# Improving Reading Fluency with Proofs

• andytoh
In summary, to improve reading fluency with proofs of theorems, some suggestions include: breaking down the proof into sections and determining the task ahead of time, drawing pictures to aid in remembering concepts, reading shorter and easier proofs for practice, and trying to recognize what the author is attempting to do in the proof. It is also recommended to read books that provide a structured approach to proofs. However, it is important to note that some authors may not conform to these methods, and ultimately, practice and experience are key in developing fluency.

#### andytoh

Any suggestion on how to improve your reading fluency with proofs of theorems?

It's frustrating to spend over 1 hour to read a proof of a theorem that is under 1 page long (or not understanding the proof altogether). Even when every subtopic within a proof is already known, I find that that one of the main stumbling blocks is forgetting the meaning of symbols and results established within the proof, causing you to spend time backtracking within the proof. But if this is the only problem, then it's just a problem of short-term memory, which I don't think you can do much about. It's definitely more than that though. Drawing pictures always helps to remember what is what, but I don't find it to be enough to be able to read a difficult proof smoothly.

Perhaps reading shorter and easier proofs of easier theorems for practice? Does doing more exercises help much in this regard?

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Many proofs involve some degree of mathematical maturity. Don't let yourself get discouraged; struggling, yet trying, to understand a proof, no matter how frustrating it is, is good for you. As you read and read, you will find yourself allot more comfortable. There is no magic trick to get good at it, at least that is what I believe.

Ah, the infamous mathematical maturity. The best thing to do if you don't understand a proof is to try to recognize what the author trying to do. Is he doing a proof by contradiction? Some kind of direct proof? So you just follow along and see what he's doing. Grant him all the claims he makes, and if they are true, does the proof work? Then if that works out, check each of his individual claims. Then re-read everything and you should be happy.

Some mathematicians have a horrible sense of mathematical prose...

Good advice, but it's an advice that everyone reading a proof already follows. I wasn't asking how to understand a specific proof, but rather how to become more fluent in reading proofs in general. But perhaps, I guess there is no magic trick for that. Just practice, practice, practice.

Considering andytoh is reading differential geometry and differential topology proofs (or is it even more complicated now andytoh?), I assume he possesses the infamous 'mathematical maturity' element. I think your question will be better answered by those closer to your level of mathematics, than those of us still working through undergraduate courses.

I am glad you mention that you often times forget the meaning of symbols or a result that has emerged previously, because that is the only issue I seem to have, when and if I do have a complication reading through proofs. I thought I might be alone but I am glad that I am not. It is frustrating when your memory lags behind your ability to comprehend and understand something.

Do they have mathematical thinking books related to higher-level maths that might help? I think someone like mathwonk or matt_graves, and some of the other mathematician's could provide to you better advice.

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Who's matt_graves?

neutrino said:
Who's matt_graves?

Good call, who knows? hahah

I botched the dudes name. I was referring to matt_grime :P he is the homie!

I have found one way of improving reading fluency, but its not a perfect cure. And that is to break down the proofs into sections. Every long proof has certain tasks and sometimes is it best to determine the task ahead of time (what is the next step that the proof is try to achieve), then jump to the conclusion after the task. Then understand the next task and so forth. Sometimes this requires jumping to the second last line of the proof to understand what the objective is.

At this point, I typically have each mini-result underlined and so the skeleton of the proof outlined. At this point I read the fine details that establishes each mini-result. It does seem more clear this way for me, and I find it much easier this way than to read the proof line by line without knowing what the next stage is trying to achieve ahead of time. Unfortunately, some writers do a poor job of paragraphing proofs, or stating what the next task is.

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Reading the book "How to Prove it: A Structured Approach" by Velleman, helped me a lot.

andytoh said:
I have found one way of improving reading fluency, but its not a perfect cure. And that is to break down the proofs into sections. Every long proof has certain tasks and sometimes is it best to determine the task ahead of time (what is the next step that the proof is try to achieve), then jump to the conclusion after the task. Then understand the next task and so forth. Sometimes this requires jumping to the second last line of the proof to understand what the objective is.

At this point, I typically have each mini-result underlined and so the skeleton of the proof outlined. At this point I read the fine details that establishes each mini-result. It does seem more clear this way for me, and I find it much easier this way than to read the proof line by line without knowing what the next stage is trying to achieve ahead of time. Unfortunately, some writers do a poor job of paragraphing proofs, or stating what the next task is.

Many authors prefer to adopt an approach that would not conform to your fragmentation of proofs, though. That is, they will proceed to several assertions that will not necessarily look relevant to you right at the moment, and will only show how these assertions make up the proof at the very end. Also some proofs are so "creative" that it is next to impossible to predict the "next step".

## 1. How does practicing with proofs improve reading fluency?

Practicing with proofs can improve reading fluency by helping individuals develop their critical thinking skills and logical reasoning abilities. Proofs require careful analysis and understanding of the text, which can lead to improved comprehension and retention of information. Through consistent practice, individuals can also become more familiar with common patterns and structures in written language, making it easier to read and understand similar texts in the future.

## 2. What strategies can be used to incorporate proofs into reading instruction?

There are several strategies that can be used to incorporate proofs into reading instruction. Some examples include having students identify key arguments and supporting evidence in a text, asking them to summarize the main points using evidence from the text, and encouraging them to make connections between different pieces of information within a text. Additionally, providing opportunities for students to write their own proofs can also be beneficial in improving their understanding and fluency in reading.

## 3. Can proofs be used for all types of reading material?

While proofs can be a valuable tool for improving reading fluency, they may not be applicable to all types of reading material. Proofs are commonly used in math and science texts, but may not be as relevant in other subjects such as literature or history. It is important for educators to carefully consider the purpose and content of a text when deciding whether or not to incorporate proofs into reading instruction.

## 4. Are there any potential drawbacks to using proofs in reading instruction?

As with any teaching method, there may be potential drawbacks to using proofs in reading instruction. Some students may find proofs to be too challenging or intimidating, which could lead to frustration or disengagement. Additionally, relying too heavily on proofs could limit students' creativity and critical thinking skills, as they may become too focused on finding the "correct" answer rather than exploring different interpretations and ideas within a text. It is important for educators to carefully balance the use of proofs with other instructional strategies in order to maximize their effectiveness.

## 5. How can educators assess students' progress in reading fluency with proofs?

There are a few different ways educators can assess students' progress in reading fluency with proofs. One option is to have students complete written assignments or quizzes that require them to use proofs to analyze and interpret a text. Another option is to incorporate proofs into class discussions and ask students to explain their reasoning and thought process. Educators can also observe students' performance during class activities and provide feedback on their use of proofs in reading. Ultimately, it is important for educators to provide a variety of opportunities for students to practice and demonstrate their understanding of proofs in order to accurately assess their progress in reading fluency.