MATHEMATICAL REALISM

MATHEMATICAL REALISM

Numbers are real. They exist. They have a use in the real world. I love starting out with statements that everyone agrees with. But my first academic love was philosophy and philosophers, as Bertrand Russell observed, start with premises that no one in his right mind could deny and end with conclusions that no one in his right mind could believe. Let’s see what happens with the proposition, “Numbers are real.”

Have you ever written a story and thought it was horrible? Sure. We all have. So you change it. This happens in fiction all the time since we’ve been hit with prequels.  You don’t like Captain Kirk being thirty-five? You change the timeline and he’s twenty-two.  Starbuck, from Battlestar Galactica, becomes a woman, Kara Zor-El finds her mother still alive, and Shaft becomes a white guy.

Uh, wait a minute. Shaft becomes a white guy? That can’t happen.  Why not? Because he’s inherently a black guy, has the experiences of a black guy, wouldn’t make sense as a white guy. (Same thing goes– dare I say it? –in spades for Easy Rawlins. James West can be a black guy, but not Easy Rawlins.}

Numbers are like that. Even more so. You can switch the racial context in a book. Ever see “White Man’s Burden,” where America was first occupied by Africans and the Europeans were the underclass? In that society, Shaft and Easy Rawlins could be white guys. But can it be the case that 2 + 2 ≠ 4? No.

This is important.  Numbers are mental, just like fictional characters. They are not things, but ideas. But numbers have inherent, objective properties as if they were things. Just like gold has 79 protons, a^2 +b+2=c^2. And there ain’t nothing you can do about it.

To name some of the objective properties of numbers:

  1. Trichotomy. Exactly one of the following is true: y = x; y < x; y > x.
  2. Transitivity. If x < y, then x + z  <  y +z
  3. Order of Addition: If x < y, then x + z < y +z
  4. Order of Multiplication. If z is positive and x < y, then xz < yz. If z is negative and x < y, then xz >  yz

These are called the order properties of numbers.  They are immutable. Even in a world where black is white, where impossibly young people command starships, and your dead mother is alive as she, per impossible, survived your planet’s destruction, these properties of numbers will continue.

As will these:

Communitive laws: x + y = y + x; xy =yx

Associative Laws: x + (y+z) = (x+y) + z and x(yz) =(xy)z

Distributive law:  x(y+z) =xy +xz

Identity: x +0 =x and x times 1 =x

Inverses x-x=0 and x times 1/x =1

But wait. There’s more.

These properties are true for all places, all times; they are non-local, not situated in time. They are as true in AD 5000 on the planet Mongo as they are on Earth in AD 2019. We know this even though we haven’t observed it.

Numerical relationships, such as those in algebra and geometry, though, are never perfect in the material world.  In calculus, the limit for a series of fractions is not 1, it’s .99999…In the material world, nothing is perfect. As Jesus observed, it’s all moths and rust. Hannibal Smith can make a plan come together perfectly, but you can’t. In the world of numbers, we may screw the numbers up, but the relationships, the real calculations are always precise.

This perfection of numbers led Plato to think that they existed apart from the physical world; that the physical world instantiated them. Numbers were forms; and forms were perfect, i.e., eternal (non-local). This led later philosophers to think that they existed in the mind of God. They were ideas, all right. But their objectivity meant they were not in the mind of beings that were fallible, could change their minds, could screw up and need an eraser. They must be in the mind of someone infallible and who doesn’t make mistakes. That explains their objectivity. Mathematics becomes the divine language, the code by which the universe runs. The world is imperfect, being composed of matter, which is corruptible. The forms are the really real.

So there you are. We have gone from a premise that no one can deny to a bizarre notion that numbers are eternal, they are more genuine, more real than the physical world, that there can be entities that are pure form without matter. It’s just a short leap from that to a belief in angels, folks. St. Thomas Aquinas thought angels were also pure form but limited by their properties. And God was a pure form, but unlimited.

In a later blog, we will address what skeptics think about mathematical realism.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

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Equations

Equations

Algebra is the science of equations. That is, it tells us how to work statements that tells us when something is equal to something else. This is tremendously convenient, especially when you are trying to figure out an unknown quantity. With algebra, you can use a known quantity to get to the unknown quantity.

There are four basic equations in algebra, corresponding to the four operations that you do in arithmetic. Those operations are: addition, subtraction, multiplication, and division.  The four basic equations involve addition, subtraction, multiplication, and division.

They are:

A+B=C

A-B=C

A(B)=C

A/B=C

The way to solve them is to do the opposite of the indicated operation. That is, in an addition equation, you subtract, in a subtraction equation, you add, in a multiplication equation, you divide, and in a division equation you multiply.

This is called the Law of Inverse Operations.

It’s a simple concept to grasp. If you have $10 and I gave you $10 more, you would have $20.  If I were to take that money back, you’d be back to $10. So, if I wanted to figure out how much money you had to begin with, I’d set up the following equation:

A+$10 =$20.

To figure out what A, the original amount, was, I’d subtract $10 from both sides of the equation and I would be left with $10.

Similarly, if someone stole $10 from you, leaving you with $10, I could restore your wealth by adding $10 back.

Multiplication works the same way.

Suppose the eight of us were to invest $8 each. A math wiz would multiply each of our holdings by $8 and come up with a total investment of $64.

If we wanted to know the original amount, we could divide the $64 by 8 and come up with it.

Similarly, if we were to divide $64 by $8 and get $8, but wanted to reverse that, we would multiply by $8.

To solve an equation, we reverse the operation that is indicated by the equation.

To solve an equation, you do to the one side the same as to the other.

You will not be able to solve anything if you only work one side. If X +4 =16, you will never, ever learn what X was by simply subtracting the 4 from the left side. You have to do it to the right to get the answer.

Equations are a form of balance. If you and your friend are on a teeter-totter and President Trump sits on your friend’s side, you will go flying. Unless you have Chris Christy, former governor of New Jersey, sit on your side. In learning algebra, always keep the teeter-tooter in mind.

Now that we got that straight, what do we do when an equation requires more than two operations?

We do them in steps. Do the steps have any order?  Yes, they do. Remember PEMDAS? (Please Excuse My Dear Aunt Sally?) In other words, we do parentheses, exponents, multiplication/division, then addition/subtraction when doing multiple operations.

In solving an equation, we reverse that. Why? Because we want to get back to the original point. So in the following two-step equation:

2X +4 =16

We first reverse the four. To reverse addition, we do subtraction. So we subtract four from both sides.

We get:

2X =12.

Now we reverse the multiplication. We multiplied by 2, so we divide both sides by 2.

We get:

X=6.

What could be simpler?  Well, actually, one-step equations which we discussed at the beginning. Can it get more complicated? Oh, yes.

We’ll deal with that later.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

 

FUNCTIONS

FUNCTIONS. WHAT THEY ARE AND WHY THEY MATTER.

Functions are a crucial concept in mathematics. You get them in algebra, in geometry, in trigonometry. You hardly ever get them at parties, but that would depend on the nature of the party, I would guess. Today, I want to explore some of the properties of a function.

A function consists of a correspondence between a first and second set. Each element in the first set corresponds to exactly one element in the second set.

Take the following:

A 1
B 2
C 3
D 4
E 5

 

This table denotes a function. For every item in the first column, there is a corresponding item in the second.

The first set is called the domain, the second, the range.

The domain is also the x-axis of a set of Cartesian Coordinates; the range is the y-axis.  This means that you can make a graph of a function!

Image result for graphical depiction of a function

These are graphical depictions of functions.

Now, suppose you were to draw a vertical line through one of the lines already on the chart. It would be a function. As the line represents a point on the x-axis and goes through exactly one point on the y-axis. This is called the Vertical Line Test of functions.

The above graph shows the use of functions. If we had two sets of data, we could see correlations.

Say the x-axis represented increased gun violence and the y-axis gun ownership. The first would show a direct correlation between gun violence and gun ownership; the second an inverse correlation.

CNX_Precalc_Figure_01_01_0122.jpg

Notice the vertical lines. The first defines a function, the second and third do not.

coulson_nat_ed_trends_chart_sept_2014.gif

This graph isn’t a purely functional graph. The y-axis is a bit crooked. But it would show a one-on-one correlation between the x-axis and the y-axis. (It also shows that as spending on education goes up, absolutely nothing happens to students’ skills, but you already knew that.)

And here’s a graph that is called a “best fit” graph, as you can’t draw a straight line.

 

mass shootings.jpgHowever, if you did draw a straight line, which is possible on the incident line without too much distortion, you would have a function. Mass shootings were 25 a year in 1980 and about 23 in 2010.

As should be clear by now, functions can be expressed by linear equations.  But instead of:

Y =2x+4

You say instead:

F(x)=2x +4.

We’ll continue this discussion in a future blog.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

 

COIN PROBLEMS

COIN PROBLEMS

 

Word Problems.  Everyone’s favorite. From time to time, I will blog on how to solve them and other algebra problems. Today’s blog will be on coin problems. You know, the ones where you wonder why don’t the people simply look in their pockets and count? But they hone your algebra skills and allow you to understand principles involved in systems of equations and other mathematical procedures.

Here’s a typical one:

Jack and Diane have 28 coins that are nickels and dimes. If the value of the coins is $1.95, how many of each type do they have?

As with many math problems, once you simplify, the answer is a lot easier to come by. We have two variables. Ah, if we only had one.  But alas, we have two.

The first step is to get rid of that pesky fact and define one in terms of the other. This way, we only have one. Don’t worry. We will later bring back the other variable.

The variables will stand for our two coins—nickels and dimes. So our variables are N and D. (Pretty clever, right? I could have picked X and Y, but N and D are so much more representative of nickels and dimes, I think.)

The problem tells us that the number of nickels and dimes adds to 28. So our first equation:

N +D = 28.

The number of nickels, N, and the number of dimes, D equals 28. With me so far?  Good.

But we need another equation to solve our problem. The solution is to define the variables and the result in the equation (in this case, the $1.95) by elements they all have in common.  They are all composed of pennies. Hey, that leads us to:

5N +10D =198.

As there are 5 pennies in each nickel, ten in each dime, and 198 altogether. Beautiful. Now, all we have to do is to define one variable in terms or the other and we are all set.

Let’s define nickels in terms of dimes. We could do it the other way, but I like nickels. But maybe you like dimes. Then, you can do it for dimes.

N + D =28.

We solve for N.

N + D-D=28-D

We add the -D to D on the left, and what happens when you add the additive inverse to a number? You cancel the number. As we are in an equation and have to do to the other side what we do to the side we are working, we get the result 28-D.

Think of an equation as like a teeter-totter. You start with your friend who equals your weight. Then Donald Trump sets on your friend’s side. You go flying. Unless, of course, Chris Christy sits on your side. You are back in balance. All’s well, the world, or at least the equation is in balance.

So, N =28-D.

We substitute that for N in our equation:

5(28-D) +10D =195

We now multiply both terms of the parenthetical expression by 5 and the result is:

140-5D+10D =195

We combine like terms.

140 +5D =195

We now have your classic two-step equation. You solve it in, well, two steps.

Step 1

140-140 +5D =195-140

There’s that teeter-totter again. Maybe a less controversial example would be Nero Wolf and Cannon, two fine but large detectives. In any case, however you imagine it, you get:

5D =55

Step 2

Now, to cancel a multiplication, you do a division. To cancel five times D, you divide by 5. Again, you do it to both sides and you get:

D =11.

And Since N+D =28, and you can now substitute 11 for D, you get:

N+11 =28

You subtract the 11 from both sides, and you get 17

Next time, we might try something really hard, like how to balance your checkbook.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

 

 

 

 

 

IS THERE A CONSTITUTIONAL RIGHT TO AN EDUCATION?

 

After the War between the States, Congress set standards for rebel States, if they were to be readmitted to the Union.  Mind you, they had no choice in being readmitted.  That was settled at Appomattox, the first time (and hopefully, the last time) a constitutional crisis was settled by force and not reason.  One of these standards was to provide public education.

Currently, there are lawsuits to make public education a constitutional right.  The argument goes that Article 4, Section 4 of the Constitution and the 14th Amendment requires this.  One needs to worry when a right already established by law is being held to not exist, as this lawsuit presupposes. The idea appears to be that the States would have to provide a curriculum that adequately prepares students for citizenship.  They are:

Martinez v. Malloy

Filed: Aug. 23, 2016, in the U.S. District Court for the District of Connecticut

Argument: The Constitution guarantees substantial equality of education opportunity; Connecticut’s policies limiting charter schools, magnet schools, and interdistrict transfers violate students’ due process and equal-protection rights.

Status: Judge Alvin W. Thompson dismissed all but one claim in the lawsuit, charging the state with failing to fulfill its duty of public administration. That claim is pending.

Gary B. v. Snyder

Filed: Sept. 23, 2016, in the U.S. District Court for the Eastern District of Michigan

Argument: The Constitution contains an implied right of access to literacy instruction; state policymakers provided Detroit students with such a substandard literacy education that it fell afoul of the students’ due process and equal-protection rights.

Status: Judge Stephen J. Murphy III dismissed the lawsuit. The plaintiffs have appealed to the U.S. Court of Appeals for the 6th Circuit.

A.C. v. Raimondo

Filed: Nov. 29, 2018, in the U.S. District Court for the District of Rhode Island

Argument: The Constitution contains an implied right to an education that prepares young people to be capable citizens, including voting and serving on a jury. Rhode Island’s failure to provide this education violates multiple constitutional rights and a section of the Constitution guaranteeing a “republican form of government.”

Status: Pending

Source: Education Week

 

 

Now, ok, I agree that there should be charter schools, magnet schools, no limits on interdistrict transfers.  All of those policies buttress up the public schools’ monopoly. And indeed, when public schools provide substandard instruction, then there is inequity.  But should we, as a nation, impose these things on States?

I should simply whisper, “Betsy Voss” to progressives who want national education and see them have hysterics.  My point is to question the whole idea of national standards.  Look at “Core Curriculum,” which has math problems that are incomprehensible, even to mathematicians.

You may ask yourself what the question is.  Beats me.

As to federal standards for citizenship?

The Medford, Oregon school district got a bargain of history books that annotated the Constitution.  (Guess which Amendments got annotations that favored one side over the other in interpretation.  You guessed The Second Amendment?  Congratulations.)  The School Board finally decided to use the books (Don’t look a gift horse in the mouth) but decided maybe the actual Constitution should be passed out as a supplement.

You can argue that education is a right.  That “republican institutions” presupposed it.  I suppose, as long as teachers stop indoctrinating their views on the electoral college and the powers of the Presidency.  (It isn’t ‘undemocratic,’ unless you think that 18 counties imposing their will on everybody else is the high point of democracy, and no, the President doesn’t get to subsidize companies or do an end run around the law.) Adam Smith thought public education was essential to capitalism.  But what of parents’ rights?

Shouldn’t you be able to raise your children according to your religion?  Shouldn’t a Christian community have a nativity scene at the local school?  Institutionalized atheism wasn’t what the radical Republicans had in mind when they imposed public education on rebel States.  They wanted literacy and arithmetic.  They wanted you to be able to read the notice that said there was a meeting to discuss public policy—and no content was imposed.  Meetings that took place including union organizing and suffragettes, but also the KKK.

Education may be a right, but “re-education” isn’t.

Dr. Fred

 

 

ENSURING YOUR STUDENT GETS THE BEST EDUCATION POSSIBLE

How do you ensure that your children are getting the best education?  There is no right or conclusive answer to that question.  There are so many factors in education.  But one thing is certain—you should never leave it up to the schools.

Don’t get me wrong.  There are excellent public school systems.  The one in Beverly Hills competes well with any private schools.  Of course, if you’re not lucky enough to live in a rich school district?  You can accept a second class education or consider private schools. But this caveat, do not leave your child’s learning applies to private schools as well.  The thing is, schools have their own agendas.

There was a study done in the ‘eighties’ that argued very well that school boards reflect the needs of their community’s businesses.  And one of the reasons for the rise in public schools was that business realized that it needed a skilled workforce and a public school system would shift the burden and costs of training to the schools.  So, if you live in a mining town, you can expect that the school’s curriculum will reflect that.  Or you can go to a private school, 90% of which are religious; the other 10%, largely military.  So, unless you’re of the school’s faith, or want your child to serve in the military, you may be in trouble.  Again, there is nothing wrong with a religious education or with wanting your child to serve his or her country.

Common core fixed the problem of the schools having their own agenda, right?  No, it standardized and gave a federal agenda to the process of your child’s education.  You may be happy with Betsy Voss and Donald Trump deciding what is best for your child, but you really shouldn’t be.

And consider this.  A survey by the American Federation of Teachers and Badass Teachers Association, released on Monday, shows that teachers are feeling especially stressed and are less enthusiastic about their jobs that before.  And the retention rate among teachers is still abysmal.  So, you want to leave your child’s education to overworked, stressed out, unenthusiastic teachers who will probably end up in another job where they can make gobs more money, right?  Again, there are good teachers, but roll that around in your mind for a while.

So how to ensure that your children are getting the best education?  Here are some tips:

  1. Consider homeschooling. You are in control of your child’s education every step of the way.
  2. Be involved. You may not have the time to homeschool.  Maybe your child is science-bent, and you don’t have the means to provide a science education.  Maybe you like the idea of public schools.  But in that case, you need to attend PTA meetings, consult regularly with your child’s teachers, research the schools, and find out what special help is available if your child has special needs.
  3. Hire a tutor. A tutor is a teacher that is responsible to you, not to a school board, a teacher’s union, or Betsy Voss.
  4. Help your child with his or her studies. If they are beyond you, see tip 3 above.

Above all, make learning fun.  You will be rewarded and so will your kids.

Comprachico Education

You couldn’t get a more motley crew together if you tried:  Paul Goodman, playwright and anarchist, Victor Hugo, novelist, Ayn Rand, philosopher-novelist, and Anthony Flew, philosopher.  But they all had one thing in common:  opposition to the current state of education.  All advocated a radically different approach to how we teach our kids, and it all centered on parents, or the community in lieu of parents, taking responsibility for their kids.

Paul Goodman, in Compulsory Miseducation (1964) argued that the fatal flaw of progressive education was that it always ate its own children.  Each generation of progressives tried to undo what the last generation wanted.  The core problem was in its centralist approach to education–school boards, large classrooms, top down education. This has just gotten worse with a federal dimension to schooling, with standardized tests and teaching to them.  Every individual is unique, according to Goodman, and some might not even need a formal education.  Children should be allowed to explore what they consider to be fruitful avenues of knowledge.  If this is done, then the spark of curiosity, so keen in children, will not go out, as it does by high school.  Goodman believed every stage of formal education had to undo what the earlier stage had done:  you unlearn elementary school in high school, you unlearn high school in college, and you unlearn college in the school of hard knocks.

Ayn Rand likened educators to “comprachicos of the mind.” (1975) Victor Hugo coined the term, and he used it to describe the child buyers of medieval times.  They would condition children for tasks and sell them for that task.  Some would be put into a pot and grow in the perfect shape of a pot.  A later industrial practice did the same.  Child labor was justified in coal mines, as their bones would adapt to the stooped environment that they found themselves in.  Rand believed educators did the same thing–only to the mind.  They would put kids in spiritual pots and they would grow into the grotesques that would demonstrate and yell, but never understand the needs of human life. There have been studies to show that school boards reflect the needs of the corporations or industry of an area and not that of the children.

Anthony Flew, in The Politics of Procrustus (1981) talked about compulsory egalitarianism.  Procrustus was an inn keeper who had but one bed.  If you were too short for the bed, he’d put you on the rack and stretch you; if you were too long, he would cut you down to size.  Flew believed that public institutions did just that, including the schools.  You dumb down the smart; you try to bring to average those who don’t measure up.

What would Goodman, Rand, and Flew think of today’s education–with standardized tests, a return to learning that requires mathematics beyond what the student is comfortable with, and imposes a particular view of history?  In Medford, the school board wrestled with history books that depicted the framers as rich white slave owners, dedicated to their own privileges, and which cut out or amended parts of the Constitution.  They decided to allow the book as long as the actual constitution was passed out and discussed. Both left and right politicians complain about the brainwashing that goes into the schools.  It seems both the first and second amendment, the one giving basic freedoms and the one protecting them, are anathema in a lot of schools.

What is the solution?  Too easy to say homeschooling, that won’t work for many.  Ditto, private education.  What will work is for parents to get actively involved in the educational process–every step of the way.  Only when you can’t teach your child what you need yourself should you involve the schools.  Far better to hire a tutor than to hand your children over to nameless bureaucrats.

Would you sell you children to a carnival?

I CAN’T READ…

In my dojo, when people were learning a new technique, if they said, “I Can’t,” they were made to do push-ups.  This was painful, and was an adverse association with “I Can’t,” and it helped build muscle strength.  Thus the punishment came with positive benefits.  Why punish “I Can’t?”

Because it tells your subconscious that you’re incapable and wires in powerlessness.  It is a form of learned helplessness.  This is behavior that is self-limiting.

What if Trump had believed all the press–“No way can he win.   He will be crushingly defeated.”  “Hillary has the numbers.”  Uh, huh.  Doesn’t matter whether you like Trump or not.  He was the ant moving the rubber tree plant.

Or Helen Keller.  Unable to communicate but became a world renown thinker.  Due to her will and the fact that Ann Sullivan did not give up on her.

When I hear “I Can’t,” I never give up on the student.  It challenges me–and I know “I Can” reach them.

I know this because I’m not just like Ann Sullivan, I’m like Trump and Helen Keller.  I went from a wimp to a Black Belt.  If I can do it, anyone can do it.

I’ve seen how attitude works.  How students who were labeled “uneducable” by their schools came to me and became successful.

So next time you say “I Can’t,” drop and give me twenty.  It will do you good.

 

Word Problems: as easy as getting to work on time.

Word problems terrify beginning algebra students.  Their lament can be heard all across the globe, and it wouldn’t surprise me if they are among the first signals aliens receive from planet Earth.  Students wonder why they have to learn these difficult problems.  Let me tell you then that not only are they important, you probably do them on a daily basis and do not realize it.

For many people, that is like telling them that, on a daily basis, they uproot trees with their bare hands.  Not possible!  Oh, but it is.  You need to get to work on time, but you’re staying over at a friend’s (I won’t ask why.), and you have no idea how long it will take you to get to work.  You look at the clock, you ask your friend how far it is to your employers.’  He or she tells you, “35 miles.”  You’re in a residential area, the speed limit is 25 mph, and you are a law abiding citizen.  You always go the speed limit (heh). You have to be there at 9:00AM.  It is now 7:30AM.  You relax.  You realize you have time.  Now, how did you do that?

Without knowing it, you applied the algebraic formula: D=rt.  (Distance equals rate multiplied by time.)  You knew that the distance was 35 miles, your rate would be 25 mph, and you rearranged the formula, by the rules of algebra, to t=d/r.  You then divided the distance by the rate, and realized that it would take an hour and twenty-four minutes to get then and you had an hour and a half.  Since you’ve done this thousands of times, you did it so lightning fast, you didn’t even know you did it.

You see how easy word problems are?  You’ve done them all your life without knowing it.  You’re like the man in the Moliere story who didn’t realize that he was speaking prose all his life.

Writing prose.  That comes next.  You’ve mastered a word problem, right?  You can do anything.

Go ahead.  Uproot that tree with your bare hands.  I dare you.

 

“I’M BAD AT MATH…”

I bet you’ve heard the following, maybe even said it to yourself. “I’m bad at math;” “I just can’t get it;” “I’ll never be able to get it.” There are so many things wrong with this approach to math, it’s hard to know where to begin. First, it’s negative self-talk; we all know where that gets us. But it also presupposes a static, fixed math ability.
We don’t think this about anything else (or if we do, we need therapy). Imagine saying, “I know I’ll be bad at basketball; I tried it when I was 10 and I was horrible at it.” Well, at 10 you were what? 5 feet tall; now, you’re over 6 feet tall. That could make a difference. Just a little.
Similarly, we go through developmental stages in learning. Piaget thought we couldn’t do abstract learning until adolescence. Yes, about the time our physical skills are flourishing, so are our mental skills. It is no coincidence that you try out for the football team or the cheer leading squad at the same time that you learn Algebra I.
Teachers are beginning to understand this. (Finally.) The idea of growth mindsets is catching on in many schools. This is accompanied by encouraging individual development and evaluating according to where the student starts, and the progress he or she has made from that starting point. My first trophy in target shooting was for “most improved shooter.” At least I had hit the target. Now, I rarely miss the 9 or 10 ring.
The idea that math skills grow makes learning math more like learning a physical skill. Any good coach will tell you that you have to “normalize failure.” In other words, as my kung-fu instructor repeatedly told me, “If something is worth doing, it is worth doing badly at first.” My first day in karate, I put my pants on backwards. Now, I can defeat Black Belts in sparring. Failure is to be expected; it is part of learning; in fact, learning presupposes it. As Socrates observed millennia ago, if you already knew the answer, you wouldn’t bother looking for it.
I read one lesson plan which said that you should love mistakes, as it’s how you learn.
Remember, there was only one perfect human being, and I bet even he had problems in learning math. I mean five loaves of bread and two fish to feed 10,000 people? That would take a miracle. All math learning takes is perseverance and the right help.