A Computational Approach to Functions

A Computational Approach to Functions

Presented by: Patrick Honner
Presented on October 8, 2019
Looking for a new approach to teaching domain and range? Or an opportunity for students to use their crossover computer science skills? Taking a computational approach to functions allows for the rigorous development of all the fundamental concepts in an active and creative way, while at the same time offering endless opportunities to extend deeper into both mathematics and computer science. If you teach about functions—and what math teacher doesn’t?—you will leave with something new to think about for your math classroom.

Recommended Grade Level: 7 – 12

Hosted by: Rana Hafiz

Watch the full presentation at: https://www.bigmarker.com/GlobalMathDept/A-Computational-Approach-to-Functions

This Week at the Global Math Department

Edited By Casey McCormick  @cmmteach
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Online Professional Development Sessions

Tonight!

A Computational Approach to Functions
Presented by Patrick Honner

Looking for a new approach to teaching domain and range? Or an opportunity for students to use their crossover computer science skills? Taking a computational approach to functions allows for the rigorous development of all the fundamental concepts in an active and creative way, while at the same time offering endless opportunities to extend deeper into both mathematics and computer science. If you teach about functions—and what math teacher doesn’t?—you will leave with something new to think about for your math classroom.

To join us at 9:00 PM EST for this webinar click here!

Next Week

Strengths-Based Mathematics Teaching and Learning:
5 Teaching Turnarounds to Build Student Success

Presented by  Beth Kobett

Explore teaching turnaround strategies that can reframe and open up students’ mathematical learning opportunities. Learn to identify and leverage students’ strengths to develop powerful and strategic learning moments that recognize and bolster students’ strengths to build mathematical success.

Register ahead of time by clicking here!

You can always check out past and upcoming Global Math Department webinars. Click here for the archives or get the webinars in podcast form!

From the World of Math Ed

Fractions: Inside & Out of the #MTBoS

First things first, S/O to Liz Caffrey (@AsymptoticLiz) and her awesome shoes, who’re redefining fashion in sequence.

Lately, I’ve been fascinated by fractions and the conversations that we, (the #MTBoS community-at-large) have about them. When I did a Twitter search for the latest/top tweets (#MTBoS and Fractions), I saw the following words used to describe fractions or the relationship that students have with them: demystifying, challenging, struggle, gatekeeper, break down, and misconceptions. As math educators, we are aware that students’ learning of fractions feels like a make or break point for whether or not students will develop a productive disposition & positive identity in mathematics.So I thought it’d be interesting to look for cases in the #MTBoS and outside the #MTBoS where fractional understanding was of imminent importance.

This week, we got to listen in on a conversation in Marian Dingle’s (@DingleTeach) classroom as her students used the transitive property to make sense of fractions being division. Fractions are Sharing; Sharing is Division; SO FRACTIONS ARE DIVISION!

Berkely Everett (@BerkelyEverett) shared a #samedifferent post to help us think about the way we count denominators.

One of my favorite resources for understanding the conceptual fluency, strategic competence, and adaptive reasoning in fractions is Graham Fletcher’s (@gfletchy) video series on the Progression of Fractions. This was a really helpful video for me, as a high school teacher who sometimes forgets how students develop their thinking in fractions.

Here are a couple of examples I found outside the #MTBoS, where the understanding of fractions played a critical role:

This week, Amber Guyger, a former Dallas police officer who shot and killed her unarmed neighbor was sentenced to ten years in prison after a criminal trial. Several weeks ago, the defense attorney in this high-profile case requested that the case be moved to a different district, to “assure a fair trial”.

In this tweet, S. Lee Merritt, Esq. (@MeritLaw) discusses the importance of having the Botham Jean murder trial heard in Dallas County, where the crime took place, instead of relocating to a different site.

How are fractions being used in this case?  How closely does the jury’s racial makeup “reflect the diversity of Dallas County?” To what level of precision (SMP.6) is the prosecutor attending to when defining the diversity of Dallas County? Does the fractional relationship in diversity change depending on this definition?

In another critical example, an impeachment inquiry against the President of the United States of America was filed this week. The President (@realdonaldtrump) tweeted the following in response:

In the thread under this original tweet, the following maps were given as counterexamples to the map provided by the President. Several Twitter users cited the number of popular votes or electoral votes for the different candidates in the 2016 election (“That’s 65,844,954 blue dots for Hillary Clinton and 62,979,636 red dots for Donald Trump.”). Find the Official Election Results here.

Source: (Mark Newman/University of Michigan, 2016 election.)

What is the fractional relationship between blue and red in each map? Why are these fractional relationships so different, even though they span the same area (the United States)? What viable argument might each of the Twitter users be trying to construct (SMP. 3) with the map they chose to post? (Tangent Time: Why aren’t Hawaii & Alaska on all 5 maps?! Talk about erasure!)

It seems that fractions are not only of critical importance for developing a positive mathematical identity for our students, but also in making sense of the world in which our students live.

Written by Lauren Baucom, @LBmathemagician

Something Old, Something New, Something Borrowed, Something Blue.

It goes without saying that there is a lot that passes through the daily stream of tweets. What I’ve picked out below come from some significant MTBoS contributors and the four highlights are merely a taste of the gold they consistently sprinkle my Twitter feed with on a daily basis. I hope you enjoy it as much as I did.

Four Stages of Using Models

Brian Bushart (@bstockus) got us all thinking about what type of thinking we might be biased towards when facilitating a number talk by sharing his thinking on the Four Stages of Using Models. As Kathy Richardson (@letkidslearn) describes, these four levels unpack how students use models to solve problems, highlighting that although we’d love all students to be demonstrating thinking at Stage 4 (solving the problem mentally), there are some important distinctions between what students are doing prior to that, and that just because students are at Stage 4, operating in Stages 1, 2 and 3 can still support their learning. If your thinking resonated with what you read on that post, perhaps you want to also check out the one Brian pushed out four days later on Multiplication Number Talks Using Models.

Esti-Mysteries!

He’s at it again, folks. Steve Wyborney (@SteveWyborney) has started releasing the first of his 51 brand new Esti-Mysteries challenges. If you haven’t seen these before, Steve’s original post is a great starting point. One particular thing that I love about them is the range that appears and adapts based on the clue. Here’s an example, which would be great for introducing inequalities:

Can you visualise this? 

There aren’t many things I love more than seeing something and immediately thinking “hmmm is that right?” then going to check it out and end up thinking, “well, would you look at that? It is right!” I went through that exact process when I saw Mark Chubb’s (@MarkChubb3latest post:

The best thing, I thought, was that I was left with the “I wonder if…” types of questions. This is part of a nice little post he put out through the week, which is definitely worth checking out. Better still, let Mark know some of your answers!

The Domino Effect

Sarah Carter (@mathequalslove) showed her generosity by contributing her files she used to create a whiteboard display of the Domino Effect puzzle. Originally from Brainteasers : 195 Puzzles to Keep You Sharp, Sarah posted this on Twitter through the week and, if you were like me and read it as any eight dominos, you would have had the extra fun of finding eight dominos that could be possible.

Written by John Rowe, @MrJohnRowe

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This Week at the Global Math Department

Edited By Chase Orton @mathgeek76
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Online Professional Development Sessions

Tonight!

Activities in the AP Math Classroom
Presented by Sharon Sterken and Randi Metz

Once we jumped on board the Student Centered Learning bandwagon, we found it very difficult to find quality, engaging, and fun activities in the AP math classroom. So, we decided to team up and share our ideas for other educators who are interested in adding pizzazz to their lessons.

This webinar was developed to have resources available to enhance instruction for fellow AP Calculus AB and AP Statistics educators.

Contact Info:
Twitter @girlmathX2
Blog: http://girlmathx2.blogspot.com/

To join us at 9:00 PM EST for this webinar click here!

Next Week

A Computational Approach to Functions
Presented by Patrick Honner

Looking for a new approach to teaching domain and range? Or an opportunity for students to use their crossover computer science skills? Taking a computational approach to functions allows for the rigorous development of all the fundamental concepts in an active and creative way, while at the same time offering endless opportunities to extend deeper into both mathematics and computer science. If you teach about functions—and what math teacher doesn’t?—you will leave with something new to think about for your math classroom.

Register ahead of time by clicking here!

You can always check out past and upcoming Global Math Department webinars. Click here for the archives or get the webinars in podcast form!

From the World of Math Ed

Notice & Wonder 
Amber Thienel (@amberthienel)\

I love a good notice and wonder. It does magical things to get students thinking and saying things I couldn’t possibly think of on my own. Seeing the connections and ideas that students come up with is impressive.

I was diving into the hashtag for notice and wonder (#noticeandwonder) and found some ideas that I would like to share with you. One of the most popular tweets right now is from Robert Kaplinksky (@robertkaplinsky) way back in June.

Another one that peaked my interest is a recent conversation from Paige Sheehan (@MrsSheehanMath) asking about how you get your students to capture their noticings and wonderings. Join in on the conversation if you have an idea to share.

 

Start a Math Club
Christelle Rocha (@Maestra_Rocha)

If you haven’t started a math club at your school, I highly recommend it. Jennifer Dao (@JDaoMath) posted about their math club and asked others to share activities for the year.

While I encourage you to look through the thread yourself, some resources were shared to help teachers get started on the right foot:

I’m grateful that Jennifer has been sharing their math club experiences, because I’ve gotten ideas on how to revamp the club at my school for the new academic year. If you are getting started or already have a club at your school, I’d love to hear how it’s going; I am shamelessly starting a hashtag, #OurMathClub with the hope that we can continue to share and learn from each other.
 

Sourcer Sorcery
Benjamin Dickman (@benjamindickman)

In this week’s GMD newsletter, I want to focus on the ways in which ideas are sourced (or not sourced). This does mean, in particular, that I have omitted material around new mathematical progress, a conference/conversation on professional norms in mathematics, and more.

Two inspirational tweets to kick it off: the first from @LBMathematician [in her quote-retweet of @mathedmatters] and the second from @ChristieNold:

Notice in the first image the concerns by generators of original ideas, especially in a case for which one’s livelihood (e.g., earning tenure as a professor) can be a function of proper attribution, and the valid concern expressed in the second tweet around conveying one’s newfound understanding along with those who supported it along the way.

I recently came across an @edutopia article (via Jerry Becker’s listserv) about the “Talk Moves” used at a particular school in Portland, ME. But, the article attributes various Talk Moves to this school – even though they appear in the math education literature! In fact, the first four Talk Moves named in the @edutopia document are precisely the first four from the Chapin et al work below:

 

Not cool, @edutopia (1.12 million followers!).

I noticed another attribution issue from @fermatslibrary (281 thousand followers!) for whom this is not the first time that work has been tweeted without attribution. In fact, that same link shows another instance in which a reddit user’s original work was tweeted without credit being given.

While these large-follower-accounts are not doing their due diligence in sourcing the content that they are tweeting out to a hundred thousand+ or a million+ people, there are a lot of great examples of proper attribution that came across my timeline recently.

So: To pivot positively, I’d like to highlight some of the quality instances of good sourcing practices in action. First, though, even when attribution and quoting go generally well, there can be another problem with journalistic practices: Picking the title of an article. For example, check out Kevin Buzzard’s response to a Vice article in which he was quoted.

Contrast the bad sourcing from @edutopia and @fermatslibrary with the @ReadPRIMUS account [managed by BK] in which attribution is provided even when the source has no recollection of their helpful remark!

Continuing with the proper sourcing theme: In linking to a NYT piece on mathematicians’ chalkboards, @nattyover credits the photographer and the writer and the person who brought the piece to her attention [the “h/t” abbreviation stands for “hat tip”].

I added links to a few more pieces about chalk, which includes three items by @MBarany.

Plus, an interesting comment about chalkboard handwriting from @katemath:

Among the many items tagged #NCTMBoston19 is the following from @beRealcoach. As to sourcing: Note that the tweet includes an attribution to the presenter, @NicoleBridge1, and that the slide includes an attribution to the writer, Zaretta Hammond.

@alittlestats links to a fascinating article about graphing calculators and their business model and cc’s @Desmos [see the article!] and includes another “h/t” [hat tip] to shout out @MathDoris.

Sam Shah writes a wonderfully reflective blog post in which he cc’s three people [spoiler: one of them is me due to my tweet here] and generously mentions the three of us at the start of the post, too.

Relatedly, @AlexPHoover posted about an Out List + Ally List from Spectra and credits @mikeahill.

Besides Ally Week, we have also been in Hispanic Heritage Month. Check out the #Lathisms hashtag, and the link provided by @zdearaujo here:

Finally, I was excited to read about David Eppstein’s efforts around wikipages for women in mathematics. I saw it in a tweet from @thegautamkamath; check out the link for more!

I hope those who have read this far enjoy the sourcing and sources in this week’s newsletter; as always, I will be grateful to anyone who wishes to notify me about other work that should be highlighted and/or amplified in and beyond math education communities. Hit my DMs, tag me, send me an email; whatever works!

Speaking of mathematical errors, I need to get one off of my chest: In the one REU math paper I contributed to there is a mistake in Remark 2.4. The constants should both be 1 [not both zero, which is what our paper says] if phi has good reduction [whatever that means]. This doesn’t affect correctness and is more of a “typo” than a “deep theoretical error.” But, it is not hard to imagine how small errors can ramify.

To see that this must be an error, one can note that cv and Cv are defined [see below] as being the maximum among various quantities that always include 1; of course, there is no way that 0 will be the max in a set that also contains 1. Logarithms are then applied to each, so it will be better not to be taking log(0)!

(Phew; I feel better already.)

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