More sets...math in our minds during the Spring break!

So today I was on the bus and the subway reflecting about the best review exercises for the exam I was supposed to give this coming Monday...right before the long Spring break. I noticed that we have just discussed the basics of sets. So today I decided that the exam will be on Wednesday after Spring break. 

I wanted to give students the opportunity to mature the different methods to prove a proposition, and also finish with the section of sets to then start such an important topic as functions. The student felt relieved after I made this announcement (I did not have big expectations since sometimes they don't want to study during breaks.) 

We had a simple class activity where two sets A and B were given (as well as a specific universal set U) and the questions consisted of finding intersections, complements, differences, etc. 

I was able to at least introduce functions and examples of a function mapping the set of students in a class to the set of final grades. They agreed that we can't assign two different grades to any particular student we choose. 

Well, this is taking the classic path: domain, range, one-to-one, onto, bijections, etc., etc. 

It brings so peace of mind to know that the class will have math preset during the Spring break. 

https://www.vecteezy.com/vector-art/16059920-easter-educational-quiz-for-kids-how-many-game-with-chicks-flowers-spring-holiday-theme-on-green-background-printable-math-activity-for-children-search-game

End of induction..."it's cold or hot, not lukewarm", sets and "The Fly in the Ceiling"

 Another Monday. Many of us were on campus before 7:30 am. The subway was not bad, I enjoyed my reading. 

Some students arrived with a cup of coffee in hand. 

We continued discussing mathematical induction. I asked them what they remember about proving by induction. Phrases such as basis step and inductive hypothesis were heard. 

We picked up on the example we left of proving the formula for the geometric series. Everyone got that P(0) was true. 

The step of using the inductive hypothesis P(k) to show P(k+1) was explained several times. We wrote the inductive step on a green box and what we needed to prove P(k+1) on a pick box (yes, with my colorful Japanese chacks.) I then stressed that if they really understand that they need to use the inductive hypothesis to show P(k+1), they pretty much got the concept.  I encouraged them vehemently to ask questions now, in the present. One student said that he was trying to understand how I used the formula of the inductive hypothesis on the sum involved in P(k+1).  I explained it, and he said "ok, so so".  I replied that "this step has to be hot or cold in terms of understanding, not lukewarm" This is something that you either understand or you don't. I believe this shocked them somehow, but in a good way. I also asked: "would be able to reproduce this proof in the future without memorizing it?". Once I got an affirmative answer from the class, I did not move on. 

I started introducing the basic definition of sets and then:

• empty set
• power set
• size of a set or cardinality
• finite and infinite sets
• Venn diagrams
• Cartesian products. Mentioned Rene Descartes (sorry for missing the accents), yes! Remember this book for children's book "The Fly on Ceiling"...it would be nice if I can get some copies for them!
• Operation of sets (unions and intersections)

The class was 100% chalk board...no a single slide projected, such a pleasure!

Spring break is approaching (they have an exam right before the break) and I told them that I felt so sad they would be so many days without math. Should I send homework for Spring break? 

https://www.mathsthroughstories.org/the-flying-on-the-ceiling-reneacute-descartes.html

Proof by induction...sometimes it takes only one geometric series

 We are in the middle of another working week and students to be more comfortable with mathematical proofs. The next exam is coming soon and I reflected on the main overall points that they have to take away from this course. One of them is understanding mathematical induction. 

I started by giving the general idea of induction. 

•  Let's think about a ladder and taking the first step/level and moving to the subsequent levels. 
• We can also think about domino pieces and what happens when the first piece hits the second one and so forth. One of them screamed that he knew what domino was because he was Dominican!  This helps the whole class to wake up 😊

https://www.uptowncollective.com/2019/05/05/dominoes-dominican-chess/

We formalized the idea of mathematical induction and move to the classical example of adding the first n positive integers.

Prove that 1+ 2 + ...+ n= n(n+1)/2

I encouraged them to always do what they need to prove. What is P(n)? 

Is the proposition true for P(1)?  What is the inductive hypothesis? What do I need to prove?

We then discussed the proof of a particular case of the geometric series. I am so glad kept pausing to ask "questions or comments"? Lots of questions about algebra and arithmetic. It made me reflect that many times we make the assumption that students remember the basic properties of powers and real numbers. If we discuss these struggles with the prerequisites, at least we are aware of what needs to be solved and take action.

We don't need to send 20 exercises so they never forget. Many times we just need that particular exercise that makes us reflect on the application of the properties. 

The lecture has a happy ending. The exercise was now proving the general formula of the geometric series by mathematical induction. Many of them were able to prove the basis step, P(0) and they were already starting to apply the inductive step P(k) to prove P(k+1)...we'll continue on Monday. 

Will they be thinking about induction during the weekend? I want to think that at least a seed was planted today. 

You can prove by contradiction at the basketball court!

 Another Monday morning discussing different ways of performing mathematical proofs. 

I asked my students if they were able to think about the proof of the proposition we left at the end of the last class. I got a timid answer..." I tried"...so we started with this class activity.  The idea was to get them familiar with proof by contrapositive.  They were to write the contrapositive of the original statement and then we discussed it together on the board. 

Next stop...proof by contradiction.  I believe the way the textbook explained it (despite being formal and correct), was a bit complicated for newcomers to the world of mathematical proofs, so I proceeded to write in my own words the meaning of the proof by contradiction method. 

Bottom line: "assume the conclusion is false and get something that contradicts your hypothesis or assumptions" 

One of the most illustrative examples: Prove by contradiction that √2 is irrational. 

Questions I asked before and at the beginning of the proof:

• What is an irrational number?
• What is a rational number?
• What is an irreducible fraction?
• What can we say of an integer if its square is even? 
• Where is the contradiction in the argument we are building? 
• Can the proposition p and the negation of p be true at the same time? 

I asked them to prove a proposition in the subway. They should take a picture of the notebook so I can see the subway seats in the background. The alternative is proving the statement in any place/context outside the classroom, office, or home (not driving!)

One of the asked me..." could it be anywhere then?", and I replied, "could be at the basketball court!". I saw the biggest smile of the day!

https://www.brooklynbridgepark.org/things-to-do/basketball/

Prove something in the subway!

 Today was a very fun class. We were able to finally dig into mathematical proofs. 

Examples such as "if n is an odd number then the square is an odd number" were discussed. 

We discussed the proofs together and one of the class activities was using a direct proof to show that sum of two odd integers is even. 

I made sure to stress that most of the propositions we have to prove would be of the form:

  ∀x (P(x) →Q(x))

What do you understand by hypothesis? For which values of P and Q is P→ Q true?

I believe this activity opened an interesting discussion. A couple of students wanted to work with particular numbers, but I told them we needed something more general. I asked them to pick their two favorite letters as the odd numbers to start using the hypothesis. They realized that as soon as they started writing the hypothesis, they know that the following step is writing how the expression of the sum would look like. 

We then recalled that proving the above conditional is equivalent to proving its contrapositive. We discussed a specific example where we got stuck on doing a direct proof and realized that an indirect proof using a contrapositive was more viable. 

The class ended by writing on the board a statement I wanted them to prove by contrapositive. 

I suggested getting a small notebook where they can write proof in the subway. I joked and said: "and you can send me a picture of you with your proof in the subway". A student asked if this was serious...now I believe this would be a good idea. Why not? Would not this be way more fun than looking out at your smartphone during a subway ride? It would be so stimulating to read and write proofs in the subway!!!

One caveat:: a student drives to school. Another student said: "do it in red lights!!!"  

https://www.imdb.com/title/tt0117069/mediaviewer/rm1372943104/?ref_=tt_md_2

https://www.imdb.com/title/tt0117069/mediaviewer/rm1440051968/?ref_=tt_md_4

Monday, Monday

 Today has been gloomy and most of the students arrived a few minutes late. Gloomy... 8 am class. 

We agreed that if the day was not fantastic, we'll make it fantastic! We focus on building valid arguments for quantified statements. Rules of inferences for quantified statements were introduced:

• Universal Instantiation (UI)
• Universal Generalization (UG)
• Existential Instantiation (EI)
• Existential Generalization (EG)

After giving a couple of warm-up examples, we came back to the Socrates example. 

Students are identifying beautifully when to use modus ponens!!!

We then moved on to solve some fun examples. Some students had questions about the inferential rule of Simplification. if we know that the hypothesis is p∧q, should we use p or q as a conclusion?  The answer is...it depends! Which statements are present in the conclusion I want to arrive at? 

We can actually use simplification more than once on different premises. 

Coming attractions: Mathematical proofs.  I mentioned keywords for the next class: 

• Theorem
• Proposition
• Lemma
• Corollary
• Definitions
• Axioms

I'll see you in class! 

Discrete structures class this week

 Today we continued talking about inference rules: modus ponens, modus tollens, simplification, conjunction, etc. 

We stress the need of building valid arguments by using the given premises and application of various inference rules to get to a particular conclusion. 

I was able to engage the students by showing different examples including the ones given in English sentences where the premise variable need to be chosen in order to translate to logical statements. 

We had a very engaging class activity. The students were able to write the premises and the conclusion in terms of premise variables p, q, r, and s. 

A few were able to build the argument step by step, applying the hypothetical syllogism to the correct premises. The hint of using contrapositive was given so they could connect the different premises. 

One of the students explained the problem on the blackboard and we had interesting discussions and questions. 

One of them asked me: "how do we know we have to use contrapositive and what to do?" I tried to be specific by saying that we have negations among the premises but that there is not a single recipe to build a valid argument. 

Bottom line: practice, practice, practice the application of the inference rules. Homework will be posted soon! 😊

Data Viz lecture for the undergrad data science seminar BYU- City Tech

 Last Friday was my turn to lecture at the data science undergraduate seminar offered to City Tech and Bringham Young Univesity (BYU) students. 

I taught a Data Viz course to Yeshiva University students a couple of times and It was really fascinating to learn so many important aspects of the field not frequently mentioned in ML/Data Science. 

The perspectives and work of Edward Tufte, Tamara Muzner, Jeffrey Heer, and Michale Bostocks are so deep and extensive. Summarizing this in a single lecture can be a difficult challenge. I did not want to focus solely on the use of Python libraries for visualization (matplotlib or seaborn), Tableau, or D3. I wanted to focus more on stressing that what matters is what we want to communicate and the importance of the effectiveness of the visualization. 

I based my talk on Jeffrey Heer's materials from his data viz course at the University of Washington. One of his first lectures is about "the value of visualization".  I asked the students why creating visualization and they were on point with many of their comments:

• Answer questions
• Make decisions
• See data in the context
• Expand memory
• Present an argument or tell a story

My favorite one: Inspire!!!

Dat encoding fundamental was mentioned while showing Bertin's diagram from 1967. 

We finally got to one of the best visualizations in history by Minard about the disastrous Napoleon's march to Moscow. 

image credit: https://www.openculture.com/2019/07/napoleons-disastrous-invasion-of-russia-explained-in-an-1869-data-visualization.html

How many variables we are encoding? Notice the temperature, the size of the troop...what else do you see? 

We also mentioned how a visualization for the O-Rings in the NASA Challenger mission had tragic consequences. 

I must confess it has been hard for me to figure out a cohesive and succinct way of writing about such a fascinating topic without feeling that I am leaving relevant information out. You might be seeing more posts about this topic in the future. I'll be taking Edward Tufte's online workshop, so more important ideas will be coming! 

For now, we can start exploring examples of visualization idioms for multidimensional data such as parallel coordinates. 

Check out the cool library of examples on D3: https://observablehq.com/@d3/gallery

Until next time!

Week 2 (somewhere in the Spring semester 2023)

 This week we continued solving exercises with propositions involving quantifiers. I talked to every single student in my class about some aspects of the problems we were solving.  I really enjoyed how a simple interaction can change their perspective and help them gain an understanding of the process. 

When discussing problems involving a combination of different quantifiers (nesting), students have good intuition to guess the truth values of such statements. 

I wrote some of them in words, e.g., "for all n, there is at least an m satisfying such equation"

We switched gears to introduce some rules of inference.  What is the argument? What are the premises? What is the conclusion?

Rule of inference:  modus Ponens, modus Tollens, hypothetical syllogism (to be continued...)

Finally, we were able to start reviewing for the upcoming test. 

For a warm-up translate to English certain propositions involving the atomic propositions:

"swimming at the New Jersey shore is allowed" and "sharks have been spotted near the shore" 

Also, we work with a set of system specifications to check if they were consistent. The students worked on translating each system specification into logical statements. One challenge was figuring out the direction of the implication for a conditional in one of them. We reviewed the different ways in which a conditional can read. 

We finish by assigning various values to p, q, and r to determine if the system was consistent. 

Overall the class was very engaged and they learned from the discussions with me and their peers. 

I'll see you soon!😉