“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.

There are many reasons for this, and I outlined some of them in a series of blogs last year in which I elaborated on ten reasons to teach STEM. It’s engaging, it promotes cooperative learning, critical thinking, and differentiated instruction. But these pale in comparison to the real reason that this must be more than a temporary swing of the pendulum.
Our society has changed, and the educational model that serves it must keep pace or we will fall further behind. Long ago, public education was designed to instill content mastery into the minds of students. The more content you could learn, the more marketable you were when you sought a career. Content mastery is no longer the primary task of education. Due to the acceleration of technology, content is fluid and grows exponentially. Even if I could teach my middle school students all the current educational content, it would do no more than ensure that they were dinosaurs when they exit high school. While content still has a value, today’s students must be taught how to learn over what to learn.
Today’s employers value cooperative skills, creativity, and problem solving over mastery of content. STEM and PBL assure students develop these skills.
Today’s jobs don’t look like the ones of the past either. As recently as my own childhood, middle class America was largely connected to the manufacturing sector. If one parent had a job at a mill or plant, the family could live comfortably on that wage. Manufacturing employed a large swath of the workforce who had moved beyond entry level work yet didn’t hold a professional degree.
Today however, many of the manufacturing jobs have moved overseas. This has left us with a dichotomized employment world comprised of low-paying and high-paying jobs with an empty midsection. As fewer and fewer people find work in manufacturing, they are forced into entry-level employment if they don’t hold a professional degree. If this doesn’t change, tomorrow’s workers who lack a professional degree will be forced into low-paying service industry positions. This requires both parents in a home to find employment in order to cling to the middle income band.
This doesn’t mean that every student must attend college. In fact, many STEM-based careers require technical and vocational education. STEM careers are the fastest-growing segment of the job market; they will take the place of the shrinking manufacturing sector that served middle class America. In fact, they will more than offset this because STEM careers offer pay in line with professional degrees. And these lucrative jobs already exist for our students.
If America does not prepare its workers for the opportunity to enter STEM fields, we must look elsewhere, for these jobs will be filled. The question is, will our students be at the back of the line, or the front?

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.

The students began the first day wiring circuits using Play Dough. They were challenged to create a circuit to light up four LEDs. After some thinking and trial and error, they learned to make a series of Play Dough balls and connect them with the LED lights. By connecting battery terminals to the first and last ball of Play Dough, all of them lit up. “However,” I showed them, “If I remove one light, they all go out.” I explained that this is how old strings of Christmas lights performed. “How can you design a circuit so that if one light goes out, the rest stay lit?”
They struggled with the task for a couple of days, but eventually a student rolled his Play Dough into two parallel snakes instead of balls. By jumping the LEDs from one snake to the other and connecting the snake’s ends to the battery, they had discovered the difference between wiring in series and in parallel.
Next we learned to drive Spheros, the programmable balls popularized in the Star Wars movie. After learning how to control them, the students designed covers for them out of tag board. A balloon was attached to the back and a straight pin to the front. Then they battled each other in an octagon made of meter sticks.
In our most recent build, students studied the principles of hydraulics. Then they used popsicle sticks, syringes, and tubing to construct battling arms. Their goal was to knock their opponent off the table.
No behavior problems, no off-task behavior. Just hard-working students engaged in the fun of learning. All without grades. How is this possible, don’t students need the motivation of grades to learn?
I’m not certain of the answers to these questions, and I think more formal research is needed, but I have some ideas based on my observations. First of all, I had to ask myself why we grade in the first place. Obviously we are trying to document and quantify learning. But I suspect that we already know who is learning in our classes and who is not.
When we give an assessment, we typically can guess who will do well and who will struggle. We also know of students in our class who bright and are learning but do not necessarily have grades that reflect that. So grading is often done after the fact in a sense as it reinforces what we already knew. Any teacher already has their hand on the pulse of the class anyway.
Sometimes we use grades as a motivation. Students who value their grades will stirve to get the A and learn in the process. I was that sort of student. It did not occur to me that the purpose of learning was to grow and satisfy our insatiable brains. For me, learning occurred as a byproduct of my pursuit of good grades.
But here was an entire class that was working and learning without grade motivation. Some of the students in my elective were also in my math/science class and were very grade motivated. This leads me to suspect that when learning is presented in a brain-palatable format, the external motivation of grading is not necessary. In fact, some of the students who I knew were learning the most were getting Fs in their other classes due to a lack of work.
This makes me suspect that perhaps the educational process has become synthetic. By that I mean that the way we help brains grow and learn does not fit its natural design. Think about it; we teach reading through engaging books, not by studying the dictionary. Yet math and science are often studied via text-based approaches instead of by engaging applications, causing many students to ask the question they do not ask in other content areas: “When are we ever going to use this?” That’s a question that is never asked in PE class. To this day, I remember the excitement of learning to do a back flip on a trampoline in high school even though I have never needed to do a back flip during a job interview.
This experiment has left me asking myself, “How can I engage my students’ brains more effectively?” As our school ends its third year as a STEM center, I believe I’m inching toward the answer.
Happy teaching,
Brad

We separated our students by gender for a day. The boys were in my homeroom, and the girls were in hers. The resulting conversations were revealing, but there was something even deeper that I noticed.
The boys said, “The guy substitutes get us. When we are off task or goofing off, they come up to us and joke with us. We all laugh and then we get back to work.”
“The girls don’t get in trouble for talking when we have a lady sub,” they added.
Though I didn’t agree with everything they said, and I certainly didn’t agree with their disrespect and defiance in the presence of a female substitute, I listened to their opinions. I explained, “Of course a male sub will understand you better than a female, but that doesn’t justify rudeness.”
I also noticed that when the boys talked that day, they were generally loud, but during a discussion, usually only one talked at a time. When the girls arrived, they spoke more quietly, but they were also more likely to talk at the same time. They seemed to have no problem with this, but it was easier for me to hear the loud single voices of the boys than the quieter mixture of the girls’ voices.
The girls were much better at working cooperatively and staying focused on a task. The boys immediately set up a leadership hierarchy that was more competitive when they worked in boys-only groups. More boys chose to work independently. Yet both classes got their work done with equal proficiency.
I wondered if we may have created a female-friendly learning environment in our schools.
We want students to work in cooperative groups and hold fair, orderly discussions.
Cory Turner of NPR wrote an article about a study at Yale recently (http://www.npr.org/sections/ed/2016/09/28/495488716/bias-isnt-just-a-police-problem-its-a-preschool-problem) that showed that teachers can tend to look for disruptive behavior in boys, especially black boys. In the study, 135 preschool and kindergarten teachers watched a series of videos and were told to try to spot disruptive behavior before it happened. In each video was a white boy, a black boy, a white girl, and a black girl. In none of the videos were there any actual incidences of bad behavior. As they watched, scanning software tracked the eye movements of the teachers.  What they found was that teachers looked at boys more than girls, and at black children more than white in searching for challenging behavior.
Ouch!
I must admit that even as a male teacher, I am probably guilty of the gender bias myself. When the students were mixed into their regular homerooms the next day, I noticed the boys’ voices dominated the class noise. There were just as many girls talking and off task, but they were quieter. I call the boys on their behavior much more than I do the girls. Part of this may be due to the fact that girls mature in middle school faster than the boys do. The behavior of the girls is usually more in line with common classroom rules, but I don’t want to penalize boys for their physiological development nor subject them to my bias.
At my school, I am the only male teacher. Many of the boys in my classes do not have their father in the home, so it is not until the 8th grade that those boys will see a male leadership figure on a consistent basis. Is it any wonder that the leadership positions in our school’s student government and the honor rolls at the awards assemblies are also predominantly female?
In our schools we are trying to get more girls involved in math, science, and engineering fields that are typically male populated, and that’s a good goal. But I wonder if we are simply displacing the boys and replacing them with girls. Could we have a more inclusive plan?
Since our gender-specific day, I have been more aware of my tendency of singling out the boys. I have tried instead to practice celebrating the difference. When the boys are loud, outspoken, or challenging, I try to laugh it off. I smile at them and let it brush off my shoulder. If it warrants serious discussion, I can do that later one-on-one.
I would love to know your thoughts on this sensitive subject. Drop a comment, and let’s see if we can make our classrooms and schools more inclusive of both genders.

1. ​​STEM has real world application
2. STEM fosters problem-solving skills
3. STEM is hands-on instruction
4. STEM is differentiated instruction
5. STEM promotes cooperative learning
6. STEM teaches creativity
7. STEM makes failure a learning opportunity
8. STEM involves high-level thinking
9. STEM requires students to be actively engaged
10. STEM is the future

Now we come to the final installment in the series: STEM is the future.Often education is the tail of the dog instead of the head. We lag behind the times in technology and in content. Within a year of my high school graduation in 1974, everything I learned to do using pencil and paper mathematics could be done by a $25 pocket calculator.Yet even twenty years later, I was teaching pencil and paper math. No one could earn a livable wage with that expertise (unless they aspired to be a math teacher) yet it persisted in our state standards and in our classrooms until Common Core came along.We are seeing the result of the old approach today. More and more high-paying jobs must be outsourced as graduating students lack the skills of their modern society. These are S.T.E.M. skills. The five top-paying jobs are all in S.T.E.M. related fields:

1. Petroleum Engineering
2. Actuarial Mathematics
3. Nuclear Engineering
4. Chemical Engineering
5. Electronics and Communication Engineering

These fields offer six-digit salaries.
That also means that a S.T.E.M.-based major will allow a person to pay off college loans faster than any other field. With the rising cost of college education and the burdensome ongoing expense of financing that education, a S.T.E.M. major is a logical and practical option.
When we also consider that S.T.E.M.-related jobs are predicted to experience more than a 60% growth in the next ten years, we see that teaching S.T.E.M. is indeed teaching for their future instead of for our past.
And teaching for the future will not even guarantee that students will have everything they need to land in a career. Who foresaw the growth in the field of drone technology ten years ago? Tomorrow’s job seeker will not only need to have a firm grasp on the latest technology, they must be flexible and adaptable to new environments. Problem-based learning promotes this characteristic.
Today I am beginning to see the fruit of this new approach to instruction. When I teach my digital photography elective, when my colleague takes on the task of teaching coding, when I utilize new software and technology in my teaching approach, it is often my students who become the experts and consultants. When someone raises their hand with an issue, I’m often calling upon one of my young experts to solve the problem.I’m no longer the sage with the content knowledge; I’m a director of a project. I keep the team focused on the goal and ensure that they work cooperatively. As often as not, I’m learning with them about a world that lies ahead – aworld they will enter and conquer. In the process, they will leave me far behind, and I’m okay with that because my ultimate goal is to create learners that are independent from me.
And one final note:For me there is an eleventh reason to teach S.T.E.M. curriculum that Wiggins never mentioned. Though I have enough years in the classroom to retire at any time, I am more excited than ever to go to work each day. I cannot imagine how retirement could be any better than what I am doing now. The excitement my students show each day rivals that look on my son’s face when he learned to walk so long ago.Being a part of that process with my students, watching their first tottering steps, seeing their smiles, lifting them back up when they fall, and witnessing their successful steps as they walk away makes this more than a job and even more than a career. I get to be part of a great adventure everyday. I’m planting synapses in adolescent brains, and the harvest is bountiful!