If you were to look at this figure, you would eventually come to the conclusion that the tacos (T) are $1 and the drinks (D) are $1.50. But in an article about mathematical reading and literacy, Gregory Beaudine shared a story about a different solution. One student offered the idea that if you bought three tacos, you would get a free drink. So looking at the first example, the tacos would be $1.50 and the two drinks would be free. Looking at the second example, you would have four tacos at $1.50 ($6) along with a free drink and the other drink would be $1. He used this example to say that books want what they want, meaning the book wanted the answer that I gave in the beginning. Students may read a question differently, therefore coming up with a different answer.
Beaudine was then curious about mathematical reading and so he looked for talks that involved that as a main point when attending an NCTM conference. In attending these talks, he asked these questions: “Why not teach students to read mathematics textbooks?” and “What would reading instruction look like in mathematics?” They came up with the solution that since reading informational text is a part of many new state standards, they could use math for this. The consensus was that they could teach students mathematical reading and fall in line with the standards.
When I tell people that I’m going to be a math teacher, the general reaction is something along the lines of: “Gross. I hate math.” Math is arguably the most disliked subject in high school. People may dislike writing papers, but they can write about things they like. People might dislike science, but they at least might enjoy doing experiments and such. Students don’t see the reason to like math if we don’t properly motivate them. I found an article recently called 21 Simple Ideas to Improve Student Motivation that has some good ideas, in my opinion.
One general idea that I like is giving students some control. This could be through letting them in on decisions like what type of assignments they do or it could be giving them responsibilities or positions in the classroom. Another idea I like is to create variation in most things. One point the author made was to change up the scenery. This could be how the classroom set up or leaving the classroom to go outside or just to another room. Another variation that could be done is in what type of lessons are done. Switching between lecture, group work, projects, etc. can be very beneficial, because students will show up eager to find out what kind of activity is being done for the day. A third idea I like is to just try to make things fun. Rewarding students both verbally and with physical objects is fun for the students. Being exciting as a teacher and making jokes is something I really believe should be done more. Students also love a little competition in class too. I think that these are some things that I will include in my own classroom so my students will be as motivated as possible.
In another class, we discussed how schools can prepare students become contributing members of society. This is an interesting topic to me so that is what I decided to discuss for this journal! I remembered reading something from one of our class books so I went to look. This is what I found: on page 233 of the book Creative Schools, the authors say, “The standards movement is rooted in competition between students, teachers, schools, districts, and now between countries. There is a place for competition in education, as there is in the rest of life. But a system that sets people against each other fundamentally misunderstands the dynamics that drive achievement. Education thrives on partnership and collaboration – within schools, between schools, and with other groups and organizations.”
I think that this offers a lot of insight into what can be changed to prepare our learners to be contributing members of a changing society. While competition is good, it can sometimes be overwhelming and negative. Instead of constant competition, we should be promoting collaboration and partnership between students. By doing this we are giving students tools to become contributing members of society. By encouraging students to work as partners and collaborate with one another, they are learning communication skills that are very important in life. Throughout college and in almost every career choice, you will be required to work with others. So, by starting this at a younger age, it will help them be more prepared for later in life. And is getting students ready for their next step in life not what education is for? While collaboration and partnership should be the main focus, there should still be a place for competition. The competition will give students that drive to want to succeed and will also help them develop a better work ethic by trying to outwork their opponent.
Recently, my math methods course just completed a lesson that we have been doing a lesson study over. When I was first told about this, I thought what the heck is a lesson study–as did some of my other classmates. A lesson study is “a form of classroom inquiry in which several teachers collaboratively plan, teach, observe, revise and share the results of a single class lesson.” When we were discussing it, it seemed like a whole lot of work for just one lesson, but I have realized that lesson study is very important for a well-rounded curriculum which is the main idea of the article, Four Ways Lesson Study Improves Teaching, by Chandler Hopper.
Lesson study originated in Japan and has brought about many good changes. People have since brought this practice to the United States. While it is not implemented by all teachers, there are some that practice lesson study to provide the best education possible for their students.
Through lesson study, the topic of the lesson and teaching methods for it are researched. Multiple teachers collaborate and communicate their ideas in order to craft a lesson they believe will give the students the deepest understanding of the content and skills they want to get out of the lesson. Once they have planned the lesson, one teacher will teach it while others observe and take notes. Teachers then get together and discuss the lesson and how it went.
I believe the most important part of a lesson study occurs during the reflection period. This allows teachers to note what did and didn’t work in their lesson, as well as make changes to make it better. With constant revisions, the lesson is always getting better based off of what the students need. This supports our rapidly changing society and better prepares students for work in the real-world because they are receiving the best education possible.
Being able to do a lesson study with others in my class definitely helped me to see this. I do not think that I would have chosen to do this right away without being pushed to do so. Now that I have experienced it, I will be able to take this to not only my classroom, but other classroom in my district. These lessons can also be published to various teachers websites to be further revised by them. While these revisions help educate students better, they also provide opportunity for the teacher. After doing a lesson study and publishing it, teachers get a sense of ownership and pride for the work they have put into their contribution.
The link above is an article by Lynne McClure on developing number fluency. This article gives the “what, why, and how” for developing mathematical fluency in students.
McClure writes about how schools are interpreting standards in a way where teachers are feeding students “a pretty rigid and boring mathematical diet”. Formal mathematical algorithms are practiced in pursuit of mastery, but giving these calculations meaning has been put on the back burner.
McClure describes number fluency three modes of criteria: efficiency, accuracy, and flexibility. These criteria lend to effective communication of mathematical concept knowledge.
In one of my classes this semester, we created a Project Based Learning unit called City Zoning. In this assignment, our group created a project where students would analyze economic and engineering data to create a plan for developing a four block area. Students are required to calculate things such as area of buildings, interest on loans, net income, profitability per square feet, how long it would take businesses to pay off start-up loans, and tax income for the city for sustainability. Students will be required to write explanations for decisions made, give a verbal presentation of their plan, and construct a model to scale of the area. By doing this, students will practice efficiency, accuracy, and flexibility of mathematical knowledge.
The difficulty of creating opportunities for mathematical fluency is multifold. Creation of these types of units is very time consuming and finding activities other teachers that address these goals is not easy. There are a lot of activities online, but units such as these are that scaffold math concepts and put them together for solving real world problems are not very plentiful for teachers to share.
As educators, we want to prepare students with maximum readiness for entering the real world. Math fluency is an important part of critical thinking and decision making because math is what I call “the language of logic”. It is a mode of communication that students need to be able to thrive in an ever-changing society. The way teachers facilitate the learning of math fluency can be the difference in improving collective math achievement.
I had the opportunity to read “Making Room for Inspecting Mistakes” (link: https://drive.google.com/file/d/1PjEE0Y8SqMDVIVrETxgLDWCpX00aeDBO/view?usp=sharing) in this month’s issue of Mathematics Teacher from NCTM. The article discusses using mistakes to help students learn. One of the examples it utilizes is choosing a homework problem that is incorrect to go over for the entire class. Choosing these problems is artful because the teacher must ensure that the problem is going to be useful to the greatest amount of people in the class. There may be people in the class that would make the same mistake, others that get a better understanding of how to complete the problem because they did not know how to originally, and even others that understand where the mistake came from and how to combat it. There are three different contexts for leveraging mistakes that the article discusses: review of homework, during a task example, and during exam preparation. In each of these context, mistakes can be capitalized upon to help students grow in their understanding of the content that is being taught. During review for exams, it is a good check to ensure that students do understand what they have learned throughout the unit/semester/year.
This article prompted me to think about how making mistakes can be useful in the mathematics classroom. We have previously discussed how mistakes can be utilized to help students. Yesterday, and earlier today I had the opportunity to attend the SDEA Student Conference in Mitchell. One of the two breakout sessions utilized Breakout EDU. The concept of a Breakout EDU is similar to an escape room, but students are trying to break into a box. They can be bought online for different content areas. However, they do cost $125 so many teachers write grants to get Breakout boxes. Although escape rooms may just be a fad, Breakout boxes can benefit the classroom. After the activity, I began to think about when I would use Breakout EDU in my own classroom. I believe that these boxes could be useful at the beginning of the year to set a standard for collaboration between students, productive struggle, and making mistakes. Furthermore, during this time at the beginning of the year, a box could be useful as a review from the previous year’s material for the students. We saw in the lesson study that students were reluctant to productively struggle, and using a Breakout box could allow the students to start the year off participating in an activity that calls for productive struggle. Additionally, in the theme of making mistakes in the mathematics classroom, students are bound to make mistakes in their search for the answers to the clues. Using an activity such as Breakout EDU would allow the students to understand that making mistakes is beneficial, especially if they persevere in opening the box. Setting a standard for the benefits listed above of Breakout EDU in the classroom would help establish a particular environment in the classroom for the rest of the school year. This environment is aided in being established because after students complete the Breakout EDU, they discuss what went well for them, what problems they encountered, what did not go well, etc. The reflection is what cements the environment. Overall, there are clear benefits to making and going over mistakes in the mathematics classroom, and Breakout EDU could be used at the beginning of the year to establish an environment that promotes productive struggle, making mistakes, and collaboration between students.
Looking through articles on NCTM’s website, I cam across this really interesting one about how a math teacher was questioning his seven year old daughter about division. During that time he asked her what 18 minus 3 was, and while his daughter was thinking his five year old son spoke up quickly and said 15. Although this is the right answer and the young boy deserves recognition for coming up with the answer so quickly, it set something else off in the little girl.
It caused her to proclaim that she was no good at math when the reality is that she is rather good at math. The difference between the boy and girl in this example is that the boy had a natural intuition toward the answer, whereas the girl had to go through a thought process to get to it. This is often time the case on many levels of mathematics. Males often have a natural intuition to mathematics and are easy to quickly get the right answer, but females are often more likely to say something more insightful about the mathematics being performed. I have seen this happen many times from elementary school to college.
One jarring food for thought the article author gave was to have every educator reading the article to think of the best math students they have ever had. There is probably both men and women in that list. The men are probably the ones who naturally got answers, were laid back about homework and tests, and generally just floated by in the class. However, the women were probably the studious, hard workers, and generally cared about how they were doing in the class. It is shocking the bias that there is between men and women!
The author also observed that women often performed better on tests if it were emphasized that it was not as important to have the right answer as it was to have the right thought process laid out. Often times I have wished that professors would grade based on effort put into the class and not just whether I could recreate a proof on an exam.
As much as I would love to spite this article as total bologna and that women learn and approach math the same way as men, it has some interesting and good points. I remember always being naturally good at math, but not in the sense that I could blurt out answers. It was more that I can follow elaborate patterns and draw conclusions from large sets of data. Maybe men and women do approach math differently?
I always said one of the big reasons that I wanted to become a math teacher was I wanted to inspire young women to like mathematics. That goal has not changed. I want to take the information from this article and use it in the sense to emphasize the fact that answers are just as important as processes. Women are more than capable of doing the same mathematics as men, and I want to inspire my students to see that.