Tag Archives: SMP 4 Modeling

Recap: Twitter Math Camp ’16

TMC LogoThe annual Twitter Math Camp is always amazing. This summer’s conference in Minneapolis, at Augsburg College. was no different. My great disappointment was only being able to stay for one full day this year, but the one day did not disappoint. 

As always, portions the Math Twitter Blogosphere (#MTBoS) rallied from around the country in genuine excitement to see and learn from each other after another year of digital friendship and collaboration. Thanks go out to Lisa Henry (@lmhenry9) for being the lead on this terrifically special event.

Michelle NMichelle Naidu (@park_star),
Saskatchewan Professional Development Unit

A packed room on the topic of intervention was surprising to both me and the presenter, because the MTBoS dialogue mostly revolves around first instruction. The large audience is a testament, though, to the need for reaching ALL kids in the era of 21st Century Standards. Michelle is leading a very successful intervention program in Canada which is focusing on some basic premises:

Differentiating for All Students is like Cowboys Herding Cats, but “it’s a good feeling having the herd [of students] arrive on time without losing a one.”

Early Intervention on the Pre-Requisite Skills (Readiness) that are required to be successful in the current curriculum is the first and most important intervention move. Pre-Assessments on prior content are then necessary to help improve students’ chances for success. Back at home we call this Boot Camp. Michelle affirmed that this work is good, and also inspired me to go back to my site and push to make it a priority.

Unpacking Standards Collaboratively serves two purposes. (1) It allows you to throw out material that is not in the standards, which buys you time for intervention/differentiation (Grade Level)  and (2) It helps you focus on the pre-requisite skills needed for students to learn the new material (Readiness).

Intervening on Readiness = Differentiated Content
Intervening on Grade Level = Differentiated Product

SnowballI also saw an interesting take on the Snowball Activity. Students write down one comment and one question about a topic (notice and wonder), then wad up their papers and throw them around the room. Each student picks up a “snowball” and adds another comment and question. This is done again, until there are three of each. After the fourth toss of the snowballs, the students do not write, but instead debrief publicly as the teacher summarizes the comments and questions on the board. This is a strong way to have ALL students reflect on learning.

Jose Vilson (@TheJLV)JoseV

Jose’s most solid point was that public conversation on math education reform often does not include educators, especially those teaching the marginalized. He accurately stated that if the medical system in America were being discussed on cable news, there would be a doctor on the show, but you never see a teacher on TV talking about education.

In many ways, Jose was calling us out to be activist on our campuses for the changes that we in the Blogosphere write so much about, particularly for students of color. He made a claim that really stuck with me: “We say that we teach math to all kids, but students of color are taught a different type of math than white students.” I know this is true on my campus, While my school is relatively diverse, the lower-level math classes are disproportionately populated by students with Hispanic surnames.

I asked a question of Jose, preluding it with a statement that prejudice on my campus tends to run more along income lines than racial lines (although, racism exists everywhere). Students are accepted and succeed as long as they behave like ‘these kids.’ So I asked, “How do you get teachers and staff to be more accepting of ‘those kids,’ so that they can remain authentically themselves and still learn?” Jose’s response was, “Teach the adults to recognize ‘different types of genius.'” I love that phrase! He went on to explain that kids in poverty are often times going to bring the norms of their own sub-culture to class, which is many times in conflict with the rigid, quite, patient, controlled environment of traditional school. If we can respect that and honor ALL students’ intellects, while also teaching proper social behavior, schools will break down a lot of walls and reach more marginalized students.

Audrey McLaren (@a_mcsquared),

Audrey showed samples of student work from her classes, in which she has students BUILD activities and graphs in GeoGebra and Desmos. The best example was Sticky Points. I love how the challenging of students to create the special points for a function like the x-intercept(s), the y-intercept, and the vertex demands that the students do some algebraically manipulation. The graph offers an immediate feedback loop until students do it correctly. This builds their algebra skills and conceptual understanding simultaneously. I’m using this idea in my class this year for sure.

Desmos Sticky points

Edmund Harris (@Gelada) and Myself (@MathProjects)

Edmund Model.png
I thought Dr. Harris asked for “mathematician modeling!”

I was honored when Dr. Harris, of the University of Arkansas, asked me to present with him on Mathematical Modeling. Edmund and I have been friends since our first Twitter Math Camp (TMC13), and I always look forward to our laughs and deep mathematical conversations. Edmund wanted to share the theoretical meaning of mathematical modeling, and he asked me to add my take on how the teaching of it manifests in the classroom.

Logo Pear DeckWe started by surveying the audience on Pear Deck, prompting for their definitions of mathematical modeling. The vast majority of the responses fell into two categories:

  • Representing a Real-Life Situation
  • Applying the representation to make Predictions.

It turns out that these are quite accurate if we include them BOTH, but the two are not necessarily a comprehensive list, as Edmund explained.

The professor started by claiming that shepherds in the field used to count sheep by using stones in their pocket by which a small stone represents one sheep and a larger stone represents 20 sheep. This, he asserted, is an example of abstract modeling. (Leave it to the Brit to bring sheep herding into a math discussion.) Then he drew this diagram on the board:

Edmunds Model Diagram

Edmund teachingHe explained that you start with “something to be modeled,” (noticed he did not say a real-life situation) and then you create an abstract representation of it. This is a back-and-forth process of verifying the accuracy of the model’s description of the something as well as the “thing” that we want to do with it. (Use rocks to keep track of the sheep). So the audience was responses were spot on… collectively. Yes, modeling is Representation AND Application, but not necessarily just Representation OR Application. Furthermore, Edmond wanted to make it clear that modeling does not have to apply to only “real-world” examples. He claims that when we discuss the transformations of a family of functions, we are also modeling… using an abstract representation to “do something” to the original parent function.

Modeling Tweet Me

In my investigating of what is expected of school teachers when it comes to modeling, I studied the common core documents and found very persistent, clearly defined attributes of Mathematical Modeling:

  1. Modeling is a process.
  2. Modeling is a verb.

In other words, using a model that is already provided is a good and healthy step in the learning process of modeling, but it is not modeling itself unless the students are generating the model themselves.

Modeling Tweet Heather

Modeling Tweet Jasmine

Thank you, Edmund. It was a pleasure working with you, my friend. You always make math appear so joyful.

I had several conversations throughout the Camp, but two that stood out were with …

Math Modeling
Edmund Harris (@Gelada), Brian Miller (@TheMillerMath) & Alex Wilson (@fractallove314)

TMC Bar ModelingThe first night of TMC16 was a huge social event by Desmos. Edmund, Brian, Alex and I had a beer-laden discussion about modeling that proved quite passionate (read as: table pounding, finger-pointing, and all in good fun B.S. calling). It was such a blast to throw ideas around with people of high intelligence, strong convictions and the deep desire to get this thing that we call teaching right. Cheers to changing the world one math lesson at a time.

Intellectual Need for VocabularyPic Dan M
Dr. Dan Meyer
Dr. Meyer completed his dissertation last year. Knowing how much those with a doctorate enjoy talking about their research, and being truly curious about it, I ask him to share his findings with me. He joyfully did, including some of the back story behind it. In essence, Dan studied the effectiveness of giving students the academic vocabulary after first posing a task that required its use, rather than front loading the terms. He called this method Functionary. His study showed that the both Functionary (using the vocabulary to communicate) and Traditional methods (making flash cards to memorize definitions) were equally effective in teaching students academic language found on traditional assessments. The Functionary method, however, showed superior results when students were asked to communicate their thinking using the vocabulary terms or to complete less traditional (more CCSS-like) tasks. You can listen to the Defense of his Dissertation on Dan’s Blog

As always, I highly recommend this event to any math teacher. I hope to see you all at Twitter Math Camp in  Atlanta, July 27-30 2017.

Interview: Dan Meyer on Using a Ladder to Clear a High Bar

Pic MeyerDan Meyer is as close to a celebrity that a math educator can get. We all owe him a debt of gratitude for making math nerds look cool. He deserves his rock star status because he is an amazing presenter, a thought-provoking writer, and an ingenious creator of math tasks.

Behind all the hype, though, is some serious substance. Dan’s ideas are akin to the spirit of MPJ’s lessons in that they seek to engage students in meaningful mathematics, and aspire to teach them high-level cognitive skills. Dan’s methods, though, have a unique twist that challenges teachers’ thinking. I know he has given me a great deal to think on. I hope he does here for you as well.

From what we read on your blog, you are about two things: 3-Act Lessons and the Ladder of Abstraction. Let’s start with 3-Act Lessons. Give us the gist of what they are and why they are an effective tool in teaching math.

Dan Meyer
We make huge promises to kids about the power of math in their world. But then we give them these problems that don’t do justice to that power or to the world they live in. Three-act math uses photos and videos to show students a more faithful reproduction of their world and a more faithful reproduction of the practices of applied mathematicians.

MPJ has preached for over a decade the need to have students create their own mathematical models (abstract generalizations). Describe your Ladder of Abstraction and how it applies to teaching mathematics.

Dan Meyer
The process of abstraction is extremely powerful and also not something I understood intuitively until I was a long way out of my secondary math education. Basically, whenever we turn the world into a table or an equation or a graph, we LOSE something. People don’t run at a constant rate. The Earth isn’t a perfect sphere. But we abstract a runner into a linear equation and the Earth into a sphere because those abstractions are perfect ENOUGH to help us answer questions. That’s an important part of modeling. Asking, “Is this model perfect enough?”

With so many teachers reliant on teaching from the textbook, do you have any ideas on how to get these practices used more regularly in classrooms?

Dan Meyer
I tell teachers what I tell myself: whatever you’re going to teach today, whether it’s pure math or applied math, make sure students have some NEED for it. A better need than “I don’t want to fail this class.” And I offer them techniques for provoking that need. I also offer teachers a homework assignment, an exercise like push ups, to get better at provoking that kind of need: take a photo or a short video and ask people what questions they have about it, if any. If they don’t have any questions, retake the photo or video in a way that provokes more questions. That homework assignment has been incredibly helpful in my own growth.

How well do your theories mesh with what is coming down the pike as the Common Core?

Dan Meyer
The modeling practice of the CCSS gets focused treatment in high school. I encourage all of your readers to study high school modeling (it’s only two pages) and ask themselves, “Are the ‘real-world’ problems I assign preparing students to clear this high bar?” Then Google “three-act math” and see if my work can help.

What do you intend to prove with your PhD research?

Dan Meyer
I’d like to understand how any or all of this translates to online education.

SMP Posters by MPJ

SMP Posters Pic 2_Page_8I created my own posters for the Common Core Standards of Mathematical Practices. I combined the best from what I found from others and added my own structure. Necessity dictated my doing this for two reasons: 1) I wanted to respect others’ copyrights, and 2) I couldn’t find any that were appealing to secondary students.

With that said, I offer MPJ’s SMP Posters for use in the classroom. (For JPEGs, click images below.) Each poster here has the following features:

The summary of the Practice straight from the Common Core documents, as listed in that famous grey box

SMP Posters Pic 1

The verbage of the Practice written in kid-friendly, first person language

SMP Posters Pic 2

A single word that embodies the particular practice

SMP Posters Pic 3

A diagram that displays an application of the practice, using Algebra as an example so as to span both middle and high school

SMP Posters Pic 4

A group of words that relate

SMP Posters Pic 5

A list of questions that pertain

SMP Posters Pic 6

A clip art image of a high school student to drive home the point that the practices are for them and not the teacher

SMP Posters Pic 7

An instructive statement that includes the word “Think”

SMP Posters Pic 8

A special shout out goes to the Jordan School District’s SMP posters for elementary schools which were the initial inspiration for this set. Other sources include: Eastern Bristol High School and Carroll County.

Multiple or Best Reps?

I find it interesting that on the day that we post our most recent lesson, 4 x 4, (sample page), Dan Meyer posts the question: Aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

I understand where Dan is coming from…Why the overkill, when one proper tool solves the problem. I have three quick responses to this.

1) If the goal of the current activity is to apply previously learned skills, then I agree with Mr. Meyer. Students should develop the savvy to choose the most appropriate tool at hand, and implement it properly. When faced with embedding a nail, is there any sense in using both a hammer and a rock?

2) If the goal of the lesson is to build conceptual understanding of the four formal representations of a linear relationship (words, equations, data, graphs), then generating the other three from any given representation develops this insight. How many students can graph a line by plotting the y-intercept and then counting the slope up and over, but have no idea that they just stated the infinite set of points that satisfy the equation?

3) If the goal of the day is to offer a point of access to the students, then the temporary representation will eventually give way to a higher level of abstraction. Look at the banner on Christopher Danielson’s blog. These multi-link cube models can represent the various ways to factor the number 24. Alongside these 3-dimensional arrays, students could be representing factors symbolically, 2 x 2 x 6, 2 x 3 x 4,  3 x 8 etc. In time, the blocks are left behind for a level of abstraction that is far more efficient. Afterall, it is faster to write the factors on paper than it is to build them with the blocks, especially when the students start factoring much larger numbers. So to push back a little bit on the original question: Are we doing kids a disservice by offering training wheels when learning to ride a bike?

The answer to the question of “multiple representations” or “best representation” is, as always, up to the judgement of the teacher at the time.