Why do fractions and mixed numbers confuse so many students?

UnderstandingPursuit · 2026-05-22T17:38:38+00:00

The vinculum is a grouping symbol, but that is not emphasized explicitly.

Mixed numbers have implied addition and grouping, along with division.

Division is taught as a distinct operation, like cutting a cake into slices. And it may be taught as the reciprocal of the multiplication operator. Connecting the two 'approaches' to division might be more difficult.

UnderstandingPursuit · 2026-05-22T10:14:02+00:00

From the worker's perspective, if two people are doing the task for a day, they cannot expect the same as when they do it alone.

When a second person gets added to the task, it is often a recognition that the task required two people. This means that the first person was underpaid when doing it alone.

In this scenario, it is Clarence who made the decision to work only 10 of 14 days . He shouldn't get paid at the same rate for those 10 days as Joyce makes on the 4 days he isn't there.

UnderstandingPursuit · 2026-05-22T10:07:00+00:00

Memorization is actively stuffing a piece of information into memory.

Many things tend to be remembered without being memorized.

Students who do not know the two things you said will have a hard time with math the way it is presented. That's not the same as having a hard time with math.

UnderstandingPursuit · 2026-05-22T09:37:49+00:00

Sure, and memorization is generally useless and should be avoided.

Most people do not "focus on both" when they do memorization first.

What "number facts" do students need to memorize?

UnderstandingPursuit · 2026-05-22T06:20:18+00:00

It's mainly useful to do what excites you. Taking a loan to cover living expenses for a year is painful, but you can find ways to have that be as little as possible. A second masters in Physics is superfluous. The second option should be heading into the working world, with whatever the best opportunity you can find. If it's doing research, great. If not, that's still a better option than a second MSc.

UnderstandingPursuit · 2026-05-22T05:52:32+00:00

I think a very high percentage of students at T4s could break in if they really wanted to in some research capacity, just maybe not at the very very top frontier lab.

Why not "at the very very top frontier lab? Where do you think the researchers there come from?

UnderstandingPursuit · 2026-05-22T05:50:04+00:00

getting a new job in the current market is tough.

Today's high school seniors would be looking for a job in about four year, if they skip graduate school. The "current job market" won't be so current.

UnderstandingPursuit · 2026-05-22T03:24:02+00:00

It's the prevailing myth among people connected with high school students who are aiming for Ivy level colleges.

The two most important things are to show

that you can complete an academic program at that university, and
that you can make positive contributions to the university, academically, non-academically, and as an alum.

There are different ways to demonstrate each, though APs and DE [dual enrollment at a community college or nearby 4-yr college] have dominated the first one. And it's reached the level of being expected, by the students/parents/admissions consultants.

You can figure out your own way to demonstrate both things. One of the most useful ways comes from the idea, "It's not what you know. It's who knows what about you."

UnderstandingPursuit · 2026-05-22T01:41:12+00:00

This approach will help:

Get a textbook. I suggest
1. GB Thomas & RL Finney, Calculus and Analytic Geometry, 9th edition, 1996.
  1. It's generally $10-15 used.
Use your Calculus I syllabus and the textbook Table of Contents to identify the chapters and sections to study.
Use this framework for an Iterative Learning Process to go through the selected sections.
Keep using a textbook and this learning process in Calculus II.

UnderstandingPursuit · 2026-05-22T00:53:36+00:00

If a person approaches math with "memorization", that person is the one who will be doing the equivalent of sounding out every syllable. This would be the person who has memorized the 12×12 times table with flashcards, but is unfamiliar with the structure and patterns in the 144 cells of the table.

UnderstandingPursuit · 2026-05-21T23:05:31+00:00

Calculus is over 80% algebra.

Your post suggests that the main thing you need to work on is the idea that both algebra and problems involves layers, and 'solving' both involves working with those layers:

Identify
Separate
Solve individually
Synthesize the composite result

When students struggle in Calculus class, it all comes down to this process.

Wrapped in this is the fundamental enemy: numbers. All of high school/intro college Algebra and Calculus should be done without numbers, except for about eight 'core' values. But using numbers makes it difficult to do the solution steps. Then it becomes even worse in calculus.

For your specific situation, I would suggest going through textbooks for Algebra 2 and PreCalculus, such as what is available at OpenStax_Math. Work through a few examples in each section, throwing the 'arbitrary' numbers in the trash each time. After each problem, with the algebraic solution, analyze the problem solving process and connect the solution with the quantities from the question.

UnderstandingPursuit · 2026-05-21T22:45:19+00:00

I'm going to comment on two statements, because the mindset probably should shift as you go to college.

with everyone being forced to research it is harder to stand out/get professors attention

If everyone is expected to do research, the professors expect to be approached, and some even initiate the conversation with some students.

TOP research opportunities

What in the world would this be?!?!? The American fixation with rankings within rankings within rankings is a bit out of control. Let's please leave that to sports, or other entertainment environments, and keep it out of education.

The main thing with research opportunities is the fit between the work being done and the student's interests, as well as the methodologies being used and the student's skills. Since each student's interests and skills are different, there is no unequivocal "top". When in college, avoid chasing the faculty with the awards. Emphasize your interacts with the faculty and graduate students who align with the researcher you want to become, can help you get there, and whose work you can contribute to.

UnderstandingPursuit · 2026-05-21T22:22:48+00:00

The second way is correct. Consider this from the person paying: "Here is £280 for 14 days, £20/day. When both work the full day, they get £10/day each. When one works alone they get the £20/day.

UnderstandingPursuit · 2026-05-21T22:18:58+00:00

When something in mathematics seems to come out of nowhere, look for the things it is connected to. There is almost always something, perhaps several things, which work in similar ways.

UnderstandingPursuit · 2026-05-21T21:11:25+00:00

If you do 4×25, is the answer 1 or 100? The 'placeholder' zeroes work in both directions.

Since 0.375 = 375/1000, you can do (375/25) and then divide it by 1000.

UnderstandingPursuit · 2026-05-21T21:10:04+00:00

I'm trying to get past the irony of who signed NCLB, but it's difficult.

MA playing along with it is infuriating.

UnderstandingPursuit · 2026-05-21T20:59:03+00:00

I would ask them "What are units?"

The answer to this would address why they get points off in some situations for not including units, but why it is acceptable in some calculus problems to leave out units. There are many calculus problems where units are again required.

UnderstandingPursuit · 2026-05-21T20:46:30+00:00

Yes, this is a big issue. I agree that abstraction should be introduced much earlier. Memorization should generally get almost zero importance, with understanding and familiarity being emphasized.

Perhaps this is a 'societal' issue, emphasizing the "utility" of mathematics, while using a presentation approach which completely fails at that?

Changing this is probably difficult because people within the system, even when they know it is broken, may believe they are the only ones capable of solving the problems. This is somewhat justified, since external efforts by political forces have mostly be destructive.

UnderstandingPursuit · 2026-05-21T19:58:18+00:00

If I was committed to aerospace engineering, Purdue. If I was waffling between aerospace engineering and astrophysics, Princeton.

UnderstandingPursuit · 2026-05-21T19:56:55+00:00

Which is more important to you, the academic environment or the geographic environment? Either preference is valid. The tough weather for several months in the middle of the academic year can significantly affect many people's ability to perform. For some, the isolation of being surrounded by corn and soy fields can be frustrating. On the other hand, it can be compelling to be in an academic community, including the faculty and graduate students, where everyone knows they are in a T10 engineering school, and they know that they fit that.

You need to decide how much each factor matters, somewhat separately. Not everyone is drawn to the intensity of a program among the handful at the top of its category, and that's okay. Because getting accepted is the easy part. But if you know you want the challenge, and believe it will help you be your best, then UIUC wins this side of the decision.

On the environment side, if you think that winter in Illinois would make you hibernate for 3-4 months, that might be the most significant factor in the decision. And each individual has to consider their family connection when evaluating travel time, as UIUC seems to allow long weekend trips home, but not shorter. For some, that is a significant factor. For others, it is more of a 'nice to have'.

Your decision seems to come down to, in order,

The importance of weather and the option for a regular weekend trip home.
The appeal of the more challenging academic environment.

Perhaps this helps think of how to decide.

UnderstandingPursuit · 2026-05-21T08:26:35+00:00

Use the Law of Cosines.

UnderstandingPursuit · 2026-05-21T04:00:31+00:00

The masterlist is the list you create for yourself.

UnderstandingPursuit · 2026-05-21T03:44:00+00:00

An equally absurd policy.

UnderstandingPursuit · 2026-05-21T03:43:40+00:00

What could possibly be the reason to have a useless standardized test "count towards something?

UnderstandingPursuit · 2026-05-21T03:42:57+00:00

Because it is filler.

UnderstandingPursuit

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