappointing that in this day and age we still must confront prejudices and preconceived notions about a child’s ability to be successful in education based on color, sex or economic status (among many others). This is a challenge that as educators we must confront daily, to fight for the rights of children.