Email: roizen@ix.netcom.com      References:      https://en.wikipedia.org/wiki/T/Maker      https://en.wikipedia.org/wiki/Peter_Roizen   Links: The Download Page           My Other Idea:  WildWords Crossword Game                    It's  $E$7CMIF#/"  Amazing!   Please pass it on. T/Maker: What do you see here?        This will end up long with details as it should for a serious proposal to turn the math world upside down.    See what you can learn in 100 seconds of a silent video?   With my lttle eye, I spy a classroom at Fairmeadow Elementary School circa mid-fifties. A teacher stands at a blackboard with chalk in their hand teaching arithmetic. An occasional screech causes some wincing. Otherwise the air is devoid of projectiles or wisecracks. Everyone is paying attention. No one doubts the utility of what they are learning. Math is off to a good start. Way back then, there were just two kinds of althletic 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 crunching numbers? What's been or appeared on that shelf? The slide rule had been around for over four centuries. It had an embarassing problem with accuracy. You could multiply 60 by 2 and then by 5 and come up with the answer of 597. Close but no cigar as they say. 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 something called "algebra." That is introduced as a branch of mathematics where numbers hide behind letters and you try to figure them out. Hide and seek but less fun. There are also formulas which require calculating from the middle out as opposed to the top-to-bottom or left-to-right presentation in arithmetic and life in general. Algebra can cause anxiety, but it is also the way scientists advance on new discoveries. Some can't live with it, and the world can't live without it. For those who are not searching for hidden things but just computing with numbers in plain sight, the notation of this algebra is a foreign language. If they want to use a spreadsheet, they must translate the native way they see things to and from an approach designed for research. That is a heavy price to pay to work in two dimensions with a screen, printer, and filing system. No one had to pass a grammer course or win a spelling bee to use a word processor. I would like to eliminate this unfairness. Algebraic spreadsheets? Excel is the name of a Microsoft spreadsheet which is probably the most recocognized one in the world. A search for "Excel with a formula" turned up the image to the right.  An average and a percent column are calculated along with four column totals. There is an arrow that points to the formula for the 10.6 value. Read it and weep but that is a formula. Were it a Rorschach Test, I suspect most people would say it's "Bingo" or swearing. Few would say "that there is a percentage!" I do sort of feel guilty using this bottom of the barrel example. But I also feel like a David battling the Goliaths of institutes and software powerhouses. I simply cannot afford the luxury of professional courtesies. Calculating that in T/Maker? Now let's do the equivalent of Excel in T/Maker. We will use three "calculation trails" to compute the 14 derived values.  When you read a bit more, you will get a full explanation. But with even a quick look at these trails you can probably guess what is going on. T/Maker works like a hand calculator. For a little clarification the exclamation point ! enters a value into the calculator and the # symbol outputs that value without clearing the calculator to zero. An equal sign = would have set the calculator back to zero. Trail 1 to the right calculates the totals of columns C and D as it might be written on a blackboard. Trail 2 below calculates Averages from the Total and the Count. Trail 3 calculates the Percent of the Grand Total (D7) for all car models. The slightly grayed green cells show the locations where the trail is used or reused starting with Step 1 at that spot. It's very in your face. These instructions were all clicked or dragged in quickly with the mouse. The keyboard was not touched. With Excel, algebraic formulas live in pigeon holes. Each calculates one value. The result of a formula is displayed over the front of the pigeon hole as are data and text. You can pull out an equation to check it or help you remember what it did. This sounds reasonable. But it is very tedious and often results in mistakes. The values in  a spreadsheet are referred to by a column letter and row number. 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. The ability to lay out the trail of operations in executable order while seeing the data at the same time is a whopping game changer. You don't have to remember what was in B7 or M11 on your way to a parenthetical formula that goes in D22. T/Maker lets you do many calculations without a single sorry cell name. It's just a click to flip between the trails and refresh your memory. Though most anything can be calculated, the "T" in "T/Maker" stands for "Table." It does a nifty job of getting those correctly specified with a minimum of intellectual stress. I hope you see the intuitively visual advantage in the above example.  I personally have no fear of algebraic notation, cosines, or logarithms. But I prefer to do things in the most convenient and straightforward way. For what I need that is not with Excel or spreadsheets like it. I also don't put on my shoes while standing on my head. BTW, I am not suggesting JP Morgan use T/Maker for their hedge funds. Actually I would not suggest they use Excel either. But you or your self-employed neighbor could make some nice invoices and estimates and satisfy many other typical life needs. Within T/Maker lives a dedicated ghost. He has a brother who is famous in the gaming industry. Both brothers have voracious appetites. The profile of the famous one shows no other facial features but a mouth. It dines continuously on nothing but dots. The ghost inside T/Maker dines on mathmetical symbols and numbers. As he eats his way through a calculation, he keeps a calculator  window always filled the current value. People call him The Enforcer or, more commonly, The Debugger. But his given name at birth was more like his brother. Thank you, Pac-Math, for all the mistakes you have pointed out and helped to resolve. Watch him gobble with a click here

I don't want to harp on the past, but whatever the good work anyone is doing, the at-large result is a decline in numeric abilities one year after another. I think it's amazing when 12-year-olds enter Ivy League universities each fall. More power to them if they really want to do that, but what about the millions more who can't compare housing options in their own lives. I worry about them. They say "If the mountain won't go to Mohammed, then Mohammed must go to the mountain." But must he? Can we not do something better than a distant mountain that calls for ropes, crampons, and ice axes to climb. Post graduation there are two basic outcomes. Some will be skilled enough to work with an algebraic spreadsheet. And the others will have nothing more that a pencil, paper, and a hand calulator. I believe I have a step in the right direction. So, why me? How did I get here? Why am I suggesting this thing that no one else has imagined? That's a long story that was six paragrahs and three pictures which I wrote but have deleted for the sake of focus and brevity.

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 fairytale of magnanamous proportions. This has been a journey up creeks, behind 8 balls, and inside pickles. Here and there, over more than four decades 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 this can all work, until I have worked it all out." The table below probably includes all the geometry many people will need to use in their DIY projects. The perimeter, area, and volume are calculated for a rectangle, a cylinder, and a triangle. Every thing is done in one trail and all values and calculations are displayed 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. It's a rather amazing presentation. I would not attempt this alone in your own home with another spreadsheet! It might be depressing. Remember: + [white cell],  +3 [constant],  +B [column named],  +C7 [cell named], +ASquared [your memory name]. Mistakes are generally less frequent and easier to find than in a spreadsheet where much of the information is hidden. Even so, Pac-Math, who was introduced above is available 24/7 to help out. His presence 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, he will take a walkabout through the calculations. Use the Pause and Play as you wish. He generally jogs in the videos. Especially interesting to watch are the values changing in his calculator window and just to the right the shrinking streams of symbols and numbers as he devours them. The Debugger At Work With Above        Velocity Required For Various Orbits         DIY: Cement For Holes With Poles        Liberation Day Tariffs

The Lists Of Life Sometimes a list lacking any serious calculation is among the most valued tables you have. Here is a list of bills similar to a realistic 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. The note is often the amount of the last bill which now suggests the guesstimate is old fashioned. Since this list ends up on paper, lines are added for easy reading. Below is a video displaying the details behind this table. The Details Of The List Shown

Best Practices Not suprisingly, over the years I have evolved some best practices. We will take a look at them. I searched online for a problem to use as an example. Below is the one I chose for no special reason. 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?  In theory this should be a problem solved by algebra, but an answer can be gotten with some trial and error. It is always easier to work forward from input numbers to a result than backwards from a result to an input number. When you wiggle your car into a tight spot, that does not mean you can back out of it. The Aproach:The first step and the most important is to lay out the space for your problem and the solution on the Data Screen. If there are important or often needed variables, put them near the top. Try to move down the table 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 feel completely comfortable with your plan of action. I was comfortable with the image to the right. I find designing the screen layout as you are writing trails works only for simple problems. Some Considerations: I would encourage you to do the math following a few principles. 1) Make a sincere and solid effort to avoid cell names.  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 the painting of a landscape. You would not paint a lake over a group of trees. Since I am striving to do this in one trail for instructive puposes, I have to be more disciplined and tidy. For my own sake, I would probably use a couple of trails on different screens. There is, however, a fun element to this gizmo that can lead you to try for perfection now and then. Moving On To Trails: This second image displays a number of good intentions. Let me spell them out in order. For variables you will use in a number of steps, you want to get them into memories with easy to remember names. The names are not case sensitive, but you can mix case if that helps you in some way. BTW, you can go back and insert new steps, delete old ones, or make modifications in any step including changing its step number. In this example, I have initially chosen the Quarts Replaced and Total Quarts to be immediately put in memories. I have also put in two steps to fill the Quarts column in the two line section devoted to establishing the antifreeze contributed by each component. Then I asked The Debugger to test what I had specified so far. A strategy of a test even after small leaps of progress pays worthy dividends because it is so fresh. It just takes some seconds. More Steps:   In this next screen shot, the other steps are added.  Steps 5 and 6 calculate the amounts of pure antifreeze contributed by each mixture. Sticking to my goal of only one trail,  I denoted values or locations 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. I would grade this piece of work an A. Trial and error suggested replacing 3 quarts 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 such a situation right now. Extra Credit: Extra credit was attempted here. You can see a list of resulting percentages for various replacement quarts noted at the bottom. That was turned into a simple graph with a few clicks. I cannot say why, but I honestly did not expect the line to be perfectly straight. When stuff is easy to do, you do it more often. And sometimes you learn a thing or two from that. The PhD Program: If you want to go directly to graduate studies in T/Maker, creating tables like the one below would be a good start. The middle of this table contains what has been accomplished to this point. A value for the quarts to replace yields the new percentage of antifreeze. Because no cell names are used, this table can be moved around in a number of ways. It can even survive inserting and deleting rows and columns as long as no rows or columns with steps are deleted. Column names are adjusted accordingly. Cell names (E12), however, are never adjusted which is why they can be used as immovable references in a trail where steps are shifted to different starting locations. BTW, the Trail Steps box which you have not seen before is a convenient way to view all the steps in a trail in one swoop. Next we are going to do a form of what is called a binary search. On row 10 I have added a cell for a target percentage. This was 50 in our previous effort but could change for another radiator, season, or owner. At the top I added cells for guessing replacement amounts. One amount should be too little for our target while the other should be too much. These were entered as 0 and 10 because zero would leave things at the 30% we have now and 10 would replace every quart to arrive at 100%. This "programmed algorithm" works by running through the table multiple times. The first step counts the number of loops and stops the calculating after 100 loops. The point is too protect against a mistake in the logic which might cause it to calculate forever. During the very first loop the low value and high value specified are placed in memories. That's steps 2 and 3. The last new step at the top takes the average of those two and puts it in the cell for the Quarts Replaced. In the previous life of this table, that value was entered by us. What we already had specified then takes over to complete the final-percent cell. When that number is known we have some new work to do. If the new value is lower than the target, we replace the "low" value in memory with the number of quarts just used in the calculation. If the new value is higher than the target, we replace the "high" value with the one just used in the calculation. Finally the latest answer is checked against the target. If it is within 0.001 (abs sets the calculator to the absolute value), the calculating stops. Otherwise, the table is restarted from the top with a new Quarts Replace that is the average using the modified high or low. The reason the word binary gets attached to this procedure is because at each pass, the possible range for the correct answer is cut in half. Luckily the process can stop when we just get very close thus avoiding Zeno's Paradox. As sophisticated as the solution is, when you look at it step by step most everyone would understand it. T/Maker's way just seems to have more in common with a recipe to bake a cake or make a stew than it does with the raw algebraic approach of manipulated numbers and letters in a hardcore formula. T/Maker leaves room for some aspect of a personality to emerge in a piece of work. If you would like to watch the Debugger move through a few cycles use the link below. To avoid a snail's pace, I will use the Next Step button instead of the Next Op button. Again you can pause and play as you wish. A Visit With The Above And The Debugger          Risk / Reward Optimal Roulette Strategy Simulation (Also PhD Material)

A Break For Both Of Us That last piece of work was exhausting! Let's take an exit from the Mathway Express onto a side road for some supplemental scenery. Like many types of software, T/Maker has a box/tool for finding and replacing text in a table. As usual, a feature that would be useful in my own household seems never to be found in such a box. That feature would be to store find and replace commands in the file. In the case at hand, it would be able to replace abbreviations with their full spellings for frequently included line items in an invoice   When a new invoice is needed the previous one with the stored replacements could be loaded and renamed, or a custom template could be made to use at the start. A Video Of The Find/Replace Box In Action

A 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 number to make that number negative. To know which is which, an underscore character is used when the intention is a prefix. See how the different cases 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

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?

When and Tally To do a demonstration here I typed in some names and used the random number generator to get a number between 50 and 95 for each name. A When-clause is a logical test with two values. If the test is in a step and true, executing the following instructions in that step continues normally. If the test is false, other instructions in the step are skipped over. T/Maker proeeds to the next step. The instruction after a when-clause might, however, be for T/Maker to start doing something else as with RESTART or EXTRAIL. The when-clause offers opportunities to treat different circumstances with different operations. The screen shown can be obtained by requesting it in the Trail Helpers menu. You use this helper by proceeding left to right through the three list boxes. First you pick the type of value you want to start with, then the test requested, and then the second value to use in the test. As you do this, a corresponding when clause is built in the edit field below. T/Maker does not build the exact test. Rather it builds a model of it based on what you indicated. In the example shown a value from another column was requested along with a less than operator and a constant number. T/Maker offered up a single letter column name for the first value and a place in the line to put the number you want. If you wanted colunn C instead of B and your number was 101, you would edit the proposed line accordingly. When that is done you can push a button to have the script put in the clipboard and attached to the cursor. Now you can click or paste in one of more steps. BUT......!  You can instead push the Tally Table button. If you do that T/Maker will run through your rows can count the number of lines for which the test is true. It will also highlight those lines. Where you go from there is your business. Because the script is not associated with a cell in this type of usage, you will end up using the cell's column name to designate the value needed. In the video below, T/Maker will determine the number of rows with a value less than 60 and the number of names  When-clauses put in steps can do a lot more but this is lickety split is you have a sudden curiousity. There is a menu item in the Misc menu to clear out the highlights. A video of using Tally twice

What the heck! Why not? We started with my elementary school days.
Let's revisit them. It twas a very different time back then.
Could this performance be the seed that grew into T/Maker's terminology?
Enjoy with a click here!

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