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.  

Screen Shot 2018-01-16 at 8.19.33 PMIt 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:

Image result for equity vs equality

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:

Image result for equity vs equality

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.