Mathematical Reasoning in the Classroom

Students today seem to only understand concepts when it is beneficial for them. They only learn the concepts for a short period of time, and seem to have a tough time retaining information. As future teachers, it is important for us to teach mathematical reasoning to help students recall information needed in future classes. We need to help create habits for our students to help relate concepts together.

We need to create situations in which the students are allowed to work through the problems. A big example I was able to experience this last week, was the complete the square lesson. The professor told us that we would be using different algebra tiles to relearn concepts we learned in high school. He began it by giving us an explanation of how to use the tiles. He then gave us a problems and let us, in partners, work how to solve them. He didn’t interject much, which allowed us students to recall past concepts he explained to us as well as our own ideas to figure it out.

As future teachers, it is important for us to bring this type of learning into our everyday classrooms. I think lectures and notes have their place in a classroom, but the biggest way students can learn is by reasoning and coming to the solution by working through the problem and critically thinking about past concepts.

According to Strategies for Mathematical Reasoning, through reasoning the student is able to evaluate a problem, understand a problem, and create a solution. This handbook provides a lot of different ideas to help create more situations for students to mathematically reason. The biggest one that stood out to me was the idea of using manipulatives in the classroom. This concept means that the teacher provides objects from the real world that the students are able to play with and manipulate to understand a concept. While learning graphing, it means seeing how to manipulate the graph by changing one variable or number. In geometry, it means bringing in objects representing each shape/theorem so the students have a visual representation of what they are learning. In the lesson explained above, the professor knew that using the manipulative of algebra tiles gave us a chance to manipulate and change how to get different equations.

They say in the handbook that “there is a proverb that states, ‘What I hear, I forget; what I see, I remember; what I do, I understand.’” I think this is a great way to prepare for lessons as future teachers. We need to provide more chances for the students to learn by using objects to understand. Instead of being the teacher that talks at the students, we need to be the teachers that allow the students to be doing the learning and teaching, while we facilitate and point them in the write direction.

Overall reasoning is necessary in the mathematics classroom. Students don’t just need to remember the different concepts. They need to learn how to understand and retain information. Through manipulatives the students will be able to mathematically reason and continue to grow in their math skills.

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Reasoning in the Classroom

When a student is asked to talk about a math lessons they usually comment about the teacher showing slides on the smartboards of important formulas while the they take notes. After they memorize the formulas students usually go through some examples with their teacher and then finally they are assigned some homework problems to take home and practice for the night. It is never a hands on lecture that gets students engaged with the lesson.

It is important for students to “reason” while learning new material. According to Focus in High School Mathematics Reasoning and Sense Makingreasoning is defined as the process of drawing conclusions on basis of evidence or stated assumptions. In other words, reasoning is coming up with solutions based off of prior knowledge. Students need to struggle through problems using their prior knowledge to come up with their own formulas and solutions. This way the new information sticks better rather than just being told a formula or steps to solve a problem and then memorizing it. Now the tricky part is to implement this in the classroom, especially in a middle school or high school math classroom. 

First and for most you need to make the math classroom a positive place to be. You can do this by the ways you answer student’s questions. There is a huge difference between saying “I just explained this, how don’t you understand” and “This topic is challenging. You just have not caught on to it yet.” You can also make your classroom a positive place by being excited to teach math and making it very known that you love teaching mathematics. 

Katy Farber came up with 4 ways to make math more relevant. Number one talks about important collaboration is and how all student working together helps with reasoning. One of the best ways to learn something is from your peers. It is key to have students bounce ideas off of each other and sometimes even argue about the ‘right’ answer. Teachers can easily have their students collaborate by saying “turn to your shoulder partner and talk about how you would start to solve this problem”. During this time, it is critical that the teacher is actively listening to the conversations going on and making sure students are working together and staying on topic. Other lessons teachers can plan for the kids in the class to work in groups the whole period. Teachers may give their students a problem that they know can be solved by using prior knowledge and lead them down a path where they can struggle through and eventfully find solutions with their group. 

Another way to make math relevant is to implement more project-based learning. I believe it is important to bring in real world scenarios into the classroom. Students are typically more engaged when it comes to math if they understand how they will use the new information. This could be another simple task by just starting that finding the area of an object could be helpful in the real world because some day they might need to figure out how much carpet their living room needs. 

Overall, it needs to be more known that students learn material better when they put the reasoning behind what is being done to solve problems. I have only touched on a few ways math teachers can bring that into their classroom, there are many more ways! It is obvious you cannot make every lesson the most enjoyable and engaging lesson ever, but the more the merrier. 

Mathematical Stories

When reading a book or a story, connecting ideas and the sequence of the story is very important. Otherwise, when you’re reading a story how can someone understand what is happening or why something is happening? A story has to be coherent for it to be understood. In the article found on the NCTM website, “Generating Student Interest With Mathematical Stories,” by Leslie Dietiker, the author explores how “mathematical stories” could better understand math concepts like stories. The author argues that stories help the student connect ideas from easy concepts to more difficult ones and help them in understanding the “why?” of a problem or concept. Otherwise, students may feel that problems are disconnected or no relation to the other. The article gives the example of these three problems:

  1. Factor x^2-6x+3and solve to find the roots of y=x^2-6x+3.
  2. Josh says that x^2-6x+3is equivalent to (x-3)^2-6. Do you agree? If so, how can you use this expression to find the roots of y=x^2-6x+3?
  3. Use your graphing calculator to display the graph of y=x^2-6x+3. Estimate the roots.

On first glance, these problems seem unrelated, unconnected to the other. However, if we do the problem in the order of 3-2-1, a “story” is made. First, the students graph the parabola and can see approximately where the roots are. Then, part 2 gives students another hint in finding the roots leading to part 1 where they end the “story” and solve for the roots.

The “story” could also be written in the order of 1-3-2. Students may not be able to factor part 1 on their own and may be unsure if there are any roots at all. Then, in part 3, students do get a see that the parabola does, in fact, have real roots. Finally, part 2 gives students a strategy to solve the problem, ending the story. These “stories” can help students see the connection between ideas and different problems along with providing surprise and excitement when it comes to solving problems.

I think this can be taken a step further. If you ask a math student what his least favorite kind of problem is, most of them would probably say word problems. Using “mathematical stories,” word problems can be made less intimidating for kids. By connecting the ideas from one problem to the next, students will be able to easily connect concepts and ideas. Mathematical stories can also be easily connected with real-world applications to answer the question of “when am I going to need this?” By implementing stories into a classroom, students can have a better coherence of old ideas into new ones. Not only that, learning math can be more exciting with twists, surprises, and suspense.

Reasoning in the Math Classroom

Reasoning and sense-making play an essential role in building a solid understanding of mathematics. In looking more into how reasoning can be implemented in the classroom, I came across a blog post by Elizabeth Masalsky, a middle school math teacher, titled “Bringing Discussion and Justification into the Math Classroom.” In her post, Elizabeth talks about how having students discuss their ideas and strategies can help students deepen their understanding and help develop their skills in creating an argument. One of the practices mentioned in the article was described using the phrase “convince yourself, convince a friend, convince a skeptic.” This method lets students establish their own ideas along with justification behind their ideas. Then students explain their reasoning to other students. What I liked about this practice was the inclusion of “convince a skeptic.” This allows students who are struggling to take more time understanding what is being explained without being judged. I also appreciate how having to provide their justification to others gives students an opportunity to see experience how other people learn and think.

Students are used to coming into the math classroom, memorizing a process, and replicating it, often without understanding why they’re doing it. It’s important that teachers know how to encourage their students to bring their reasoning skills into the classroom. One of the easiest ways that teachers can do this is by moving away from yes or no questions. Instead, present questions that prompt students to think more about the process behind their thinking. Some examples would be “Why do you think that is?” or “Walk me through your thinking.” Simple questions and prompts like these allow students to say what they know without necessarily having to know how to do the whole problem. Developing this kind of thinking is important because it allows students to gain more understanding of concepts and will eventually help students understand how to provide justification when they begin to learn how to prove mathematical concepts.

I think that the practice of “convince yourself, convince a friend, convince a skeptic” would be helpful for a review day. I would split the room in half and give each side of the room a different problem with concepts that they’ve already been introduced to. Students would work on the problem individually for a few minutes and then would have a few minutes to collaborate with someone who had the same problem as them. Then, I would pair students up so that each pairing has a student from each group. Students would then “teach” their problem to the other student. This idea lets students develop a deeper understanding of the concepts because, instead of just understanding how to do the problem, they have to understand the process behind completing the problem and how to explain their reasoning to others.

Link to the blog post:

https://www.matific.com/us/en-us/blog/2015/07/30/bringing-discussion-and-justification-into-the-math-classroom/

Equity in the Classroom

Equity in the classroom can be defined as emphasizing what students need in the classroom. In other words, this refers to the principle that all students are offered equal educational opportunities that encompasses all students individual learning styles. Earlier this week, I was tasked with the assignment to dissect the case study at Southwest High School and whether or not equity is being attained. With that being said, at Southwest High School students who do not pass math in 8th grade must start by taking Pre-Algebra. This course uses curriculum that is mostly computational review in order to prepare students for the rigorous Algebra course. Students must score an 80% on the skill assessment in order to enter the Algebra course.

After doing some research, it is evident to me that an equity issue is present in this case study. First and foremost, whether or not the math in 8th grade was taught in a way that encompasses individual learning styles and highlights areas in which the students need improvement. The method implemented at Southwest High School set their students up to fail before they even have a chance to succeed. The equity issue presented in Southwest High School is evident in the schools description of the Pre-Algebra class: “This course uses curriculum that is mostly computational review in order to prepare students for the rigorous Algebra course”. This description is insensitive to students who struggle with 8th grade math. Set aside from students finding difficulty in the material, the description makes it seem as if the students are expected to find the computational review easy. In addition, students that find 8th grade math and the computational review difficult, may be inflicted with anxiety and become disheartened when the description highlights the following Algebra course will be rigorous. Consequently, it almost sounds as if the students are expected to find the course difficult before they have even taken it. It is important that students are aware of the benchmark that needs to be met in order to continue with their education, but when Southwest High School states that the students need to meet the requirement of 80% in order to advance onto the Algebra course might create unnecessary pressure before they have even taken the test.

Southwest High Schools lack of equity reminds me of a time when I was in high school and my Algebra teacher didn’t enact equity in the classroom and personally hindered my advancement in math. Prior to that class, math is a subject that I have always found interesting but found myself struggling to understand the material immediately. Nevertheless, I was still getting good grades but had to put in the extra work. With that being said, I was always asking my teacher questions to improve my understanding. Consequently, one day during worktime my teacher asked me to meet her outside of the classroom. It was to my surprise that she suggested that I go into a different math class for students’ who struggle with math called ‘Math Star’ because I asked ‘too much questions’ and she ‘didn’t know how to answer my questions’. Reflecting on that experience, I know that her heart was in the right place, but it doesn’t take away the emotions that I was inflicted with at that moment in time. Set aside from the fact that I felt like a burden, this experience created a new found distrust in my teachers going forward. As a result, instead of asking questions when I was confused, I elected to stay quiet and try and figure it out myself. Consequently, my grade started to drop.

As a future educator, I know that I will be faced with different challenges similar to this experience; however, one thing that I will always strive for is being there for all my students and creating an equitable environment that promotes questions and learning at a comfortable level. With that being said, I decided to research strategies that endorses equity in the classroom. This led me to the article on edutopia by Shane Safir, 6 Steps Towards Equity, this article
is a very helpful tool and sheds light to the fact that equity is hard to embrace in the
classroom; however, it can be achieved in time through time. The points that I
will be highlighting are methods I will conduct in my future classroom.

1.     Know Every Student – The more a teacher knows about their students, the more they can build trust and differentiate instruction in a way that is tailored to individual strengths and struggles. Hence, promoting trust and comfortableness.

2.     Become a warm demander – Teachers need to convince students of their potential and brilliance. With that being said, I will carry out this in my classroom by holding students to high expectations. This will instill confidence in students and their capability.

3.     Practice lean-in assessment – No standardized test will provide teachers with quality data on students understanding of the material. Lean- in assessment will help diagnose students’ learning needs. With that being said, I will implement this in the classroom by carrying a clipboard around while students are working, and take careful notes on what I observe. Additionally, students should be assessed by the teachers in whether or not the teacher thinks the student is capable and ready to move onto the next subject. Standardized and benchmark testing can be overwhelming and set students up to fail before they have even taken the test. For example, students may be so stressed out during the test that their thinking and logic may be hindered, when in actuality they are intelligent on the subject.

4.     Flex your routines – More often than not, curveballs can happen. Teachers need to be willing to set aside well-laid plans in individualize instruction. I will enact this in my classroom by never allowing my students to see my discomfort in the classroom. My confidence in the situation will ultimately rub off onto my students.

5.     Make it safe to fail – In an equitable classroom, there is no need to struggle and failure. In actuality, struggle and failure should be normalized in the classroom and even celebrated at times. This will promoted in my future classroom by having my students meet in groups once a week and share something they struggled with and what they learned in the process. This strategy can help students understand and discover the subject in a new sense.

Complex Instruction

When I was reading the book Strength in Numbers, I came across a phrase I had never heard of. Our assignment was to read chapters 2 and 3, but I was confused by the layout and accidentally read chapter 1 as well. However, I am very glad I made this mistake. While reading chapter 1, the author mentioned “complex instruction”. Upon first read, he tried his best to explain a little about what complex instruction is. I’m sure he also figured readers would be able to infer for themselves what complex instruction really means. I mean it should not be too difficult right? It just means classroom instruction that is more complex, right? Well, that definition is insanely vague, so I knew I needed to do more research on the topic.

I came across a research journal titled “How Complex Instruction led to High and Equitable Achievement” by Jo Boaler from The University of Sussex that helped to break down complex instruction into a couple different categories. The first is multidimensionality, which is described as an approach with multiple ability treatment. This allows for more success for each student to talk different sets of abilities to avoid the usual system where some students rise to the top of the class, while others quickly sink to the bottom. The second is roles. Under complex instruction, Jo claims that assigning roles in group work, such as a facilitator and other roles, helps the group hold each other accountable. However, I was trying to think a little beyond this claim by Jo and compare it to something Kevin said in class about how giving everyone a specific role does not necessarily help student success. Rather have the students work together while making sure everyone is contributing. I am going to transition to the fourth component as it relates more to the second, and that is teaching students to be responsible for each other’s learning. Jo said, “group work brings with it an element of student responsibility for others, but the teachers went beyond this to ensure that students took their responsibility to each other very seriously.” The way the teachers would grade group work would help the student realize how important it is to hold each other accountable. A teacher would possibly grade the conversation by the group and not necessarily, how much each student spoke, but specifically the language used in making sure each member is understanding, contributing, and succeeding. The third component is assigning competence. Assigning competence is the job of the teacher to make sure each student feels valued with the ideas and thinking they brought forth to the group. If a student who was being dominated in group discussion made a solid mathematical connection but was inevitable overlooked, the teacher would chime in to praise the student for their thinking and make sure their thoughts and points are clearly heard by the rest of the group.

I have already bookmarked this article to easily find to use in my future classroom. There are solid examples of how to use complex instruction, but it is also left open for any teacher reading to be able to modify the examples to better work for each teacher. For instance, I would find a way of modifying the roles portion to make sure students still feel like equals during group work.

https://nrich.maths.org/content/id/7011/nrich%20paper.pdf

“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:

https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/%E2%80%9CThis-is-easy%E2%80%9D_-The-little-phrase-that-causes-big-problems/