Math - The "sorter"

I just got done reading a post by Geoff Krall titled Math and the Message.

As I was reading, I was thinking about the chapter on equity in math in Jo Boaler's book Mathematical Mindsets. My mind is spinning. As I read the story of Geoff's daughter, I kept thinking about the ideas around tracking and testing that Jo addressed in her book and how that impacts who considers themselves as mathematicians, who is given the message they can do math and who is given the message they aren't good at math.

Unfortunately, I too, have had similar experiences to Geoff's.

Background information: The University of Wisconsin system has a math placement test that all students must take as they enter any UW system school (all 2 and 4 year schools). This placement determines their entry into college level math courses, and ultimately, determines who gets in to math/STEM related fields.

Student 1
In the advising role for the secondary math education program, I was working with a student last year who expressed interest in becoming a high school math teacher. He happens to be from Illinois and came to UW Oshkosh for his schooling (out of state). I met the student after reaching out to students who had indicated an interest in the secondary education math program (becoming a high school math teacher). He contacted me with concerns about the math class that he was in and he started to realize that not being in the calculus sequence was going to add at least a year to his college program.

He explained that when he came up to register, he was told he had to take the math placement test (as all students do). The placement test is only given on campus. The student was finishing taking Calculus I at his school (got an A-) and figured he didn't need to study, that surely he would be allowed to sign up for Calc I at the college level. Long story short, he didn't do all that well on the placement test. He wasn't aware of the ramifications and wasn't about to make an additional trip to retake the test.  The consequence: He was placed in to a BASIC college algebra course.

By the time he contacted me, it was out of frustration that he was in this (his words) "too easy" math class when he knew that he needed to get to upper level math courses. The clear message our university sent to that student was that he likely wasn't a candidate for upper level math BASED SOLELY ON THE MATH PLACEMENT TEST. His prior academic courses didn't even count.

Student 2
My son graduated from high school just over a year ago. Since he was in middle school, his goal has been to become a pilot. He wanted to fly. During his junior year, we found a flight program for him at a local technical college that was his number one choice. During his senior year of high school, he decided to begin taking college level courses at the technical college under course options in order to get a jump start on the program. He took mostly general education courses such as communications and math. He had taken high school math courses through precalculus (in his junior year) and always done well. However, the technical college required what they call tech math for all aviation students, so he took that class at the tech school his senior year. His words.... "this is super easy".

A bit more information to round out his background.... with no studying to speak of, he received a score of 30 on the math portion of the ACT. The classes he took at the local high school included band (he had always enjoyed music) and AP Physics. Yep, a true physics geek.


As many college students do, he found that aviation wasn't the career he wanted. His second career choice, "mom, I want to be a physics teacher." Honestly, I was thrilled. There is such a lack of physics teachers and I've seen him work on physics. He loves it. He has such a great way of explaining it. Watching Interstellar with him explaining all the physics of different dimensions was telling.

As he began enrolling for courses at the 2 year college in order to get started on this new path, the advisor at the school suggested he take introductory college algebra because that was the next class in the sequence after tech math. When I saw he was enrolled it that, I was enraged. Why would I pay tuition for that class? I work in the field. I know to be a physics teacher, you need 3 semester of calculus, linear algebra and a few other higher level math classes. On this road, he would have to complete two years of college Algebra before even attempting Calculus. I suggested he call the college back and ask to be enrolled in Calculus I so that he could begin preparing for his future career as a physics teacher. That would be worth paying for.

Denied.

You have to take the placement test. (And if you don't do well because you haven't had math for a year, too bad. We already know you won't succeed in Calculus.)

REALLY?

A student with a 3.0 in all other college level classes? One that has a private pilots license, an ACT in math of 30, a history of success in all high school math classes, success in AP Physics. You need more evidence than that?

Why are we shutting students out of upper level math? Why not let these students try? These are students who have taken challenging courses in high school and succeeded. If they take Calculus and fail are they any further behind than paying for a bunch of classes they really don't need?

Unethical. 
As Jo Boaler explains,
"The fault lies with our culture, which has favored a role for mathematics as a sorting mechanism and an indicator of who is gifted. There is an imperative need for mathematics to change from an elitist, performance based subject used to rank and sort students (and teachers) to an open, learning subject, for both high-achieving students, who are currently turning away from mathematics in record numbers, as well as the low- achieving students who are being denied access to ideas that they are fully capable of learning." (Boaler, 2015)

Testify - Grading in Math

This morning I read Dan Meyer's blog entry titled "Testify". I have been inspired by all the blogging Dan does, and many others that I follow. I also can't tell you the number of times I have told myself I was going to start blogging also. I'm not much for writing like this, but today Dan's entry ignited me to sit down and write a bit about what I'd like to testify about.

GRADING.

From Seneca, IL High School website. 

When I started my teaching career as a high school math teacher, I was great at collecting "grades". You know, those numbers that you put into the squares in the gradebook (or now into the computer program). Being a math teacher, I must admit, I was really good at it! I had no trouble computing averages, tweaking what things were worth and looking at how that changed grades, adding points for things like the "I like you" factor.

In the last few years, I've been really thinking and wondering about this notion of grading. Why do we grade? What are they for? Do we need them? Why do we need them? Why do they drive almost everything we do in classrooms? And.....

Are grades to culturally ingrained in what we do in education that we can't live without them?

I have spent the last 10 years of my career working with teachers to improve math teaching and learning through professional development of pre-service and in-service teachers. I must be honest it is always fun and rewarding. Teachers begin to see that math can be taught in a way that is exciting and fun for both them and their students. That transformation almost always happens.

However, I often hear, "Well those changes are all good and fine, but how do I grade that?" In other words, how do I put that number into that cell and tell it to compute at the end of the semester? If I do one of Dan Meyer's 3-Act math activities, how do I grade that? If I do some tasks like Jo Boaler suggests, how do I grade that? Are those activities formative or summative? How much should they be worth?

I think we can propose all the great teaching that we want, but unless we really address the GRADING elephant in the room, changes will limited by this beast.

Hopefully more to come...... and that the spark that has been ignited will continue to burn for a few more posts on this idea!

My Brilliant Husband - The Artist (Mathematician)

A great friend of mine (who is a math specialist in a school district) and I met late last week for lunch. During our visits, we often discuss different mathematical problems and new things we have learned along the way. This time was no different. After getting caught up a bit, Jenny asked me about a problem she had seen where people were asked to mentally do the problem 600 ÷ 48. Jenny explained that in what she read, there were 3 common ways that folks attacked that problem. How would you do 600 ÷ 48 mentally? What answer do you get? How do you get it?

True confessions....that one stumped me. It wasn't that I couldn't get it, but on-the-spot pressure blocked me from moving forward. I knew 10 something or other.

Jenny shared with me 3 common ways that people approached this problem mentally that were mentioned. There are a ton more, but these the three methods below were mentioned in the book. 

600 ÷ 48 

• Think of the problem as 100 ÷ 8
• Think of the problem as ½ of 100 ÷ 4
• Think of the problem as 600 ÷ 50 + ½

For the first problem, Jenny and I agreed that was easy to figure out.....make it a simpler problem by dividing both 600 and 48 by 6. Now your problem is 100 ÷ 8. And if you know that you can do that and not change the value of the solution, it is easier to do 100 ÷ 4 first and simply take half of your answer. But the last one....hmmmmmm.

Is 600 ÷ 50 an easier problem, sure. That gets you an estimate but where does that darn + ½ come from? I wanted to be able to provide Jenny with a visual model of what that might look like because to be honest, if we could figure that out, it would make it a much much easier mental math problem.

The next morning while my husband David and I were walking. I was thinking out loud and said "ohhhh...I think that would work." He asked what I was talking about and I told him I was working on a math problem.  Not impressed. We didn't talk about it any more. When we arrived home, I put my ideas on paper and created the video to send to Jenny explaining my thinking.

BUT HERE IS THE KICKER!!

As I spent my weekend working on math problems, my wonderful husband David simply shakes his head. He is an artist and musician at heart and often doesn't see the beauty of math as I do. He tolerates my endless talking about an "I love math generation" and at least acts interested when I get going on math problems. He is a self-proclaimed math hater. In all honesty, that makes me very sad.

So I posed this problem to David. 600 ÷ 48. (He had not seen what I had been working on.) I asked him to share how he would attack that problem. You know what he said.....

"I'd take 600 ÷ 50 and get 12. But I know that is 2 too many in each group so I have to do something with the 2's."

HE IS BRILLIANT MATHEMATICALLY! Why didn't I just ask him (the self proclaimed math hater) to explain the darn thing to me! Well Mr. Smarty Pants, what about those 2's?

Below is the visual model of what happens in that problem and a quick visual explanation of the other 2 methods as well.

My point is this. In order to become an "I love math generation" we need to foster creativity in thinking instead of forcing memorization and algorithms. I think many people out there who hate math only hate math because of their school experience. After this experience with David, I think that is why he hates math. 

We can change that. We MUST change that!

I love you David!

NOTE: The video is "raw". If I was producing these professionally I would clean up some of my errors in how I say things mathematically. Perhaps I will do that someday. In the end you see I say "divide by a half" when actually I meant "divide by 2". Because I know Jenny knew what I meant, I didn't fix it. You'll get the point.

Talk Less, Listen More

My reason for starting this blog was to try to influence mathematics education in the United States. As many of you know I have a deep passion for mathematics. I really believe we can create an "I love math" generation. One of the biggest roadblocks to changing the culture of math education is that we, parents and educators, don't understand the need for the change and what that change might look like.

Last summer I participated in a Stanford online mooc facilitated by Jo Boaler. Like any blogger, I pass this along because she is an inspiration to me. She recently posted this video to help teachers and parents understand the reasons we need to make some shifts and what those shifts might look like. It was a time for me to talk less and listen more.

The 20 minute video is very worth the time it takes to watch. If you don't have time for the full 20 minute video, start at 9 minutes, 30 seconds and listen for about 4 minutes. Those 4 minutes will be enough to paint a good picture for you. It may entice you to watch the other 16. 

Make sure you catch the quote from Sebastian Thrun!

I hope you enjoy the video. If you do, PLEASE pass it on to every parent and educator you know and encourage them to watch it too!

What in the heck is "ALGEBRA" anyway?

A t-shirt with this saying was circulating Facebook and I couldn't resist making a comment. This is another indication we need to become an "I love math" generation!

I'd also argue that you probably did use algebra, but you didn't know it. I say that because in order to know you used algebra you'd have to know what it is.

Do we really know what Algebra is anyway? Go ahead....try to define it! Give it your best shot!

This has become one of my latest pet peeves. We use this word "algebra" like we understand what it is and what it means. but even for those of us who are trailblazers in the "I love math" generation, this word should give us pause.

What I know from working on 4K-12 math curriculum is that algebra begins actually as early as 1st grade. Some may say kindergarten, but I'd have to say developmentally one-to-one correspondence has to occur first and for some 5 and 6 year olds, that isn't happening quite yet.

So here is the Oxford definition of algebra, "The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations." I'll summarize it as the transition from doing math concretely (such as counting) to more abstract representations. When I was trying to find a ale price and I knew the percent discount but not the dollar amount that would be taken off the original price, I had to use algebra. (Read Ann Taylor post)

I think for many of us the definition of algebra is what happened in our first year math class in high school that just so happened to be called "Algebra". (From this point forward in this blog I will capitalize the course title Algebra, but when referring to algebraic concepts it will be written with lowercase a - algebra.) Perhaps in that year you spent a lot of time factoring quadratic equations. We think that is what algebra is. I'll agree it is part of algebra, but algebra is so much more than that.

Folks, please always remember that Algebra I and Algebra II are course titles. That is it. Course titles that contain the teaching of mathematics. Some of which will be using algebraic concepts, some will use graphing, some geometry, etc. They are course titles. They actually could be called anything like HS Math 1 and HS Math 2. What you learned in high school in those first two courses isn't what you should tag algebra as being.

Remember that age old argument about should we teach Algebra in 8th grade. Guess what!!! We do! We teach algebra the concepts, (as stated above as early as 1st grade) but perhaps we didn't title the course Algebra.

This article is an example of an article that uses a random course title Algebra to make an argument without talking about what algebraic concepts are being referred to. The problem with this article is they don't ever get to the details of what is taught in each of these so called courses. In my experience working with 8th grade CCSS, much of what used to be called 8th grade Algebra or Algebra I in high school is now being taught in regular 8th grade math. In many districts however, they don't change the name of 8th grade math to reflect that. The first course is high school is still Algebra I, but the content that is taught within that course will be shifting to the topics normally found in Algebra II (or the second year course). Without going into tons of detail....the bottom line is this....

Algebra is the name of a course. It doesn't describe the content taught within that course. So you really have to be careful.  And if you step back even further to say what IS algebra....it is much more than a course. It's the way you abstractly use mathematics to solve problems that you can't do using your fingers and toes. It is taking an idea and using a formula or equation figuring out how to solve it. That could mean mentally also, especially since most of us do math mentally on a day to day basis.

So.... did you use algebra today?

Ann Taylor - Savings HOW Big?

After finding all the great deals at Ann Taylor, my next hurdle was to convince my husband that all all the items were a HUGE savings! Of course he would believe that the best sale is when you don't even go to the store. But remember, these were 70% off of the lowest price...such great savings!

I take the items up to the register with Dave in tow. He glances at the "stack" of clothes. They start scanning each item and it shows the sale price, discount, and price we are paying as each items is scanned. However, I could tell he wasn't impressed yet. The total came out to about $93. (Whew, kept it under $100 thinking that was great too.)

I was waiting for the sales person to hand me the receipt and to circle the amount saved. You know, like they always do at Kohl's.

The moment to impress the husband with how well I did. To my great disappointment, she didn't say a word. NOW WHAT?!?

Ta-da! Math again. My brain starts in high gear. How can I quickly figure this out before we get to the car? Otherwise it might be a very long drive. I begin quickly thinking.....

I just paid 30% (see Ann Taylor - the Purchase) of the total. I saved 70%. Well if $93 was 30%, I would add $93 + $93 + $31.

In my head I could easily get to $186 ($93 doubled), but how can I quickly add $31 on to that. Remember folks, I'm on my way to the car. No stopping to write out the problem in standard algorithm form. I'm trying to redeem my expenditures and I'm losing the battle with every second. $186, $196, $206, $216, $217!

Here is a picture of what I did in my head.....(they call this addition using an open number line)

I saved $217!!! Wowy Zowy!


To take you back to your math class memories, your teacher might use an example like this to have you set up a proportion in order to solve this problem. Something that might have looked like this:



I know...you say I'd never do it that way. But that is an example a middle school textbook would have for that type of problem. That method does work great, but I believe when there is an easier way, find it. For that problem I would have needed, pencil, paper, and a calculator problem to tackle it that way.


Not needed. FLEXIBILITY and FLUENCY. That's what we need for the I Love Math generation. People can do this stuff in their heads!

It doesn't end there.....

I frowned when I saw my husband wasn't quite impressed with the $217 savings. All the way home I was thinking about it. Then the smirk started coming across my face. DUH.... The $217 was off of the sale price. What kind of savings would be reflected off the original price? I couldn't wait to get home to add up the total prices (which were not on the receipt) and give him the even better news.

For that task, I did revert back to my high school math teacher ways and using a calculator, found my answer. I added up all the original costs, subtracted the amount I spent, and proudly announced I had saved $505 on this purchase. I added....that meant I only paid for less than 20% of the original cost of the clothing. IS THAT CORRECT? Time for you to do some math.....

For those impressive numbers I got the response, "That is great! How come they never have men's clothes on sale like that?" (I consider that a win!) To which I responded, "They do, but you have to go shopping to find great deals like that, silly."

Ann Taylor - The Purchase

I always wondered what my first blog entry would be about - if I decided to write a blog. The first blog entry after the introduction that is. Here goes..... my trip to Ann Taylor.

Last week as my husband and I were returning home from meeting with his retirement specialist (yes, he is retiring and I have many years of work left). On our trip home we drove past the outlet mall.  Thinking about being on his retirement income, I wondered if I should even ask him to stop and go shopping. I asked, he accepted.

We went in to my favorite store at that mall, Ann Taylor. Lucky for us (well...me), they had just started the "70% off the lowest price" sale! YIPPEE!

As I started going through the racks and racks of sale items, I noticed the signs. You know the ones. The signs that tell you if the item is this much, the sale price is this much. They always have them at Kohl's too. That got me thinking....Do they think we need the signs? Can we not figure out 70% off an item without help? I suppose that is the case sometimes. In an "I love math generation" all people will be able to figure 70% off mentally. Without the signs.

NOTE:  Perhaps the way we learned to do this in school is part of the reason why we don't do this in our head. See an example here. But hang on to your hats if you decide to watch this video. It will take you back in time to "Algebra" class! 

I picked up a pair of dress pants, checked the tag....$59.99. Wow.....$18. I found the coordinating suit jacket, in my size, checked the tag...$49.99. REALLY?!? A suit jacket for $15? That's an entire dress suit, light wool and lined, for $33. That's a steal.

Then it got me thinking about that math. For the jacket, I estimated quickly by taking 5 x 3 or $15. Well, not exactly 5 x 3, but that was my shortcut for finding 70% off of $49.99.  I rounded the cost to $50. Then I took that $50 times 30% (because if it is 70% off, I am only paying for 30% of the cost of the item). Then because I love math, I know I can translate that into 5 x 3.  (I hope that makes sense...)

But why is it that I know that works mathematically? It is called FLUENCY with numbers. In school we tend to think fluency was knowing your math facts. You know...flash cards...timed tests.  But fluency is more than that. Fluency is about having strategies (mathematically correct strategies) that help you quickly find answers.

When I was telling my son about my adventure, he laughed at me and told me his much quicker way to think about this. (He thought my way was longer and harder than his.) He said, duh mom. If the item is $50, just double it. That is 100. If you are only paying 30%, you would pay $30. But because it was only $50, you divide the $30 by 2. Ending up with $15.

Well, he is correct. But I like my method better because if the original price isn't 50, or something that easy, the problem is a lot harder. So there, Ronny!

Having more than one strategy or learning from others helps build FLEXIBILITY and FLUENCY. It is important that all of us build strategies and recognize when they work and when they don't (because that happens). For instance, what if the sale was for 65% off. I'd likely use a different strategy than the one above to help me figure out my costs. Not impossible.....but I'll leave that for another blog entry.

Enough for now but stay turned for part two...how I impressed my husband with how much I saved!!!!!!