On Teaching Math Podcast, Episode 2
The Failures of Remediation and How to Fix Them
Remediation is typically focused on training students how to get answers. This is an overt focus on the development of procedural proficiency without conceptual understanding. Such an approach has students able to repeat the steps they were trained to perform under strict conditions. However, the understanding is minimal and students will not recall what they've "learned."
In this episode we discuss why remediation fails and what to do about it. Finally, we discuss a specific application of how to remediate effectively.
Be sure to read the show notes at the bottom of the page. Click here to download the show notes.
The last item up for offer here is a PowerPoint on why the order of operations works. Download it here.
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E2: The Failings of Remediation
What To Do About It
Hello, and welcome, to the On Teaching Math Podcast. This is the podcast where we discuss all things related to developing mathematical fluency in your students. I’m your host, Philip Brown.
Even though this is just our second episode, I’m super excited to announce our first sponsor! Our sponsor has been a HUGE player in math since long before any of us even knew what math was!
Today’s episode is brought to us by the number 5. That’s right, just like the old days with Sesame Street. Hey, did you know that the number five is the only prime number that is the sum of the previous two prime numbers? For bell work, or extra credit on a quiz, perhaps ask students to explain why that’s true. It won’t even matter if you don’t know why, or if they’re right in their reasoning.
This is about changing “how,” not “what.”
Before jumping into today’s episode, I want to make clear that this episode, like the last one, is about sharing ideas on changing how math, now what math is taught. A great lesson that comes from a great curriculum, with fantastic assessments and feedback is not very good, if the delivery is not on point. Teaching math has very little to do with sharing information and almost entirely to do with two things. The first is helping students integrate the information into their body of knowledge correctly. The second is encouraging them to do so.
While a new curriculum or delivery style, fancy technology and helpful new displays might be catchy, and even impress an unknowing administrator, they are no better than having students copy from the book. Without adjusting the focus, intent, and ways in which you engage with students, the outcomes will remain the same.
Today’s Episode is About:
In today’s episode we will be discussing a very controversial topic in math education. It is controversial because some elements of the education industry demand it, they say it is a staple of education. An overwhelming bulk of research suggests that at the High School and higher levels, it just does not work.
We will be talking about remediation. We will see why it doesn’t work. Then we’ll discuss how it can be set up to be effective. Then, in part two we will see an example of how remediation can be done effectively with something woefully misunderstood by students, the order of operations.
Defining a Successful Outcome in Math:
Let’s begin by establishing what a successful outcome from a mathematics education would be. It would make sense to start here because it is when this outcome is not realized by a student that student would need remediation.
First off, passing a test is not necessarily an indication of a successful outcome. Of course we want students to pass. But, did the test really assess their knowledge and proficiency? Let’s not go down that rabbit hole … yet.
Perhaps we can boil a successful outcome down to two things: Number 1: A student is successful in mathematics when they possess the literacy of mathematics. That literacy means they can read, write and communicate at the level appropriate for the class being taken. In order to do those three things, the conceptual understanding must be solid. With this ability comes recall, and the ability to build upon prior knowledge.
The second marker of a successful outcome is the one most commonly measured; procedural fluency. That is, can the student manage the tasks appropriate to this level of math in a timely manner?
Ideally, proficiency should be an outcome of conceptual understanding. But with an increasingly narrow focus on specific skills, all of the conceptual understanding is bypassed and procedure is taught to the exclusion of concept. It is not at all unlike memorizing phrases in a foreign language and being deemed fluent when your master the accent.
What Happens When Students Don’t Get It:
Typically, a student is identified as needing remediation because of the second marker…they cannot perform. Since we get what we measure, and we focus on measuring the procedural proficiency, remediation and tutoring develop this with laser focus. Motivated, or just compliant, students will develop a roadmap of steps to follow when the problem looks like this. By “this,” I mean the students find false meaning in the way certain math looks on paper. When they see the false pattern they’ve keyed on, they follow their if-then statements.
“If I see a square root symbol, I factor the inside number to find pairs of prime numbers. For every pair I write one of them outside and get rid of the other. Whatever else is leftover I leave inside the square root symbol.”
False Marker of Success:
So long as the testing device fits their experience, the students will show growth! SUCCESS! The problem is, they’ve been trained to pass a test to validate the efforts of the educators. It leaves the student feeling victorious, the teachers feeling accomplished, and both parties, over time, flummoxed. When this style of remediation is done in Elementary schools, say with fractions, and it looks like it works, all involved parties wonder why it doesn’t work at the high school level.
The answer is quite simple … it didn’t work at the Elementary level either. The methods and the measure of success were inappropriate and not valid.
Students did not forget later on, they never understood at all. They would likely pass the same test that was used to evaluate their progress, but they lack the ability to apply what they know. The laser focus on skill in exclusion of conceptual understanding and mathematical literacy is the heart of the failure.
Let’s take a step back and exam exactly what we’re talking about here with remediation. Remediation could be considered anything that reviews previously discussed ideas and procedures. Even covering prerequisite material at the beginning of a course is remediation. But, we’re talking larger scale here.
Formative Assessment and Formal Reteach:
A popular model works like this. Students receive instruction and then are given a common “formative assessment.” If students fail to earn an acceptable mark, they’re corralled into a re-teaching program. In small groups, the instruction is repeated to this clan of reluctant students. At the end of the session they take another formative assessment, often the same one. If it is different, it is only superficially so.
The students will of course do better on the second assessment, at which point they are deemed to have been caught up!
Another model, which is more aggressive, populates entire classes with those needing remediation. The class goes at a slower pace, should be small class sizes, and students are taught the basics, again.
Then there is the staple of mathematics education, the prerequisite review at the beginning of the course. In Algebra 2, for example, they’ll deal with factoring, the quadratic formula and the like. After all, how can you move onto polynomials beyond quadratics if they do not understand quadratics and basic factoring? In Algebra 1 the remediation starts with factoring numbers, signed number arithmetic and fractions.
Homework Review and Tutoring:
The time spent at the beginning of each class reviewing homework is remediation. One-on-one tutoring is remediation as well. Tutoring typical starts with the questions or problems in an assignment or on an assessment. The focus is on how to get answers, how to pass that assessment of procedural fluency.
And then there are the remedial courses, like the one I taught for the past three years at a community college. Math 081 is basic math for adults. Of course, if a person wants a college degree, they’ll likely need to take and pass College Algebra. That is out of reach for a person that cannot perform arithmetic. It is reported that up to 60% of first year college students need a remedial math course! And the failure rate in those classes is astronomical.
They All Do This:
In each of these models of remediation the students are trained on exactly what to do. “Here’s how you add fractions with unlike denominators.” “Here’s how you find the slope of a line given two points.” The focus is on training the students to follow a procedure, often without context or concept.
Why It Fails:
I have a fundamental philosophy that I try, diligently, to live by. If my actions are not aligned with my goal, my efforts will be futile. A slight variation of that philosophy applies directly here. If a solution to a problem does not address the source of the problem, it will be an anemic approach.
So this begs the question … why do students fail to make adequate progress? Did they forget the steps? Did they not understand what to do? Was the class too fast?
Let’s go one step deeper. Why would students forget steps? Why would they not understand what to do in the first place? Why would a class be too fast?
Have you ever heard a student say: “I get it; I just can’t explain it.”? That’s a good indication that the student does not possess the quality of understanding required for retention. They likely connect procedures, but have considerable conceptual flaws.
Students struggle in math for many reasons, and too often, they’re social or personal. We will not consider those at this moment. Let’s keep our focus on those ready to learn. The reason those students struggle is because they’ve not made sense of it. There are too many random, unassociated things to remember. They have a seemingly endless list of “If – Then,” statements to follow when they perform the “steps,” of mathematics.
Let me share with you a story that is truly the compliment of the endless list of “If – Then,” statements.
Perhaps the single most powerful demonstration of the efficacy of conceptual understanding I have ever witnessed happened in college. I was an aspiring math major. I thoroughly enjoyed the challenges, and I was particularly drawn to the over-all professionalism, organization and passion the math professors had. They were focused and purpose driven, which was a stark contrast to many other classes that felt like the professor was filling time and assigning busy work.
The first day of my Probability class during my junior year, the professor showed up late! I’d never seen that in a math class at the University of Arizona. I was shocked. But there he was, about 15 minutes late, disheveled, sweaty and malodorous. He was well into his 60s, and like a sooty outline of frustration, he had smeared outlines of his fingers on his bald head, which was ringed in stringy grey hair. He did not exude confidence!
The board was covered in writing from a previous class and the tray under the board did not have an eraser or marker. The professor reached into the trash can and grabbed some used tissues and erased the board. Then, with TSA thoroughness, he patted himself down feeling for a marker.
He left the room, heads swiveled around nodding to validate the common experience of the students once the door shut. When the door reopened, all eyes were fixed on the professor. He had a marker in hand.
He moved to the front center of the room, tugged down on his shirt, took a long breath to calm himself, and finally spoke. He explained he does not own a car and had a series of flat tires on his bicycle on the way to work. It was Tucson and August, all of which explained his appearance and tardiness. He apologized profusely.
Then he said, and I’ll never forget the words, “I guess I’m teaching you probability. That’s a bit of a tricky one for me. I never use it in my research, and I have not taught it since I was a grad student at MIT.”
“But, I remember this …”
He wrote on the board what he understood to be the essence of probability. Then he explored, tested and expounded upon that single conceptual rock. This is how the entire class unfolded. It was so powerful that I began to think it was an act.
It was only later, when I’d visit him during office hours for help in another class that I realized it was not an act. He really knew, intimately well, the foundations of all sorts of mathematics. From those basic pillars, he was able to construct the rest.
Why It Was Powerful:
Of course, he was a brilliant adult, with a Masters from MIT and a Ph.D. from Stanford. But the take-away is that because he knew a handful of facts, he was able to fill in the gaps. That’s what conceptual understanding does. It allows a person to rediscover lost knowledge and ability.
It seems logical to me that the more intelligent a person is, the more they can keep track of. The less intelligent someone is, the less they can keep track of. If a student is struggling in math, it would be a wonderful gift to compress all of their “If – Then,” statements into a handful of facts.
That compression takes a lot of intelligence. That’s why my professor was able to do it. I could not compress his knowledge. The reason I could not seems like an issue of bandwidth. I just can’t keep that really abstract math me knows from disintegrating into a seemingly endless set of facts. That’s likely how our students feel. Those with the ability to compress their knowledge into a handful of conceptual ideas have already done so. Those that need help with the math, would likely greatly benefit from some assistance with compression.
Remediation, in the forms discussed here today, bypasses this entirely. The focus is too narrow and the scope, the time range used to claim efficacy, is too short.
Let’s Starting Pulling It Together:
There is a lot of research that says remediation does not work for mathematics students after a certain age. Perhaps this is true, but perhaps it does not work at any age … if the focus is on developing procedure to the exclusion of conceptual understanding.
My challenge to you is to dig, ask creative questions, and determine if the extra help you are providing for your students is focusing on procedure. Or are you guiding students so that procedure is an outcome of concept?
Have you identified possible conceptual errors students might possess? Think of our sponsor, #5. If students are just taught that 2 + 3 = 5, making 5 the sum of two previous primes, and that 1 + 2 = 3, but 1 is NOT prime, then we have lowered the expectation. Students no longer have to identify a key fact and use it as it applies. Instead, they have to recall information.
But there’s a second half to this question… why is this property only possessed by the number 5? So even if you told students about 1, or ask too leading of a question, there’s more opportunity for students to engage their thinking and communication.
Getting at conceptual understanding is a tricky thing to do. Most likely you never received that type of instruction as a student. You’re almost certain to come up empty if you search for publications that approach math education this way. So it is up to us to be creative.
One Last Common Practice of Remediation Styles:
One last thing before we dive into teaching the order of operations to Algebra students. There is one more major commonality between all of the remediation techniques and practices that we didn’t mention. It is typical that when re-teaching old content it is contextualized and delivered at the same level the student would’ve originally been exposed to the content.
For example, it would be common to teach fraction arithmetic to high school students at the same level, and in a similar fashion, as was done when the students first saw it in fifth grade.
This is ineffective. Students have seen the information before and there are no magic words to make it “click,” this time. It is better to teach the old information in a context that is relevant to the current class. It is often when students see some old, misunderstood idea applied in a new way that they finally can understand.
Apply it Specifically:
With that said, let’s get into teaching the Order of Operations to Algebra 1 students.
The Order of Operations is certainly prerequisite material. Teaching it in Algebra 1 is certainly remediation! To make it even trickier, it isn’t a discoverable fact, but a concept rooted in convention, like grammar rules. The order of operations is the law of the land for how we write math. It is a commonly agreed upon notation system that allows us to couch increasingly powerful methods of arithmetic.
That makes this a double-tricky topic for kids to master in remediation. Here’s what I do.
Get the basic fact out there, from students:
Students already can recite PEMDAS, or whatever acronym is popular. They can explain that multiplication and division are done from left to right, and then the same goes for addition and subtraction. If you give them a set of problems that are stylistically similar, they’ll master them. But this likely has more to do with false clues than literacy. Change things up and it gets tricky for students!
Have them use that knowledge in a novel way, in a new context:
Instead of just giving students a worksheet of problems to calculate, it is more effective to challenge their thinking by asking the same old question in a different way. After reviewing the facts I like to give them mathematical statements that are NOT true, and have them either add arithmetic signs or group things with parenthesis until the statement is true.
A simple example is 2 times 5 plus three equals 16. Students would have to group the 5 + 3 in order for the statement to be true. This is of course simple, but it needed to be in order to be described verbally! I’ll put a link in the show notes so you can download a worksheet of these problems.
Another way to do this is just give two strings of numbers with an inequality. 2 5 12 1 < 4 2 9. Students need to add arithmetic signs and parenthesis until they get a true statement.
This does a few things. It helps them to learn to read and consider more carefully through trial and error. It is engaging and challenging. But perhaps most important, it isn’t the same old thing they’ve done over and over again. By breathing fresh life into the topic, students are willing to try things, make mistakes, and work with each other to learn.
Introduce the old material as it is applied at this level:
The next thing I do, usually the next day, is introduce some common formulas students will see and use through-out high school math. Students already know how to substitute and evaluate expressions, so this is similar, but embedded in high school level material. A great place to start is with the quadratic formula. The surface area of a cone or pyramid is good, as well as the distance formula and volume of a sphere.
For more advanced students you can always give them the volume of a sphere, or cylinder, and have them solve for one of the dimensions.
This gets the students practicing arithmetic and thinking about the order of operations in a way that lays the groundwork for future learning.
After this exercise, which is usually about 15 minutes or so, I introduce students to the basics of function notation. I only want them to know about the input and output, and how to read it. By having them read the function notation, it feels like new material for them. But evaluating a function at a given input is not significantly different than being told to substitute a value written out in common language.
But again, this also gets students applying their knowledge in a new way, which will expose misconception and misunderstanding that can be addressed, and does so in a way that aligns with the purposes of their current class.
It is well worth the two days spent!
Summary of Effective Remediation:
- Start with what the students know (from them)
- Ask them to use their knowledge in a new way, in a new context.
- This is not demonstration, they have to explore, take risks, be willing to make mistakes. The moment you demonstrate is the moment remediation becomes about procedure.
- By exploring and trying, students will uncover misconceptions!
- It is when the misconceptions are uncovered that students will begin learning.
- Show how the old concept is applied and embedded in new content in a way that is accessible.
In the show-notes I’ll place a link where you can find some lessons on Order of Operations for High School. This is my most popular selling lesson on TeachersPayTeachers, but I’ll give free copies to the first twenty people that email me. My address is: firstname.lastname@example.org
I hope you found this episode interesting and challenging, as well as informative and useful. If so, please let me know. Send me an email, reach out to me on Facebook.
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Thank you once again, and have a great week teaching. Please tune in next Thursday for the next episode.