Critical Thinking in the Mathroom

Recently I had the opportunity to attend the SDCTM/SDSTA joint conference in Huron, South Dakota. While there, I attended many different sessions that gave tips and trick about keeping students engaged while still learning the material. As society moves away from traditional teaching, I have been trying to think of ways to incorporate more lessons in a form other than lecture. I have also thought about the need to get my students to think critically and struggle productively in the process. While in one of the sessions, I was introduced to a method called 3-ACT math tasks. These were quick lessons that force the students to think and problem solve on their own and in groups. Each tasks contains 3 steps (where the name comes from):

1. Introduce the central problem of the task with as few words as possible.
2. Have the students determine the information they might need to solve the problem and have them guesstimate a logical answer and reasoning for their answer.
3. With the information at hand, the students are able to solve for the solution–then set up a sequel or extension to make sure all students grasped the concept.

The 3-ACT we did in our session was over surface area. First, we watched a video of a guy who had a filing cabinet and was covering it in numbered post-it-notes. We did not get to see him finish, but were then asked to determine what he might be doing. While there were many answers, the obvious was trying to see how many post-it-notes it would take to cover the cabinet. We all guesstimated an answer for this question. After this, we were asked what information we would need to know to solve this without physically covering the cabinet. Being a room full of math teachers, it was not difficult to know that we needed the dimensions of the cabinet as well as the post-it-notes. However, in a classroom just learning about surface ares, it will take longer for the students to decide on the information they need. After waiting for the students to decide, the teacher gives the students only the information that was asked for–making them think a little extra if they get it wrong the first time. Once the class has decided on the information needed, they solve the question mathematically. Once we all had our guess, we watched a final video showing all of the sticky notes going on and revealing the final answer. We then discussed errors made (I will not give it away in case you want to try this yourself).

These ACT’s are a great to get kids engaged in the material. Here is a list of 3-ACT math tasks created by Dan Meyer, an officer for Desmos which is an advanced calculator application that is being implemented in current testing.

Connecting Mathematics to Real Life Through Pictures

In our 7-12 Math Methods course last week, we discussed an article about reasoning and sense making, which was very good and informative. Reasoning and sense-making is very important in our math classrooms. The thing I want to discuss in this post, however, is preparing students for using math in real life. The reading we discussed in class gave three purposes for high school mathematics. We use high school math to give students “mathematics for life… the workplace…” and “the scientific and technical community.” It may seem obvious, but we need to give students the skills to use math in their daily life. Not enough students can look at objects in the world and use mathematics skills to figure out the area or circumference or something of that nature. One way to teach students how to use mathematics in the real world is by having them take pictures of things and use mathematics to figure out problems involving these objects.

The article I read that suggested this method is Mathematical Selfies: Students’ Real-World Mathematics by Kathy Jaqua. She is an associate professor at Western Carolina University. She uses this method with her students. The idea is that students take pictures of things in real life that are similar to what’s being discussed in class. For example, if a student sees a ladder leaning up against a building, they can take a picture of it, bring it in to class, and maybe figure out the length of the ladder or high up on the wall it is.

Even your typical story problems don’t relate to students in the way that them taking pictures can. Stories of buying twenty watermelons, walking twenty miles, or figuring the length of a guy wire aren’t going to relate to students the way that this method can. Students aren’t going to go tell their parents about story problems they did, but if they take a picture of something they encounter and use it in a math problem, they may go home and tell their parents about what they found, which means they’re excited about it.

An important part of this method is to realize what students are interested in so we can suggest things for them to take pictures of. Students may not realize what they know until we suggest it for them. A student who rides their bike may not realize that they can figure out how far they’ve ridden a bike just knowing the diameter of the wheels and how many times they’ve gone around. We need to suggest thing. One suggestion that Ms. Jaqua gave was using video game examples. If students are having trouble focusing, but know vast amounts about video games, then using math as it relates to video games can be a great way to reach these students.

Geometry is probably the easiest subject to use this for, but there are other mathematics that can be used. As mentioned with the ladder problem, it can relate easily to the teaching of the Pythagorean Theorem. There are plenty of parallel lines and shapes in the world that can be used when teaching geometry. There are other easy real-world applications of math. Going back to video games, students could use the amount of money gained on a business venture in a game over a few days and use that to find the slope, so they can figure out how much money they’ll have gained months from now. They could bring in pictures of the money differences from day to day. The same goes for real life finances. While high school students may not be buying their own food, they can still bring pictures of nutrition facts and figure out whether one type of snack is healthier than another when they eat the same amount of each. There is plenty of math in the lives of students that they can take pictures of and take to class.

Students need to learn math to do well in the world. They will be better prepared to be adults if they can recognize what math to do and when in real life situations. Using pictures that they take of things in their life will increase their ability to recognize math problems in real life. Also, this will make sure that the students can relate to the mathematics being taught, and so they will be more willing to listen and learn.

Reasoning and Sense Making Article

file:///C:/Users/brett/Downloads/mt2017-09-054a.pdf

Building Procedural Fluency from Conceptual Understanding

The effective mathematics teaching practice that I was assigned to read about from Principles to Actions was building procedural fluency from conceptual understanding. Conceptual understanding in itself is a functional understanding of concepts, operations, and relations presented in the mathematics classroom, whereas procedural fluency is the ability to execute procedures accurately, efficiently, and flexibly. The idea is that teachers should strive to build this procedural fluency in their students from the conceptual understanding that the students have through making connections between concepts, or introducing multiple ways to approach a problem. Reading about building procedural fluency through conceptual understanding prompted me to reflect on my own procedural fluency, and how I can build procedural fluency in my future students. It helped me to realize not only what procedural fluency is, but how important it is in the mathematics classroom. While students may understand a concept, they may not be able to apply it to another problem. Procedural fluency aids in problem solving skills, which are at the heart of mathematics. I can use the information found in this article, and others to help my future students learn procedural fluency in the math that they encounter. This can be done through encouraging students to look at different ways to approach problems, connecting the mathematics learned to real world applications, and focusing on group work after students have had the opportunity to approach a problem individually.  The suggestions made in Principles to Actions for building procedural fluency from conceptual understanding can be found in the table below.

One article that I found on the subject procedural fluency concerns practicing frequent mental math in the classroom (link: http://www.nctm.org/Publications/Mathematics-Teacher/2015/Vol108/Issue7/Five-Keys-for-Teaching-Mental-Math/). Although mental math may not be exactly procedural fluency, it does help in building it. The way that I think of mental math may be different from another student, and there are different ways to approach each problem. Reading this article, I began to reflect on my own use of mental math since beginning college. My first college course was Honors Calc II, and no calculators were allowed. This was a stark difference from my high school classroom where calculators were heavily relied upon. Since beginning to  rely on my own calculations rather than a calculator’s, I have found that my mental math skills have drastically improved and I believe my overall mathematical ability has as well. This has helped to build a procedural fluency in my mathematics skills, and I expect it would have a similar result in a high school classroom. Furthermore, in my internship last fall, I found that the majority of students in the sixth grade classroom that I had the opportunity to observe still used a multiplication table, and were unable to perform the calculations on their own. These students did not have a conceptual understanding of how to perform multiplication, addition, subtraction, or division, thus hindering their procedural fluency. I would like to implement no calculator quizzes and exams (not all) in my future classroom to help students build a procedural fluency in their mental mathematics.

Another article that I found was a study done in an elementary school involving utilizing a conceptual understanding versus procedural fluency. They split students into two groups, keeping track of previous ability in mathematics, and administer a conceptual understanding intervention to one group and a procedural fluency intervention to the other group. The study found that the students who received the intervention considered most appropriate to their abilities received the intervention better, i.e. students who had historically struggled more that received the procedural fluency intervention did not respond to it as well as students who were at the same level but received the conceptual understanding intervention. This shows how important it is for students to understand what they are learning before they are able to develop procedural fluency. The results of this study are indicative of what I should remember in my future classroom: that my students must understand the material fully before they will be able to build procedural fluency. It is because of this that conceptual understanding must first be established before procedural fluency can be built from it.

 

References

Burns, M. K., Walick, C., Simonson, G. R., Dominguez, L., Harelstad, L., Kincaid, A., & Nelson, G. S. (2015). Using a Conceptual Understanding and Procedural Fluency Heuristic to Target Math Interventions with Students in Early Elementary. Learning Disabilities Research & Practice (Wiley-Blackwell), 30(2), 52-60. doi:10.1111/ldrp.12056.

National Council of Teachers of Mathematics (2014).

Olsen, J. (2015). Five keys for teaching mental math. Mathematics Teacher, 108(7), 543-547.

Modeling in the Mathematics Classroom

 

The primary focus of the chapters that we read from Strength in Numbers was creating equity and access to mathematics in the secondary education classroom. In the theme of making mathematics accessible to all students, I found an article about using modeling in the high school classroom, called “Mathematical modeling in the high school curriculum” (link: http://www.nctm.org/Publications/Mathematics-Teacher/2016/Vol110/Issue5/Mathematical-Modeling-in-the-High-School-Curriculum/). Mathematical modeling is centered around “using mathematical approaches to understand and make decisions about real-world phenomena.” Utilizing this type of instruction, the teacher will give students a real-world problem that they will come up with multiple solutions to. In the article, the example given is centered around comparing different prices of gas at gas stations, and if it is more economical for a driver to drive outside of their “usual” region to find gas. Problems such as the example given are what the concept of modeling focuses on. Using this type of instruction in the mathematics classroom reminded me of the chapters from Strength in Numbers in the essence that utilizing modeling can help to make mathematics more accessible to students. Modeling focuses on making connections from mathematics to the outside world, making it more meaningful to the students. Furthermore, it aligns with the principle of asking students to see themselves in the mathematics that they are learning. In modeling, students are asked to use and develop problem solving skills to investigate a given scenario that applies to their day-to-day lives. The concept is reminiscent of project-based learning (PBL) but on a much smaller scale.  

It asks students to perform tasks similar to that of PBL, but from the description in the article, modeling should take place over one to two class periods. The concept of modeling in itself asks students to apply their learning to real-world situations, deepening their understanding of the material. A portion of the article focuses on the teacher’s role in modeling. It addresses questions that the teacher should be asking him/herself before the lesson begins, such as what other resources students may need access to in order to properly address the question that the model gives the students. This indicates the preparation that should go into preparing a modeling activity for the students in the classroom. The article prompted me to consider how much of an influence giving students the opportunity to integrate their learning into real world problems can have on their learning. At the end of the article, a is quoted who describes her appreciation for having the opportunity to model in her mathematics classroom because it helped her to “remember the math.” Modeling gives students to apply what they have learned in their classes outside of the classroom, as they will eventually do as adults.

In my own classroom, I can use the information gathered in this article about how to model, and the benefits of modeling in the mathematics classroom to integrate modeling into my curriculum as a teacher. As we discussed in class, I would be sure to give students the opportunity to work in small, random groups to exploit the skills of each individual student. Giving students an opportunity to apply their thinking is a common theme in recent articles read, and in the assigned reading for class, as well as what research has supported in the past. It gives students real-world applications to what they are learning, answering the perpetual question “when are we going to use this in real life?” This article was further support for me to ensure that I create a classroom centered around applications for the mathematics that students will learn. Doing this will not only give them an opportunity to apply their learning, but will help them gain a deeper understanding of the mathematics that they are learning, and thus retain the information gained for a longer period of time. Altogether, this makes mathematics more accessible to the students, as it aligns with the ideal in Strength in Numbers. Modeling in itself can be changed to fit what the teacher utilizing it needs for their classroom (i.e. a model can be made shorter or longer, what the model is will depend on what is being learned in the classroom, and how frequently the teacher uses models to apply student learning).

Paired with ideas that I noticed in the article that I wrote about last week, as well as what I learned in my Curriculum and Instruction (C&I) class, I began to wonder about the impacts of modeling, or PBL on students in classes. In my C&I class we visited New Tech in Sioux Falls, and those students had significantly lower standardized test scores than other schools in Sioux Falls in the mathematics subject area. Modeling gives students an opportunity to use problem solving, but not to the extent that PBL does. It also inherently employs aspects of an equitable classroom, aiming to make the mathematics more accessible to all students. Thus, I questioned how deeply modeling affects students in the classroom. I found a study through an online database where modeling was utilized in one differential equations course while another professor used a traditional lecture technique in his differential equations course. The study found that on the same final exam, students in the class that used modeling as a instruction technique had a mean score 12.4% higher than the students in the traditional classroom. Although the study admits that it was “quasi-experimental,” it still gives serious implications to the usefulness of modeling in the classroom. Link: https://www.sciencedirect.com/science/article/pii/S073231231630147X

References

Hernández, M., Levy, R., Felton-Koestler, M., & Zbiek, R. M. (2016). Mathematical modeling in the high school curriculum. Mathematics Teacher, 110(5), 336-342.

Making a Math Classroom Equitable

This week in class the concept of mathematical classroom equity was introduced, which is a concept that immediately elicited my attention. I have often contemplated the idea of equity, but in the concept of equity vs. equality. It was a topic introduced to me a while ago when I found this image:

At the time I was considering the argument of equity vs. equality in a political sense, because in our current political climate many groups campaign for equality when they really wan equity. I had not thought about it in an educational sense.

So, when the topic came up in class that we would be looking at Case Studies and deciding whether they were equitable or not, I was immediately interested. My main item of discussion and knowledge for equity in the classroom comes from our assigned reading of chapter two Equitable Mathematics Teaching from Strength in Numbers Collaborative Learning in Secondary Mathematics by Ilana Seidel Horn. URL: https://drive.google.com/file/d/0B1zVoWMFl08-TGN2NWxQMXVPVGc/view

I found the chapter intriguing to read and took out many good points and concepts from it.  The first thing is how the book defines equity in math as “equitable mathematics teaching involves using models of instruction that optimally support meaningful mathematical learning for all students.” Meaning that teachers should be using a variety of methods and techniques in order to reach students of various learning styles.

The second thing I found most helpful was the three practices they listed for collaborative learning environments that influenced equitable math teaching.

1. What counts as math involves how mathematics is presented to students and the messages about what success means.
2. Pedagogical practices focus on the work of teaching.
3. Relational practices address the relationships that students build with others in the school and classroom.

Finally, the four principles for equitable math teaching where:

1. Learning is not the same as achievement.
2. Achievement gaps often reflect gaps in opportunities to learn.
3. All students can be pushed to learn mathematics more deeply.
4. Students need to see themselves in mathematics.

There are many things here that I would love to implement into my own classroom. Like, using group based work in order to help build the classroom as a community of learners so that way they feel part of a collaborative effort. They could take on a role in their team that meets their strong suit. Also, having across classes activities. That is to say, the algebra students work on the calculations to some 3-dimensional shapes the geometry students are making. That way they feel more connected as a school.

Having students see themselves as mathematicians is also so important. I am a firm believer that everyone can do math, because it is a universal language that can be taught in many different ways. If one way is not working for a student, then it should be I as a teacher to make my classroom equitable so that that student can find a way that helps them learn math. Every student deserves a fair opportunity to learn such that they can be at the same level as all of their peers. Constantly berating students with quizzes, homework, and tests when they are doing poorly does not mean they are going to learn math. Students learn math in many different ways, but they can all learn math.

After learning more about equability in education I like to see equity more like this:

After learning all this information about how I can make my classroom equitable, I wanted to be able to see it in action. What are the different ways equity can be incorporated? So, I went to the National Council of Teachers of Mathematics and found this video that showed me a lesson of how a teacher used equity in her math classroom. URL: http://www.nctm.org/Conferences-and-Professional-Development/Principles-to-Actions-Toolkit/Equitable-Pedagogy/

In the video the students are learning about finding the area of a square, and they all have to go to the board to present how they found their area. They are all in partners by the looks of it, and the partner groups have varying degrees of difficulty in their square. What I mean by this is some students have a regular square that has a flat side on the bottom, but other students have their square skewed so a corner is touching the bottom. Like this:

The students who may struggle more would be given the square on the left so they could count the grids from top to bottom and left to right then multiply to find the area. While the students who understand it more get the square on the left where they have to combine the areas of a square and triangle to find the total area. This way both groups of students are given materials at their level of understanding.

Finally, in the video you can see a great example of peer collaboration. When two students with the harder square are at the board presenting, one student asks them how they got the area of their triangle. This forces the two students presenting to explain their rational, allowing the teacher to check their thinking, and the student who asked the question gets to learn something new.

One adjustment I would make to an equitable classroom would be to find a better way to mix students at the upper end and lower end of the class to see how further growth and understanding could be resulted from that. Right now, to me, it seems like equity is giving the “smarter” kids material for deeper understanding and other students material to try to reach the “smarter” kids. Equity has been an interest of mine for a while now, and I am excited to learn more about how to implement it into my future classroom.