It is common for algebra students to be confused about the difference between unknowns and variables.  After all, they are both often represented by the letter x. But they are not at all the same things.

An unknown in algebra is simply an amount whose value is not stated or that you have yet to figure out.  

A variable is an amount that changes.

Sometimes unknowns are variables. Sometimes they are amounts that do not vary, also called constants.  Sometimes variables are unknown, and sometimes they are known.

The relationship between the two types of amounts looks like this:

Why does it matter? Because the techniques of algebra can be used to achieve two quite different objectives: 1) determine the value of an unknown, whether it is constant or a variable, and 2) determine the mathematical relationship of two or more variables, whether or not they are known or unknown.

And that means that the solution of a problem can take very different forms.   Some problems call for finding the value of an unknown amount, and the answer is of this form:

       x = ______

Other problems call for determining how two or more variables are related, and the answer might be of this form:

       an expression involving x = an expression involving y

If students don’t understand the difference between unknowns and variables, they will be unable to understand which kind of a solution a problem calls for.

In the academic world of manipulating algebraic expressions as modeled by a textbook or teacher in order to get a specified result, this understanding may not be essential to getting a right answer and passing an exam.  But in the real world of solving actual problems in engineering, business and finance, graphic arts, health and fitness, and a host of other fields of activity, knowing what you’re trying to do is essential to success.

This confusion may or may not be one of the reasons many students have so much trouble learning algebra.  But it certainly doesn’t help.

By the way: When I asked ChatGPT to give me an example of a problem involving unknown variables as well as unknown non-variable amounts, it failed to understand the difference five times in a row.  I then explained the difference and it got one right, then failed the next six times in a row even with the modified prompt.  It’s important to recognize that AI is simply a highly sophisticated search engine and pattern recognizer with a natural language interface, there is no actual intelligence there.  See the Real Algebra blog “AI and Mathematics Success”.

Some of the proposals I’ve seen for AI applications that are supposed to help students learn mathematics are deeply flawed, because they fail to take into account a fundamental principle of learning, a principle that applies to all education and training activities that are intended to develop proficiency and judgment.    

I’ll illustrate with a personal example. A couple of years ago my grandson was pursuing a rigorous college program in food sciences, which involved a considerable amount of mathematics, chemistry, and other science and engineering courses.  Every so often I’d help him on his homework.  Most of the time it was material that I hadn’t seen myself in decades, so we worked our way through it together. But physics was a different story, since that was my major field of study.  As a result, I was able to give him a lot more help with that homework.

Fast-forward to final exams: I believe this was the only course that my grandson took in that rigorous program for which he did not get an A.

I should have known better.  One of the things I had learned very well in my “on-the-job” training as an educator at The Delphian School almost half a century ago is that giving a student too much help is as damaging to their education as giving them too little—and probably even more so.  Students can often find their way through to a solution with too little help, but there is no such cure for too much.

I’ve written in other posts about the value of productive struggle to the learner.  I imagine we have all experienced its benefits, not only in terms of learning but also in terms of the self-confidence generated by success in the face of adversity.  That experience is so common that there’s probably no real need to explain why it works, but for what it’s worth in my own opinion it comes down to a basic truth of learning theory: the learner themself must create the mental connections between a new idea (or datum or principle) and their existing ideas, the connections that are the essence of understanding.  No one else has the power to create those connections in the learner’s mind.   And those connections are forged in the course of a student’s often challenging journey from confusion to understanding.

Whatever the explanation, productive struggle is very valuable. If we agree on this, then we have enough common ground to consider the proper constraints on the role of AI in mathematics learning.   There are four things that AI can and should do for the math learner:

1) Aid them in developing conceptual understanding of concepts and principles, when that understanding is being properly measured (which precludes multiple choice assessment).

2) Diagnose failures and direct the learner to resources that address the concept, principle, or skill that they need to give more attention to in order to be successful.

3) Walk the student through model problems.

4) Pose appropriate exercises and problems which the student must learn to do independently.

And there is one thing AI should never do for a math learner who is in the process of doing those exercises or solving those problems: tell them exactly what they did wrong and guide them to do that step correctly. That is simply too much help, because rather than making the struggle productive, it's eliminated entirely.

Bottom line: If a student cannot successfully and independently do homework exercises or solve math problems, AI should help with 1, 2, or 3.  And that’s all.

There are already too many obstacles to success in learning mathematics for many of our students. Let’s not also undermine their ability to overcome those obstacles.

The first Real Algebra blog explained the need for a widely effective solution to the problem of learning the course of mathematics starting with algebra and continuing through calculus. Other blogs discussed the importance of developing conceptual understanding and how to do that, and why it is essential to provide resources that make it easy for students to understand the purpose and use of each math topic.  We also considered the fact that all of mathematics, and particularly algebra, is a form of language, and some of the implications of that in terms of applications in the real world.

Along the way we pointed out that existing solutions to the problem of learning high school level math have three common flaws: they do not develop conceptual understanding, they do not make it real to students what each topic of that math is for, and they do not develop mastery.  

This blog focuses on mastery.

If we’re going to talk about mastery, we’d better start with understanding what the word means. Oddly enough, the master of mastery, Dr. Benjamin Bloom, chose never to define the word because (according to In Search of a Useful Definition of Mastery) he didn’t want to alienate educators that had different definitions.  Fair enough, but that won’t really work for us in developing a product that is designed to enable students to achieve mastery.

My colleague Dr. Anastasia Betts, who worked with me to create Age of Learning’s early learner mastery mathematics programs, defines mastery this way: “a high degree of independent proficiency demonstrated consistently over time that transfers and doesn't decay.”

That gives us a goal to aspire to, but we need be clear about what we mean by proficiency, particularly since there are different types of learning objectives and what we mean by proficiency depends on the type of objective.

For Real Algebra, we have adopted these meanings:

Proficiency in the understanding of a concept means that the student can easily relate and compare the concept to other concepts already understood and can recognize and generate instances of the concept in various contexts.

Proficiency in the understanding of a principle means that the student 1) has a proficient level of understanding of the concepts that are elements of the principle, and 2) the student can recognize and generate examples and counter examples of the principle in various contexts.

Proficiency in the execution of a skill means that the student can carry out the actions of that skill in order to generate the desired (correct) results, can recognize situations in which the skill can be usefully applied, and can explain and generate examples of the execution of the skill.

Proficiency in the demonstration of an ability would include proficiency in understanding the relevant concepts and principles and in executing the skills that add up to being able to apply knowledge in order to solve a problem.

How do we develop such proficiencies so effectively that the student can apply them in different contexts (“transfer”) and they do not decay?  The answer is contained in the phrase “productive struggle,” which describes a student’s experience as they actively build knowledge, skills, and abilities by reading and listening for understanding, evaluating information, thinking of examples and counter examples, solving problems, and in general executing tasks which they find engaging, challenging, and achievable.  

In Real Algebra, the productive struggle occurs primarily in the process of playing a series of learning games—activities in which the student’s response requires demonstration of the targeted understandings, skills, and abilities.  To “win” a game, the student must complete a task successfully several times in a row, independently and fluently.  Scaffolding is provided as needed, but to reach the goal mastery the student must demonstrate that they can repeatedly succeed independently.

The essence of this approach is to prompt the student to do something, provide appropriate feedback, and adjust the next prompt as needed based on the student’s response.  The theory of learning is straightforward: learners construct knowledge in the process of observing, making decisions, and taking action.

The set of games provided and the sequence in which the student engages in them is guided by a detailed map of the relationships of concepts, principles, skills, and abilities that make up the algebra knowledge space.

These first four Real Algebra blogs have sought to lay out the rationale and general framework of the program. Future articles will fill in details.  

In Real Algebra blogs #1 and #2, we addressed the need for a program that makes algebra and other grade 7-12 math topics accessible to many more students.  We also discussed some of the characteristics of such a program, focusing the development of conceptual understanding.  This third blog deals with another element that is essential to any math education program that seeks to be widely effective:  resources that demonstrate to students the value of learning what we are asking them to attempt to learn.

First, though, we need to clear up a common misconception about mathematics itself.

It is generally understood that mathematics is a type of language. Specialized, certainly, because of its built-in logical relationships and the ability that it confers to describe objects, phenomena, characteristics, and relationships with great precision, but still in essence a language.

However, although It is generally understood that mathematics is a language, people often do not make the leap to the direct implication of that fact, which is this: as a language, mathematics can be used to describe anything in one’s environment or experience, just as the language of English (or French, or Russian, or Chinese, or Esperanto), can be used to be described any experience.  

Imagine that you share an experience with a friend, and then you ask them to tell you about it. If they are unable to do that, you would not conclude that it was because the experience “did not have any language in it”. You would conclude that your friend didn’t have the vocabulary to describe the experience. Similarly, when someone says that there is no mathematics “in” some particular aspect of human experience,  what they are really saying is that they don’t have the mathematical vocabulary to describe the experience.

Wherever you are right now, as you read this, everything in your environment, and for that matter everything that’s happened to you today, can be described mathematically. That’s not to say that mathematics can describe the entirety of an object, phenomenon, experience, or relationship, but it does mean the language of mathematics can always be employed to describe some aspect of it. It’s also true that the more of the language of mathematics you know, the more aspects of an object, phenomenon, experience, or relationship you can describe mathematically.

Each of the branches of mathematics—arithmetic, geometry, algebra, statistics, trigonometry, calculus, and so on—offers additional “vocabulary” and the capability of more complete descriptions of objects, phenomena, experiences, and relationships.  And once you understand that, you’re ready to understand, and become able to explain to others, exactly what it is that makes algebra valuable.

We’ll illustrate with an example.  Here is an arithmetic description of a relationship:

         12 = 3 x 4

This could apply to many specific situations, such as how many applesauce servings you get by buying three cartons  contain 4 servings each. But what if you wanted to describe how many servings you would get if you bought an undetermined number of cartons, say anywhere from 2 to 8 of them? For example, if you are having a dinner party and aren’t sure how many guests will arrive.

The vocabulary of arithmetic does not allow you to describe that relationship.  But algebra includes in its vocabulary a way to represent a quantity which is not always the same, usually in the form of  a letter of the English or some other alphabet. If we use n to represent the number of cartons, and s to represent the total number of services, we can describe the relationship like this:

         s = n x 4

This equation does not apply only to one specific purchase, but rather is true of all such purchases. In other words, it’s a generalization. And so, algebra can be defined as follows:

Algebra is the branch of mathematics for describing and solving problems that involve generalizations about relationships between quantities.  

Admittedly, the above discussion is a little too abstract to be a useful explanation of the value of algebra to a middle school student. But it does make it easy to think of examples of that value in virtually any human endeavor: just imagine any useful generalization of a relationship between amounts that applies to that endeavor.

For Real Algebra, we are creating a series of one minute videos that draw on this strategy to provide examples of the uses of algebra in a range of endeavors. As these are produced, they will be made available here.

Real Algebra, Blog #2

In the previous blog I talked about the importance of providing a reliable route to success in math, and specifically in algebra, for all students.  Such a route does not currently exist.  

There are three fundamental flaws in existing algebra curricula:

• They do not effectively get across to students the value of learning each topic.

• They do not develop conceptual understanding.

• They do not facilitate true mastery of skills and abilities.

I’ve listed these in order of importance as well as the sequence in which they should be accomplished for each topic.  Despite the fact that having a reason to learn whatever you’re learning always comes first, in this article we’ll talk about conceptual understanding.

Fundamental to the development of conceptual understanding is recognition of the difference between a word or symbol and the concept that it represents. This idea goes back to the Greeks, and very likely earlier than that, but it doesn’t take a philosopher to see the truth of it. If a concept were not different from a word or symbol, then we wouldn’t be able to communicate about the same concept in different languages, and by using different symbols. The fact that we can do so demonstrates that concepts are mental ideas that are represented by words and symbols.

This may seem a little abstract, but it is essential in the development of any curriculum resource that seeks to guide and help students to achieve true understanding, because for many students the development of conceptual understanding goes beyond providing students explanations involving words and symbols.  It requires experience.

The question to ask ourselves as curriculum developers, instructional designers and teachers is this:

How can we provide students with experiences that develop their comprehension of a concept?

As we do this, we need to keep in mind that when you try to explain a concept with words and symbols that the student doesn’t already understand (that is words and symbols which are not connected in the student’s mind to comprehended concepts), you make that concept inaccessible rather than accessible.

The usual process for teaching a concept to a student is to start by explaining it with language that the student doesn’t understand, and then have them take some action to engage with the concept.  That’s backwards.  

The correct sequence is to first develop understanding of the concept, and only then introduce the language (nomenclature and/or symbols) associated with the concept.  In that way the student can easily attach the language to the concept that already exists in their minds.

This is not theoretical. It’s been the basis of my curriculum development work in many different contexts for decades.

As an example, here’s how Real Algebra treats the concept of slope:

Stage 1: Get across the value of learning the concept.

Using language that is likely to already be part of the student’s working vocabulary, describe with plenty of visual support how controlling the steepness of a path or line is used to achieve goals in engineering, art, and finance.  (Here are examples of short videos we’ve created for that purpose.)

Stage 2: Establish conceptual understanding through experience.

Student a) identifies and creates lines which have greater or lesser slopes than given lines, b) identifies lines which have positive versus negative slopes, and © extends lines without changing their slope.

Stage 3: Develop mastery of mathematical language and actions related to the concept.

Demonstrate to the student how slope of a straight line in a coordinate system can be measured as a ratio of the differences of vertical coordinates and horizontal coordinates for two points on the line.  Provide a ladder of  exercises of gradually increasing difficulty and diminishing scaffolding that result in mastery of the ability to calculate the slope of a line in a coordinate system or create a line of given slope.

You might think that this concept-first approach takes more time, but in truth it’s faster over time, because each conceptual understanding adds to the foundation on which further conceptual understandings can be constructed.  Understanding of language and symbols accumulates right along with that.  And once an element of language is closely associated with a concept in the student’s mind, that language will forevermore summon that concept, which means it can be used in the development of conceptual understanding in more advanced topics.

There are many definitions of “mathematics.” The one that I have found to be most useful in teaching and curriculum development is, “a language that enables us to describe and solve problems that require precision.” While that’s true, it's also true for many students that the learning of mathematics IS the problem.

In the United States, the journey of students as they attempt to learn mathematics from preschool through 12th grade is akin to Napoleon’s “march to Moscow,” a march which began with roughly 400,000 soldiers and ended with about 100,000, due to horrific losses along the way from a variety of causes. Based on NAEP proficiency scores, that ratio is pretty close to the ratio of students in the US who "survive" their study of mathematics all the way to 12th grade.

In future posts, we’ll talk about why this is the case and what can be done about it, especially in relation to the learning of algebra. But first, it’s important that we recognize how much of a tragedy it is, not just for our nation, but more importantly for all those students who do not survive.

It is the purpose of education, in my view, to empower individuals to create their futures.  There are a lot of ways in which education can help or hinder learners in that respect.  But perhaps the most useful thing that education can do to put them in a position to create their own futures is develop their language skills, the ability to give and receive communication. Language opens the door to just about anything one might wish to accomplish.  

For a moment, imagine what your life would be like if you did not have the ability to communicate in your native language.  If you’ve ever visited another country where your native language was not spoken and you didn’t speak theirs, you’ve had a taste of the feeling of isolation and frustration that would result.

That is the exact feeling of isolation and frustration that many students feel in their math classes, and especially in their algebra classes.  That isolation is not imaginary; if students do not learn to communicate in the language of mathematics, including algebra, there are many futures that will forever be closed to them.

Benjamin Bloom famously said, “What any person in the world can learn, almost all persons can learn if provided with appropriate prior and current conditions of learning.”  It’s a truth that I have experienced.

Over the past several decades, a lot of very knowledgeable people have engaged in sincere and often well-funded efforts to provide “appropriate prior and current conditions of learning” in order to address the problem of learning the language of math. Unfortunately, there hasn’t been much progress in terms of results for students. There are specific reasons for this, reasons which have proven difficult or impossible to address in the context of a math class of 30 students with 30 different sets of learning needs and interests and one beleaguered teacher trying to somehow stay on schedule.  

It's time to change the context.

It’s funny how one can work in a field for many years and in many contexts and discover that some of the most basic concepts of that field are not clearly elucidated.  Let’s consider the concepts of “student” and “teacher.”

Everybody knows what a student is—it’s someone who is going to school.  But is it?  After all, there are lots of people who are being students who are not in school.  One well-known example is found in “hole in the wall” experiment conducted by Sugata Mitra, who embedded a computer in the wall of a slum in New Delhi with no instruction and observed as students taught themselves how to use it, and then taught themselves many other things. His conclusion? “Children will learn to do what they want to learn to do.”

Perhaps a student is “someone who is pursuing a course of education”?  But that implies that the course being pursued is something that someone else has created or developed, a pre-existing pathway that the student travels.  That’s a common enough experience, but Mitra demonstrated that children can and will learn without being provided with such a pathway.

It took me more years than I’d like to admit, working with students of many varieties, to come up with a definition of “student” that seemed to be to be universally applicable and also answered some of my own questions about how to design and improve effective learning experiences.

A student is, simply, “Someone who is trying to learn something.”

This concept includes the children in the New Delhi slum, the person who is pursuing a degree in a university, the 3 year old who is learning to read from bedtime stories, and the learning architect who is trying to figure out how to set up their email.  But what’s most interesting about it to me is who it doesn’t include.  

Often, individuals sitting in classrooms are not actually being students—they aren’t “trying to learn” anything. Of course that can change in a moment, if the teacher says or does or presents something that catches their interest and causes them to want to know more.  And it can certainly change in the 5 minute interval during which they move from one classroom to the next—they may not be a student of math and yet be an avid student of poetry.

I must admit that I’ve led Professional Development workshops for teachers in which not all of them were being students—there were often teachers present who had not had wonderful prior professional development experiences and before they entered the room had decided that they weren’t going to be learning anything.  I did my best to change their minds about that, but was not always successful.

I’ve always believed that it is the first and continuous responsibility of a teacher to make sure they have real students present—that is, to get everyone in the frame of mind of being a willing learner.  I’ve known many teachers who think the same way, and a ways back I was involved in producing classroom video content which had the sole purpose of helping math teachers turn the individuals in their math classes into people who saw the value of learning what they were being asked to learn—that is, to make real students.  It was called FUTURES with Jaime Escalante , and it turned out to be the most popular video content ever distributed by PBS Learning.

Which raises the question, what do we mean by the word “teacher”?  Mitra, again, provides an unexpected example. He says,

I found that they had a friend, a local accountant, a young girl,  and they played football with her.  I asked that girl, ‘Would you teach them enough biotechnology to pass?"  And she said, "How would I do that? I don't know the subject."  

I said, "No, use the method of the grandmother."  She said, "What's that?"  I said, "Well, what you've got to do  is stand behind them  and admire them all the time.  Just say to them, 'That's cool. That's fantastic. What is that? Can you do that again? Can you show me some more?'"  

She did that for two months.  The scores went up to 50,  which is what the posh schools of New Delhi,  with a trained biotechnology teacher, were getting.      

A teacher, then, is “Someone who helps a student learn.”  This concept, too, is as useful for what it excludes as what it includes.  Lecturing to a group of people, even if they are trying to learn something, doesn’t make you a teacher unless it actually helps them to learn.  Explaining something to a student that they already understand, or that they do not have the prerequisite knowledge that would make it possible for them understand, is not helpful and is not, therefore, teaching.  On the other hand, just giving the right book at the right time to a student who has decided they want to learn more about a subject can be a profoundly powerful act of teaching.   As is demonstrating to educators how a hole in a wall can become a university.

One aspect of these concepts of student and teacher and that they are not binary—one can be more and more of a student, depending on how hard they are trying to learn, and one can be more and more of a teacher, depending on how helpful they are in that process.  The nice thing about these identities existing on a spectrum is that the ability to be a student or a teacher can be gradually improved.

Along that line, there’s quite a bit of interesting research on the topic of motivation to learn.  And there’s also research on the topic of the helpfulness of a teacher—perhaps the most dramatic result being the impact that Bloom found of expert one-on-one teaching.  That’ll be discussed in a future blog, but in the meantime you can find out more in his paper, “The 2 Sigma Problem.”

I do not claim that any of this represents brand new thinking, but it does strike me that if schools, curricula, and administration kept these notions of “student” and “teacher” always at the center of their efforts to educate, it would keep them on a path towards providing education that’s not only effective, but also a pleasure for everyone involved.  Given the amount of their lifetimes that children spend in school, let alone the amount that teachers spend, that wouldn’t be a bad thing.

Occasionally people ask me why I decided to name my company Real Curriculum. It’s kind of a long story.

In an education career spanning 5 decades, I’ve carried out or guided curriculum development work in a range of widely different contexts: private schools, non-profit foundations, public schools and districts, professional development, ed-tech, tutoring, adult health education, textbook editing, documentary and instruction television production, textbook publishing and more. And for the first 25 years of that work I have to confess that I wasn’t really certain of the answer to the question “What IS curriculum?”

It's not that there aren’t plenty of definitions available. A state Department of Education or school district might think of curriculum as the expectations they have for students—the standards they’ve established for learning in all of the disciplines and all of the grades.

A textbook company is more likely to think of curriculum as what’s included in the products they produce—that is, the content materials that they are hoping students will learn from: student texts, teachers’ guides, workbooks, practice exams, copy masters.

A college professor probably thinks of the curriculum as the topics they’re going to cover, aka their syllabus.

For a K-12 teacher the word is likely to bring to mind their lesson plans— what it is they intend to have happen each day in class, and what they hope students will learn from it.

What each of these meanings have in common is it allows educators to think of curriculum as something that they can fully control. But what if that’s just hubris?

What has never sat right with me about the widely accepted meanings of curriculum was that all of them describe what we as educators were doing, but not what our students are doing. Yet one thing above all others that I have learned about education over these 5 decades is that even when they all read and do exactly the same things and answer the same questions, each student has a different learning experience.

Every well-intentioned, responsible, professional educator that I have known has developed the sensibility and habit of thinking about what happens during learning not just from their own point of view, but from their students’ points of view. And I believe that most if not all curriculum developers would agree with me that the essence of successful curriculum design is the ability to put yourself in the shoes of the people you wish to serve through that design.

A while back these thoughts led me to a definition for the word curriculum that has served me, and I hope the students I work for, ever since:

Curriculum: The experiences a student has in the process of learning.

To differentiate this definition from the others, I came to call it the real curriculum—the curriculum that actually exists in our classrooms and wherever students are learning. It is the real curriculum that we seek to influence through all of the others—standards, instructional guidelines, textbooks, lesson plans, and digital learning experiences. Those efforts are all worthwhile, but it is only when we recognize that they are also somewhat artificial and do not represent what is happening in the minds of the students we serve that we fully understand that education is not an activity in which we are the leaders and our students are the followers, rather it is a collaboration between us and our students. Our job is to provide opportunity; theirs  is to take advantage of that opportunity as they travel their own paths toward the achievement of learning goals that are meaningful to them.

And that’s how I came to call my company Real Curriculum—to remind myself every day that my job is not to control, but to help.