Kimberly Jacobs, M.S., is a teacher of more than 30 years and holds three teaching credentials. Currently, Kimberly provides intervention and instruction to students in 5th to 8th grades at a public school in California as well as being an adjunct professor in the Teacher Credentialing Program at Humphrey’s University.
You can read more about supporting struggling students on her website: https://lessonsinabubble.com
Why do students struggle to be successful in Algebra Order of Operations? There is no short answer but, there are common themes: skill, attention, mindset, process. Within these themes, many factors can be identified as inhibiting success in a unit on Order of Operations.
- Students don’t know their math facts
- They don’t or can’t focus on all the important details
- Math anxiety causes a fight, flight or freeze behavior
- Students give up believing they can’t do math
- Misunderstandings of concepts
- Failure to follow the process steps correctly
Basic Math Facts
Students may have diagnosed memory issues, social-emotional trauma, or simply did not prioritize learning math facts in the earlier grades. The good news is you can support all of the students by giving them tools to get the right answer every time.
In my intervention, we implement the use of touch math, number lines, and multiplication charts. Doing the operations in the correct order will not matter if the calculations are wrong.
Seeing the Details in Order of Operations
Students need to SEE and understand all the elements within an expression or equation. Knowing and seeing when they have a term, a coefficient, an exponent, a negative integer, or a grouping is required for success.
When a student has a visual processing deficit, they don’t automatically see the spacing or size of letters and numbers. They honestly might not notice at first glance the exponents in the problem. These numbers look identical to their untrained eye.
Demonstrating how size and placement are important will benefit so many it is certainly worth 10 minutes at the beginning of the unit.
Students with focus and attention issues struggle to follow specific processes and routines in all areas of their life. By ensuring a deep understanding of the language and precision of mathematics can help communicate the importance of details. Building the habit of process is not easy but will be key to success for all the students.
During intervention practice, often we will start by just counting all the operation symbols, including parentheses to multiply and exponents so we can predict how many steps it will take to solve or simplify. If we get done before then, it is a signal to check the work. This focused analysis addresses more than one stumbling block.
Thus, when beginning teaching Order of Operations, you may consider previewing all the problems noting all operations, groups, and exponents then, counting and noting how many steps each problem will take to simplify or solve.
You can drop this support for most after a couple days however, some students who have focus or attention issues, will always benefit from adding this step to their process.
Reminders
- Students have most likely been on autopilot during math practice for years. They get practice pages with a single operation or skill so do not stop to analyze what is in front of them.
- In younger grades students are often told, skip what you don’t know and move on to the “easier” problems. This Does NOT work in Algebra Order of Operations – but, you already knew that.
Math Anxiety
Students can come to us with generalized anxiety. Meaning, they struggle or suffer with anxiety in many areas of life. Outside of that, there are students that have been lost in understanding math for so long, they have math anxiety.
Due to misunderstandings, focus or a myriad of other reasons, students have worked and gotten the wrong answer so many times they believe they, “can’t do math.”
This is more profound during Order of Operations as the problems in front of them are generally longer. Just as a young reader might be fearful of a long word, students with math anxiety feel fear with anything more than a simple calculation.
To support these students, it is key to work slow. Given tools and strategies students can learn to see the little pieces and learn to approach one piece at a time to arrive at a correct conclusion with any Order of Operations task.
Once they trust you, the tools or strategies student confidence and effort will increase.
The Emotional Investment of Algebra
Quick story. My oldest was pulled out of class in 2nd to 4th grade to work on reading. She missed math instruction. As oldest children go, she really wants to please people.
When she arrived in Algebra in eighth grade, there were tears every night. She didn’t want to try because she knew she would get it wrong. She didn’t want to do the thing she hated most for two hours.
It would take two hours at this point as she had missed so many foundational concepts. I don’t share this to embarrass her but to spotlight the emotional waters many of our students navigate when reaching Algebra.
Knowing you don’t know makes you feel bad about yourself. As young adults developmentally, students are looking to feel acceptance and belonging. If we don’t address this and just move on. They won’t learn, they won’t try and opportunities will be lost.
The grit needed to persevere at a hard endeavor is not present in all students.
One solution, in this setting, is to have students highlight, annotate, or analyze the expression/equation before they begin any work. Provide them tools to make correct calculations. Give time and immediate feedback so perfect practice is inevitable.
Understanding Math Language and Concepts in Order of Operations
Concepts students struggle with include adding and subtraction of integers, distributive property of groupings, variables, what exponent notation actually means and how to interpret nested groups.
Adding and Subtracting Integers in Order of Operations
Positive and negative integers come along in a students career when teachers have move past concrete models of math in the real world. Those that are not ready for the true abstract nature of math may be able to get by but have not internalized the real meaning of working with integers.
Distributive Property in Order of Operations
Distributive property across a grouping is difficult for three reasons.
One, students don’t understand the terms and operations as being a set.
Two, if their earlier teacher did connect this to repeated addition it was most likely a quick side note without concrete building of understanding.
The third struggle again, is with attention and focus. Students who have to stop to use tools for calculation will lose their place in the distributive process and make errors repeating or skipping steps.
Variables in Order of Operations
Next, when letters begin to show up in math, students do not understand the variable is just a mystery value. They don’t realize different letters will represent different mystery values.
By reviewing the empty box math from the early years teachers can remind students they do know how to solve mysteries and variables are nothing to fear.
Exponents in Order of Operations
Finally, the concept of exponents, if not fully understood, will have students treat them as simple multiplicand and multiplier. If you chose to add an analysis step at the beginning of each task, having students annotate 7 ^ 2 as 7 times 7, could be beneficial.
Order of Operations – Groupings
In language arts these groupings might be compared to commas. In my experience, students are confused by both! The prevalent behavior of rushing, guessing and assuming has left many students without a sense of precise communication in language arts and math.
By analyzing the task students can learn to look at groupings with a critical eye. The nesting can be understood if they know what they are looking for in the shape and placement of each parenthesis or bracket.
PEMDAS as Order of Operations
Initially, I would like to suggest we share this acronym as the mnemonic devise that it is. These devices are helpful when learning to tie your shoes or getting through med school. The biggest error in PEMDAS as an acronym is the suggestion that multiplication is before division and addition is before subtraction.
When students deeply understand these operations to be inverse of each other, it makes sense that one would not take an advanced position over the other.
So, we use the device to remember the operations, but Multiplication and Division are to be solved in the same step, working left to right. Addition and Subtraction are to be solved in the same and final step, working left to right.
Action Items for Teaching Order of Operations
- Create an anchor chart showing PEMDAS where multiplication and division are equal as well as addition and subtraction.
- Stop, give time, and support students in analyzing all the tasks before beginning to solve or simplify.
- Listen to the questions and answers students provide during instruction to identify their personal struggles to appropriately support their success.
- Provide time for students to practice perfection. Best practice would be to check in on struggling students at least every other problem. Continuing to practice incorrectly digs a larger deficit than not practicing at all.