Tuesday, June 28, 2016

Go Shawty... It's My Birthday... In A Super Magical Square (6)

Well it's not my birthday today... so I'll celebrate a couple weeks late.  I was totally in love with and fascinated by these magic squares, especially by using your birth date.  I love Sudoku puzzles and this reminded me of that, but using more math!  I know, here I go again... but Jo B says that puzzles are a perfect way to engage your mind in mathematical thinking.  I started by using my own birthday and honestly, that one was pretty easy.  Like many others who have explained their thinking, I started by doing the diagonal first (obviously after the first row) and once I got that to add up to 120 then I worked down from there.  However, I quickly realized my other diagonal was not adding up to 120 and was much lower.  So I had to do some rearranging.  Since I had two low numbers in the first two columns as the top square, and one much higher number in the last column, I was able to use a nice range of numbers to get to my sum (120).  After I got the whole square set with each column, row, and diagonal adding up to 120... I decided to add a little extra challenge and make it a SUPER magic square.  So I focused on making the 4 corners add up to 120, then the first, second, third, and fourth quadrants to also add up to 120.  Lastly, I calculated the middle square to see if it would add up to 120 and it already did!

 So I decided to try more birthdays... because I'm a puzzle addict.  So I started to work on Emma's, one of my littles, and since she was born 4-2-20-11... I didn't have a huge number range to work with and I didn't have a very big sum to work with either (37).  The amount of combinations I could use to total 37 without repeating was very tricky.  So I gave up for a little while and did my boyfriend's birthday instead.  Even though he was born in January, eliminating early my ability to use the number 1, the rest of his numbers gave me a good range: 25, 19, 80.  I tackled his the same way I did mine by starting with the diagonals but for his, since I had already worked with quadrants for my square, I decided to try those first.  I think this actually made it easier... because once I got all the quadrants and middle squared away... with very minor adjustments the rows and columns were already in place.

Now for my crazy little Emmeline, I had to get creative.  So like Nick did, I also used a negative number.  I felt like this was probably an "illegal" move also... but it was the only way I could get it t work.  Especially since she already had 2 and 4 in her birthdate, I was limited about which small numbers I could use to adjust the sums.

So THEN I wanted to do something kind of crazy and see where it went.  Because I'm not adventurous enough, or experienced enough, I stayed in a 4x4 and used the first 4 letters of my name [MICH].  Then I assigned those as a number value [13, 9, 3, 8] and then developed kind of a "mod" style number association where numbers could be negative and also higher than 26.

That is actually the finished square once I got everything to be SUPER magical and equal to 33.  I tried to stay within 1-26 but it was basically impossible.  I mean, is it actually impossible?  Because the total was 33 so I need 24 combinations of addition expressions equal to 33 using only numbers 1-26, without repeating.  So since I could NOT figure out how to do that (I'm sure there is a formula... I just can't figure out how to figure it out lol) I went to using negative numbers and numbers above 26.  This led me to the above square... FINALLY without repeating - Well, technically I did repeat but when I tried to change it... I evened out all the rows, columns, and diagonals but then my quadrants and were not equal to 33 :( So I went with second best.


I subbed in the letter equivalents.  However, you will notice that many of letters ended up repeating.  So does this have something to do with the fact that I cycled them in a modular form... which I technically didn't even do correctly because I started A as 1 and not 0.  At this point, you know I had to go back and try to fix it to be done the way it is supposed to and with a correct modular formation (because my growth mindset makes me).  In this square, for the number portion I could NOT get it to be correct without repeating one number; same problem from the first time as well.  However, with the letter portion, I was able to eliminate one of the repeated letters!

By default, the repeated number (12/M) would be a repeating letter but the only other letter that repeated was U when it was used as 20 and -6.

While these puzzles took a fair amount of time, going forward I would love to further research the patterns or formulas that go into making the squares "magic"... if there even is one.  Based on my understanding of what math is though, I assume there must be a pattern.  Reflecting on what was taking place during the activity was actually a lot of adding and subtracting.  I roped my 7 year old into helping me with her birthday square, while she loves puzzles and math... it was a little over her head.  She WAS however at least practicing adding 4 numbers up to equal a certain sum.  So in that aspect, I believe there was educational value for her.

Thursday, June 16, 2016

The F word. (5)

Word, phrase, whatever it is.  Dun dun dun... The Fundamental Theorem of Calculus.  The bane of my existence.  When calculus is put in front of me... this is what happens on the inside.

And outside really.  Don't tell my best friend Jo B, but I "have yet to" [i.e. can't] understand calculus.  If I were in a battle and I was propositioned with "Give Me Calculus or Give Me Death," it was nice knowing you.

So I activated my growth mindset and after another failed attempt of understanding any form of calculus thought, I said to myself "Lulu, you are capable of understanding this and you WILL get a deeper understanding."  Here I am, back at it again.  First, I went to to try to explore The Fundamental Theorem of Calculus for dummies.

Here is my now [much deeper] understanding of the (1st) FTC:
                               F(x)=xaf(t) dt

  1. It's important. Phew! That was a tricky one at first.  You did it! Press on...
  2. Then... we have a function "f(t)", okay I'm totally with you.  The function creates a line on our graph.
  3. Now... they want me to find the area under the curve? This is called F(x).  So this F(x) IS the area under the "curve" or the function.  I still feel okay, pending I'm not completely wrong.
  4. Now, the area that we are measuring starts at value a and ends at value x.  Makes sense.
  5. Now I'm getting lost... dt represents the derivative of f(t) which is the amount that it is increasing or decreasing?  
  6. And that's it!
                         \int_a^b f(x)\, dx = F(b)-F(a).
  1. Okay... so now we tackle the 2nd FTC, which I guess solves for all definite integrals?  Well they switched me from f(t) to f(x) but I guess they're the same thing?  Also, F(b) - F(a) is apparently F(x) evaluated from a to b... but where did b come from?  Now I am lost.  But hey! I got through at least half of the explanation without crying.  I call this a win all around.

Sunday, June 12, 2016

My Bff Jo B. (4)

If I haven't already said this... Mathematical Mindsets is the best education "text" book I've ever read.  Jo Boaler, "Jo B", starts by taking you on a journey through how the brain works and what we can do as educators, parents, peers, etc.  The brain is such a complex being but the most fascinating piece was that our brain never stops changing or growing.  Inherently, we think that our "smartness" is what we are born with and we simply activate our preexisting knowledge by learning in classes.  Now if you actually think about that... it sounds ridiculous.  That's exactly what Jo points out.

Jo follows up this exploration of the brain by how the brain reacts when we are making mistakes.  I think so many of us have gotten used to knowing or finding an "easy" way out of many tasks and activities.  This has been evidently true in my learning and even my teaching.  Students are always wanting the "answers" and never interested in figuring out how to get there.  Students are TERRIFIED of making mistakes.  That is the standardized testing, cookie-cutter education that we are bringing students up in today.  Jo fights strongly to combat that, explaining that it's a total waste of time to teach and learn that way because your brain is not actually learning.

I've been politely reminded by Jo that math is an art, a beautiful art that is for the creating.  She spends a lot of time (and I won't lie... gets a little repetitive) explaining how to bring back the art and creativity of math.  Not only does she give many, many creative and realistic ideas to apply to the classroom but also apps and assessments.  This definitely one of the best books I've ever read in the field of educator education.  Not only do I plan on implementing a growth mindset into the classroom for it's overall benefit;  I also see mathematical mindsets in my students' futures.

The one awesome thing about this book is that I have been able to relate much of this discussion to what I have experienced with a few teachers, and in this class.  I find myself learning much better when the activities are "hands-on" and I'm able to try to understand it through my own mind first and then get an explanation.  As I have mentioned many, many times, I do still battle with this fixed mindset that I have had since at least high school if not before.  I wish I knew when my learning started to change but I could relate to so many of the negative thoughts and experiences that Jo expressed in the book.  Even thinking I was "good" at math because I did well on timed tests or that I didn't have to try to understand things until very high level math.

Sunday, June 5, 2016

The Dog Ate My Homework (3)

After reading Mathematical Mindsets (Boaler), I was intrigued by what she stated about the research on homework and why homework (or at least the kind that teachers give today) is NOT beneficial for the student.  How crazy is that thought?  I've always thought students need to practice what they have learned otherwise they won't remember it or understand it, but apparently, I'm wrong!

I decided to invest some extra research into how learning math is related to homework.  I'm pretty much a "type A" person so I decided to make a list of the negatives of homework that is being assigned and then the positives (if there even are any!)  The points I will reference refer to mostly the type of homework that I mentioned, problems presented after a lesson to practice those new skills.

1.  Differentiation: There is no varied level of ability to the homework.  All students receive the same homework no matter what level of understanding they have that day.  This can be detrimental to all levels of students encouraging some students to be bored and others to be defeated.

2.  Amount:  Students are being assigned "too much" homework or "too many" problems.  Once a student understands a concept, the repeated practice of it does not help the students understand the concept any better than they already do.  So students who struggle are not going to learn anything more than they did during the lesson that day.  Students who understand it do not need to keep doing it for 10 or 15 or 20 problems because they already have an understanding of what is being asked.

This quote from the journal below describes perfectly what goes wrong when students are assigned too many rote practice problems.

"The first gets frustrated and quits, the second gets bored and quits, and the third might get frustrated and bored by all the time it takes to get done or hastily complete the work with errors.  Some may copy each other's work along the way, too."

These are not the emotions we want our students to go through when they are trying to practice or continue learning at home.

3.  Parental Involvement:  Parents are either too involved or not involved enough for their child to successfully complete it.  There really is no winning.  Parents, while I totally love them and value their importance, are not trained in education or teaching of a child (in most instances).  Most parents are used to the way they grew up and the way they were educated and we are not trying to recreate the 70s, 80, or even 90s...  we are trying to revolutionize education and learning.  So what sense would it make to spend the whole day putting a child in one mind set, to send them home and ask their parents for help who have a completely different understanding and mindset about learning.  This does more harm than good.  Other parents may just want their child to succeed and think that good grades will mean that... while robbing their child of understanding, they rush their child into answering.

4. Home-School Relationship: Students are encouraged by parents, teachers, and friends to be involved in many "extra-curricular" activities.  Not only are these activities great for learning and their self esteem and to teach the child many aspects of appropriate social interactions.  When we assign homework to be 30 minutes for this subject and this many problems for this subject (which could vary from child to child in being 10 minutes or 2 hours!), we rob children of these opportunities outside of academia.


One of the most interesting things that I've taken away from this class is our "homework".  I've made the connection already between Jo Boaler's book, this website, and the class about the reflection piece of learning, like how in class we make a quick reflection on every discussion topic to kind of wrap up any thoughts or take-aways.  The blog acts in a similar fashion to be a reflection of something we explored in class or in our "homework" time and to take a step further.  While I have honestly found the blogging a little painful (which I really normally love) I think I was definitely trying to actively stretch my thinking about math.  I'm also a little hard on myself by not thinking my work is good enough... can't pin point if that is from my fixed mindset admitting defeat or my growth mindset always wanting to do more.

Sunday, May 29, 2016

...My {Math} Life Is A Lie (2)

I can't even believe how great this book is (and I'm a math-y... not a read-y) Mathematical Mindsets.  I have easily spent over two hours of outside class time reading, highlighting, rereading, note taking, and side-researching everything the book talks about... and I haven't even gotten to the good stuff! All of the "which I write about in Chapter yada yada" that she has said so far.

Why has my whole life been a lie, you wonder?  Well, that is because "math education" or math in schools today (and in my day) is not actually what math is! Timed tests, repetitive procedures, rules and memorization, all of it... We've been forced to think that that is what math is, and it's not!  I'm here to tell you [or at least pass on from Jo] that we need to take back math.  As math teachers, elementary teachers, mathematicians... all of us.  We need to put the conceptual process back into math so kids, students, adults, anybody can explore math and realize that it is about more than just answering as fast as you can and using rote procedures.

Math is patterns!  I feel like most (if not all) math can relate back to patterns.  The book talks about how babies and infants (which I think are the same thing... no?) are obsessed with math and patterns.  I didn't even think about this!  So we just shove math at kids from the time they are born all the way through preschool and they love it.  They just manipulate, create, build, learn and conceptualize and then as soon as they get to kindergarten we say "Math is a test and you have to be the fastest or you suck at math."  Math trauma anyone?  Math anxiety anyone?

I don't know about you, but the school I taught at had a SCHOOL WIDE GOAL for timed tests.  Like hearing that the first time, made my heart hurt.  But now, with Jo B in my life, I literally want to die thinking about it.  #stopthetimedtests #mathisnotarace Another thing I heard all day long was "[the students] HAVE to know their facts first or they won't understand this"... the lie detector determined that that is ALSO a lie Jerry.  I know how you're feeling right now, breathe through it.  Jo B herself did not memorize any facts growing up and she went to become a famous mathematician and author.  This is crazy, right?

So HOW do we teach math facts then.  HOW do we teach math if we can't use the same rote methods that really haven't been working for years.  Number sense - of course!

What is number sense you might ask? Teaching kids to memorize numbers right? Wrong-o.  Teaching kids to be flexible with numbers, to PLAY with them, that's right I said PLAY with numbers... while learning.  It can happen people.  This website tells us that the five components of number sense are: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers, and referents for numbers and quantities.  Basically what you need to know about number sense is that it's the foundation for math and it helps kids make sense of math.

I bet if you talk to someone who specializes in ELA or who has seen a dynamic reading program they will tell you all about the fundamentals of phonics and how phonemes has revolutionized learning to read.  Well, guess what?  Number sense will do that same thing for math and your kids will LOVE it.  What it comes down to is that kids need to be able to learn numbers without structure, without rules, and without stress!

The other step towards freeing students of math anxiety by doing the REAL math is to do so much modeling our faces fall off.  And I don't mean stand up there and talk at the kids and show them method after method after method.  I mean put that pencil in their hand, that array on their desk, those chips or coins or blocks or whatever in front of them and let them do it THEIR way.  Then, you have 26 little modelers in the room who will completely engage their classmates by showing their thinking, their process, and their creativity.  Exploration is the answer, what was the question?

Sunday, May 22, 2016

Math Toys (1)

So since I've been introduced to this Math Toybox... I've been addicted to these Pattern Blocks.  I started a tessellation in class and spent a good chunk of time continuing it.  The problem is, I will be honest, I think at some point I didn't copy it correctly so it's really (maybe) not a true tessellation... very sadly.  But overall, it looks totally awesome! In addition to it being very addicting, time consuming, and fun... this could also be an awesome tool to use in the classroom! (Everything I do gets connected back to the classroom some how :-P )  One of the things that I love most about this website is that it defies the teacher complaint of manipulatives.

"They take too long to get out" - This takes about 33.4 seconds to load on a classroom iPad/computer/laptop/chromebook/smartboard/whatever!
"The kids play with them and don't do what they're supposed to" mixed with "They're too distracting" - This is on the interweb so you can easily have them "put it away"  or out of sight and pull it right back up!

Manipulatives are SO beneficial to math.  Don't believe me?  Well I did a little research...

By using manipulatives in mathematics, it allows children to be create their own models and commit it to memory.  If we think about the popular "gradual release model", we think about how first the teacher models a new skill.  After that, the teacher will walk through the skill supporting students and asking for them to support her.  Then, the students are to explore and practice the skill on their own.  Just like the practicing of the skill, students must also be practicing new concepts.  And by practicing, many students learn best by using their hands to discover and manipulate.  This is even more important in today's instantaneous environment for students to be required to explore the process before getting to the solution and a "rule" for solving a problem.

This ties in with the book I'm reading, "Mathematical Mindsets."  So far, Jo (Boaler, the author) has talked in depth about the negative attitude toward mathematics that so many people have.  This is including the influence that is being put on students in and outside the classroom.  Where I'm going with this (and Jo) is that overcoming the instantaneous environment that society has created for today's student ties in with Jo's focus on utilizing the growth mindset specifically focusing on mathematics.  Empowering students to learn their own process is key for encouraging growth mindset and also in encouraging a love and understanding of math!

Tuesday, May 10, 2016

What even IS math? (0)

What a fabulous question.

At first, my answer is what I knew math to be growing up.  Which is what I consider a system of rules, equations, formulas, etc.  I always loved my math, and my brain loved math (even thought I'm a girl!) I loved math every single day until it was conceptual and applied.  This is how we teach math today.  When I was growing up it was about "knowing your facts" and memorizing formulas or using a "formula sheet" and just plugging in numbers.

Math is SO. much. MORE!  As a classmate described working in math, he says doing math involves wearing a tool belt.  If you think about when you're working on a project (or a problem) and you use different tools for different things.  The tools are put in place to make the project easier or faster.  Sometimes you have to try to different tools if you are inexperienced with the project until you find the right tool that works.  Sometimes you have a new project and you have to learn what tools to use to do the project.  This is how math works.

Math also explains the things that are going on around us.  Like when I go to the grocery store and I need to know how much food I can buy.  Well, first I have to know how much money I have.  To figure out how much money I have, I have to know how many hours I worked and how much money I made for each hour I worked, or the total amount of money I made.  Doesn't make sense?  Okay, well I want to play outside with my friends after school.  But my mom says I need to do chores for 45 minutes and then I need to do my homework for 20 minutes and I also have to read for 30 minutes.  And if I want to play with my friends the whole time then I won't be able to watch my normal 30 minutes of TV, but if I want to do both then I'll have to factor that in too.  So how long can I play outside in order to make sure I get all my things done... well that's do some math! Now you're getting it!

Math can be proven... and it has been.  That's why when we are working with problems today, we get to utilize our tool belt! Because there is proof in the proofs :)