Wednesday, February 6, 2019

Rolling Dice Microworld and Learning Fractions

Common Core, no more? The new governor of Florida has decided to eliminate their version of the common core standards and replace it with something else. But he doesn’t know what that will be. All he knows at the moment is that he is committed to teaching the basics (reading, writing and mathematics). 

Meanwhile a teacher posting on My NCTM’s community forum suggests that teachers should not use a particular textbook because it isn’t aligned with the standards. 

Another 4th grade teacher who was falling behind her common core pacing chart decided to skip a chapter on probability to try to catch up. My reaction was what a missed opportunity. I’m sure she wanted her students to learn their fractions but skipping the probability chapter is not the best way to do that since fractions can be taught using probability as a motivator. 

She may have known that, but her concern that a poor performance by her students on standardized tests would reflect badly on her teaching forced her hand. 

Probability would enhance student interest in learning fractions, but this teacher needed to save time and go directly to the fraction skills chapter. So how can an elementary teacher get the best of both worlds?

The answer is to blend the learning of fractions in the context of probability using technology in the same lesson unit or chapter. It’s too bad textbook companies don’t do a good job with that. 

Here’s my attempt to help with solving this problem. I’m working on a mini probability project that includes three dynamic microworlds written in Scratch

These three links will get you an initial peek at these three draft activities.

Friday, January 11, 2019

CLIME - Technology in Math Ed - A Year in Review

Dear friend of CLIME,

Here are some of the blogs CLIME posted in 2018.

Another Rags to Riches Story
What to do with a Broken Calculator that's Broken...
Technology in Seattle - a Showcase of How to Use it Effectively(?)
High School Math is Not Working
Will Blended Learning be a Game Changer?
On Making the Ordinary Extraordinary in Learning Math with Technology
It's the Pedagogy, Stupid
Focusing on the M in STEM in the High School Math Curriculum

Your feedback on any or all of the above posts will be most appreciated.

It's a new year and a time of transition for CLIME. We will be discussing that issue and others related to math & technology at our annual CLIME meeting this year in San Diego.

If you are planning to be in San Diego next April please note the date, time and place for our meeting. More details to follow in future blogs.

CLIME's Annual Meeting
Thursday, April 4, 2019
7:30pm - 8:30pm
Hilton San Diego Bayfront Hotel
Room: Aqua 310

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.