Better Homework

In yesterday’s problem we took apart a poorly designed math homework. Essentially the math textbook asked the students to practice a highly sophisticated method of addition.


The strategy for breaking one number and adding it to the first to make a 10 then mentally adding the rest is great, but probably should come naturally as it occurs to students as opposed to being forced on them. The real problem is the students who need it the most probably wouldn’t come upon this strategy naturally. So what we have to do is teach this strategy to our students so they can add it to their arsenal of weapons to use when solving math problems. We want to do all of this without actually walking them through a step by step process.

Why not just teach a strategy straight out? Two reasons: First teaching a procedure doesn’t always lead to “ownership” of the procedure. Second, because that isn’t the hard part of math. The hard part is recognizing when it might be the best strategy to use. (Which I suppose is kind of the definition of ownership.)

So for homework (and I am really against homework, but if you insist on giving it at least make it painless and force the parents to be involved as more than a checker of correctness) I might take these same problems and then ask them to choose one and talk out a strategy. They could use a phone or iPad to record the strategy, or call my google voice number and leave a message, They could tell it to their parents, or in any number or methods. The one caveat might be, if they are leaving me a recording it has to be less than 15 second long. (Do you ever notice how much you ramble when you are unsure of yourself?),  The next day I might ask two or three or even four students if I could play their recording or if they would like to explain their method. Then I could ask the rest of the class if they tried a similar or different method.

Another alternative, I could ask them to ask their parents to solve one of the problems in their head and teach them the steps they took. Then the student would have to do a different problem and explain the steps back to their parents.

A third alternative, I might ask the students to choose one problem and ask the them to solve it in 3 different ways. Explaining their work each time. I like to encourage a voice recording the when a procedure is new, because it is easier for the kids, I just want them to keep redoing it until it is short enough so that I can listen to 30 in less than a week.

Yes the textbook and I would like to teach the students all of the great strategies for addition, but I think they are going about it the wrong way. They are pulling each strategy out and teaching it explicitly, so instead of learning one way of “doing” addition students are forced to learn a plethora of ways to “do” addition. Completely missing the point of understanding the concept of addition and choosing the best strategy based on the situation.

By asking students to talk about how they solve a problem in their head, especially with others like parents, students are exposed to a variety of strategies for “doing” math.  By choosing to have students explain a few different methods the teacher can then make sure each child is exposed to all the strategies she feels the students should know. Now instead of asking students to solve a problem by the “making tens” method we can ask students which method did they choose. Did you choose Bobby’s method, Sarah’s mom’s method, etc… and why did you choose that method?

The point is not to make students practice problems, but to give them an arsenal to choose from and the knowledge of which weapon works in which situation.

Math Homework

What is wrong with this homework?

Nothing really. Actually, it showcases an excellent strategy for addition.

What you are supposed to do is make a ten, which makes it easier to add the rest.

Take the example 29 + 52. Look at the first number 9 + 1 = 10, take a 1 away from 52 and add it to the 29 to get 30. Then add the remaining 51 to 30, which can be done in your head.


The publisher even made it simple for you by putting a nice helpful line underneath the number they want you to break apart.

helpful hint

Lets try the first problem. Now go back the the first number and ask yourself 5 plus what equals 10? Yes, 5 + 5 = 10, so I need to take a 5 from the second number (27) and add it to 35 to make a nice round 40.


Then we finish the problem with the left overs from the original addend. I hope you didn’t add 27, because we took the 5 from the 27 leaving ourselves with just 22.


40 + 22 = 62.


Do you understand how to do the math now?


Good, because this is an excellent strategy for addition. To use this strategy requires you to be fluent in your addition facts up to 10, which also happens to be one of the common core standards for 1st grade math.

Then you should be able to add by tens (also a common core standard). It wasn’t explicitly asked for on this sheet, but my son’s teacher was nice enough to give out number lines on which they had practiced adding two digit numbers starting from a ten.

Again, I say this is an excellent strategy for addition, especially addition of two digit numbers. When I shared the picture I asked “What is wrong with this homework?” There is nothing wrong with the math, but everything is wrong with the homework.

What is happening is they are taking an advanced addition strategy and teaching it explicitly, then going back and asking students to practice it over and over again. This is no different than going back to the old days and requiring students to line up the number one above the other and adding down the lines. It is actually worse because that strategy is often the most effective way to add any two random numbers on paper. The strategy above is probably one of the easiest if you were asked to add two numbers in your head. (The second easiest for me at any rate.)

Instead of teaching students how to do this strategy it would be better to contrive a method for discovering this method in the classroom and hope that someone brought it up during a number talk. Even if they didn’t come up with this specific strategy I wouldn’t force it on students, rather the goal is to get them comfortable in discovering and using new strategies and as they progressed through the years they will discover it. You will see in the series of videos some ways to use number discussions in a classroom. Even those non-teachers should watch the first video at least.


Practicing someone else’s strategy for solving math doesn’t teach us how to do math, it teaches us how to follow directions.
Now my question is, “How would you make this problem better?” My suggestions tomorrow.

(edited for typos and readability)

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?

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