The learner climbs a series of ladders. The lower ladders are fairly easy to get onto, but there are gaps that the learner has to traverse to reach the higher ladders. The ladders are grouped in isolated sets. To climb a new set of ladders, the learner has to drop back down to the ground and start climbing from the bottom again. The ladders don't go very high. Some learners can't figure out how to cross the gaps and get stuck. Some learners get frustrated at getting stuck all the time and quit climbing all together. Other learners get sick of climbing short distances over and over again, and also quit. It takes years before the learner climbs and reaches someplace interesting.
Here is a standards-based math curriculum viewed through the same lens:
The lowest ladders are a bit higher off the ground. It is easier to jump from ladder to ladder and the learner has more freedom to explore. The ladders also reach a bit higher. But there are still large gaps so few learners are able to reach the higher ladders. Some learners aren't able to pull themselves up onto the lowest ladders. Some learners get frustrated and give up when they hit a dead end, especially if they are used to a traditional math curriculum where the path from ladder to ladder is clearly marked. After clambering on a cluster of ladders for awhile, the learner still has to drop back down to the ground to climb a second cluster.
Just to clarify, a standards-based math curriculum is a curriculum based on the Curriculum and Evaluation Standards for School Mathematics published by the National Council of Teachers of Mathematics (NCTM) in 1989. Examples include Investigations in Numbers, Data, and Space (TERC), Connected Mathematics (CMP), the Interactive Mathematics Program (IMP), and Everyday Math. The "math wars" is a debate over traditional and standards-based math philosophies.
Here is a math curriculum that uses vertical learning:
Like the traditional math curriculum, the lowest ladders are easy to climb onto. Like the standards-based math curriculum, the ladders are linked together and interesting to explore. However, there are no difficult gaps to cross, there are solid platforms to stand on (making navigation clearer and easier), and the ladders go much higher. More importantly, once the learner is on the ladders, there is no need to drop all the way back down to the ground to reach other ladders. The learner is free to climb onward and upward.
When most people see a math curriculum that uses vertical learning for the first time, they love it because it enables learners to climb much higher. But they assume that the result will look like this:
But having taught math with vertical learning before, I know that the result looks like this:
If you give learners an accessible ladder that goes someplace interesting, they will climb it. And they will become stronger, faster, and more confident climbers in the process.
The standards-based math curricula of the 1990s grew out of a desire to make learning in math classrooms more constructivist. According to constructivist learning theory, we build our own understanding by developing theories (mental models or schema) of how the world around us works. Some people associate constructivism with discovery-based learning, but I am still building my own understanding whether I am reading a book or playing with blocks. First, I experience something. Then, I try to interpret it (make sense of it) using my existing mental models. If I have to modify an existing mental model in order to make sense of the experience, then learning has occurred.
In a constructivist classroom, the role of the teacher is to probe the mental model of the learner. When you ask the learner to read a book or play with blocks, you can't know what meaning the learner will make from it. You need to probe their mental model to see how they see things. If the learner's mental model is faulty, the teacher then needs to create learning experiences that generate cognitive dissonance, causing the learner to recognize the fault in his or her own mental model and fix it.
The mental models that we want students to construct for themselves should be functionally similar to the mental models that experts have.
An Expert Mental Model of Order of Operations
In this expression, there are two operations (one addition and one multiplication):
If I evaluate the addition first, I will get one value for the expression:
But if I evaluate the multiplication first, I will get a different value for the expression:
To make sure that an expression only evaluates to a single value, mathematicians created rules for deciding which operation should be evaluated first. In sixth-grade, these rules are:
If there is a tie, then you go from left to right.
These rules are somewhat arbitrary, but everyone has agreed to follow them. Two common mnemonics for remembering the rules are PEMDAS and "Please excuse my dear Aunt Sally."
In this expression, there are five operations:
We do multiplication/division before addition/subtraction, so we are going to start with either the division operation or one of the multiplications. To decide which of those to do first, we go from left to right and start with the division operation.
Now there are only four operations left. Again, we do multiplication/division before addition/subtraction, so we are going to do one of the multiplication operations. Going from left to right, we do the first multiplication operation.
Using the rules for order of operations, we evaluate each operation in the expression until we arrive at a single value.
For most sixth-graders, order of operations is used to decide which operation to evaluate next when evaluating an expression. But that is not how math experts understand and experience order of operations. For a math expert, order of operations is used to group parts of an expression.
So, when I see the expression:
I know that I can multiply 5 times 4 first. How do I know that? I know that because I know that when I do follow the rules for order of operations and get to that multiplication operation, I will be multiplying 5 times 4, so multiplying them out of order doesn't change anything.
Likewise, I know that I can't add 2 + 10 because before I ever get to that addition operation, I've done:
So I am really adding 18 ÷ 3・2 plus 10, not 2 plus 10.
This is also how I know that I can add 2x + 7x first in this expression:
Technically, I am performing operations out of order, but I know that I can get away with it because, if I did each operation in order, I would be adding 2x + 7x at some point anyways.
Building an Expert Mental Model
A sixth-grader doesn't need to have the same mental model for order of operations that I have. But a seventh-grader does need to be able to chunk expressions in order to add like terms, apply the distributive property, and identify the bases of exponents. Without that foundation, a seventh-grader would have to memorize separate rules for each of those skills (jumping all the way down to the ground just to climb a new ladder).
By tenth-grade, anyone who is fluent in algebra has figured all of this out for themselves. I would say that there is an extremely high correlation between students who can use order of operations to chunk an expression and students who are successful at algebra. Yet, we don't invest any effort in helping students go from point A (a sixth-grade understanding of order of operations) to point B (an expert understanding of order of operations). We don't even bother assessing whether a student is building the appropriate mental model or not, which is the primary role of a teacher in a constructivist classroom.
A curriculum that uses vertical learning is explicitly designed to help students build expert mental models. In order to access this curriculum, students need to be able to pull themselves up onto the first rung of the lowest ladder. From there, they are able to climb anywhere. In the case of order of operations, that first rung is applying the rules for order of operations to identify the next operation to evaluate in an expression.
When I was teaching sixth-grade, I liked to start the year off with order of operations because it let students know that they weren't in Kansas anymore. You may have gotten C's and D's in math all through elementary school, but once you had one foot on the ladder, it was a quick and easy climb to evaluating expressions that look like this:
I have worked with hundreds of students on order of operations over the years, and I have yet to find one that wasn't able to climb to this point in two or three days. To evaluate this expression, you don't need to be able to chunk it. But it does help if you can isolate or ignore parts of the expression. For example, students quickly realize that when they are evaluating an operation inside of a set of parentheses, they can finish evaluating all of the operations inside of the parentheses without having to go back out and look over the entire expression.
Chunking the expression also makes it easier to copy the expression down to the next line. (Think about copying a long sentence that is gibberish versus copying a sentence made up of simple phrases.) As long as students want to evaluate these complex expressions (which they do — it gives them an immense sense of accomplishment), then they will start chunking.
Another place in the curriculum where students can apply order of operations is when simplifying expressions with exponents. Before they know or are fluent with the laws of exponents, my students expand these expressions out with multiplication:
In order to expand an exponent using multiplication, you have to be able to identify the base. While that might seem trivial, it really isn't. Some students never understand how to identify the base and guess each time.
If a student isn't sure what the base of an exponent is, he or she can figure it out by applying the rules for order of operations.
The first operation you would do is the exponent in the inner parentheses. This means that the base of that exponent is just the a.
You would then complete the operations inside of the inner parentheses (two multiplications) before doing the exponent on the inner parentheses. This means that the base of that exponent is the contents of the inner parentheses.
After completing the operations inside of the outer parentheses (two more multiplications), then you would do the exponent on the b. This means that the base of that exponent is just the b.
Over time, students begin to notice patterns and soon they are chunking the expression. It helps if you can recognize that the base of the first exponent is just the b without having to even think about the contents of the parentheses. Then you can tell at a glance that there are six b's as factors in this expression:
And thirteen a's as factors:
In a curriculum that uses vertical learning, students don't develop fluency and automaticity by doing repetitive practice, they develop fluency by applying skills and concepts in a wide-range of contexts and they develop automaticity in order to reduce cognitive load (the more they can do on autopilot, the more they can focus on problem solving).
The structure of an expert mental model is similar to the structure of a curriculum using vertical learning. This means that students learning math using a traditional or standards-based curriculum must do a significant amount of re-mapping in order to build the understanding they need to be successful. But, regardless of the type of curriculum you are using, you can't just assume that a student will absorb a curriculum as it was designed and presented; everyone builds their own understanding as they go. Therefore, it is the responsibility of the teacher to assess the robustness of each student's mental models and to provide learning experiences that will generate cognitive dissonance if a student's mental model is faulty.
In a traditional curriculum, the emphasis is on procedural learning. Students are expected to develop conceptual understanding later. Conceptual understanding is important because that is how a math expert understands and experiences those procedures, not in isolation but as part of a larger conceptual framework. Unfortunately, in a traditional curriculum, students are generally left alone to develop a conceptual understanding outside of the curriculum and without any help from the teacher. Evidence suggests that this doesn't work for a significant number of students.
In a standards-based curriculum, the emphasis is on conceptual understanding. Critics complain that these curricula do not enable students to achieve automaticity because there is not enough of an emphasis on procedures. I disagree. I believe that the lack of automaticity arises because these curricula are generally unfocused. They provide a wide-range of learning experiences in the hopes that every student will get what they need, but students and teachers are given little guidance in terms of what they should be getting out of them. Teachers are not assessing a student's mental model against any kind of standard. Again, evidence suggests that this doesn't work for a significant number of students.
In a curriculum that uses vertical learning, the emphasis is also on conceptual understanding, but the curriculum is designed to help students develop expert mental models. Just because there is a specific target in mind does not mean that students are taught procedures. Students still learn by doing and problem solving, it simply means that the teacher is assessing those mental models and providing targeted learning experiences. This leads to a curriculum with ladders that are easy to climb onto, but still reach great heights and can be explored freely. In fact, the ladders go higher and provide more room to explore because the teacher is guiding the students to develop mental models that we know lead to a greater understanding of math.
In turn, ladders that are easy to climb onto but go high and reach interesting places quickly are incredibly motivating and empowering for students. Once something makes sense, you want more things to make sense and you never look back.