Writing Math

What is the cost of, the simplicity of writing with a pencil, vs learning to write math for a computer – with the more powerful responses that come with it? http://blog.mrmeyer.com/2014/a-response-to-the-founder-of-mathspace-on-the-costs-and-benefits-of-adaptive-math-software/

The conversation started with feedback. Research has shown that feedback is important. -> Many math software provide simple right or wrong feedback quickly. -> Too often this feedback is more the fault of syntax errors than actual math errors -> programmers add hints, or expand the possibilities of correct answers -> kids still hate it.

Teachers follow-up saying, basically feedback has to start with what the student is doing and thinking and start customizing from there.

The holy grail is for a computer to recognize the typical mistakes, often mistakes can be put into general categories, and send a standard, but custom, message to each students as they need it, preferably in the form of a question so that students actually solve the problem instead of waiting for the computer to do it for them.

I saw a company trying to do something like this through hints, but I think they needed to get more data on which mistakes meant what hint to actually give.

While the cost of learning to write for a computer program is high and the feedback so far doesn’t seem that great, I start to think of programs such as Geogebra and Desmos and the feedback they give, even though they don’t advertise their feedback.

When I was learning Calculus, before the widespread use of Maple (I remember struggling to input equations correctly then waiting minutes for the graph to load), I never made the connection between functions, tables, and graphs. I got lost in the calculations, which took too long (and I was faster than most at the calculations).

Fast forward fifteen years and I’m teaching middle school Algebra and we are given a class set of graphing calculators, suddenly relationships between numbers and graphs are evident. Which, brings us to the age-old question, “How much of teaching math is teaching understanding and how much is teaching the mechanics?” or “How com we never really understand something until we try to teach it.”

The question for teachers I suppose it:

“How long do I let my students practice on the computer with the limited feedback, and how much time do I have to work with individual students?”


I find myself spending a lot of time on imgur. It fascinates me how much of the new language of young people is changing from words to pictures.

The world is a changing at a fast pace and our language needs to keep up with the language of images.

Where once upon a time, the only time when the average person cared about or used citations was in an English paper for school, we are now starting to see them pop up all over the place.

It is not terribly uncommon to find someone asking for a citation in a Facebook argument. It is even more common to see someone cite a debunking of a meme, on Facebook or G+ or any social media. That isn’t to say we have a lot of well educated populous politically. There are still a lot of people who will believe almost anything. There are also a lot of websites who are more than happy to create their own semi-legitimate proof of their own half-truths.

With the rise in the use of citing a source to prove a point, and the more visual aspects of the Internet, (imgur)we are actually seeing a change in the method of citation. In an English paper teachers still expect to use the traditional form of citations, APA, MLA, or Chicago style. On the other hand, on social Media and blog posts we more often see the hyperlink to another article as opposed to a bibliography at the end of the post.

Getting even more popular is the infographic, Pictochart,  This will usually have a couple of citations written in small print at the bottom, but the modern writer still prefers inline citations, like hyperlinks. So the next invention that I have been seeing is the Thinglink.

Similar to a infographic the Thinklink can insert a pop-up for more information. Most of the time this is used to give someone more information, but I love the idea this author uses. He uses Thinklink to add citations to his writing.

So, the question is, “Is our education system keeping up with the changes?”

A conversation

You should comment more on blogs.

Sure pot calling the kettle black and all, I don’t comment as often on blogs as I should, but still you should.

Not because I want to feel special, but I do every time you make a comment.

You should comment simply because it furthers learning. It forces me and my blog, and you by extension because you are reading it, to think, grow, and evolve.

And no please don’t comment here just for the sake of commenting, go find another blog that has few comments and just say something, even something that is critical.



A comment on this blog post just seems to say to me that being within the Zone of Proximal Development (ZPD) is a motivating factor in education.  It isn’t, it’s a scaffolding tool. The ZPD is basically that sweet spot where a question is just hard enough to make a student think, but not hard enough that they cannot figure out the answer.

“What is important for personalisation is the ZPD where a student has enough support and incentive to become independent and make their own connections. Before that can happen, it is up to the teacher (or virtual teacher) to teach students to make connections by example or open questioning.”

Sure, you can interpret that comment differently, but I’m not so sure many of the EdTech companies actually do, especially in math. Too many of these companies seem to think that what constitutes math education is the ability to solve a problem so they come out with adaptive learning, hints, pre-testing, post-testing, alignment to common core, and anything else they can find to make the teacher as superfluous as possible.

France:The Tyranny of Mathematics by Sylvain Labeste

Of course the company line will always be, “Nothing is better than a real teacher”. When in the end what they are selling is the ability for a human being to supervise the work of 100 or more students at one time. Face it, the hard facts of education is that 70 to 85% of every school budget goes to teacher salaries.

Think about that for one second. Here let me help you, a school district near me just passed a bond to build a new school, $40,000,000, and still 80+% of their money goes to salaries.

Carl Schurz High School by Teemu008 (not this school)

If an edtech company wants to sell to a school they will do so by making it possible for teachers to do more work.

I’m getting off track, I don’t want to talk the money, but the money is one of the reasons for the tortured justification. the Zone of Proximal Development is not a motivation tool, it is a scaffolding tool. When questions are posed at just the right level students can, if they choose, figure out the answer and hopefully learn. Magic right? If enough of these questions are strung together eventually the student will progress up the ladder to knowledge.

Total poppycock, when a student is motivated he or she will work to answer the questions posed. If those questions are within the ZPD then the student will be able to work more independently and is less likely to give up. See, the motivation comes first.

Not only that but we can expand the ZPD of our students, but teaching them how to learn. Giving students tools and instructions for breaking down questions into manageable parts, teaching students skills for researching information, connecting with experts, etc… all can help make what was once an impossible question, into maybe we can figure this out.

Shooting the inspector by edrabbit

None of it however happens without the motivation. Motivation can be external like a basketball themed question, but it can also be internal. “Why do I hear the sound of fireworks a second after I see the flash?”

Teachers can sus out the educational value of the themed questions. This is a lot of work and requires a lot of creativity as well as subject knowledge. Technology can help, Open Educational Resources (OER) can be a way to crowdsource the knowledge of thousands of teachers.

Teachers can also encourage the internal drive of students. Instead of drilling procedures on children day after day quashing any and all questions. Teachers can slow down and encourage questions, “Hey, that sounds like something I would like to know. What is a good first questions to start researching?” or “I’ve got this questions and I hope you can help me design a solution?”


edited for clarity

The Love of Science

What if kids love Science when they are young because they get to play around with it. You know the hands-on stuff is fun and occasionally gross.

Image from Wikimedia.org








Then as we get older we lose some of the hands-on stuff so we start to hate Science. The fix is easy right? Just add more hands-on stuff!

Photo by looking4poetry

What if the fix isn’t that easy. High school Science is more that just hands-on. It requires a bit of precision; quality measurements, repeatable experiments without impurities. This isn’t, throw some corn starch and water together and look it’s a magic substance.

Rigorous experiments in Science require creating experiments that remove all but one variable. They require students follow complicated directions and make precise measurements. Then they have to collect the data correctly and feed it into tables and graphs, perhaps even perform some magical mathematical functions. Then after all that hard work they get a conclusion that might, just might, resemble the correct conclusion in the book.

Image from Wessex Archaeology

Not quite real experimental Science of discovering the cure for cancer, but closer than those semi-magical demonstrations in Kindergarten.

So what would be the better way of teaching Science?

  1. Nose to the grindstone. Here is your worksheet and a virtual experiment don’t make any mistakes and pay attention to those details, especially in the math.
  2. Do the experiment, fail, do it again because you were sloppy. Then do it again because you were sloppy, then do it again because you were sloppy, oh forget about it here are the numbers you should have gotten to use now pay attention to those details.

Yeah, I don’t like the choices either, though honestly if I were forced to pick I would choose the second. The real trick in teaching, if you want a silver bullet, Science, is to Be Less Helpful (I think Dan Meyer can be credited with coining that phrase).

So really, the question is, “How do we keep the interest of our students, in this really cool experiment, while simultaneously requiring them to take excellent measurements, and controlling for variables in experimentation?

Maybe this is the way – http://www.fastcodesign.com/3032886/innovation-by-design/this-is-the-physics-lesson-of-the-future-and-it-looks-insanely-fun#12

Image from Matthew and Tracie

This is, at least as I see it, a kind of middle ground. Students are still excited by Science, they are just getting bogged down in the details. Quite honestly I think those details will bore just about everyone except a Scientist. On the other hand understanding that there are details and this rigour is important are also takeaways we really need to have. However, if you want to collect excellent data and import it correctly into graphs and tables so you can examine the actual physics this seems like an excellent way to do just that.

Blended Learning

Blended learning is new for teachers as well as students. Most teachers don’t have a frame of reference because they have never experienced it in their own lives. Therefore it is easier to find examples of blended learning opportunities for teachers than it is for students. As teachers learn by doing. Some schools around here are creating a new position called a chief innovation officer or something similar. I would assume they do things like help implement blended learning and alternative professional development methods.

Like many other innovations online learning filled a niche, the cheap, easy way to get unimportant learning. (Think compliance training) Many online schools and classes continued that tradition.  Blended learning, I think has become popular as a way to add rigor to those online learning opportunities. It is just so much harder to cheat or slack off in real life.

Blended learning can be so much more:

There is a cultural shift that needs to happen for true blended learning. Many teachers I work with see a web site as a place to put up the newsletter and post homework. They don’t know what they don’t know and never imagine the interactivity they could have in the classrooms.

Blended learning happens when we shift thinking to something along the lines of – Here is how I teach math in the classroom, here are six other methods you can practice with your children (look ma no homework). Or here are the shared notes we took in class today, add some comments. Or any number of ways teachers extend the school day not just to the students, but also to bring in or increase parental involvement.

Some teachers do this shift on their own, and sometimes get in trouble for it. Some teachers wait for the OK before doing anything, some will not do it because they don’t see it as part of their job. (Certainly this would take up more than the 1/2 hour prep time most teachers get in Illinois, not to mention the fact that many teachers would end up checking work online late into the night, weekends etc…)

But what I can tell you from visits to blended classrooms and schools, in both traditional public and charter schools, is that students tend to find what exists thus far as fairly dull, lacking both the community and the accountability that comes with good face to face learning. A number of students told us at one highly celebrated blended school that they liked everything about the school except for the online learning!

Fifteen years ago I was studying education at Northwestern and the technology professor was talking to me of the advantages of online work. That students who didn’t like to speak up in class could, and often would, be active participants in teaching and learning online. But even then it was never about teaching online, it was about creating and environment for learning that was appropriate for the students. It wasn’t expected that every student would participate online, or even if the participation would be better, just that it was a way to extend the learning and reach more students. Blended learning works when you think like that.

If blended learning is extra work to be done online, or a substitute for traditional homework, it is just the same junk in fancy wrapping. When it adds something new, giving parents a method of being connected, extending the learning that happens in class, giving a voice to deep thinker, or shy student, then and only then does blended learning enhance education.