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The Downward Spiral of Math Avoidance

Okay, so imagine I’m a student who is having a tough time in math. Because I’m having a tough time with the content, doing the homework is a struggle, and that struggle is uncomfortable. Homework is bad enough as it is, but what I’m experiencing is much worse: homework where I don’t know what I’m doing. Hence, I often choose to not do the homework.

So because engaging with math is uncomfortable, I avoid it. Because I’m not practicing, I fall further behind. I never completely understand any of the topics being taught in class, so I’m unprepared for whatever comes next, which means that the next topic will be even more uncomfortable, so I’ll be even more inclined to avoid it.

This only gets worse as time goes on. In class, I feel less and less inclined to engage with what’s being taught, less and less inclined to take notes and ask questions. At home, I conveniently “forget” to do my homework. On tests, I feel like I’m drowning. In other words, I’m experiencing the downward spiral of math avoidance:

This downward spiral leads me to believe that I’m “bad” at math. But because I can’t see this feedback loop – because I don’t understand the pattern – I think that I’m the problem. I think there’s something wrong with me, that I’m “not a math person,” or that I’m simply not smart enough to learn the math.

And because each new math topic is more uncomfortable than the last, I start to really dislike math. I decide that math just isn’t my thing. I don’t recognize that I just don’t like things that I have a hard time with. It’s a natural human tendency to prefer doing things we do well and avoid things that don’t feel good. If I always get praised for the way I sing, I’ll keep on singing. If I crash my bike and get hurt every time I go for a ride, I will quickly decide that I don’t like riding bikes.

This downward spiral of math discomfort and math avoidance sets me up to have fixed-minded beliefs about my relationship with math that actually have nothing to do with my own abilities and everything to do with this feedback loop.

Reversing Course

The good news is that feedback loops can be reversed. If I take notes in class, ask questions, learn techniques, use resources to figure things out, and practice math on a regular basis, I’ll start doing better. This will feel good, making me less inclined to avoid math. As I continue engaging with math, I will find more and more success, and math will be more and more enjoyable.

Thus, the happy opposite of the downward spiral is the following feedback loop:

Reversing course may be simple, but it’s not easy. If I’ve been in the downward spiral for a long time, the behavior pattern will have some serious momentum behind it. It’s always better to intervene early. The further I fall behind, the harder it is to catch up.

The longer I’ve been in the downward spiral, the more toxic my relationship is with math. Once I’m really deep in the pit, I won’t be able to see the light of day, and I’ll lose hope. If I’ve fallen to that depth, I’ll need a great deal of help climbing back out.

I will need a steady stream of growth-mindset reframes from coaches, parents, tutors, and teachers. Because I don’t believe in myself, I will need to hear these people repeatedly express a certainty that I am capable of figuring the math out and capable of getting caught up. But this must be realistic optimism, backed up by concrete support and my own hard work. I might simply need to be reassured that I can, in fact, reverse course and build positive momentum.

I will need content tutoring to support me as I struggle with math concepts. I will need academic coaching to learn techniques to manage cognitive load and form stronger memories, such as making written product and spaced repetition.

I will need to steadily engage with math by paying attention in school, completing my homework, and doing extra practice beyond the homework. This will be very uncomfortable at first, so I might need procrastination coaching to help me get started. I might need help overcoming perfectionism.

Seeing Where You Are

Most students who are caught in the downward spiral of math avoidance are unaware of it, and the few who are aware usually can’t see a way out. Author and YouTube educator John Green said in his TED Talk that “You very rarely go to a place that isn’t on your personal map.” So a student needs to at least know that there’s an alternative if he’s to have any hope of going there.

Students who live in the downward spiral are like hikers stumbling around in the fog without a map and compass. They’re lost, and they don’t know how to get unlost. Our job isn’t just to show them how to do the math, which would be like showing them how to hike, our job is to help them see the territory and navigate it deliberately. Our job is to clear the fog, so they can choose the right direction. We cannot make them walk the path, but we can provide them with the necessary tools and present them with a choice.

Sometimes I have a conversation with a student that goes like this:

Student: “I’m really bad at math.”

Chris: “Oh. Let me ask you something. Aside from the homework they give you, how often do you practice math?”

Student (brow furrowed): “Um, never.”

Chris (happy tone): “Okay, that makes sense. I never practice basketball, so I’m really bad at basketball.”

Student: “So you’re saying I should practice.”

Chris: “Nope. I’m saying you could. You don’t have to, and I won’t be upset if you choose not to, but you could.”

Then we might discuss something they are “good” at, like a sport, a musical instrument, or a video game. And then I map out the downward spiral, so they can see that the issue is structural rather than personal. This pairs well with a discussion of how math is an upside-down pyramid and the mastery path, which are two more structural reasons students struggle with math.

I’ll explain that there is no quick fix, but there is a way forward, and that I would love to support the process if they’re interested.

This applies to more than just math. The downward spiral of discomfort and avoidance shows up in reading, writing, science, the study of foreign languages, and even school as a whole. It’s important that we recognize it and respond to it as soon as possible.


Image Credits

Title Image: Creative Commons Public Domain. Courtesy of Pixabay.

Feedback Loops: Loper, Chris. 2018.

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Teach to Learn

“He who teaches, learns.”

This classic dictum is normally used to explain how those who teach learn a surprising amount from both the act of teaching and from their students. But it also offers a prescription for effective studying: If you want to comprehend and remember something, teach it.

When you teach something to another person, you have to really understand it. You have to be able to simplify it and summarize it. You have to be able to explain it several different ways. If you can’t restate the concept using the phrase “in other words,” you don’t really get it. The ability to teach something distinguishes mere fluency in concept from true understanding. Attempting to teach something, then, is an excellent way to see how far along the mastery path you’ve come.

Teaching often requires the use of analogies to help people connect new concepts to previously understood ideas, and the metaphors we use to teach improve our own understanding. Creating such analogies demands creativity, thereby offering a chance to exercise those mental muscles. Both this creative challenge and the other difficulties of teaching require so much effort that the act of teaching does an excellent job of convincing your brain to care, thereby greatly enhancing learning.

This means that one of the best ways for students to help themselves is to help their fellow students. If a classmate is struggling to understand something, and you teach it to him, you both benefit.

Because studying is so often a solitary act, teaching to learn remains one of the most underused study techniques. But you actually don’t need anyone else around to use teaching to enhance your learning. You can simply pretend you have an audience. When I engage in mental recall practice, this is usually what I am doing. I have an imaginary conversation with someone about the topic I’m trying to comprehend or remember. I try to present the ideas in this imagined conversation in a clear, logical order. I imagine what questions the person would have, and I answer them. If I can do all this way from memory, then I know I’m well on my way to remembering it for good. If I can’t, I’ve got more work to do. Teaching, whether to a real audience or an imagined one, is a powerful form of self-testing.

Greg calls this his “crazy professor” talk. It is no accident that we often see scientists portrayed in movies mumbling to themselves. When a scientist does this, she is holding a Socratic seminar in her mind, and in doing so, she is deepening her understanding. You can do this while waiting for the bus, walking the dog, doing the dishes, or getting ready for bed.

A Note for Teachers and Tutors

This idea has also spurred one of the biggest shifts in how I interact with students as an academic coach.

Here’s a well-meaning question I used to ask all the time: “Does that make sense?”

And at least 90% of the time, students nod and say “Yes.”

But here’s the problem: They say yes even when they’re confused. Most students aren’t willing to admit that your explanation of a concept didn’t make sense to them. And unfortunately, this is especially true for the students who have the hardest time making sense of academic ideas. Many students with learning difficulties have developed this behavior over time: If an adult asks them if they understand something, they can just say “Yes,” and then the confusing topic gets dropped, and the conversation ends. This is a subconscious strategy of avoidance. It does not facilitate learning, and we shouldn’t encourage it by asking the question.

So, instead of asking students whether or not they understand something, I now insist that they prove it. If the student can explain the concept to me in her own words, then I’ll believe she understands it. If all she can do is regurgitate the exact language that I or the textbook used, her understanding might be no better than that of a well-trained parrot.

This turns out to be a big deal. The vast majority of test questions don’t use the exact same wording as the teacher or textbook. Plus, many test questions force you to apply several concepts at once. And you’ll often need to apply what you know in later courses or in real-world problem solving. You can’t do that if all you’ve achieved is rote memorization. True understanding is required.

Students need to practice combining the vocabulary and the concepts they’re learning into “the story” of the content. Doing so taps into a key aspect of natural learning – storytelling – thereby improving both understanding and memory. It is critical for educators to insist that students express full thoughts and complete sentences that form complete paragraphs and a coherent narrative.

Now, in practice, what I’ve just described is very difficult for most students, and it’s something that’s normally only achieved in a later stage of the learning process. At the beginning, when a concept is brand new, we need help. One very helpful tool is the “word bank.”

Word Banks

The first time students attempt to say back to you what was just explained, they might appreciate having a list of the vocabulary that they should be using in their explanation. They get to use this word bank as a reference when they attempt to explain the concept to you. Because this lightens their cognitive load, putting the puzzle together is easier. Their only job, at first, is to put the words into a logical order and make sentences.

You can ask prodding questions to get them to say more. You can give hints. Sometimes we put the words onto little notecards so students can manipulate them on the desk, trying to arrange them in the proper order before verbalizing their explanation. If this is all still too challenging, you can explain it together and ask the students to finish your sentences, or you can generate a fill-in-the-blank exercise.

Early on, many students won’t be able to tell you the story, even with a word bank, so we sometimes begin this process by having students answer questions using the word bank. Later, they’ll need to be able to answer without a word bank. Eventually, you can start insisting on more details. If the student answers a question correctly, push her toward additional depth by saying “Yes! What is that?” or “Yes! Tell me more about that.”

Eventually, they should be able to teach it from scratch with no resources.

“In other words…”

As I mentioned earlier, the ability to explain the same idea in multiple ways is the true demonstration of understanding. This means that, at some point, students need to challenge themselves to move beyond the vocabulary of the word bank. When students can answer your questions with the correct vocabulary, they may have simply memorized the right responses without actually understanding what it all means. The next challenge, then, is to have the student explain the content in other words.

This is something that students can do in class and in tutoring. The game is this: The instructor explains something, and then the student replies, “So are you saying that ______________?” putting it into her own words. By doing so, she’s either proving that she gets it or discovering that she’s misunderstood something. And she’s tapping into the power of asking questions and practicing taking healthy risks.

Often, a good ‘in other words’ statement will include figurative language. In math, we might say that distribution is like dealing cards to players around a table – a simile. In chemistry, we say that chlorine desperately “wants” an electron and will “steal” one from an element like sodium – a personification.

One important thing for students to be able to do is connect the concepts to related or parallel ideas that aren’t being mentioned by the book or the instructor. If they can draw connections between what they’re learning now and what they’ve learned in the past, they’ll form a deeper understanding and a stronger memory of both the old and the new concept. If students don’t mention these connections when they’re explaining new concepts, you should ask them how the new information fits into their preexisting web of knowledge.

I get to experience the power of teaching all the time because I teach for a living. What many people don’t realize is that, as a tutor, I’m also a student. There remains a huge amount of academic content I don’t know, so my job requires constant learning. I routinely arrive at better understandings of the things I’m learning when I teach them to my students. I strengthen my memory and deepen my comprehension every time I think of a different way to explain something.

In other words, the best way to be a good student is to become a good teacher.


Image Credits

Title Image: Creative Commons Public Domain. Courtesy of Pixabay. Text added.

Parrot: Creative Commons Public Domain. Courtesy of Pixabay.

Mind Map: Creative Commons Public Domain. Courtesy of Pixabay.

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Start With The Summary

A chapter in a typical textbook will present all the information, in detail, and then conclude with a chapter summary. The logic behind this is that, after reading all those details, you’ll want to refresh your memory with an overview.

The science of learning, however, shows that this standard format is backwards.

It turns out, we comprehend and remember the details better if we read the summary first. Luckily, there’s no law against flipping through the pages of a textbook out of order. Any student can easily flip to the end, read the summary, and then return to the beginning and read the chapter. Of course, this requires actually having a textbook, which we are strongly in favor of.

But why is this better?

Why does it help to start with the summary?

One reason is that, because a summary provides only the big picture, it is digestible. We can understand it. And once we’ve discerned the lay of the land, it’s much easier to examine the details without feeling overwhelmed and confused.

History textbooks are perhaps the best example of this. A typical history chapter will present a deluge of information that inundates the mind. Most readers don’t know which details to focus on. What really matters here? Which facts are most likely to show up on the test? But when readers get a sense of the big picture first, they navigate through the chapter more adeptly. Because they understand what story the information is telling, they have an easier time deciding which details to pay attention to and which ones to discard.

Often, a chapter summary will briefly mention how this chapter’s content fits into the textbook as a whole – how its ideas are connected with the other ideas in the course. When we see how it fits into the larger puzzle we’re building, it makes more sense and sticks in our minds more easily. The new knowledge has a home, a place where it belongs. Conversely, ideas are much more difficult to understand and remember if they’re presented at random with no explicit connection to things we already know.

Normally, the big picture offers a sense of the subject matter’s significance. A good summary will explain why the content is important. When we know why something matters, we become more motivated to learn the details. We also remember the content better because significance is one way to convince the brain to care.

What if there is no summary?

Unfortunately, many textbooks do not contain a chapter summary. Dr. Barbara Oakley, author of A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) and co-creator of Coursera’s Learning How to Learn, offers a solution.

She suggests that we flip through the chapter and read the section headings and sub-headings before reading. When we do this, we get a sense of where the chapter is going and what the big ideas are, which is almost as good as reading a chapter summary. This process also creates metaphorical bookshelves in the mind, complete with boxes and folders. Learning a mess of details is easier when we can sort the information into categories and sub-categories.

In many cases, however, this won’t be enough. For example, the most commonly used textbook for AP World History is Strayer and Nelson’s Ways of the World, which contains a staggering amount of detail and no chapter summaries. The amount of ground covered and the level of detail make reading the section headings in each chapter woefully insufficient. Students need to seek out summaries and additional teachers online to make AP World History easier to navigate. Here are some resources that I’ve found:

And it’s not just textbooks that struggle with this. Sometimes teachers neglect to provide an overview before diving into the details. I see this most often in biology, specifically with the topic of meiosis. Meiosis is the special type of cell division that produces sex cells: eggs and sperm. The process takes many steps, introduces a bunch of new vocabulary, and is, well, pretty overwhelming:

Students benefit enormously from seeing a highly simplified version of meiosis before attempting to digest this Thanksgiving feast of biological detail.

But just as no one needs to be a victim of a textbook without chapter summaries, no one should be a victim of a teacher who fails to deliver an overview before teaching the details. In most classes, it’s fairly easy to predict what topic is going to be taught next, and seek out a pre-lecture summary of the topic that will be covered in tomorrow’s lecture. And if you’re not sure what’s coming, ask questions.

Students are in charge of their own success. Preparing for lectures by finding summaries the night before and seeking out additional resources to improve your comprehension of a textbook are both excellent ways to practice being an active agent. The ability to go find what you need and use it to succeed is the most important skill students can learn in school.


Image Credits

Title Image: Creative Commons Public Domain. Courtesy of Pixabay. Text added.

Map: Creative Commons Public Domain. Courtesy of Pixabay.

File cabinet: Creative Commons Public Domain. Courtesy of Pixabay.

Meiosis: “Meiosis.” Encyclopedia Britannica. 2010.