Houghton Mifflin iPad Algebra App

I have not been a huge fan of Houghton Mifflin math programs in the past, but I am also against these huge, heavy and overly busy algebra books that come to be the first and most lasting impression our students have of what should be a beautiful algebra experience.


That being said, the very idea of have a dynamic algebra app for the iPad or some other tablet seems to me to be a winner. The trick is being able to create an interactive and vibrant app. I wish I could try this one out to see its strengths and dilemmas. 


"There's no shortage of difficult news when it comes to California schools. But amidst the budget shortfalls and funding challenges is a tremendous achievement. California students are making history, introducing America to the future of instruction.

Since last September, over 400 eighth grade students have been part of a pilot study, using Houghton Mifflin Harcourt's award winning Holt McDougal Algebra 1 core curriculum on an iPad. In Long Beach, San Francisco, Fresno and Riverside Unified School Districts, students are learning algebra with the fully interactive HMH Fuse^TM: Algebra 1 App for iPad."


more info:
http://www.publicceo.com/index.php/local-governments/151-local-governments-publicceo-exclusive/2870-21st-century-algebra-theres-an-app-for-that

Define algebraic thinking.

According to some, algebraic thinking is: 

Kaput (NCTM, 1993): Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture.


Greenes and Findell (1998): The big ideas of algebraic thinking involves representation, proportional reasoning, balance, meaning of variable, patterns and functions, inductive reasoning, and deductive reasoning.


Herbert and Brown (1997): Algebraic thinking is using mathematical symbols and tools to analyze different situations by (1) extracting information from the situation...(2) representing that information mathematically in words, diagrams, tables, graphs, and equations; and (3) interpreting and applying mathematical findings, such as solving for unknowns, testing conjectures, and identifying functional relationships.


LUMR Project (Driscoll, 1997): The facility with algebraic thinking includes the ability to think about functions and how they work and to think about the impact on calculations a system’s structure has.


NCTM Standards (5-8) - Algebra (NCTM, 1989): Understand the concept of variable, expression, and equation; represent situations and number pattern with tables, graphs, verbal rules, and equations, and explore the interrelationships of these representations; analyze tables and graphs to identify properties and relationships; develop confidence in solving linear equations using concrete, informal, and formal methods; investigate inequalities and nonlinear equations informally; apply algebraic methods to solve a variety of real-world problems and mathematical problems.


NCTM Standards (5-8) - Patterns and Functions (NCTM, 1989): Describe, extend, analyze, and create a wide variety of patterns; describe and represent relationships with tables, graphs, rules; analyze functional relationships to explain how a change in one quantity results in a change in another; use patterns and functions to represent and solve problems
.
Usiskin (1997): Algebra is a language. This language has five major aspects: (1) unknowns, (2) formulas, (3) generalized patterns, (4) placeholders, (5) relationships. At any time that these ideas are discussed from kindergarten upward, there is opportunity to introduce the language of algebra.


Vance (1998): Algebra is sometimes defined as generalized arithmetic or as a language for generalizing arithmetic. However algebraic more than a set of rules for manipulating symbols: it is a way of thinking.




As a math teacher and thinker, I am pretty ok with most of these definitions, but I do not agree with reducing algebra to a simple language, as Usiskin tries to do. It isn't the language or the symbols that matter, but rather, the abstract way our minds work with ideas of generalized arithmetic and pattern description. 

What Algebra and When?

What Algebra should we teach?

Algebra is fundamental to understanding mathematical thought. It encompasses an understanding of patterns and functions, ways of representing and analyzing mathematical structures and situations, models that represent quantitative ideas and relationships, and approaches that analyze change in a variety of situations.

Most people recognize that algebra is needed by scientists or engineers, but algebraic thinking and reasoning are also used in many other occupations, including health care providers, graphic designers, and home builders.

Algebra in the Early Grades
Algebra is not an abstract, difficult school subject intended for the few who choose to study advanced mathematics and science. It is fundamental to a basic education of all students, starting in the earliest grades. Young children's understanding of algebra builds on their understanding of number ideas and arithmetic concepts and properties. For example, to help students learn the basic facts of addition, students should work with missing addends, such as + 4 = 12. Here, the box represents a fixed unknown the students need to identify. This is one way to help young students understand the concept of variables.

Another example of building algebraic thinking is the zero property of addition. Students learn that adding "0" to any number produces a sum equal to that number, as in 0 + 5 = 5. In working with variables, this property may be represented in expressions such as 0 + = or + 0 = . Later, letters like "a" and "b" may be used as variables, resulting in statements similar to 0 + a = a or b + 0 = b. Important algebraic concepts like this can and should be taught in the early grades.

Middle-grades instruction must build on and continue to develop such concepts in preparation for formal instruction in algebra. For example, in middle school connections to data analysis and geometry play a role in helping children grasp patterns, functions, and ways of representing mathematical situations algebraically. A good illustration of this is helping a box manufacturing company determine the dimensions of a carton that must hold 35 cups stacked on top of each other. In order to find the dimensions of the carton, students make a table to record the number of cups and their stacked height. Using different types of cups reveals various patterns that can be represented on a coordinate graph. In this type of activity, algebraic reasoning about patterns and functions and the use of variables strengthens students' understanding of algebra and make a connection to real-life situations.

A Variety of Courses

All children must have access to algebra. It is critical when considering how best to prepare students for algebra to underscore the importance of an algebra content strand throughout the middle grades. However, the approaches described below clearly have benefits for many students.

A two-year algebra course. A two-year course in algebra spreads the content of a regular year-long algebra course over two years. It is designed for students for whom the content would be too challenging at its normal pace and gives them needed time to strengthen their understanding of patterns and functions, relationships among arithmetic operations, mathematical structures, and algebraic properties.

Exploratory algebra. An exploratory algebra course, also called hands-on algebra or algebra investigations, centers around students constructing their own understanding of algebra by uncovering facts and relationships. A course offering this approach requires careful management and monitoring by skilled and knowledgeable teachers to guarantee that students meet the goals and objectives of instruction.

Pre-algebra. A pre-algebra course, in preparation for regular algebra instruction, is designed for students who have adequately mastered arithmetic knowledge, skills, and procedures, but have not made the necessary connections between arithmetic and algebraic ways of thinking and reasoning. A pre-algebra course can give students the opportunity to bridge arithmetic and algebra with activities and experiences that deal with variables and functions, connections between arithmetic and formal mathematical structures, models of mathematical situations, and ideas about change.

Comparing Slopes of Perpendicular Lines

This is a particular concept in 8th Grade Algebra that I find difficult to teach. From my perspective, it can easily fall into the category of "memorization" rather than conceptual understanding. The CPM text we use has a pretty nice way of leading the students towards a "discovery" of this relationship. They draw a pair of perpendicular lines on transparency sheets and then place the lines on graph paper to determine their respective slopes. The students collect data in a table and then analyze it.


Perpendicular lines are complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down


I find that the "opposite" nature of slope (positive or negative growth) is not really a problem. But determining the slope accurately can be problematic. Many students are inaccurate or wishful in determining the slope of the lines. They lack a certain ease with estimation: either they over generalize or they are nit picky about when a line crosses a lattice point on the graph paper. This can lead to them not really "discovering" the reciprocal nature of the slope ratio. 


When they make this error, I usually ask them to show me the orientation of the lines on the graph paper. At that point, if their error is not self evident to them, I find myself telling them how to see the slope correctly. I am cautious with this final step because it tends to take away some of the "discovery" nature of the activity and disempowers the student. But if I never intervene, they may never really figure out the lesson in the first place.


This year, with the addition of the SmartBoard, the lesson was far easier to present and analyze. I can make far crisper and interesting lines and graphs and manipulate them with grace and meaning. Thanks to this technology, many more of my students were able to "get" the opposite reciprocal concept easier. I was able to save the graph and put it up on the school website as well. This has been an important addition to the conceptual teaching I strive for.


Perpendicular lines are a bit more complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down

8th Grade: Meaning of Slope Activity

The 8th Grade worked on several iterations of a tricycle racing problem. They used clues from the problem to determine different lines. Using graphing skills and systems of equations, they found out who won.

from College Preparatory Math: Algebra Connections

The Meaning of Slope.

It seems to me that one of the most central ideas in a beginning Algebra I course is slope. When I first picked up the text book I was planning on using as the base for my 8th grade algebra course (College Preparatory Math), I noticed that slope was dealt with in several different locations during the year. With little experience actually teaching algebra, I found the idea of repeating a topic as "simple" as slope (rise over run) ridiculous.

Being of independent mind in a private school, I was free to reject, and so I did. I skipped the unit.

While I cannot directly correlate the results to the action, it did become evident, later that year, that my students' algebra education had taken a turn towards the "mechanistic". While many of them did successfully adopt the "rise over run" mantra, they began using funny language, such as identifying "x" as part of slope (think y=mx+b) or not getting the whole "change in y compared to the changed in x" concept.

I am in an unique position of being able to direct own mathematics program since I am also the 7th grade teacher. It wasn't until my second year into the job, though, that I realized the GRAND importance of "ratio" in a middle schooler's conceptual map of math. If they had a fluid notion of comparing two quantities, they were well positioned to simply transfer that idea to slope and move on. But, as is the case of many if not most of my students, they had a very creaky understanding of numerical comparisons and if they maintained a decidedly "fractional"view of "ratios" (part to whole), than indeed, the whole concept of "y" changing according to something done to "x" is confusing and meaningless.

As I study calculus with one of my super advanced 7th grader, I have finally come to see the supreme importance of slope in differenting equations. In fact, without an extremely solid foundation in the ratio known an "slope", calculus will remain, as it did for me so many years ago, a mystery wrapped in an enigma (but one in which I needed to get a passing grade).

This year's 8th graders are exhibiting even more "slope-itis" than in year's past (though this could be entirely due to my almost singular obsession with "figuring out what they know"). It was with this concern in mind that we embarked whole-heartedly into the once skipped unit on slope.

We hit a wall.

Indeed, lots of confusion. Many requests for clarification. A whole lot partial understandings or flat out misunderstandings.

It wasn't pretty.

I said to myself, "This can't go on. What to do?"

"Close up your books and let's go out on the green top" (thanks, private school, for the freedom to be really in the moment and not driven by outside schedules). All we took was a piece of paper and my iPhone (for the timer).

I asked for several volunteers to show us different style walking or running. Some sprinted, other ran backwards. Some strolled, others sashayed (and not only the girls, btw). In the end, we measured off 30 meters, chose 5 volunteers and had them, one by one, walk the distance. We timed them according to my iPhone stopwatch.

10 minutes later, we had a collection of data. We came into the room and created a table with the five volunteers' times. I asked "What is speed?" Many knew it was distance over time. I said, "So, that is a slope of line." Many of them looked confused. We made a big graph together  and briefly reviewed that time would have to be the independent variable, always placed on the x axis, while distance would be the dependent variable, placed on the y axis.

From the table information we were able to create five different slopes. They graphed these lines and we talked about how steepness tells us something very important about "rate of change": steeper means faster change.

The class looked at me matter of factly. They claimed to know this already. In fact, judging from the discussion, they did know this. But I am convinced that they were not applying this sort of real world experience to the slope of lines we were analyzing in class.

On Friday afternoon I tweeted to my PLN:  "it was a long week, but my students understand more about slope than when we started."

It makes teaching worthwhile, don't you agree?

High-School Calculus: A Mathematical Dead End?

Advanced Placement calculus classes deter students from taking the subject in college, according to an article by MAA President David M. Bressoud in The Chronicle of Higher Education.

Over the past two decades, enrollment in college calculus courses has decreased dramatically, a trend that coincides with the rise of high school AP calculus classes. According to Bressoud, the AP courses are the reason students choose to abandon mathematics. He points to a lack of synchronization between high school and college calculus curricula as the source of the problem, claiming that the lack of sophistication in AP classes leaves students too experienced for Calculus I, but unprepared for Calculus II.

College professors like Bressoud see problems with the AP courses, while high school teachers blame the worn-out lesson plans and lack of hands-on activities in college courses. "We have very little evidence of what has caused the general decline in enrollments in mathematics at the level of calculus and above," said Bressoud.

In the article, Bressoud suggests ways to improve the situation at both the high school and the college levels. First, schools should collect more information about why students are avoiding college calculus. For their part, high schools should establish and enforce guidelines to standardize expectations of the calculus classes they offer. Colleges, in turn, need to take into account the amount of material covered in high-school calculus, and offer courses specifically designed as a transition from high-school to college-level mathematics.

Source: The Chronicle of Higher Education (January 17, 2010).

What is Algebraic Thinking?

Algebraic thinking can be organized into two major components: 


1.  The development of mathematical thinking tools and 


2.  The study of fundamental algebraic ideas. 


Mathematical thinking tools include:


Analytical habits of mind: Problem solving skills, reasoning skills, and representation skills. 


Fundamental algebraic ideas represent a domain in which mathematical thinking tools can develop.


They are the content for study. 


What I am interested in discovering and discussing with others is this topic of algebra content. What do we consider central to the study of algebra? How do we determine when this is learned sufficiently? Can the content be learned satisfactorily if the accompanying algebra thinking tools are poorly understood by the student? Conversely, can the algebraic thinking tools be acquired and refined without delving deeply into all algebra topics.


Examples I am considering:


Do 8th graders need to know, for example, how to determine the slope of a line perpendicular to another line with known slope?


Do 8th graders need to know how to multiply binomials even if there is no practical use other than preparing their skills for future math. In this sense, is learning binomial multiplication a valid example of "algebraic thinking tools"?

Collaboration via Twitter and blogs.

On a Monday after Winter break I came back to my 8th grade algebra with the topic of multiplying binomials. We had spent about two weeks on the topic before break. Most of the time was devoted to creating specific rectangles using Algebra Tiles, but also we moved onto to the use of "generic rectangles", which in our CPM course, are simple graphic organizers to multiply polynomials.


Before re-visiting the topic, though, I wanted to know when such a skill is actually used. I conducted a quick Google search: why multiply binomials. The only results I got were "how to multiply binomials". So I posted my query on Twitter. Gary Davis (@republicofmath) took up my question and started to write about it based on an email I sent him with my initial thoughts. Later, I read what he wrote and responded. 


Several great things happened for me. First, I got to think deeply about a topic with someone who knows so much about it. Secondly, I got to bounce my ideas off of his expertise. Finally, we collaborated on a blog post. The collaboration via Twitter and blogs was fantastic. Thanks, Gary.


This is the blog post:

http://republicofmath.wordpress.com/2010/01/05/why-multiply-binomials/




a

The hills are alive...with the sound of healthy confusion

Thinking about the analogy of marching learners into confusion, and then marching them out, we embarked on this problem from the Algebra Connections series from CPM:

First, most of my 8th graders "got" the veiled references to "Sound of Music". They also "got" why xylyphones and yodelers (x and y).

Oh, but what a storm of confusion upon writing the equations related to this problem.

They initially want to write: x = 2y .

That is perfectly understandable because it is mimicking, at least in order, how the English language lays it out for them.

A few students, though, saw a problem in this. Debate ensued. No agreements easily reached.

I suggest that one way to determine whether this equation would work would be to assign values to the variables. They all understood that there are twice the number of yodelers as xylophones. So to speak, two yodelers carry one xylophone onto the gondola. I assigned the value of "1 to x, but that was too hard to conceive. So I upped the value of x to 2, and some students started to get that y would have to be 1.

This means that for every 2 xylophones, there would be one yodeler. But the problems says it is the other way around. Still, many students were not convinced.

So I assigned a value of 1 to y, meaning 1 yodeler. The value for x would be 2. Still not coherent to the problem.

A student suggested we switch around the variables to write y = 2x . Many students were unsure about this, but I overtly supported the idea, so they started to pay attention. By assigned a value of 1 to x (meaning one xylophone) we find there are two yodelers: matching the problem as stated.

While I thought the case to be convincing, several students still questioned the logic. I tried assigning new values to the variables showing them how it would work.

In the end, I would say that the conversation was rich. The problem really elicited the difficulty of translating oral or written language into algebraic form. I do not believe this was a moment of "confusion in vain", but rather, a deep discussion of the meaning of variable in algebra. This is just one of many different discussions we have had and will continue to have as we move forward in this journey out of confusion.

a

More thoughts on Algebra for the 8th Grader:

I teach 8th grade Algebra (for the past 5 years), so the Algebra II list is not immediately relevant to my anecdotal experiences. It has been ages since I have thought about logrihms. I do think a WHOLE LOT about Algebra I, though and have slowly been developing a love/hate relationship with certain topics:

Love:

I love the "magic" of algebra in describing so many different type of patterns of growth. Most of my students enjoy working with this as well. They are developmentally intrigued by the ways algebra helps them describe these patterns outside of simple tables or vague words. In other words, they like the elegant way equations and graphs work together. Linear growth is a natural for them: it is easily seen and predictable. They like that!

Data is a topic Leinwand put into his talk: I think it very very important, particularly logic around reading graphs, tables, and such. I also see that my students need a lot more work with interpreting graphs and basic statistics.

Less Love:

Exponential functions, particularly quadratic, are fun to a point and then quickly become tedious, particularly when dealing with quadratic formula. We spend a good amount of time in Winter and Spring on factoring quadratic equations, really like working out puzzles and trying to make sense of them in various "real world" situations, but I feel I start losing a lot of my students around quadratics. It all starts to feel increasing procedural, even when I emphasize algebra tile models and pattern recognition. Attention wanes.

Then we come to inequalities, which are relatively easy and accessible for them.

Hate: 

We end the year in a whimper with rational expressions. All the procedural knowledge a student got during our work with factoring quadratics comes into play, plus concepts of fractions. Some students are totally "there", some struggle but "get it" and many simply "give up". Add it some absolute value stuff and it all gets messy for them. I have yet to find a way outside of thinking of these expressions as "puzzles" (Sudoku) to make this topic approach real to them. I also am ignorant of their use in higher math. This is an area of growth for me, to be certain!

I HATE teaching rational expressions in the 8th Grade

I wish this were the reality of my 8th Grade Algebra Program:

Steven Leinwand proposal for Algebra I and Algebra II curricula, paced at one chapter per month.

Algebra I

1. Patterns.
2. Equations.
3. Linear Functional Situations.
4. Representing Functional Situations.
5. Direct and Indirect Variation.
6. Data.
7. Systems of Equations.
8. Exponential Functions.
9. Linear Programming.

Algebra II

1. Review and Reinforce Big Ideas and Key Skills of Algebra I.
2. Quadratic Functions.
3. Polynomials and Polynomial Functions.
4. Patterns, Series, and Recursion.
5. Exponential and Logarithmic Functions.
6. Rational and Radical Functions.
7. Probability and Statistics.
8. Optimization, Graph Theory, and Topics in Discrete Mathematics.

What is Algebra, anyways

It isn't a jumble of x's or y's. 

Learning algebra is a lot like learning to drive a car, or swim, or ride a bike. You must learn concepts at the same time you are developing skills. There are pedals to push, gears to shift, lights to turn on and off, turn signals to manipulate, and a steering wheel to wrestle. Those are skills.

It’s also useful to learn what the brake pedal does, why you need to use your turn signals, what the clutch is good for, and which direction to steer when the car begins to skid. Those are concepts.

Most of the math you have been learning until this year is based in concepts of arithmetic. Arithmetic relates directly to real life. We add apples and oranges, we subtract money, we discover the area of a wall to learn how much paint will cover it. We might assume algebra is going to be like that, too.

But algebra is different. While it can, indeed, be useful in real life, much of it is simply a game that has no direct relationship to anything we can touch or count in our everyday life. Like many games, algebra has game pieces, moves, strategies, goals, and its own vocabulary. Usually, the object of the game is to discover some specific unknown by using available clues. Other times, the object is to translate recurring events into an equation.

Why must we learn this stuff? One answer is because it’s fun. There are many side benefits to many games. Tennis improves your eye-hand coordination. Racquetball is great aerobic exercise. Golf is meditative and social. Chess teaches concentration.

A side benefit of the algebra game is that it may allow you to become a chemist or electrical engineer. It may allow you to calculate interest rates and design cars and computer systems. Many people say that the thought processes they developed by learning algebra were more useful than any direct application. And, of course, if you want or need to learn more advanced math, algebra will be a prerequisite.

So I invite you to join me as we learn the rules and strategies that will make you winners in the game called algebra.

Why Learn Algebra?

Because it is a great thing to KNOW!


The truth is that you really do not "need" algebra unless you plan to teach it or use it in a scientific profession. Just like you don't need to lift weights or shoot baskets unless you plan to be a basketball player. But before you put away your algebra books to shoot some hoops, let me give you some good reasons "for" learning algebra.

Algebra is a very unique discipline. It is very abstract. The abstract-ness of algebra causes the brain to think in totally new patterns. That thinking process causes the brain to work, much like a muscle. The more that muscle works out, the better it performs on OTHER tasks. 


In simple terms, algebra builds a better brain (as do other disciplines such as learning an instrument, doing puzzles, and, yes, even some video games). When the brain is stimulated to think, the hair-like dendrites of the brain grow more extensive and more complex enabling more connections with other brain cells. We often hear that we use only a small percentage of our brain's capacity. The study of algebra is a way to increase our use of this marvelous muscle. By studying algebra, more "highways" are "built" upon which future "cargo" is transported -- cargo other than algebra.

My favorite analogy is comparing learning algebra to the construction of the railway system in the United States in the 1800's. When railroads were built, surely those men never conceived of the items that would be transported on those rails more than a hundred years later. They could not have imagined home appliances and computer equipment traveling over that railway system. But they knew that building the transportation system was important. 


So is it with the study of algebra -- you learn algebra by transporting numbers and variables -- later, those variables will change and you will transport something useful for your purposes.

Why Algebra Tiles?

Algebra tiles provide a concrete, visual manner to "see" equations. They are part and parcel of the approach to math that exalts the value of multiple representations and a variety of approaches towards solving problems.

They are not a crutch, but they do serve students who have significant difficulties dealing with algebra on a purely symbolic level. For students who are able to navigate the symbology, these same tiles stretch the mind to "justify" why the moves they have learned "work". For all students, the tiles continually reinforce the concept of negatives as "opposites" because each time they cross a region on the mat, the students is obligated to flip them, thus showing their opposite. That is a concept that is easily lost to beginning algebra students as they rush to "solve" equations.

Later in the course, when we look at factoring polynomials, these tiles will no longer feel "backwards" but will actually provide key support for all students to feel successful at this difficult task.

Finally, these tiles reinforce the very true interconnectedness between algebra and geometry as well as arithmetic in general as the rectangle model we are using (known as "array" model) is an excellent basis to understand multiplication/division and fractions.

I notice that students who accept the tiles as a part of algebra are more successful down the line when we come to use them for more complicated concepts. In particular, many MANY 8th grades make simple errors as the “rush” along the path towards “solving” equations. The tiles kind of make them slow down and justify their moves.




And yet, I must admit, my students resist the use of tiles year after year. They often say they are more confused about what algebra means when they use them. Usually when I ask them to take out the tiles, there is an adolescent groan in the room. 


This year I made my best effort to ignore this because I know that the tiles present algebra concepts very concretely. In particular, I like how the tiles, on tile mats, oblige the students to consider the real meaning of negative numbers (as opposites). I have begun to see my students make intuitive decisions about how to solve for variables rather than procedural ones. However, they still complain. 


I have my theories why, but what do you think is happening with them?

Great Site for Virtual Algebra Tiles

http://nlvm.usu.edu/EN/NAV/frames_asid_189_g_3_t_2.html

A great start on the year

We are well into our second week of classes. It has been very gratifying to see how my 8th graders have stepped up to bat, so to speak, and really put great effort into learning algebra this year. They come well prepared, happy and enthusiastic. Many of them come to class a little early to drop off their materials so they can come in right after recess and begin the warm up. I am very excited about how the year is shaping up.

We started the year with my presentation of a new metaphor for learning algebra. Last year I tried out the idea that learning algebra was like learning the rules to a game. The example I used was backgammon, a game I enjoy and enjoy teaching to my students. This year, I came up a different metaphor as I was driving down long, endless roads in central Oregon over this summer: Learning algebra is like learning to drive. I am the driving instructor and my students are working towards their driving license, but first need to master the basics to obtain their drivers permits. To learn to drive, one must learn skills, such as turning on the car. These skills may or may not make sense per se, but they can be readily learned and applied without a deep understanding of why they work. In algebra, this could be live solving equations for a variable.

In order to drive, you must also learn concepts. For example, to take a left turn, you must learn why to use the left turn signal, when to use it, what to look for ahead and behind, how to slow to correct velocity and so on a so forth. Without a deeper understanding of why you need all this and how it helps, you may find yourself unable to make this turn safely. In algebra, I was thinking of the meaning of slope and how that is applied to word problems to represent aspects of growth that help us make reasonable predictions. If one sticks with a very mechanical understanding of slope, there is a very real limitation to using it in a variety of contexts.

Finally, there is the aspect of practice. Of course, in driving, there are many hours of practice, usually guided by an adult. In algebra, I also believe in hours of practice, either guided by an adult or done independently with adult guidance as needed.

After completing a driving training course, learning all the laws and regulations determined important by some authority, and sufficient practice, you can take the theoretical and practical tests. With a passing score, you receive your license, which is to say, it is legal to drive alone. Of course, the learning will continue on for a long time and mistakes will be made and certain laws will become vague (what is the difference between a double yellow line and single white one?). But with that license, you can travel far and wide and have many more options open to you.

I believe that algebra is very similar in its scope and its ability to open up options in the future.