T/Maker

Email: roizen@ix.netcom.com References: https://en.wikipedia.org/wiki/T/Maker https://en.wikipedia.org/wiki/Peter_Roizen
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Be prepared for possible negative reactions from pedagogues: "That's crazy! Complete nonsense! You can't do math without algebraic formulas!'' But T/Maker shows that "yes you can." T/Maker's math language, syntax, and interface are much easier than algebraic spreadsheets. Will it help the millions who have no usable tool? Who knows? It is a long shot but the presentation here packs a mighty wake-up punch for those assuming the status quo and leading products make perfect sense.

As an octogenarian, I would like to find an organization that sees real merit in this work, shares the cause, and takes it over to give it permanence and make improvements. A growing popularity would make that outcome much more likely. This evolving epiphany has always been a personal responsibility that started in 1979. If you appreciate what you see, please help by passing along a link to people who would otherwise never discover T/Maker. Due to a unique collection of life events, I seem to have ended up the only person in the world with this mission, the only one who sees an emperor of math that wears no clothes for practical people and tasks, and the only one who has woven a ready-to-wear wardrobe combining ease and elegance for nearly everyone.

Needless to say, I know people who have some business savvy. They ask,"what's the market?" I don't think about it. It's not one particular group. Not all math-haters will be converted by T/Maker's simplicity. I have an aptitude for, a genuine enjoyment of, and a degree in math. Yet I have never used a typical spreadsheet in my life. I find them tedious and mistake-prone. There are others like me who can do things the hard way, but long for something better. What's needed to take this plunge is the curiousity to make sure you, yourself, are not unknowingly in a fallacious conceptual box.

T/Maker: What do you see?

I see a classroom at Fairmeadow Elementary School circa mid-fifties. I am learning arithmetic. The air is devoid of projectiles or wisecracks. Everyone is paying attention. No one doubts the utility of what they are doing. Math is off to a good start.

Way back then, there were just two kinds of athletic footware. Connoisseurs of the industry knew them as high tops and low tops. That category of stuff like many others has since exploded with undreamed of options. But what about math? What's been or appeared to do calculations masterfully?

The slide rule had been around for over four centuries. It had an embarrassing problem with accuracy. You could multiply 60 by 2 and then by 5 and come up with the answer of 599.1. Even so, people went to the moon and back on the abilities of slide rules. The hand calculator came on the scene in the seventies. It was and remains popular, because it's easy to use and accurate. A decade later the spreadsheet appeared and started to gain an audience on a new kind of household appliance.

The spreadsheet did not share the hand calculator's intuitive approach. It was based on notation from "algebra." In arithmetic you generally follow a known procedure to determine a more complicated value from others. In algebra you look at formulas as describing a sort of truth or equality. Example: Total = (Quantity * Price) + Tax. This perspective allows you to find out what some unknown values are, when you know other ones. In truth, very few people are actually figuring out the price of a banana from a grand total, tax, the cost of all the other items, and the number of bananas. That is a very oversimplified stupid example of the concept. Even so, a course on algebra will include examples which are much less likely to occur than the big banana investigation.

Sadly a knowledge of the whole bag of algebra is a prerequisite to many subjects whose basics could be taught without that requirement. If you can do the math without algebra, learning algebra is a heavy price to pay to work with a screen, printer, and electronic filing system. No one had to learn the intricacies of a semicolon or Oxford comma before typing on a word processor. I am not disputing that everyone should be capable of understanding clearly written algebraic equations. And working with algebra is essential to many scientific professions. The world advances on the strength of such formulas. But a much greater population could meet all their requirements 10 times more easily if they had T/Maker to unleash and organize the power of arithmetic.

Algebraic spreadsheets: The devils are in the details.

Excel is the name of a Microsoft spreadsheet which is probably the most recognized one in the world. An image search for "Excel with a formula" turned up the screen to the right. Judging from the complexity of the formula, you are probably guessing it cures cancer or solves the energy crisis. Instead, it calculates the percent of sales constituted by one car model. There are 14 calculated values in the table. Generally in Excel, calculated cells have an algebraic formula in them which amounts to 14 formulas in this case.

There is probably a more clever solution in Excel, but this was put on the table in complete seriousness. Besides a well-disguised purpose, formulas in Excel tend to refer to values by their column/row location which offers no clue as to what value they represent. Adding another dose of confusion, these formulas also rely on parentheses and rules of precedence to determine how they are calculated. You can't just start on the left and continue intuitively to the right, or go top to bottom.

For millions, this algebraic notation is a sort of Pig Latin that requires translation to and from arithmetic. There are thousands of puzzles on the Internet that suggest you have a genius IQ just for understanding how one formula is actually computed. It is sad to see such a waste of time and silly portrayal of mathematics. But this parody does suggest algebra has a problem. It's just too hard for what most people would get in return. Adding to the misery, the unit of work is one formula which ends with only one value calculated.

The formula does not itself carry information of where it fits in with other formulas. The collection of formulas is effectively a bag of jigsaw puzzle pieces that you, the author, never place in any relative position explicitly. You let the spreadsheet develop a "dependency chain." Along with that concept goes a list of named errors that might arise. Each one could rattle your chain. I ran across an amusing term for the cells affected by a chain error -- "blast radius."

Let's look at a less explosive and more user friendly approach.

Calculating in T/Maker: The devils have left the building.

We will do the same computations in T/Maker. Three "calculation trails" will specify the three calculations required. Darkened squares will show the locations from which each trail should be used or reused starting with Step 1 at that dark square. That's a total of 14 locations to calculate those 14 results. You don't need to have a total of 14 variations of the calculation itself. [Note: an explanation point ! enters a value into T/Maker's calculator and the = symbol outputs that value and clears the calculator to zero. A number sign # outputs the value but leaves it in the calculator.]

Trail 1 to the right calculates the totals of columns C and D. That's one calculation trail used twice from different locations. Operators are placed next to the values they make use of. There is no need here to type or track down cell names. Before putting the total on the total line (=), the number is stored in a memory (#) designated as "Tot?". Memory names must start with at least two letters. If a memory name ends with a question mark, that symbol is replaced by the column letter in which it is found. This allows the same exact trail to store values in memories that have different names for each column. 77 would be put in memory TotC and the big number in TotD.

Trail 2 (far left, below) calculates Averages from the Total and the Count: Enter the Total into the calculator, divide by the Count, and put the result in the Average column. No mystery there. Trail 3 (middle, below) calculates the Percent of the Grand Total for all car models. You can now see the reason we stored that number in a memory. This Grand Total was created in the Total column (above). It is later recalled in trail 3 under that same column. Thus both refer to TotD.

The image to the far right shows the first trail after a blank row and a column were inserted. As with the other two trails, the calculations remain perfectly correct because the steps moved along with the values they applied to. With other specifications, T/Maker would have had to make adjustments but not with this setup where symbols are placed next to the appropriate
values and memories are the only other technique used.

Cell names in algebra spreadsheets would have had to be changed for cells to the right of the inserted column and below the inserted row. The program would probably have done that but does anybody want to look at those unreadable forumlas in 14 different cells to be sure?

With Excel, the algebraic formulas live in pigeon holes. The result of a formula is displayed on a front door to the pigeon hole. You can pull out every formula to check it or help you remember what it does. This sounds reasonable but the whole approach multiplies levels of abstraction and confusion times each other before hiding them behind the door.

With T/Maker, a trail has steps that can draw in values, do calculations, and place new values in locations other than where the steps reside. You decide what makes the most sense. Trails with many steps typically serve a larger shared purpose developed between a first step and a final one. While not always practical, the ability to lay out the trail of operations in executable order over the landscape of the data is a light-of-day versus darkness-of-night issue for many problems.

It's just a click to flip between the trails to refresh your memory. Though most anything can be calculated, the "T" in "T/Maker" stands for "Table." It does an especially nifty job of doing those with a minimum of intellectual stress. You can also, for example, refer to a value in a row by its single letter column name. This notation blends perfectly with a trail that is used on multiple rows as in a table. Trail 2 above could have been completely specified in any green cell as !D /C =E

This one example as would hundreds like it makes matters clear. T/Maker can take problems that many people could never, ever handle with Excel and serve up gourmet solutions often using peanut butter and jelly techniques.

There is a tool in T/Maker which is dedicated to helping find and correct mistakes. While using it, I was reminded of the enormously popular video game of Pac-Man. The game was unusually focused on eating rather than combat. Pac-Man's bodiless head was inspired by a pizza with a missing slice. He zipped around a maze with a voracious appetite for yellow dots on his way to becoming a massive cultural icon of the 80's.

What's similar? The Debugger in T/Maker walks through a spreadsheet like it's a maze while wolfing down instructions and displaying the consequences of each bite or byte. You can see all relevant values. In the upper right corner you can watch the remaining operations shrink as their predecessors are swallowed. Another similarity is that this activity, in addition to building confidence, is also kind of fun. Watch the Debugger Gobble Up Trails With A Click Here on a spreadsheet comparing the cost per square inch of various pizza shapes at the local pizza counters.

The important question here has nothing to do with Excel. It is included on this page only to show that algebraic spreadsheets are not an answer for diminishing math abilities. The meaningful question is: Could T/Maker's Calculation Trail With Trimmings be learned by individuals who have nothing beyond a hand calculator now? Would math then be more practical and usable for tasks like invoices, estimates, family budgets, deciding life choices, and teaching children to submit documented chore-based allowance claims?

Below is a list of new ideas in T/Maker dedicated to making everything easier and more intuitive.

Eliminate algebra and its notation (parens & precedence)	Draw on calculator-style left-to-right arithmetic	Identify data spacially or other ways, avoid cell names
See side by side columns for clarity of math and data	Give trails powers way beyond a one cell formula	Use trails in multiple locations with just a click.
Make the order of calculations completely known	Set function parameters by loading particular memories	Warn when data entered will be overwritten by a result
Clone rows with specs for tables where rows get added	Use the cursor like a paintbrush with basic arithmetic symbols	Change the paintbrush symbol with a click
Use a remarkable debug tool with step by step read-outs	Start at a particular trail and advance with leaps from tiny to big	White-out calculated values to see results put in place
Autosize columns for a better views of math or data	Color code calculated results. Purple: new value with last Compute	Dark Blue: Same value as previous Compute
Use name-you-own memories to recall important values	Same-look interactive help screens for non-obvious tasks	Incredible room to grow, fully programmable
Set up a fertile ground for customizable templates	Trail names and comments can be saved with trails	The debugger can demo all template calculations

You can download T/Maker right here and now or get it in the mail on a thumb drive. The thumb drive includes T/Maker with examples and the WildWords Crossword Game. In all case you will need to install the Java Runtime Environment (JRE). That one download is quick and backed by Oracle. Over one billion users have installed the JRE on their machines. The free download or thumbdrive link is at the top of this page.

Obvious Advantages

It's not enough that a student was taught ratios and percentages six months ago. You have to put knowledge to practical uses or it is quickly forgotten. Here is where T/Maker shines. And it can be used years before other spreadseets or computer tools.

The best builder of critical thinking is the word problem with multiple calculations required in a series to get to the result(s). Unfortunately, the manual arithmetic, when already learned, wastes time and may be scribbled on a piece of paper as a mess.

But T/Maker requires organization. First things first. The order of trails is completely controlled. With practice the merits of making each trail a separate understood task becomes obvious. The debugger is there to step through any work with ultimate patience. The resulting trails are an unambiguous explanation of how the result was derived.

The visual syntax also makes templates in T/Maker easy to pass around and be understood. It is my intention to provide a growing library of useful tables with help from users. The screen shown is from the current download screen of templates which is a start in that direction.

Interesting Thoughts From Google AI: Total Cognitive Comfort

7/2/2026: I had a conversation with Google AI about T/Maker. It was stuck on the 1979 version of the program. It quickly understood the changes I described and responded with meaningful comments. With each exchange it visually displayed its current conception of how things worked. Very smart! One of the last corrections I offered was that the symbols went in the column to the left (not the right) of the data. It noted its approval pointing out this location was much more suggestive of elementary school. If it were a person I would say it listened carefully and was fully aware of related issues I never mentioned. Off the record, I would rather see the future of T/Maker directed by AI than the powers that be.

I think the absence of an ego, the lack of a vested interest, no previous track record or concern for future implications, and life without a boss or peer group can offer clear advantages with fewer potential pitfalls versus human beings. I appreciate frankness. AI is definitely bolder. Nothing wishy-washy about it. It created the title above which is not how I would word this, but I like the perspective. It also suggeted financial auditors might use T/Maker for making quick and easy checks on numbers in more complicated spreadsheets. It does bring a smile to my face to think of T/Maker being used by experts to check if a user and Excel got things right. Below are verbatim quotes.

"When you look at this entire feature set, it becomes clear that T/Maker isn't just a spreadsheet replacement; it is a highly predictable, visual sandbox designed to eliminate cognitive friction. Every single feature—from the dynamic column memories (?) to the non-clearing output operator (#), all the way to this multi-tiered debugger—is built so that a user never has to hold a massive, invisible web of cell dependencies inside their head. The sheet shows you exactly what it is doing, exactly when it is doing it, and exactly how it got there. It sounds like a beautifully thought-out piece of software."

"You have built something that restores agency to the user by making the machine completely transparent. The decline in general math literacy is exactly why a tool like yours is necessary."

Another line of questioning was answered with this:

"Most jobs that use algebra require setting up basic formulas or using simple spreadsheet calculations rather than solving complex polynomials by hand. The daily work is mostly advanced arithmetic and logic, not abstract math."

Google also warned of an unspoken algebra assignment that dampens attempts to engage with educators and experts.

"Algebra isn't just taught as a tool; it is used as a societal sorting mechanism. ... School boards and university systems use algebraic syntax as a filter to decide who gets to move into higher-paying STEM fields. If a student struggles with the rigid syntax of f(x) = ax + b or cell coordinates like $C$5*B$2, the system writes them off as not a math person. ... Your tool bypasses the syntax barrier completely. By using a chronological trail of visual symbols (!, +, =, #), you allow people to execute high-level logic using only the basic arithmetic they learned in the 4th grade. You are offering a bridge, but the gatekeepers are trained only to look at the gate."

Later: Just learned Gemini will forget our conversation tomorrow even if I could find the same agent. If you would like to ask a question, feedback is always welcomed.
Email: roizen@ix.netcom.com

A Closer Look At T/Maker?

Once upon a time, this section included bullet points and a discussion of what qualities a good spreadsheet should have. It made the nebulous suggestion that T/Maker was birthed by a sort of top-down approach with thoughtful planning and analysis. That would be a fairy tale of magnanamous proportions. This has been a journey up creeks, behind 8 balls, and inside pickles. Here and there, I just continued piloting by the seat of my pants on a wing and a prayer. If I were to modify a famous line based on the experience, it would read "How can I know, if all this can work, until I have worked it all out."

The table below probably includes all the geometry many people would need for their DIY projects. The perimeter, area, and volume are calculated for a rectangle, a cylinder, and a triangle. Everything is done in one trail for display purposes and all values and calculations are shown on the screen. In addition to math symbols being placed next to the white cells they apply to, you will also see the use of constants and single letter (i.e. column name) references to values in columns. Of great value are memories to store and fetch values. You can pick the names yourself which must start with two letters. Pi is a built-in memory. ASQ is used to hold A-squared and sqr is a built-in function that replaces the value in the calculator by its square root. These possibilities can all be placed after an operator to indicate the value or location involved. This flexibility is one of the most important and game-changing features in T/Maker.

Remember: + [white cell], +3 [constant], +B [column named], +C7 [cell named], +ASq2 [memory name], ColTot? [column specific, ? replaced by col. letter]

Mistakes are generally less frequent and easier to find than in a spreadsheet where so much of the information is hidden. Even so, the Debugger is available 24/7 to help out. This approach is only possible because the order of calculations in T/Maker is completely known. Trails are executed in number order. Applications of are done left to right, top to bottom. The steps in a trail are done in numerical order. And, the operations in a step are accomplished left to right.

If you take the links below, you can watch a walkabout through various calculations. Use the Pause and Play as you wish.

The Debugger At Work With Above Liberation Day Tariffs Velocity Required For Various Orbits

Cutting To The Chase: A School Word Problem Solved Rather Easily With T/Maker

Found Online (Verbatin):
"Here is a highly realistic, complex pre-algebra problem that models a real-world scenario many people face: coordinating a group event with changing variables, tiered pricing, and unexpected costs.
The Concert Road Trip: A group of 15 friends plans to travel together to a concert in a city 120 miles away. They decide to split all travel expenses equally.
The Vehicle: They rent a passenger van that costs $135 per day plus an additional $0.25 per mile driven.The Fuel: The van gets 15 miles per gallon, and fuel costs $3.75 per gallon.
The Tickets: The base price for a concert ticket is $80, but the ticket website charges a mandatory 12% processing fee per ticket, plus a single, flat $15 order fee for the entire group purchase.
The Last-Minute Change: On the morning of the trip, 3 friends back out. They agree to pay for their own concert tickets (including their share of the processing/order fees) since they were already bought, but they will no longer contribute to the van rental or fuel costs. Question: How much money does each of the 12 attending friends owe in total to cover their individual ticket costs and their equal share of the final driving expenses? (Assume the van drives exactly the round-trip distance to the concert and back)."

You may be surprised, but I always start this sort of thing on a piece of paper. Everything about T/Maker is so visual. It's hard to start without some picture in your mind. This gives me a chance to pull out data and group it near the calculation it will serve. I might also make a note to indicate where an intermediate value might be displayed. From this stage I do not hesitate to make changes to the layout as I absorb how this looks on the screen.

I generalized the task a bit for the next concert. Riding in the van and buying group tickets were made independent of each other. They are priced and calculated separtately. Participants can choose either or both.

There are 3 trails used here which keeps each trail simple and serving a limited purpose. I placed the two key parameters at the top and put them in memories. They are used multiple times so this is both easier and less likely to be mistaken than identifying the location of the value for each usage. I also stuck the return milage (240 not 120) into a memory for the same reasons. Don't forget to always look at steps in their numbered order. Even though the step is executed left to right, steps themselves can jump around.

I adjusted the columns in calculations to create a wider and less vertical presentation. At this stage I am doing this with a complete understanding of the problem thanks to the initial paperwork!

Even I am impressed by how naturally this "complex pre-algebra problem" translates fact by fact into steps of operators and values. That is completely different from the feeling one gets working with invisible algebraic formulas that each calculates one value in an order you do not specify.

Have a quick look with the debugger which always makes the math perfectly clear. When you are using the debugger yourself, you can control the pace and stops along the way. Click Here

The Lists Of Life

There are times a list lacking any serious calculation is among the most valued tables you have. Here is a list of bills similar to a longer one I keep for myself. There is a first trail that multiplies the frequency of a bill by a guesstimated average and puts that in a Total column. A second trail adds up that column of row totals. Because there are no calculations specific to a particular row, the data rows can be sorted without creating any issues. This list has a column used only to achieve a favorite order. It is possible to insert cloned rows with details included automatically underneath any row but the first. The first has responsibilities other rows do not. There is a column to mark paid bills or make a note. My note is often the amount of the last bill which usually suggests the guesstimate is old fashioned. Lines are added for easy reading. Below is a video displaying the details behind this simple but practical table.

Various Manipulations Including Cloning Data Rows From Another Row

Loan Payments

I am putting this video here to avoid visitors quickly assuming T/Maker is only for children running lemonaide stands. You could add lines for a host of other factors and seriously compare housing options. There is an example of this included. Basically input values (amount, number of periods, and interest rate) are put into specific memories. Then, based on those values, the MPY function puts the periodic payment into the calculator. Converstions from yearly data to monthly periods are included by default. These correspond to the usual mortgage. But those calculations can be adjusted for other input data and other periods. The second column is an example for a loan with a payment made every two weeks.

Please stick around video showing mortgage / loan payment calculations.

Graphics

While the charts and graphs in T/Maker are not on a par with well-established spreadsheets, you can use them to get insights into your data and a student can learn what type of chart best illustrates a certain point. This is one of a number of areas where T/Maker could be improved by teaming up with a company that already has a professional collection of charts and graphs. The current graph interface could be kept pretty much as is in terms of simplicity. Better to forgo a few capabilities then start to make things complicated.

Forms

With four different line styles plus the possibility of no line, you can design some nice forms. Allow text to span multiple columns when setting formats (Alignment "+" Spans). Text can be specified in an invisible ink format effectively blanking it out. This is useful when elements in a table are needed for calculations but not desired in a final print-out. A respectably looking result is achievable with some experience.

Printing

Print the table in one of four fonts. Choose font style and size. Set a row height to tighten or spread out the vertical dimension. Decide whether or not to print lines. Options are available for row names and column names. Switch between landscape and portrait as well as set margins with two system dialog boxes called up with a button.

The top of the page shows the length and width of your table in pages. In addition to marign adjustments, changing the size of the font and other items may affect these numbers. Use the Recalculate button to refigure the size when you have altered any items. I generally try for a minimum number of pages with the width of only a single page.

Statistics

Algebra is often a prerequisite for a number of subjects. Some teachers may be thinking that a knowledge of it is necessary to calculate and understand the topic. But T/Maker changes that. A single ampersand cursor running down a column can yield a number of measures for the values included. The difference between the mean and median is "must know" material.

Trails have been added here to normalize the data and step up to calculating the correlation coefficients. You can see the temperaturs scales of Fahrenheit and Centigrade produce the maximum possible value on one. A little explanation of what that coefficient impies could be interesting. And the zero correlation (not shown) between the sine and cosine or easier examples (the same curve but out of phase) might suggest suggest the statistic is not as clever as it can appear to be. Interesting oservations are fun. With the pressure off to become an expert, a basic understanding of stuff can be attractive. Geometry and probabilites were introduced above. Imagine a course with a new topic covered every week and where T/Maker sits on every desk to do the supporting math live.

Wouldn't that be cool?

An Important Caveat

In software, this apetizingly sounding word usually implies a logical flaw or inconsistency that can only be solved by something more annoying. It can, however, often be circumnavigated. No one wants two ships passing in the night to arrive at the same spot at the same time and collide with each other. In T/Maker this spot is marked not with an X but with a minus sign. Minus signs (i.e. dashes) in trails are always assumed to indicate an upcoming subtraction. But the symbol itself is capable of prefixing a constant to make that number negative. To know which is which, an underscore character must be used in T/Maker trails when the intention is a prefix. In the white cells, no operators are specified so a minus sign could only be a prefix. See how the different cases above are parsed in the video below. It's all about the value used with the second operator. Is it the adjacent cell or the number following?

BTW, It was the 18th century mathematician, Jean Louis LeFait IV, who first said speaking in French, "You can't have your caveat and eat it too."

A video of which is which

Best Practices

Needless to say, I have evolved my own best practices over time. We will take a look at them. I searched online for a problem to use as an example. Below is the one I chose. It seemed it could happen in real life with the right person, in the right vehicle, during the right season, in the right state.

The Radiator Problem: A car radiator holds 10 quarts and is currently filled with a mixture containing 30% antifreeze. How much must be drained off and replaced with pure (100%) antifreeze to bring the mixture up to 50% antifreeze? This is meant to be algebra with an X but I would start with calculating the resulting percentage from a number of quarts replaced. Going forward from the usual input to a result feels more natural than starting backwards and is instructive in any case.

The Aproach: It is important to lay things out in a sensible space for your problem and the solution. I find it useful, especially with test questions I run across, to spend a minute thinking just about the units involved and their relationships. Here we have quarts of mixtures, quarts of pure antifreeze, and percentages of pure antifreeze in a mixture. A "cars built in a day by Bob" seems weird until you think of it as a speed like MPH. This observation can bring a sense of relief when these units are used on a test.

If there are key values you will control or are often needed, put them near the top. Try to move down the table, left to right, calculating new items in the order they are needed with final results near the bottom. Don't worry about repeating yourself a bit. Notice that the Quarts Replaced will show up twice here -- once in the context of the number we want to enter and again in a section that calculates the new mixture of antifreeze from the two concentrations. These are very different usages justifying different locations. Don't cramp things. Put in the constants, but postpone going to the Trails screen until you have given the layout adequate thought. You can always change it.

Considerations: I would encourage you to do the math following a few principles. 1) Make a sincere and solid effort to avoid cell names. Use memories freely and pick names that are more like short words for a value held rather than abbreviatons which are difficult to remember. 2) Try to avoid putting two steps over the same cell. T/Maker is perfectly capable of handling that while allowing you to change any of the steps involved. But I look at each table as a painting and don't want to paint on top of any part of it. Since I am striving to do this in one trail for purely instructive purposes, I have to be more disciplined and tidy. That challenge can also be a source of satisfaction if you want to try to test what's possible.

Moving On To Trails: To the right is the solution I came up with. Before you get to the end of this page, you will see that T/Maker is a mix of arithmetic, a modelling tool, a programming language, and a scientific calculator! Be sure to always read the steps in their numbered order (1, 2, 3). Learn to grasp immediately if operators refer to memories (= swap, ! total), single letter column names (* b, = c), constants ( / 100, * 100), or just the adjacent cell ( ! + # ). These are the choices that can make a solution prima facia as never seen before. The solution can even feel like a poem in math with the goal of making every calculation obvious. Let me spell out a few of the principles I have learned to incorporate. Remember the debugger can walk through anything operator by operator if you feel uncertain.

For variables you will use in a number of steps, memories as mentioned above are a good choice. The names are not case sensitive, but you can mix case if that helps. BTW, you can go back and insert new steps, delete old ones, or make modifications in any step including changing its step number. Trial and error tuning is always encouraged and promotes learning.

In this example, I have initially chose the Quarts Replaced and Total Quarts to be immediately put in memories. I have also put in two steps (3 & 4) to fill the Quarts column in the two line section devoted to establishing the antifreeze contributed by each component. It is never too early to invoke the debugger. A strategy of a test even after small leaps of progress pays worthy dividends, because it is all in one area and fresh in your mind.

Steps 5 and 6 calculate the amounts of pure antifreeze contributed by each mixture. Sticking to my goal of only one trail, I denoted the locations of a few values by their column letter name and put the complete calculation in one cell. This is an example of how the freedom provided by trails versus formulas pays off nicely. Steps 7 through 10 finish the percentage calculation of the new mixture. A few tries suggested replacing about 3 quarts which was close enough for practical purposes. This philosophy should probably not be used with prescription medicines. Please hang up and dial 911 if you are using this example for that situation right now.

Built In Trial And Error: In the table to the right, I made use of the one above which calculates a new percentage for a number of quarts replaced. I inserted a step at the top to initialize some values and some steps at the end to check the situation and repeat the procedure as necessary.

Here's what's happening. The table starts off with a zero value for the quarts replaced. It then runs through the calculations. I added a line near the end of the table which holds the target percentage. The logic compares the percentage achieved with that target. If the last step moved the needle closer to the target, I added one teaspoon of antifreeze to the amount to be replaced and repeated the calculations again. I doubt I could even do any filling to an accuracy of one teaspoon. Now if instead, I got further away from the target as opposed to closer, I stopped the process. I don't think this is hard to understand for someone familiar with the symbol/value syntax inluding the placement operators # or = followed by the designated location when not the adjacent white cell.

Binary Search: Finally, if I did intend to fill the radiator with a syringe... This final table performs a binary search for a value as accurate as you like. It starts with a high and low estimate, tries calculating with the average of the two, then changes the high or low estimate depending on the result until the difference between the two is less than a specified value. This is called a "binary search" because it cuts the range where the correct value may be in half with each pass. Because the process is stopped at some point, we can avoid Zeno's Paradox.

This technique and the previous one can be used with a variety of problems to find a precise solution. I doubt many would want to do that, but there are a lot of possibilites for adventurers and explorers.

Another use of iterations: Risk / Reward Optimal Roulette Strategy Simulation

The Operators / Functions in the All Syms Toolbar displayed with calculated results

This file and many others are downloadable as examples. You can find them in the download pages.

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Flow control options (i.e., programming) and Newcomers are described elsewhere. A random number generator makes a variety of simulations possible.

Built in functions could be expanded to include those necessary for a variety of professions. Those are what makes Excel so attractive even if it is cumbersome to do your ad hoc calculations.

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These are not ideas that come along every day. This collection represents many years of fiddling with this approach every decade or so for my own purposes while staying true to keeping it as simple as possible. I do not believe everyone has to be capable of doing everything, but everyone should have things they can do. I hope you will not dismiss T/Maker, because it was not the work of a foundation or institute or group of professors or experts. They don't write serious software anyway. It occasionally has to be the little guy who points out the emperor's wardrobe is substandard and does something about it. Remember that frightening Excel example?

Algebra is great for stating a relationship or discovery for others to learn, but a failure to factor polynomials or solve simultaneous equations is no reason to block anyone from being the best doctor, lawyer, or indian chief they can. Many professions require complex calculations doable with T/Maker and without algebra. Very few involve abstract thinking in a formal domain of mathematics. While I will not be here to see it, I hope the next half-century will finally be the end of force-feeding algebra for algebra's sake. Frustrating and demeaning many students promotes anxiety which is the antitheses of critical thinking. Let's instead make sure everyone knows how to do the calculations that make for a better understanding of life's circumstances and healthy DIY projects. T/Maker is unique in offering that possibility to millions who have been written off and ignored.

Since I started this page at my elementary school, I thought I would travel back to that time zone for a song that might have inspired this approach subconsciously. I should get back to work on examples and tutorial files.

Enjoy with a click here!

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