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)


I’m reading through this “Implementing Common Cores State Standards: The Role of the Secondary School Leader Action Brief” by direct pdf found here (

When I come across this section.

fluency in mathematicsI read the number 3 there and I think to myself. That’s just wrong. I finally put in my notes that it is an incomplete definition of fluency. There can be no doubt that fluency includes being able to recall simple math facts quickly and accurately, but it means so much more. (quickly recognizing common factors, fast estimation, finding patterns etc…)

Then I read number 4 and I think I have to blog about this. Teachers who read number 3 will naturally assume that math is practicing calculation. The same thing we have done for years. But going to number 4 is says specifically that math is ‘more than “how to get the answer” ‘ “more than a set of mnemonics or discrete procedures”.

So should I have students “memorize, through repetition” or not?

Sure this is an action brief for high school teachers. Students should have their basic math facts down pat, but are we expecting third grade teachers to spend most of the year forcing students to memorize the facts?

If education with Common Core State Standards  CCSS is a spiraled curriculum and “The CCSS require educators and school leaders to make fundamental shifts in practice…” shouldn’t that shift be universal? Shouldn’t we teach fluency through “…profound changes in the way students learn and are assessed, in the way teachers teach,…”

Why should one area be expected to “…memorize, through repetition…”? Be a little more clear in your writing. Instead say, something to the effect of, “Students will have enough practice through real-life experience that they will memorize, through repetition the basic facts of mathematics.”

Or should third grade teachers teach memorization?

Teaching Math

Most math instructional software that I have seen tends to fall in two categories, either they want you to follow  procedure and check your answers (occasionally checking the steps along the way) or they play some games and try to allow you to discover math concepts along the way.


Mathematics (Photo credit: Daniel Morris)

This coincides with the two dominate teaching philosophies, teacher led examples or some form of constructivist/discovery/inquiry teaching.  (yes I hate lumping them together but they aren’t the subject of the post)

There is a third way that most good teachers have used naturally, but few have considered it a separate teaching method. That is teaching from misconceptions. When your math teacher asked you to show your work, they wanted the opportunity to follow your thinking and then either explain in writing or during a conference the exact part of the problem where you made your mistake. That is, they were looking for your misconception.

It turns out that this is a very powerful method of teaching, or more correctly reteaching a concept. Now to the point of this post. I haven’t seen a math software that teaches this way. I’ve seen a science dissertation on teaching through misconceptions (this explains it quickly), and now I’ve seen an English theory on teaching through misconceptions. When are we going to see a math program that teaches from misconceptions?

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Review of Math Tutor

Dear Kevin,

Thanks for asking readers of Dan’s blog to chime in on your website.

What I don’t understand is how adaptive programs are better than a worksheet. Sure it’s nice to know where you went wrong when solving a math problem and a teacher can’t give every student the individual attention s/he deserves, but that can also be considered also be considered robbing the student of persistence in problem solving. (Individual v Personalized)

When a student comes to me, or a table mate, and asks for help we can backtrack his or her attempts. I can see the mistakes and make a quick guess at what s/he was thinking before asking a question.

I’ve often thought that for these adaptive computer programs each problem should have a list of common mistakes, and these mistakes should lead to questions the computer should ask after the student learns s/he is wrong. It’s just that students often have to input information into fields and the computer has to read the field and interpret what the student means. It become a huge AI problem.

I worked for a company that was paying teachers to write hints and clues along this manner. I thought it was a good idea, but I couldn’t seriously add the information I wanted to each question in the time allotted. I also thought it would be better idea to have teachers bounce ideas of off each other to make each problem better.

I also want to ask how we can incorporate different methods of solving a problem? (And perhaps that is my biggest problem with the last company and yours) Sure if I’m teaching multiplication I can give a 3 digit by 2 digit multiplication problem and check each step in the standard algorithm, but what if my student wants to use the lattice method? What if s/he wants to add repeatedly? What if s/he wishes to expand the numbers first then multiply? How do we write this into a computer program?

Ok you’re teaching students to use the most efficient procedure of multiplication (I would type it into Google or Wolfram Alpha) Where do we learn those other methods? Why are they valued less? If I can multiply do I understand? I get it this was all taught in class before we started using this method of practice (Why are we paying so much for a method of practice?). Is that the way the software will be used?

Yes, you are building software as an aid to teachers. A way for students to practice problems and be told immediately when and where they make mistakes so they can self correct. So I ask, “Is that the best way to practice?” “Will, that be the way your software is used in schools?” “How can students practice multiple ways of solving a problem?”

I get the temptation to use adaptive software to teach math. If you know certain sub topics then you can learn a specific new topic. Math of course is pretty well mapped out too. To multiply we need to know how to add, to add we need to know how to count, etc….. I don’t know how we can go from doing math to knowing math from a computer program, at least not without some quality guided discussions with real live human beings. Maybe Watson can do it, we should try?

What happens and what is happening on this website, at least in my opinion, is that the emphasis is on the skills and not the knowing.

Stills captures from using Skitch and the practice problems

Persistence is important in math. Many students will just type in numbers until they go green.

The second hint was better. It’s true though students don’t read the questions they just flop around until they hit upon the right answer. My three favorite questions as a teacher are: “Did you read the questions?” “Why are they asking you/?’ and “Why did you do that?”

I think your instructions are nice and clear, but they are teaching how to solve the problem, not teaching math.

What happens if I the student has a question that is not specifically about solving the problem, but is related to the fundamentals of the concept?



The Problem with Standards

Some people suggest that the medium in which we present mathematics is the problem. And I think that is true. However, as with all things that is only one part of the problem.

The Department of Education sees a lack of high standards in schools as the main problem in education.

Politicians, parents, schools boards, and millions of other people see unified standards as a method of solving this problem.

It certainly is tempting. The idea that if everyone would just teach that same stuff then at least we all have a base of knowledge to build upon, to depend on.

If we raise standards by requiring schools to teach specific standards how do we make sure this is being done? The obvious answer of course to raising standards in the quality of education is to set standards and then measure whether we are meeting those standards.


Let’s follow the logic:

When people think that a test is the way to measure a students mastery of a standard we think it is a good idea to develop a better test.

When we try to develop a better test that measures specific standards we spend a lot of time looking at those standards.

We write questions with those standards in mind.

It is very hard to write a question that meets a specific standard and only that standard.

We modify the question so that it only includes information or questions for that specific standard.

These modifications change the question from a fair description of real life into some mutant cyborg that scare little children.

Mutant Cyborg Costume front

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Kids Say the Darndest Things

I was listening to the radio the other day, WBEZ, and they were interviewing two remarkable 17-year-old musicians. (don’t quote me on anything I am blocked at school from going back and checking my facts)

The first boy had a learning problem with reading music, he just couldn’t do it. It wasn’t that he didn’t want to read it he just couldn’t. Yet somehow here he was as either a finalist or winner of a $10,000 dollar prize for being a musician. Can you imagine a writer who couldn’t read winning a prize?

I can actually, with the help of books on tape and voice to text software.

The second girl was even more interesting. She said she transferred from private school to public school in 5th grade. She had never done math without hand on activities and was confused. She said, “I don’t know how kids can learn like this” (I’m pretty sure those were her words, they kind of stuck in my head)


Make of this what you want, but I thought it was very interesting.

The Mobius Strip

We are starting a new math club online. I look forward to learning some different ways to introduce math to my children while at the same time getting them excited about learning math.
We start with a Mobius Strip. I simple piece of paper with a bit of a twist. Watch and learn.

The babysitter was still here waiting for here ride and she had never seen a Mobius Strip before. She picked up the idea very quick, (quicker than I ever did).

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Material-less math and questions

Playing Piano

As a support person I often find myself with a class for a day, or a period, or even just a few minutes while the teacher is gone. I need something to keep the students occupied with something other than gossip. So when the question came up “Need games children can play without any material to improve mathematical skills for thousands of slum area’s children.” I paid attention.

The first suggestions were games of NIM, which is a game played with stones. Any sort of counter will do and they don’t have to be uniform. Basically the game is played by making a pile of stones then picking up a number of stones in turn eventually forcing your opponent to pick up the last stone. Rules can include putting the stones in various sized groups and picking from one group at a time. Having a minimum and maximum number of stones that can be picked up, or really anything you can think of.

The second suggestion was playing “20 questions”. The answer can be as simple as a number and increase in difficulty such as rules or functions, to equations of lines, or just about any sort of concept in math. Imagine guessing a number but not being allowed to ask if it is higher or lower.

When I teach 8th grade math I basically like to make sure my students can recognize each function from the graph, the equations, and the table. So this fits in nicely. Actually anything we define in terms of properties should, theoretically, be a good answer for a 20 questions game. The game can and should be a vehicle for teaching students how to think critically about the properties of an object.

The last suggestion was Bizz Buzz. I’ve played Buzz a lot, which is a simple game. The rules are: students line up or sit in a circle and count up saying Buzz when they reach the number or its multiple. Bizz Buzz is a variation using two numbers and their multiples. Too add even more difficulty try using numbers from different bases. After playing this in the classroom a few times I increased the difficulty one my time by asking students to say Bang when they reach a number that is a common multiple. Playing with factors and common factors should also work.

I might also recommend ideas such as which I think is a great method to learn math. Creating patterns of dance or stomps with your feet.

I was also talking to a music teacher a few weeks ago. He was trying to teach his students the relationship between fractions and notes using the old pizza method. I suggested he stay with what is natural and use the timing of the notes. Whole notes, half notes, quarter and eights are fractions of time not pizza. Sustained notes are simply adding fractions. Students would obviously practice with their instruments, but drums can be easily created. I would assume that difficulty could be increased with various time measures.

If you have any other suggestions please add them to the comments below.


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I Think I Need Some Knee Pads


Image by nkayw via Flickr

At the moment until MAP testing is finished I am substituting and assisting as needed.

These classrooms are generally quiet places. The teacher talks, the students write, then in the second half of the class students get to work some practice problems. I generally hope the lecture isn’t too long, I get bored pretty quickly.

When the students are working I finally feel like I can be helpful. They raise their hands and I come over and answer questions. There’s no such thing as a quick answer from me however. I don’t lean over a student to correct mistakes.

What I do is kneel down, read the problem carefully, then read their attempted solution. I try to find where they went off track. Then I ask questions. Why did you do this? What is happening here? How does that help? What would happen if you did this?

Sometimes, I see students begin to raise their hands then put them down when they see me coming. They would rather a quick answer I suppose.

Earlier in the year I had my own class. I choose to leave my classroom to do interventions instead. Sometimes I think I should have stayed in the classroom. I forget how much I love teaching. It was frantic and chaotic, but fun.

No one else seems t teach like me around here.

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