Wednesday, December 12, 2018

A Few Things That Annoy Me

Andy Rooney (1919-2011)
As I notice and wonder about blogs that I read, I'm finding two themes (memes?) that are very annoying. They are (1) instructional and (2) notice & wonder routines. Now that I’m retired and become more like Andy Rooney of 60 Minutes fame, I find my personal routines becoming more and more distasteful. After my morning coffee (which I look forward to) I do the following: (1) Clean the litter and litter area (we have 5 cats) (2) Make the bed assuming of course that my wife is up by then (3) brush my teeth with a fancy battery powered toothbrush (Quip) that takes 2 agonizing minutes to complete and (4) take my mouthful of vitamins which takes a while to sort out and swallow. Then I finally make my second cup of coffee and go to my desk to think creative thoughts (like the ones I’m writing now) and time begins to fly by. So needless to say I’m not fond of chore routines. Unfortunately, they are mostly necessary evils.

My concern about using instructional routines in math classrooms is that they can take over and become boring to students. Speaking of boring there are the notice and wonder routines. The ones I notice and wonder about are happenings in classrooms of bright, bushy tailed students that will notice and wonder excitedly about anything including spilled milk. But I wonder about the students who are less enthusiastic noticing, but only wondering and joking about why the teacher is asking such a boring question.

Now don’t get me wrong. If notice and wonder routines work for you, then by all means continue.

I found out from working at an adolescent kid’s camp in Maine back in the 70s that anti-routines where the best anecdote for boredom. At the end of each day the counselors would gather and plan the events for the next day. They were different each day. I know that camps are supposed to be about fun, but they are also educational. The lessons I learned at that camp (when I was in my 20s) served me well. I included an element of surprise in my math classes every day especially for those students who suffered from routines paralysis.

Friday, December 7, 2018

Another Rags to Riches Story

New Version of Fraction Tracks (NCTM)
In a previous blog I lamented over the demise of the Broken Calculator. Well there is another program - Fraction Tracks (FT) - that appears to have disappeared as well for the same reason (unsupported plugins). Why does FT need resuscitation? If this simulated board game is used with care it can be a part of the solution to the number one challenge in elementary math education: learning fractions.  Before we discuss this NCTM FT "miracle" app, a little background may be useful.

If you search for a fraction app on the Web you will be amazed at how many there are. FT was/is a golden needle in that haystack. It encourages students to explore equivalent fractions in order to continue with their moves and win the game. From a teacher perspective it can improve their knowledge base and perspective on teaching fractions.* But, alas, the NCTM version which was developed with funding help from the now defunct Marcopolo project suffers from the "unsupported plugin" plague.

(Put the plague story on hold for a minute or so.) An effective way to introduce the game to students is to use a different modality: a board game version of FT that doesn’t require a computer. Here’s a description of one version of the game.
The Fraction Tracks Game, used in fifth-grade classes at Tollgate Elementary School of Expeditionary Learning in Colorado, provides opportunities for students to use equivalent fractions, break fractions apart into unit fractions, and mentally add and subtract fractions with denominators 2, 3, 4, 5, 6, 8, or 10 as they try to move their game pieces from 0 to 1 along tracks labeled in various fractional divisions. Included here are the game boards, rules for a basic game play, and “extension” rules that provide more advanced practice. 
Another way to introduce the game (and eventually the app) is to play it cooperatively. 
Start the game by placing one chip on each track, at the beginning (0) of each track.
A student then chooses a card from a deck of fractions. The fraction on the card is the total move a player must make using one or more chips.
The first fraction chosen in this game (see figure on left) was 8/10. On the board the bottom chip is moved to 8/10.
 The play can move on one track or on several tracks, but the total must equal the fraction on the card. What if the next card is 4/10? The student may be stumped as to what to do unless he realizes that 4/10 is equivalent to 2/5. Thusly, the fifth's chip can be moved to 2/5 or the student can realize that 2/5 = 2/10 + 1/5 and thus score a point on the 10th's line as well as move the 5th's chip to 1/5. (See figure below.) The ultimate object is to get all 7 chips to wind up on the 1. (Here's more on strategies for making moves.)

If a play cannot be made, the game is over. The team score is the number of chips that made it to 1.
Highlight: Watch this video** of one teacher’s way of teaching FT to her students. 
The app version has been resurrected (written in Flash) and is now available here. It is a two player game and has a timer. Have fun!
** The video is 16+ minutes but well worth it! Watch it soon because it too might disappear. (A different kind of plague: lack of support.)

Wednesday, November 28, 2018

Technology in Seattle - a Showcase of How to Use it Effectively(?)

The Seattle regional conference opens today.
According to my search results there are 262 sessions of which there are 22 presenters that indicated that their session has a technology focus. That’s 8%. A really low number based on my years of counting sessions at previous meetings. I hope the regional meeting’s low number does not portend what will happen at the annual next April. I don’t think there is causation here since the committee for San Diego is different and well on their way to planning what’s happening there.

So why am I so fixated on tech at conferences? A conference could be great without tech sessions, right? Well there has been a lot of progress at conferences on the peripheral side - more Wifi availability - speakers have Wifi access as well and it's less dangerous for a speaker to use now than it was in the wild early days of the Internet. However the way speakers use technology for the most part is just a replacement for transparency slides that were ubiquitous when I first started in the 1960s. That's a shame since sessions should be promoting how to use technology in creative, cooperative ways. Let see if that applies in Seattle.

This link will get you a list of all 262 sessions. If you choose this category type: technology and tools (22) you will see a listing of the 22 sessions - including 2 that have been cancelled.

Here are the descriptors of some of these sessions:
Learn to leverage adaptive, game-based challenges and gamification principles to provide joyful, "just in time" activities that personalize learning for all students. Teachers, coaches, and administrators will experience practical strategies for 1:1, computer lab, and BYOD device settings to move K-12 students to deeper levels of understanding.
A great task engages students with an interesting problem involving essential content. It builds deeper understanding, supports discourse, and provides the opportunity to persevere. Technology alone does not solve it--nor does technology replace thinking. Learning is enhanced when we leverage classroom technology to guide students to deeper math.
See how handheld technology promotes algebraic thinking and a deeper understanding of sequences, functions, and limits to help students move from algebra to calculus.
With access to technology, K-5 teachers now have opportunities to promote exploration and inquiry, all while encouraging collaboration. Keep the devices out and help students really engage with the content.
Come learn with a Desmos Fellow! I will be sharing activities and teacher moves that have created deeper learning, more relevant and thoughtful discussions, and a greater level of engagement in my classroom through Desmos' Activity Builder. These free activities can be used immediately in your classroom.
Graphs programmed with adaptive "Cold, Warmer, HOT" hints allow students to play hide-and-seek in precalculus and calculus. These dynamic interactive graphs, created with Desmos, purposefully guide students as they explore concepts, make conjectures, and build intuition. Many graphs will be shared. Stop in to check it out--you're getting warmer!
Fractions are complex mathematical concepts that children struggle with. We will present findings from two types of innovative instruction to fourth graders highlighting the use of number lines, scaffolding, and gesturing to enhance children's knowledge of fractions. One intervention used paper-and-pencil and the other used gaming technology.
Desmos and GeoGebra offer free access to all with top-of-the-line, highly reviewed websites. Integrating them into the classroom makes math more relevant, engaging, visual, conceptual, and fun for today's digital native students. This interactive session will cover the basics of each website and include strategies for learning.
Technology can amplify student thinking and creativity, helping students create and understand like they haven't before. It can also think and create FOR students, dazzling them without inspiring or educating them. We'll look at technology (including handheld calculators, Sketchpad, and Desmos) that thinks WITH you and your students, not FOR them. - Eli Luberoff, Desmos 
Augment your calculus teaching by using Desmos to animate its greatest hits! We will share ready-made examples, plus lift the hood to show how to dynamically visualize such classics as secants approaching tangents, derivative sketching, related rates, Riemann sums, the fundamental theorem of calculus, Taylor polynomials, and polar curves.
Learn to integrate benchmark percents with Desmos tables and graphs to develop a percent backstory for a given situation. The number lines, tables, graphs, and equations discussed highlight the proportionality of percents and present students with a mathematical backstory to guide their decision making when solving percent problems.
Sounds like if you go to these sessions you will be inspired. Hopefully so.

Tuesday, November 27, 2018

What to do with a Broken Calculator that's Broken...

Duncan Keith's app
What to do with a broken calculator that is really broken? Back in 1989, Judah Schwarz developed a software package named "What to do with a Broken Calculator?" It was a simple, but brilliant idea. Take a simulation of a calculator that allows the teacher to break (disable) some of the keys and challenge the student to produce an answer, of say, 50 with most of the number keys disabled. This can be a challenge and an important learning experience for young students.

Unfortunately, because of the vagaries of Java-based applications all of the really good versions including Cut-the-Knot, Seeing Math and Mathcats don’t work any more because of “unsupported plug-ins.”  Alexander Bogomolny who developed writes: “While a good deal of the site has lost its interactivity (Java failure), most pages remain interesting and informative, whereas many others that do not use Java, are still interactive. The situation is unfortunate for which I am deeply sorry. Alex had a working version of Broken Calculator which we may never see again. But then who knows. Stay tuned. If there’s a way and time to do it, Alex will figure it out. (Unfortunately, Alex passed away on July 7th, 2018.)

In the excellent, but ill-fated On-Math NCTM publication, Broken Calculator was one the issue’s themes. Here’s a quote from the article:
Broken Calculator is, in effect, an inexpensive, versatile number laboratory. Using Broken Calculator to solve problems forces students to explore operations on numbers and representations much more deeply than does rote application of algorithms. Most important, working with Broken Calculator shifts the responsibility of finding and expressing the answer to the student. Solving problems means inventing a wide variety of strategies to work around the limits set by the disabled keys or functions. Students deepen their number sense, problem-solving capacity, and understanding of the number system— and have fun at the same time! 
So is the idea of Broken Calculator lost forever? Or is it alive somewhere? Alas, there is a working remnant(?) of Broken Calculator alive and well on Duncan Keith's site. I say remnant because it's not the open ended one that was developed by the staff at Concord Consortium and others, but rather by an independent software developer Duncan Keith who somehow manages to keep this site going despite the unpredictability of Java updates. If you go here you will meet Eric (below).

The program has a built-in library of challenges (levels). The one drawback is that it has a timer that may be intimidating for some students. Also it's designed to be used by students independently so it would be difficult to use with a whole group of students to analyze strategies. (Read this.) Also, the calculator does not do order of operations.

So until we hear from one of our Broken Calculator champions of the past (or future) Eric smashing his calculator is our best hope for now.

Unless, you prefer...

There is a low tech version provided by Dan Finkel developer of Here's a New York Times article about his Broken Calculator approach.

Thursday, November 8, 2018

High School Math is Not Working

Here we go again. We've seen similar charts like this before. What are we the math education community doing that's contributing to this trend? Our new NCTM president Robert Q. Berry recently posted a message to the community in response to articles such as this: MATH SCORES DROP TO A 14-YEAR LOW AS ACT SHOWS MANY HIGH SCHOOLERS UNPREPARED FOR COLLEGE.

He writes:
"The decline in recent years in the mathematics score on the ACT exam has many educators and policymakers concerned. There is apprehension about whether these scores suggest a negative impact on science, technology, engineering, and mathematics (STEM) readiness and the potential impact on America's economic, social, and political security. While I understand why these are causes for concern for many, I see the discourse about the ACT mathematics scores as an opportunity to broaden the discussion to include issues of equity, curriculum, and assessment. [...]  
Critical conversations are necessary for knowing and understanding not only the indicators for mathematics and STEM readiness but also the inequities that contribute to the factors that offer advantages to some learners while disadvantaging others."
The inequities problem continues as does the call to do something about it. Holding conversations does help, but the majority of educators feel helpless to do something that would make a significant difference to what is mostly a complex political problem. 

In my opinion the complicated problem we can solve that keeps things at status quo is the continued use of flawed textbooks that not only do not adhere to the Common Core but also do not offer help in effective pedagogy. Many teachers who have the freedom to use substitute lessons that they find on the Internet or develop collaborately with their colleagues at school or on the Internet (ala #MTBoS) help improve student's learning. For example, some creative schools like SLA (Science Leadership Academy) in Philadelphia use projects to motivate the learning of conventional topics from algebra I and II and geometry. The teachers also create their own lessons which are cooperatively developed. Unfortunately too many teachers just follow the textbooks lessons which turn too many students off to math. There are of course exceptions of textbooks that are well designed and conceptually well grasped by students. An example is EDC's Transition to Algebra (T2A) which is designed to make the student's experience of learning Algebra more understandable and interesting. Also, using the Heinemann book, Making Sense of Algebra: Developing Students' Mathematical Habits of Mind help teachers to dive more deeply into the goals of T2A so they can provide an optimal learning experience for their students.

A more radical (creative) approach* to writing textbooks is to make lessons more like stories which are intrinsically interesting to kids. There are videos that tell stories that could be used as part of a lesson: the 3-act kind that Dan Meyer likes. My 3-part lesson is: 1. Set the Stage 2. Do the Activity 3. Debrief.

Three examples are: The Weird Number, Murdered for Math - Making sense of Irrational Numbers, and 13 x7 = 28

Also, STEM should be a math credited course in High School. We moved Algebra 1 to the 8th grade in my lifetime, why should seniors have to wait till college to experience STEM? It doesn't make any sense not to do that. See my previous post on this topic.

*More on "radically" creative lessons in future bog entries.

Monday, September 24, 2018

Regional Meeting in Hartford, CT

The first of three regional conferences is almost here. You can register here. As far as technology is concerned NCTM has highlighted Technology as a Tool sessions in its preview as an essential topic. Here's the description.

Technology and Tools

Discover ways in which tools and technology enhavce meaningful instructional experiences with presentations that highlight opportunities to support students' increased discourse, strategic thinking, and engagelment with importan mathematical ideas.

The opening session sounds good for an overview of issues concerning teachers today.

OPENING SESSION | Thursday, October 4, 5:30 p.m.–7:00 p.m. AMY LUCENTA
Making Equitable Practices Routine
Students face a constantly changing, data drenched world filled with fake news and powerful technologies. Learning concepts and skills will not suffice, and leaving students behind is not an option. Each and every student needs to develop mathematical thinking and reasoning. This can only happen when students
are talking together to make sense of important mathematics and each and every student is contributing to the conversation. So, how do we ensure that all students develop as mathematical thinkers and communicators? Leverage the predictable nature and uniform design of instructional routines to support students and teachers alike.

You will find a list of all the sessions here.

To see all 25 of the Technology and Tools sessions click on the Search Sessions image above. Then choose Technology and Tools from the Category field.

You should see the following list for Friday:

And for Saturday:

There are a total of 195 session so 25 Tech sessions are about 13% of the total.

I was surprised to see that almost all the tech sessions were geared toward high school teaching. Desmos, Geogebra and graphing calculators were the application most used in the sessions. Since my personal interest is middle school, here's the ones I would attend (in no particular order).

Session 27: Geometry Explorations: From Drawings to Constructions, Discovery at Your Fingertips
Workshop leader: Karen Greenhaus
Experience the power of exploring geometry concepts hands-on, starting from simple drawings of geometric objects and going right into manipulating their attributes, testing conjectures, and developing geometric properties. Participants will participate in a hands-on workshop working with technology and geometric problem-solving activities.
Session 103: D^3: Discourse, Differentiation & Desmos: A Deeper Look at Technology through an Equity Lens
Workshop leader: Allison R Krasnow
This workshop will explore how to integrate Desmos with your curriculum to deepen discourse, differentiation, and formative assessment. You'll experience several strategies for more robust mathematical engagement with Desmos including using word banks, sentence frames, and supporting students to do error analysis on common misconceptions.
Session 142: Ideas for Math Class on Twitter: Sharing, Exchanging, or Lurking
Burst Speaker: Robin Schwartz
Twitter is a great place for math educators to find tasks, routines, and camaraderie. In this workshop, we will visit the #mtbos (Math Twitter BlogOSphere), #elemmathchat, #observeme, #iteachmath, and other Twitter hashtags and people for inspiration and motivation for both teachers and students! 
If you attend any of the above sessions (or any others for that matter) please let me know how it went.

Friday, August 3, 2018

Will Blended Learning be a Game Changer?

Here’s a picture (circa 1968) of me pretending to teach while my students were pretending to learn. But how could I say that given that this was my best Algebra class and most of the kids got As and Bs? Well, what I discovered many years later was that the students were just doing what I asked them to do to please me and no one was very inspired to explore algebra beyond what I fed them. But that was good enough then. My model was the textbook: Mary Dolciani’s Algebra 1. For those of you that remember the Dolciani method you probably recall that it was a hard-nosed, traditional approach. For the kids it was a series of hoops that they struggled to jump through. But to what end? Algebra II of course! I didn’t question this approach at the time, though I did think about alternative ways of teaching math in my university days. My school’s culture was traditional, so I didn’t dare to deviate.

According to an Edutopia article by Beth Holland the peril of such an approach handicaps students. They remain “consumers of teacher-directed content instead of becoming creators of knowledge within a context that they can control.” Will Richardson (during an interview with Beth)* said that he went through a standard school experience and he turned out OK. So what was so bad about that? Beth responded that that model was OK for then, but not for today. We need a new model. This new approach shouldn’t make the old model “wrong” but through a gradual process move teachers to a new way of teaching and learning given all the new resources that are now available. Teacher buy in to new models is crucial. One reason that the introduction of technological devices into the classroom is a good way to start is because most teachers understand that technology tools are important in the lives of students so integrating them into the classroom is an accepted norm by most teachers.

In the aforementioned article in Edutopia Beth writes:
“A few months ago, I noticed an increased amount of discussion around the notion of blended learning. Many of these conversations started on a similar note: “We’re blended—all of our teachers use Google Classroom” (or Edmodo, Schoology, Canvas, Moodle, etc.). However, in probing further, I often discovered that these tools had merely digitized existing content and classroom procedures. […] While blended learning [e.g.] brings with it the promise of innovation, there is the peril that it will perpetuate and replicate existing practices with newer, more expensive tools.” 
“True blended learning affords students not only the opportunity to gain both content and instruction via online as well as traditional classroom means, but also an element of authority over this process. […] blended learning could fundamentally change the system and structure of school, and provide students with a more personalized, active learning experience.”
Previously, Beth interviewed 3 instructional coaches from Bellevue, Nebraska about their 1-1 iPad initiative and move to blended learning.
“These coaches saw blended learning as providing students with control over how they learn, the pace of the learning experience, and where they might choose to learn within the classroom.”
Supporting student agency is one of the main tenets of the blended model (that is described in detail in Michael Horn’s book “Blended”) that districts like the one in Bellevue, Nebraska have adopted. But unfortunately blended learning can become just a buzzword when teachers and administrators don’t understand it very well.

Horn writes on page 34 “Some element of student control is critical; otherwise, blended learning is no different from a teacher beaming online curriculum to a classroom of student through an electronic whiteboard.”

*iTunes podcast “Modern Learners, Podcast #47”