How to Cure Math Anxiety with Modeling Mondays?
“I hate Math!” Almost every math teacher has heard and dreads this phrase. Many teachers learn to ignore the statement and continue focusing on their students who love math—the students who make their job fulfilling. Rookie teachers would ask follow-up questions such as, “Why do you hate math?” But the answers were always similar: “It’s too hard!” or “When am I going to use this in real life?” Yet the only responses the teacher could think of were— “You just need more practice.” “Well, you need math on the ACT/SAT if you want to get into college.” or “It could be useful if you want to be a scientist of engineer.” But deep down the teacher knew those answers were unsatisfying and that her beloved subject would only be useful to a handful of students that go into STEM careers. So, overtime she learned to just ignore the statement and continue teaching.
But is that correct? Is higher level math only useful for those entering STEM careers?
The solution to a negative math identity (and the anxiety that comes with it) is to incorporate more mathematical modeling into the classrooms.
I’m Trying to Love Math
I’m Trying to Love Math by Bethany Barton is a good beginning of the year book to introduce mathematical modeling. In the book, a human is explaining to his/her alien friend that math is not very lovable. However, the alien keeps pointing out all the times the human is unwittingly doing math while doing all the things he/she loves. It turns out the human doesn’t dislike math, but dislikes calculating. What makes the difference for the human is the engaging real-world contexts that disguises the mathematics.
Here are some examples the author uses to give you a feel for the book:
Real-World Contexts
Using real-world contexts in the classroom builds relevance and motivates students as they can identify with the problem. It shows the usefulness of mathematics. In reality, no one solves algebraic equations in isolation. The context is what drives the problem. Yet too often in schools, we remove the context (or give a pseudo context) and wonder why students hate math. A context lends to understanding. Without a context, the problem is mostly meaningless or at least demotivating. Why should they care? Why should we care? No wonder kids struggle with math. Without a context, there is nothing for a student’s brain to make connections with.
Many teachers want to use real-world contexts, but they don’t know how or where to find them. After all most teachers have only been—well teachers—which means they’ve never used mathematics in the real-world. They’ve only used mathematics in an academic setting. This is where Mathematical Modeling comes in.
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What is Mathematical Modeling?
To be informed citizens and make wise choices people need to understand mathematics. Mathematics literacy is so vital, understanding it has been deemed the new “civil right.” It’s not the calculating or the manipulation of equations that’s so important (although these are necessary skills), but it is its application in real-life situations.
GAIMME Report
According to the Guidelines in Assessment and Instruction for Mathematical Modeling and Education (GAIMME) Report,
“Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena.”
Modeling simplifies a real-world context into a mathematical form (equation, diagram, graph etc.) that can help solve the problem. It is open-ended and messy—just like real life! It uses mathematics to answer big, reality-based questions that may have multiple solutions and multiple variables.
The GAIMME Report is a must-read for anyone interested in incorporating modeling in the classroom, but it’s a little hard to read. So I would suggest starting with the Preface, Chapter 1: What is Mathematical Modeling and Chapter 5: What is Mathematical Modeling the Art and Flavor. It also has a lot of great problems by grade level that you can use in your classroom.
Common Core
Modeling is so important it’s even one of the standards of mathematical practice: MP.4 Model with Mathematics. According to the Common Core,
“Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.”
The description surrounding MP.4 Model with Mathematics explains what modeling could like at various grade levels. In early grades modeling could be as simple as writing an equation to describe a real-world context. In middle school students might use mathematics such as a proportion to plan a school event. In high school students could use Geometry to solve a design problem.
Modeling vs Model
A Mathematical Model is different than Mathematical Modeling. Mathematical Modeling is a verb; it’s a process. A Mathematical Model is a noun; it’s an object such as a diagram, a manipulative, a symbol, a figure, an equation etc. that represents a real-world phenomenon. A student will make a mathematical model as part of the modeling process. It’s also possible for students to use a model such as manipulatives but not actually be mathematically modeling. Remember modeling is grounded in authentic real-world contexts.
The Modeling Process
Common Core
There are several ways to think about mathematical modeling. The Common Core has as diagram for the modeling process it promotes, but it is can be hard for students to remember.
GAIMME Report
The GAIMME Report also has a process, but it too can be hard for students to remember.
Spies, Analysts, Model
However, I think Robert Kaplinsky’s Spies and Analyst Model is friendlier for students. This is the modeling process we choose to use for Ohio’s Mathematical Modeling and Reasoning Course, but we modified it a bit and added some elements from the GAIMME report. I recommend starting the year introducing this model as one of your classroom routines. You can even pull some examples from Robert Kaplinsky’s website.
Spies
Have students become observers and question-askers. People are naturally curious. Provoke and direct students’ curiosity by filling their minds with mathematical substance. Present an interesting problem, and then implement The Notice and Wonder Routine. This routine is a great, easy way for students to become spies. After the Notice and Wonder Routine, I would have students identify the problem, identify their variables, and list any assumptions they are making and incorporate all those elements into the Spy stage of the modeling process.
Here are some resources to help you implement the Notice & Wonder routine:
- MPIR – Notice and Wonder | OER Commons
- Notice_and_Wonder_Process_and_Resources_4.25.19_AiKcYJD.pdf (oercommons.s3.amazonaws.com)
- Notice/Wonder – Formative Strategy (smartertoolsforteachers.org)
Facts vs Assumptions
Since modeling problems are messy students will have to make assumptions in order to solve them. Students should be able to differentiate between facts and assumptions. It’s important for them to identify the assumptions they are making when problem solving. An assumption is anything that a student may assume about the problem. In order to solve the problem, they may assume that gas costs the same at all gas stations in the area. In a sharing problem they may assume that each person will eat an equal amount (although in real life that may not necessarily be the case). It’s important that students make genuine choices, so they need to be the ones to decide the assumptions. As students gain more skills in modeling, they may be able to articulate how their model can change based on the different assumptions they make.
Analysts
This is where students analyze the problem and create a model (an equation, graph, diagram, etc). At this stage, they need to figure out what information to keep and what to discard. They need to figure out what tools they need. The students need to actually do the math!
Model
At this stage, students need to create a model and analyze it. Did their model do a good job at answering the original question? Is their answer reasonable? Are their assumptions reasonable and relevant? Does their model follow their assumptions? Is their model easy for others to understand? Does it communicate what they intended it to communicate or is there a better way?
If their analysis is good, then they can report the results. If their analysis reveals that their model needs to be reworked or improved, students need to revisit the Spy Stage and start the cycle again.
Note: Assessment should be focused on the product not the process because in modeling communication (not content) is king. Can students communicate quantitatively? That’s the skill they need in real life! Although there still needs to be mathematical accuracy in calculation and terminology, but it’s a process and it’s ok for them (and you) to make mistakes along the way. Give students a lot of grace in the beginning but push for accuracy and correct terminology. Give yourself grace as well; It’s ok for you the teacher not to know the answers when starting the problem as there probably will be multiple solution pathways.
Why Modeling Mondays?
Modeling is vital, but it can be hard to start (both for you and the students) as it’s a new way of thinking. So start small. Try instituting Modeling Mondays. Instead of kids dreading Monday morning Math class, leverage modeling problems so students look forward to Monday as the favorite part of their week.
To get started choose a small modeling problem that can be completed in a class period. To save time find problems that complement your curriculum. If the problem runs over by a little you can always revisit it later in the week when you have time. Maybe in a few years, you may teach your entire course through modeling!
Questioning Creates Rigor
The key to successful implementation of the modeling process is good questioning. Never give the students and answer, but instead answer their questions with a question. This is hard at first (for teachers and students), but everyone will reap the rewards over time as students realize they have to think for themselves. When doing classroom observations, I’ve seen the same lesson taught at a sixth-grade level and at a Precalculus level. The biggest difference was the type of questions the teacher asked.
The rigor in the class is driven by the quality of questions you ask. Ohio defines rigor by,
“Students use mathematical language to communicate effectively and describe their work with clarity and precision. Students demonstrate how, when and why their procedures work and why they are appropriate. Students can answer the question, “How do we know?”
To maintain the rigor in your class through modeling, you need to get your students to this level. Obviously how an elementary student answers this question is vastly different than a high schooler with more math knowledge and experience under his/her belt.
To improve your questioning, have a colleague come in, observe you, and take note of the types of questions you ask. Or you can try taking a video of yourself and then analyze the types of questions you ask. Is your questioning on grade-level? Are you pushing kids to think for themselves or are you giving answers away? Does your questioning convey belief that the students are capable of finding the answers?
Collaboration
Collaboration is an important part of the modeling process. Remember mathematical communication is key. Students can work in teams or work individually and then share out. At times they may need to go do research and then come back to report to their group. Everyone should be involved, and everyone should have some type of role. Communication is a two-way street. Groups and individuals need to be able to explain their thinking to the rest of the class, and the class has to be able to understand the models that their classmates present to them.
Classroom Environment
The classroom environment is important for modeling to be successful. Choose engaging tasks that require just the right amount of productive struggle. Students need to be comfortable making mistakes and asking questions. Overtime they need to become independent learners using the teacher as a resource but not as the knowledge source. Encourage research. As in English class, revision is the name of the game. As there are typically several solutions to a task, there are typically many ways to find an appropriate solution. Students need to constantly be evaluating their model and revising as necessary.
Where to Find Good Modeling Problems
There are a variety of place to find good modeling problems online. Here are a few that I found:
- Math Modeling Can… Archives – Robert Kaplinsky
- Past Problems – M3 Challenge (siam.org)
- GAIMME Report
- Tasks Archive – YouCubed
Three-Act Tasks are also a light, easy-to-implement version of modeling that you can start with. Some good websites for Three-Act Tasks are the following:
- Dan Meyer’s Three Act Tasks | Math (newvisions.org)
- The 3 Act Math Beginner’s Guide | Spark Curiosity to Fuel Sense Making (tapintoteenminds.com)
- How Many Ducklings Are There? – Robert Kaplinsky
- 3-Act Tasks | Questioning My Metacognition (gfletchy.com)
The GAIMME Report has some examples of modeling by grade level. It also gives advice on how to transform a word problem into a modeling problem. So if you have some favorite word problems, you can restructure them to become modeling problems. But remember, they need to be open ended and have several solution pathways.
I’m Trying to Love Math (Revisited)
You can use I’m Trying to Love Math by Bethany Barton to launch some modeling problems. Here are some ideas:
- What would you put on the Golden Record that was sent into space? For this question students have to figure out how much digital space was on the golden record. (This is also a great tie in to my The Care and Feeding of a Pet Black Hole post.)
- How much would it cost to make cookies for aliens? (There would be a lot of fun assumptions here!)
- Choose a stanza of a song and model it with mathematics. Can your classmates figure out your song?
- How much would it cost you to drive to Disney World (just the drive not the destination)?
- How much fuel is needed to get a rocket into space?
- How much pepperoni do you need to make enough pizzas to feed your school?
How to Cure Math Anxiety?
Mathematical modeling won’t cure all cases of math anxiety, but it will help many students! As the classroom becomes a safe place for discourse and productive struggle based on real-world contexts, students will enjoy coming to class and learning. This will help their confidence to grow and improve their math identity as they become active doers of mathematics!
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