Productive Disposition and Modeling

The assigned reading for outside class from Strength in Numbers focused on creating tasks and lessons that align with the values expressed in the first few chapters of the book. This mirrors our shift into creating lessons for the high school. A particular part that stood out to me in Strength in Numbers is the section on mathematical fluency. Strength in Numbers states that mathematical fluency has five strands:

  1. Conceptual understanding
  2. Procedural fluency
  3. Strategic competence
  4. Adaptive reasoning
  5. Productive disposition

The productive disposition strand stood out to me this week, as I tutor students on the football team enrolled in MATH 095 and MATH 102 four hours a week. Productive disposition is defined as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.” What I have noticed most in my hours tutoring these students is their lack of productive disposition. While reading Strength in Numbers I began to think about why these students are missing a productive disposition. The chief complaints that I hear from students is that they are teaching themselves with videos, their work does not matter (and thus no partial credit is given), and the amount of homework is excessive. Keeping what I have learned in our class in mind, I began to look into the implications of having a productive disposition for mathematics courses. I was able to find one study that looked at mathematics in the middle school, and how mathematical teaching materials impacted students’ productive discipline, called “The Impact of Mathematical Models of Teaching Materials on Square and Rectangle Concepts to Improve Students’ Mathematical Connection Ability and Mathematical Disposition in Middle School” by Irfan Mufti Afrizal, and Jarnawi Afghani Dachlan. The study found that students who learned by using mathematical teaching materials through modeling had an improvement in their productive disposition.

Link: Mathematical Modeling

In the future, I hope to use this information in my own teaching practices. Not only has reading about this study been beneficial to me, but my consistent interactions with students who struggle in mathematics has been beneficial. It has helped me to realize the true implications of having a productive disposition, the importance of a teacher in the classroom, and to consider how I can work toward creating a productive disposition in students. The study found that productive disposition in students improved when they learned material with more modern methods of teaching mathematics, rather than traditional. It emphasizes the importance of mathematical modeling as an instructional strategy In my own classroom, I can utilize this information when constructing lessons and tasks for my students. In the past, I have written about how I hope to use mathematical modeling in my own classroom. Cultivating a productive disposition in students is at the core of mathematical fluency and helping students to be successful in mathematics. Much of what we have discussed previously in this class indirectly aids in creating productive dispositions for students. In the study it was shown that using modeling in instruction does help to bolster productive disposition, but I believe that tasks focused on reasoning and sense making, coherence, and other topics discussed can help as well. The emphasis of the process rather than the answer is inherent in each of these topics, can boost productive disposition because students themselves will begin to shift their focus to the process of finding the answer, and their ability to problem solve rather than finding the correct answer. Furthermore, there is frequently more than one approach to these problems so this also aids in building a productive disposition in students. These mostly aid in building the belief in one’s own efficacy. Regarding the belief of the usefulness of mathematics, tying in real world applications is imperative. How students view the mathematics will impact how they approach the problems that are presented to them. If they do not see it as useful, their productive disposition is hurt and they will not be able to sufficiently build their mathematical fluency. This particular strand is an essential part of building mathematical fluency, and a great consideration when developing tasks for students.


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.

Narrowing Participation Gaps

One thing that we touched upon this week in class was found in the chapters we read from ‘Strength in Numbers’ that participation in your classroom can be tricky. It can be difficult to make students who seemingly do not care answers questions and volunteer. This creates a “participation gap” where the students who participate, participate a lot and the students who do not participate, never participate. It is no surprise then that this correlates almost one-to-one with achievement as students who participate are the students who gain a deeper understanding material and know it (that is why they are volunteering answers).

The ‘Strength in Numbers’ story about the teacher who drew out participation from all of her students really intrigued me to find out more about how to get your students to participate. Especially when you have students that are all very different in terms of personality, race, backgrounds etc..

I found this article called, “Narrowing Participation Gaps” by Victoria Hand, Karmen Kirtley, and Michael Matassa that does just that. The url is:

The article discusses three specific ways to encourage and increase participation by all students which, in turn, will narrow the gap. First,

  1. Organize Mathematical “Contributions”
  • Participation is not only talking or answering questions
  • More than one way to get math answers
  • Prompt students unclear answers with directing questions

The first bullet point makes the point that teachers too often ask a question and evaluate the student’s knowledge based off of who answers and how they answer the question. It is important to remember that math is all about process and teachers need to evaluate how the students got to their answers. There are other ways of participating in class other than speaking as well. An important quote that goes along with this explanation is, “This orientation also prioritizes correct uses of academic language over students’ sense making”.

The third bullet point is one that was not said, but I observed from the example in the article. These directing questions can be uber-focused on the words that students use such as “length” and “width” as it may have a different meaning to the student than it does to the teacher. Thus, these clarifying questions help the teacher know what the student is saying and it helps the student understand what he/she is saying. This can also help narrow the participation gap when students aren’t afraid that they will be “wrong”, but rather they will get guided to the right answer. It is important not to simply give the answers to the students as they are learning absolutely nothing there. Confusion is the best way to learn.

2. Expand “Smart-ness”

  • Expand perception of who is “good” at math
  • Complex Instruction in groups
  • Assigning roles
  • Reward different ways of thinking

Group work, in general, causes more participation, as each member is responsible for their own work. This is very true for what roles are assigned and each student has a responsibility. In my opinion, group work sometimes creates less participation as the “smart student” does the work. Assigning roles changes this and allows students to see the way others think and it will expand what they deem as smart as well.

3. Engage Instead of Motivate

  • Takes away the blame on students
  • Change the classroom as opposed to the student
  • Don’t label students

While I agree it is important not to label students, I do not necessarily agree with “engage instead of motivate”. Yes, there is a time and place where you do the work along with the students, but giving motivating problems is good as well. I believe there must be a balance of both to really get the students to participate. The article talked about how teachers should allow foreign language speaking students to solve the problems in their first language and then to explain it in english and I think this is a great method. Bottom- line I believe you still have to motivate your students and it is not all about doing the work with your students because then they will start to bank on you for the answer and that will only hurt their learning.

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: 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:


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