“I’m not good at math. I failed it once.” We’ve all heard that, but here is something we never hear: “I’m not good at video games. I tried it once and I died on level one.” When a student fails in a video game, it is engaging. He or she says, “That was cool. I’m going to start again.”
But that’s not what we hear in a math class. This tells us that there is something radically different about how video games are designed and how our math textbooks and our state standards are designed. And it all comes down to three factors:

• Intentionality
• Incremental development
• Ongoing feedback

Every line of code in a video game is intentional; it’s there for a purpose. Secondly, the skills you develop in conquering level one prepares you for what will happen in level two. And lastly, you are getting constant reinforcement on your progress.
Compare that to the typical textbook layout. On the left page are a few examples, and on the right page are practice problems followed by word problems. Our state standards mandate that the problems in that lesson are at grade level, but are our students? If we teach directly from the textbook examples, any students who are not at grade level are already lost. It would be like starting a video game at level 28.
And do the practice problems build incrementally like the levels in a video game? Does problem 1 provide you with the skills to tackle problem 2? There’s a chance that the problems in the practice set were created randomly by computer software. Imagine if the levels in a video game were randomly sequenced.
Also, the student may not see math in a real-world context until those word problems at the end, and they may not know how they did until they get their homework back the next day.
We need to model our mathematics lessons after the design of video games: intentional and incremental development with quick feedback. For example, if we want to show why subtracting a negative is the same as addition, we can ask students to complete the patterns of problems below:
3–3=0
3–2=1
3–1=2
This starts simply. It is level one. All the students who can subtract one-digit numbers are on board. But what comes next?
3–3=0
As the subtracted number is lowered by one, the difference increases by one. And then what would we write?
3––1=4
But wait, it’s also true that 3+1=4. Therefore, the two negatives make a positive. Notice that we started simply and built incrementally in an intentional way. The simplicity of the first few problems gave the students feedback as they worked.

​Here is another example. This is how I help my middle school students practice operations with integers. I give them the two numbers on the sides of the X in Example 1and ask them to find the upper number and lower number. In this case, the upper number is 12 and the lower is 7. Because I begin with such a simple example, all of my students realize that the upper number is the product of the side terms and the lower number is the sum. Then we can begin to “level up.”
    
Once they understand how the puzzles work, I can introduce examples that have integers as in Example 2.
Now they are practicing multiplying and adding integers, which was my goal. Later, we level up again by giving them the top and side numbers as shown in Example 3 Now they must divide and add.
And then I give them the bottom and side numbers in Example 4, so they have to subtract and multiply.
They have practiced working with integers in all four operations. And finally, we meet the beast at the highest level shown in Example 5.
Here, the side numbers are 9 and –4. The student has truly leveled up as this puzzle is more demanding cognitively. Students approach these puzzles like levels in a video game. The brain likes to have moderate and sequential challenges like they see when gaming or doing lessons like this.
Notice that I’ve intentionally sequenced these problems to intentionally bring the student to this level. If this is an assignment, I provide an answer bank, so the student gets the critical immediate feedback.
And what is my intention with this lesson? First, it provides critical ongoing practice in all four operations with integers. More importantly, it provides an incremental lead-in to factoring quadratics. If we want to factor x2 +5x–36. We can use the previous puzzle. The solution, 9 and –4, are the solution:
x2 +5x–36=(x+9)(x–4)
Teaching math with intentional and incremental development coupled with ongoing feedback mimics how our students’ brains interact with the video games that so engage them. Our textbooks then become a resource much like the dictionary in a language arts class. It has practice sets and sample solutions, but it is not an instructional tool. The greatest instructional resource in the classroom is the teacher presenting intentional and incremental instruction.
For more examples like these, explore some of over 100 activities in my TeachersPayTeachers store like the ones below.

 “I don’t get it.” the student complains.
“What don’t you get?” asks the teacher.
“Everything!” answers the student.
We’ve all heard that, especially in an algebra class. But what if I told you there is a way to avoid that conversation? It’s true. I found a way to introduce the following algebra concepts in a way that it just makes sense to students:

• Working with variables
• Translating word problems into math
• Combining like terms
• Understanding the properties of algebra
• Solving equations

How do I know it works? Because I have used this unique approach in my own classroom for over 30 years. And teachers across the U.S. have tried it with success as well. In fact, this lesson was piloted with over 100 students in 4th and 5th grade classes, and they not only “got it,” they enjoyed doing algebra as well.
And the best part was that I never told them how to do algebra. It made so much sense in the format I presented that they taught themselves.
I began by showing them the fast-food menu shown here. Then I told them I was going to make some math problems based on it. They were expecting to see numbers, but I wrote:
h+f=
You should have seen their confused faces…and then the immediate change as the light came on and they shouted, “$2.90!” We did a few similar examples, and then I threw them this curve ball:
3f =
Immediately, they said, “$3.15.” I asked, “How did you get that?” They said, “We multiplied.”
“Why?” I asked, “It doesn’t say to multiply.”
“It just makes sense,” they replied. Without me telling them how, they began to see how letters could be used to replace number and how to calculate with them.
Next I gave them orders that customers had placed and asked them to write them algebraically. Notice that each one adds an incremental increase in thinking.

• I'd like four hamburgers, six orders of French fries, a large soda, two medium sodas, and an extra-large soda. (4h+6f+l+2m+x)
• I want three cheeseburgers, one hamburger, a small soda, two fries, a medium soda, and another hamburger. (3c+h+s+2f+m+h or 3c+2h+s+2f+m Some students combined like terms.)
• Let's see… I think I'd like three hamburgers and a cheeseburger, three fries, a large soda, two medium sodas, and an extra-large soda. Add another order of fries on that and make one of those hamburgers another cheeseburger. (3h+c+3f+l+2m+x–h+c or 2h+2c+3f+l+2m+x Changing one’s mind introduces subtraction into the expression.)

This led naturally into combining like terms. I wrote this expression on the board and asked the students to write it more simply for the cook.
(x + c) + (2f + c + x) + (m + 2f + c) =
Again, I didn’t show them or tell them how to do this, yet they wrote:
2x+3c+4f+m
We worked more examples with increasing difficulty until I wrote this;
(3h + 2f + x) + (c + f + m) – (h + m + f) =
Interestingly, the students were able to distribute the negative sign across the terms – a commonly missed skill in algebra.
This led naturally into exploring the properties of mathematics:

• Commutative property: h+f=f+h
• Associative property: (2h+f)+(c+m)=(2h+m)+(c+f)
• Distributive property: 2(h+m)=2h+2m

For solving equations, I asked them to imagine that they were the cook, and someone had written down the wrong letter in the order. Could they figure out what the w is?
h+6w=$8.15
Interestingly, even the 4th and 5th graders were able to do this even though they had not been taught to solve equations. They used subtraction and division in steps 1 and 2, even though the problem has an addition sign and implied multiplication (6•w). What was more amazing was that these students had been taught order of operations, yet they all subtracted before dividing.
That left me thinking, if there is this much algebraic intuition residing in their brains at age 10, how can we believe that our students can’t learn algebra?
In my 8th grade algebra class, we also used the menu to approach other concepts such as solving systems of equations.
If you’d like a free copy of the handout, click the “handout” button below. And to watch a video explaining the whole process, click the “video” button.

Mastering the Multiplication Table
 
So many times, I’ve heard middle and high school teachers lament, “My students don’t know their multiplication facts. Didn’t their elementary teachers cover that?”
Yes, they did, but it got stuck in the wrong part of their brain.
It turns out that there are two types of memory: taxon memory and locale memory.[1] Taxon memory is what we use to store things we’ve memorized. Locale memory is for information that makes sense – information that we understand.
The advantage of taxon memory is that it’s fast. If I ask you what is 6x8, you recall it quickly from your taxon memory. You don’t have to add six eight times or draw a six by eight array. The disadvantage of taxon memory is two-fold. First it is finite in scope. There is only so much you can memorize. If I ask you to go to the store and start listing items, after about five or six, you are reaching for a pencil and paper.
Secondly, it is finite in duration. It requires a lot of ongoing practice. Much of grades three and four is invested in memorizing the multiplication table, but after that, we move on to other skills. That’s why our middle school students – who once knew their multiplication facts – now struggle to recall them quickly. Taxon memory fades.
Locale memory is the opposite. It is not fast. You must think about it. Instead of being effortless, like taxon memory, it is effortful. But it is locale memory is also opposite in the other regards as well. It appears to be infinite in scope and duration. There seems to be no limit to how much the human brain can remember when things make sense. And locale memory is “sticky”. It lasts forever.
We’ve all experienced the situation in which we were trying to remember something like the name of a restaurant. We can’t recall it and say, “I forgot the name.” Later, while trimming our toenails, the name jumps back into our head! We didn’t actually forget it; we forgot where we put it. Locale memory lasts forever. Even if the road gets washed out, our synapses can reroute and find a different way to get there.
All that to say, when we tell students to learn their multiplication facts, what they hear is, “I have to memorize 144 unrelated facts.”
My approach is to show students the meaning behind the multiplication table. This encourages storage in long-term locale memory. Then, through consistent practice, it gets stored in taxon memory as well for quick recall, but the anchor always holds in locale memory.
Nice theory, but does it work? The good news is a resounding YES! In just five minutes a day, I was able to help my struggling eighth graders master their multiplication facts. (Boy did that make it easier to teach factoring of polynomials!)
And as a bonus, I showed my students how to use their multiplication table to reduce, add, subtract, multiply, and divide fractions, and how to solve proportions with it as well. They became my fraction experts!
You can learn all about it in my video, “Fast Facts and Fractions” available on my EdStream site. Simply click the button below,

[1] Caine, Renate Nummela, and Caine, Geoffrey. Making Connections: Teaching and the Human Brain. Alexandria, VA: Association for Supervision and Curriculum Development, 1994.

​Today I asked on Twitter, “If you could start your career over, what one thing would you do differently?” I expect some interesting responses, but for me, there is one change that stands out. I would be more relaxed and personable with the students.
After over three and a half decades in education, I am finally getting close to a sweet spot. I look forward to the start of the school year and wish I could indeed start my career over with what I have learned.

I’m conducting a bold experiment. Six weeks ago I started teaching a new elective for our third trimester. Even though Mistletoe Elementary is a STEM school, I have more ideas than time, so I offered to teach an elective simply called “STEM”. On the first day, two dozen 6th through 8th grade students arrived, and the first one in the door asked, “How are you going to grade this class?” I answered honestly, “I don’t know; I haven’t thought about it.” Now a month and a half later, I have not issued any ​grades, yet the students rush to class each day, work from the moment they arrive, complain when they have to stop and clean up, and I have had zero students off task and no behavior problems.

​I’ve noticed that when I have a female substitute, the boys in my 8th grade classes get in trouble. When I have a male substitute, they do not. I don’t think the issue is with the substitute (nor their gender) as these are good teachers with successful track records. So the other 8th grade teacher and I did a bold experiment. ​