Learning Discrete Mathematics and Computer Science
via Primary Historical Sources

Student comments on post-course questionnaires concerning the benefits of learning from historical sources reveal:

• 'The benefits are that I learned how math came to be and why. I believe that I need the background to make a stable foundation of my mathematical knowledge, which I am just truly learning. If I can understand the why, then the how it works should come easier.'
• 'To link together different ideas that occur over time. It also enables us to grasp a better understanding of why that historical period thought about a particular mathematical subject.'
• 'Understanding the origins and foundations of mathematics is useful in understanding why certain assumptions are true and gives me a more solid foundation to understand the `bigger picture' and higher levels of mathematics.'
• 'The benefit that I see is that it helps to understand a subject more deeply if you know how that idea originated.'
• 'Love it, good to see where mathematics evolved from.'
• 'Learning mathematics from historical sources allows the student to understand how it is we have arrived at the mathematics that we have today. It also gives perspective to how long it has taken and how difficult the task was in order to arrive at modern mathematics.'
• 'It is easier to understand the progression of thought that occurred as the ideas were being developed.'
• 'You get to see how the people who generated these ideas thought and did their work.'
• 'A benefit would be that it creates motivation of how some math ideas came about rather than just introducing it.'
• 'It is extremely useful to see and understand how a mathematical concept has been developed and to see its progress through time.'
• 'You learn the way they did. It also gives you valuable context and connections that would not be made otherwise. Give you a personal connection with those who discovered the great theorems, which makes them easier to recall and to apply outside of the orthodox uses.'
• 'You see that mathematical concepts and uses transcend time, and the evolution of thought and acceptance of those concepts.'
• 'In my opinion learning from historical sources helps reinforce concepts.'
• 'The original sources can be debated to form new interpretations.'
• 'You can understand how ideas are thought up.'
• 'Historical sources will help us to learn a technology or a topic from basic steps. So, it lays a strong basic foundation.'
• 'One can learn that not everything learned in CS has to deal with programs or computers but rather a higher level of thought based on some problem. Another benefit would be the ability to compare and contrast different ways of solving the same (or similar) problems. Exposure to this can lead to new ways of thinking and learning.'
• 'Lots of things to learn from old ideas and models. Learn the basics.'
• 'It is interesting to see the train of thought of problems and notice that the idea may not have changed much even though the technology has.'
• 'It my view learning CS from historical sources has given me a different approach for solving and thinking different ideas in a different way.'
• 'It brings another aspect of the CS science. Sometimes it is good to learn with respect to the time in which some important discoveries have been made.'
• 'It is fascinating to see what people thought of, and proved, without having any of the machines we have today. In that context, it is interesting.'
• 'We can gain knowledge of several things and a possibility of applying them.'
• 'In my opinion, I think students can get benefits of learning logic and practicing the thinking. For me, I used to be scared at programming. (Because my basic concepts and syntax are not so good.) After the class, it seems like my logic is getting more clear.'
• 'It helps to put things in perspective and gives a more rounded approach to learning mathematics.'
• 'It is great for understanding math concepts on a deeper level because we can follow the thought processes of those who invented the math we study.'
• 'You get to see the reasoning and logic that was used to figure out many of the equations and theories that we take for granted now.'
• 'Understanding the human/cultural element of math helps me to understand the background of, remember, and relate the mathematical concepts.'
• 'Learning from historical sources helps us to link math topics together.'
• 'It helps you understand why a math concept is the way it is today and why we use it in the ways we do.'
• 'It makes me care more about learning.'
• 'I feel that we can better understand how mathematics came to be today.'
• 'Provides one with better understanding of underlying principles.'
• 'You begin to understand the basics.'
• 'We can see an ingenious method totally different from before.'
• 'Inspire our mathematical intuition.'
• 'It links modern ideas and the CS way of thinking to the way of thinking of the past.'
• 'Getting to the roots of computer science is useful because it makes you think differently and find other ways to solve and go about problems.'
• 'You get to see the connection between math and CS and how everything fits together. It is very beneficial for us to do this.'
• 'You see the progression in thought and it is also enjoyable to understand.'
• 'You learn more about the concepts behind computer science.'
• 'It's helpful to fully understand how something was derived.'
• 'Being able to understand where ideas come from.'
• 'It helps me understand the reason why things were put together like they are.'
• 'Shown basic ways of doing it, simple explanations.'
• 'Many important algorithms can be taken from historical sources and applied to code of a relevant nature.'
• 'It allows one to understand much of the background of CS and why computer scientists do much of what they do.'
• 'I really enjoyed it. I found it to be very intriguing.'
• 'I think that it gives the student a more deep understanding of the subject.'
• 'It is easier to see how certain concepts fit together.'
• 'Past mathematicians seem to have a much greater appreciation for math.'
• 'Learning how people thought is valuable, because you can get a path to help you out when coming across similar problems.'
• 'Learning about computer science when it was starting is a bit easier, since things were simpler than they are now.'
• 'You get answers to questions like 'where did all this come from?'
• 'It can help you understand future ideas.'
• 'It helps one understand the why along with the how.'
• 'I think learning the context of the math is valuable in that it makes the concepts being learned more interesting and meaningful.'
• 'It is very helpful, because you get to understand the basics.'
• 'For me, being able to see how the thought processes were developed helps me understand how the actual application of those processes work. Textbooks are like inventions without instruction manuals.'
• 'We get a better understanding on where the methods came from.'
• 'Walks the student through the process by which modern mathematics achieves new facts.'
• 'Makes learning the material easier.'
• 'I think it's easier to understand certain areas of mathematics when we can see where they are derived from.'
• 'You get to learn the roots of math.'
• 'It allows one to see how the modern, axiomatic approach to mathematics actually came about.'
• 'It benefits me in understanding the process of coming up with ideas of proofs.'
• 'You get to experience first hand how mathematicians came up with proofs.'
• 'Projects can be made more interesting than a typical lecture.'
• 'Helps me to link concepts and ideas more effectively than just being blindly told the result.'
• 'I like being able to 'see' how the formulas were derived and what process got people in the past to their claims.'
• 'Provokes more self discovery.'
• "One can learn how math works and how it was derived."
• "We develop an appreciation for the progression of mathematics."
• "Shows the process from an observation to a theorem."
• "As a student you get to see where the math we do today came from and engage in the kind of thinking that was necessary to create it."
• "You get to learn not just the rules and theorems, but how and why they were developed."
• "You learn the concept."
• "See why a certain type of mathematics was created and it's easier then to see its purpose."
• "I like to see where everything comes from and how it works, especially when I am able to make sense of it."
• "It gives me an insight into the concepts we now know as established. This helps me understand the mathematics better."
• "We learn that just because we haven't found the answer to something, that doesn't mean it can't be done."
• "It allows me to obtain a broader perspective of the foundations of computer science."
• "Shows us how we reached the advanced stages of computing we are in today."
• "Understand the ideas behind these monster proofs and theorems."
• 'It's very important, because it gives us a good place to start learning, and then develop our own ideas.'
• 'You get to see the progression of ways and techniques of thinking and seeing how other people approach problems.'
• 'You learn a lot more than just looking at examples, because you see how an algorithm, etc., was formulated.'
• 'I like the discussing in the class, so I can learn more from others' perspective and enlarge my insight into mathematics.'
• 'It provides excellent insight into how discoveries are made.'
• 'To see the original not contaminated by intermediate interpretations is nice.'
• 'Historical sources are good for understanding computer science as it gives alternate views and ideas, because to them it was not computer science, but reasoning and understanding.'
• 'Learning from historical sources shows how people developed the principles of CS before computers existed.'
• 'One benefit is that it gives a person a better view on how computer science worked in the past in comparison to now.'
• 'It helps us to see how computer science developed over time and how to develop easier and faster techniques in computer science.'
• 'You learn what exactly was the thought process for coming up with the way our computers think now.'
• 'You get to see where modern mathematics is derived from. Showing the evolution of math is very interesting and you get to see how it has become more advanced as well as more simplified. Very cool!'
• 'The benefits are, particularly with math, that you can get a better idea of the foundational skills involved in doing more advanced math.'
• 'You get to know where the ideas that we use today came from. To know something's history helps you better understand its present and future.'
• 'I think benefits would include being able to explain why this is the answer rather than saying the formula says so.'
• 'The benefits are that you learn how and why different math concepts were created and have evolved. It also expands your way of thinking about math concepts.'
• 'It gives more context to material. It takes you through a different thought process, and puts names with ideas.'
• 'It shows you how equations started and who came up with them. Origins are learned in other areas; why not math?'
• 'In other [math] classes you just do problems and learn formulas. But by learning the reasons why math was invented, etc., it make the material more interesting.'
• 'In my opinion it is extremely important to know where something came from and this is why it is beneficial to learn math from historical sources.'
• 'A really big benefit is actually seeing what important mathematicians wrote, to see their material as they intended us to see it. I also like to see the contrast between what they claim is true and what is modernly true.'
• 'I think that understanding the methods used by historical mathematicians and the modern methods used to describe the same concept provide a much more firm foundation for the student to apply the concept in other areas and in other contexts.'
• 'Learning directly from historical sources not only gives us the mathematician's ideas, but also how he/she comes up with the ideas and how he/she proves it.'
• 'Historical sources are beneficial in that they allow the student to learn about the backgrounds and meanings behind the concepts being taught.'
• 'Its a great way of gaining perspectives that can help mature my mind mathematically.'
• 'One can see how mathematics was first developed without the modern dress-up.'
• One gets to see how modern formalizations arose.'
• 'It's a perfect way to give math some context in the world. Historical sources teach you the math while simultaneously fitting math into history and give you meaning for why the math was and is important. Historical sources also break up the monotony of textbooks, making math more accessible for a wider variety of students.'
• 'The major benefit to learning this way seems to be the guidance we are given, rather than taking shots in the dark and feeling uncertain about the results. There is a clear marked and milestoned path laid out before the student. Another benefit may also be the sharing in the accomplishment of the historical reference -- in this case Henkin. A sense of understanding a 'mastermind' can be very rewarding.'
• 'To understand the evolution of thought and a certain topic in mathematics. To understand the origin of a branch of mathematics. To examine classical methods in mathematics and their short comings, and hence reinforce the modern view of an approach to a subject in mathematics.'