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.

Equity in Education: The Relationship Between Race, Class, and Gender in Mathematics for Diverse Learners

Equity in Education

Reading about a particular case study for class caught my interest. It was the case study in which a particular school was placing the majority of non-white students in below grade level courses, while just over one-tenth of white students were placed in these same classes. I was curious to learn what could possibly make such a disparity occur. I found the article “Equity in Education: The Relationship Between Race, Class, and Gender in Mathematics for Diverse Learners,” which was written Debra Rohn from the University of North Carolina at Charlotte for The Urban Education Collaborative. The article addresses how race, class, and gender affect learning outcomes and gives some theories on how to fix it.

The first topic discussed is “race and mathematics.” There is huge differences in average test scores between African Americans and Whites and Hispanic Americans and Whites, to put it plainly. This could be the result of direct discrimination, such as placing students of other cultures in classes below grade level just because the teachers do not expect them to do well based on their race. The article seems to suggest, though, that the more likely scenario is the expectations put on minorities by society as a whole. They are likely not expected to do well, and they realize this. The article suggests that many of them probably give up on their math abilities, since mathematics is widely viewed as a “White subject” in America. The article makes the claim that even when African Americans succeed in mathematics, some would describe themselves as “acting White.” Now, at first, this may seem to be more about equality than equity, but the solution that I gathered from the article is to give extra psychological help, of sorts, to children that are minorities. In an equitable classroom environment, students would be taught that a student’s race does not take them out of the running for a career in mathematics. One major issue is that the students are believing that they cannot pursue success in mathematics simply because of their race, and in an equitable classroom, they would be taught to think otherwise.

The second topic of discussion is “class and mathematics.” This has less to do with the idea that someone is behind someone else and more to do with the idea that students from lower socioeconomic status are set behind. When a child is raised by a single mother who may not have even graduated from high school, that mother is probably going to work long hours, which leads to her not being able to be with her children as much as parents from higher socioeconomic status. This leads to less development for the children of the lower socioeconomic status family, since they had less chance to communicate and do learning activities such as reading. This does put them behind from the beginning. A solution to this is for teachers to quickly realize this, and, rather than put them on a lower track for the whole education career, give them added supports and multiple modalities to work with.

The third topic was “gender and mathematics,” which focused on how many people think that men are naturally better at mathematics. There have been studies that show that this is not the case. As with race, the differences in test scores most likely comes from them thinking that they are less able to do mathematics, so they give up on being as successful as they could be. Also similarly to race, the solution is to eliminate these ideas in our classrooms, so that everyone gets a clean slate and can pursue whatever career choice they would like.

My view of this is that the solutions to all of these are to either reinforce positive thinking so that everyone believes that they can learn mathematics or help those that are behind so they can catch up to everyone else. Of course, one shouldn’t assume that a person that is African American or a woman isn’t at a disadvantage, because they could be someone who has a difficult past. Even if a classroom was all white males, we as teachers should know what past they come from, so we can potentially help them come out of it with hope of a better future. I can use this information to help my students overcome racism and sexism to pursue their dreams. I can use this information to help students who come from difficult beginnings get to a place where they can learn more information easily. The point is, teachers should learn as much about their students as possible, and cater to that. That is the point of equity. That is how it can be used with race, class, and gender.

What are some best practices for teaching high school mathematics?

By Kevin Reins

This week I read the responses of a two part series in Ed Week by Larry Ferlazzo. The question of the week was, “What are some best practices for teaching high school mathematics?” This intrigued me as I was preparing the 18th revision of my secondary mathematics methods course here at USD.

“…there are a zillion different instructional strategies and practices that math teachers can use in high school.”

The focus of part 1 was on these ‘instructional strategies and practices,’

  • teach to big ideas (see image above), it allows students to have interconnected schema
  • focus on the processes and connections between different processes
  • use instructional routines (see Why instructional routines?)
  • keep a record of conversations when you orchestrate full group discussions
  • be selective and cautious in your use of technology
  • incorporate high leverage long-term strategies
    • -David Wees
  • embrace mistakes, normalizing mistakes, safe space for discussion and correction, utilize error analysis
  • formal error analysis through test corrections
    • -Jillian Henry
  •  provide relevance and contexts for the mathematics
  • engage students in a variety of practices and strategies
  • provide scaffolds for those who need the extra support when working with challenging content
  • plan intentionally and deliberately so your instruction is impactful, consistent, and effective
  • develop a community of learners where group participation and interaction is expected
  • employ student-centered teaching and learning
  • provide opportunities for students to develop and strengthen their skills of mathematical communication (including vocabulary)
  • make the development of a variety of problem-solving techniques a priority
  • eliminate the blank paper; require students to write (1) determine a strategy that could be used to solve the problem, (2) write a question that you have about the problem, (3) record everything you know about the content related to the problem.
  • develop their ability to ask good questions during problem solving phases, Entry (getting started), Moving (when stuck), Reflection (thinking about thinking), and Extension (deeper thinking).
  • utilize graphic organizers to help them employ processes independently
    • -Tammy Jones & Leslie Texas
  •  What works at elementary or middle level works for high school
  • pose interesting problems or set the stage for students to pose interesting questions/problems about the situation
  • encourage investigations, experiments, collaboration, and discourse as students explore problems
  • expect representations or models for the problems being investigated
  • engage students in discourse, creating mathematical arguments and critiquing the reasoning of others
  • proving their work with both formal and informal proofs 
    • Anne Collins

Part 2: Students must ‘engage in math problem-solving’ and not just ‘follow procedures.’

The acquisition of best practices for teaching high school mathematics is necessary for student academic success.

The focus of part 2 was on engaging students in problem-solving. The following was said by the experts interviewed,

  • you must have as your guiding philosophical principle the belief that all students can learn
  • you must provide opportunities for them to fall in love with learning
  • Standards for Mathematical Practices can serve as a guide for the ways students need to be engaged in mathematics
  • choose open-ended problems
  • focus more on the process rather than the correct answers
  • challenge them with mathematically rigorous tasks, choosing a Higher-Level Demand Task
  • learn how to anticipate student responses and misconceptions for tasks
  • ask students to find multiple strategies to the tasks you present
  • learn how to help students learn from mistakes
    • -Wendy Monroy, LA math coach
  • math learning should be developing conceptual understandings of the mathematics
  • focus on the conceptual relationships
  • create a synergy between the lower levels and higher levels of thinking through inquiry
  • create a social environment that promotes team work and collaboration
  • provide an open, secure environment that allows for mistakes as a part of the learning process
  •  use an inductive teaching approach (vs deductive)
  • reduce teacher talk time (increase productive mathematical discourse)
  • differentiate by content, product, and affect (Tomlinson)
  • use all types of assessment; visible thinking routines, “I use to think… Now I think…” (Harvard University’s Project Zero)
  • use a flexible fronts layout of the classroom which encourages more collaboration
    • -Jennifer Chang Wathall, educational consultant in concept-based mathematics/curriculum
  •  give challenging problems that build patience and persistence in their maturing problem solving skills
  • then spend ample time in joyful struggle
  • create rich mathematical dialogue that leaves the building
  • 12 challenging problems that 5 of which will appear on the final, and give them time in class to work on them (e.g., A point P, inside a square, was 3, 4, and 5 units away from three of the corners. Find the length of the side of the square.)
    • -Sunil Singh, author of Pi of Life: The Hidden Happiness of Mathematics
  • sufficient time to make sure that students know how to solve problems using different methods
  • look for opportunities for students to have multiple entry points or strategies for solving a problem
  • take time to discuss strategic choices
  • find flaws in short cuts and when certain methods won’t work
  • open their mind to new and different approaches
    • -Matthew Beyranevand, author of Teach Math Like This, Not Like That: Four Critical Areas to Improve Student Learning.

So after bolding all of the big ideas of the laundry list of instructional strategies and practices that were provided I compared it to the content that I normally would teach in my secondary mathematics methods course. The result was two ideas, one new, and one that could use a deeper focus. I would like to incorporate more ideas on how to utilize math mistakes in the classroom as learning opportunities. I also would like to explore instructional routines a bit more.

To think more deeply about embracing mistakes, normalizing mistakes, and creating a safe space for discussion and correction, I think it is important to start off knowing what some common math mistakes in high school are. I found Math Mistakes website that does just that. This should be a good start for a discussion on how to utilize some of these mistakes when they pop up as a learning opportunity.

With respect to instructional routines, I read Why instructional routines? It turns out I know what they are and how one should utilize tasks in teaching. One instructional routine that David Wees talks about is, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem.

“Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” Kazemi, E., Franke, M., & Lampert, M. (2009)

I’m looking forward to expanding my teaching and learning opportunities to include both instructional routines and normalizing mistakes.