Teaching HFA - executive function limitations

[CliffStamp]

Show a problem with an example of what constitutes a correct answer.

They have solution sets.

I have a lot of experience with kids with various language issues, so generally I have found showing them what is needed works better than telling them, especially if they have multiple examples to review. They also have rubrics which will state things like :

• one point for a fully detailed free body diagram showing ALL forces
• one point for proper decomposition of all forces
• one point for F=ma properly evaluated in x and y
• two points for solving the two equations simultaneously

(a sample solution is also marked according to this rubric, which circles the relevant part and notes 1, 1, 1, 2)

I normally avoid jargon, but they do entire units on "simultaneous equations" so it is a term they are well familiar with. Now to be clear I do vocalize it, keeping it simple, each sentence is one requirement. I normally also try anagrams for some kids who enjoy it, such as :

FDFS (Free body diagram, Decomposition, F=ma, Solve)

Frogs Don't Fly Silly.

So they can remember each of the four steps they have to do to get full marks. If you don't do these, even if you get the answer right you will NOT get full credit on the exam. If you leave enough of them out you will fail the question even if you have the right answer. If you right down nothing but the final answer at most you will get is 1 and you could get 0.

...if they can't adapt then that is not an area in which they would be able to excel.

Yeah, but they want to, I can't stand kids wanting to do something and being told they can't. There has to be something I can do or say to convince them that sometimes you have to adapt even if the requirement seems (or actually is) unfair.

 

[Bella Pines]

I tried to give some background information in the introductory post, but it seems my intended point was lost in the process, I will summarize. I have two concerns :

1. they will attempt full mental solutions of problems and only write down final answers

Hmm, okay for for this one.

As an HFA, I spend a lot of time in my head, having full conversations with myself, working through every permutation, extending forward to each possibility. I only ever relay my final conclusion to anyone for many reasons. Examples include:

1. Short term memory deficit. I often won't recall the steps that led me to the conclusion
2. Permutations. I consider every possibility, examine every fact from first principles, it's difficult and time consuming to write all that down.
3. Communication. Sorting through the jumble of an HFA mind is a task unto itself, I often need hours of silence to do so.
4. Loss of detail. I don't really store the facts or the steps, what I hold in my mind are many facets of insignificant information that form a pattern. It's the emergent pattern that I can recall and relay, not how I formed it.
5. Insignificant facts. When I was studying physics, I recalled Newton's laws, memorized the formulae but also recalled that he was born on Christmas day and remember all the schools he attended. However, I would forget to put my name on the front of the exam paper. It's difficult to pull out the "working" and logical steps from that jumble.
6. Aversion to process. I can't follow more than 2 instructions and I can't repeat a process more than twice without messing it up. I hate being boxed in with structure. I would much rather understand general relativity and approximate it down to Newtonian mechanics than I would follow steps to answer a question.
7. Concentration. I can hyperfocus if a subject interests me, but I can't concentrate lightly on anything. Maybe more "won't". But I either do it to death or not at all.

1. if they can't see where to start, they will not do anything, they won't explore

This is more difficult. Perhaps this is the hyperfocus part where we are all or nothing? I do perceive the world in black and white, I either do something really well or don't even try. I also work in logic, not fuzzy or subjective. If there is no clear path then I won't make one up, if there is no question then I won't answer.

Maybe for this one, see the text like those 3d diagrams. You know, the ones you stare at until a picture pops out. If an HFA looks at a block of text, patterns will appear. Repeating words will jump out. So get them to recognize keywords like "AREA" or "PERIMETER". Once the patterns pop out, they have their hook.

formal structure alone :

1. draw a diagram
2. write down the unknowns
3. identify what is being asked
4. do all unit conversions

Yeah, I really don't like the phrase "formal structure" so I'm not even going to try to answer this one

 

[AloneNotLonely]

Are you absolutely certain they understand? Even if you explain "Oh you do this and this" it might not be understood exactly. My math teacher told me to show the steps I used in solving the problem, and even when I did it was still wrong because I have my own unique way of writing it down.

Yeah, but they want to, I can't stand kids wanting to do something and being told they can't. There has to be something I can do or say to convince them that sometimes you have to adapt even if the requirement seems (or actually is) unfair.

Well tell them what you told us. That when they will start working in a lab that they will have to write exactly what methods they used and what the results are because in science other people need to be able to reproduce results. It's not about the result, it's about the method. If they can't do that on a test then they won't be able to do that kind of work.

You can't help everyone, so if explaining in 10 different way doesn't work stop wasting your time and focus on the ones that are capable of learning it. But it is absolutely important that there is no "uncertainty" in what they need to do and what syntax needs to be used. They might just have no idea where to start, even if it's obvious to you. If they have no idea where to start, they won't "try". So don't ask them to "try". Ask them to execute the assignment exactly according to the way you show first by example. So the goal is not to solve the equation but to show the method used to solve it.

 

[CliffStamp]

Well tell them what you told us.

I have.

They might just have no idea where to start, even if it's obvious to you.

The basic procedure I stress is :

1. write down what you are given
2. draw a diagram
3. label it
4. write down what you are asked to determine
5. ask yourself what can I calculate (not how to I get the answer)
6. repeat step two until you get the answer required

Even if you can't do 5/6, just doing the first four can often get you a large percentage of the grade. If you only could fully solve 25% of the problems and just did 1-4 on the rest you would likely still pass.

If they have no idea where to start, they won't "try". So don't ask them to "try". Ask them to execute the assignment exactly according to the way you show first by example.

I have fully worked out examples with them, usually color coded, (knowns, unknowns, solution, verification) for them to follow, including a graded rubric.

So the goal is not to solve the equation but to show the method used to solve it.

Yes, they are aware of that. If you just ask them for example on the problems they will do what mark they will get they will say openly they won't get full credit because they didn't show where it came from. They just believe that is unnecessary and unfair so they won't do it.

 

[CliffStamp]

I only ever relay my final conclusion to anyone for many reasons.

I appreciate the detail, but the problem I face is that they simply have to do it. It doesn't matter if they even get the right answer, they will fail every question where they just write down the answers even if they are correct (plus since they are not perfect, some of them will be wrong just due to minor calculation mistakes). I need to find some way to move past these.

On a side note, one of them likes to play DnD, but he gets kicked out of almost every group for the same reasons. In DnD you can do long chains sometimes of combinations of activities to produce some result. But he won't actually spell out what he does, just the final result and that frustrates everyone else as they can't understand how he did so much damage, or avoided getting killed, etc. .

This isn't really a specific issue to math/physics, that is just where I am seeing it, it is more of a general behavioral issue.

Maybe for this one, see the text like those 3d diagrams. You know, the ones you stare at until a picture pops out. If an HFA looks at a block of text, patterns will appear. Repeating words will jump out. So get them to recognize keywords like "AREA" or "PERIMETER". Once the patterns pop out, they have their hook.

In general they will understand the concept. Like for example they would say "It is refraction, you have to use snells law." But you can make a question so layered it isn't obvious how to do that. Multiple materials, multiple equations, conversions, etc. . They will attempt it, try to hold it all mentally, start to unravel, get frustrated and then stop.

Yeah, I really don't like the phrase "formal structure" so I'm not even going to try to answer this one

How do yo work in a situation where it is required? For example my first lab job there was a very rigerous structure, you had to log in/log out, record every activity and time stamp, every calibration was a step wise procedure you had to use checklists, etc. . I hated all of that, it seemed pointless but if I didn't do it then I didn't work.

 

[Tom]

Cliff teaching special education is a very long learning process. Trying to use the same methods as the general population are not quite doomed, but close. Every spectrum kid is unique in certain senses and similar to others on the spectrum in others. One can waste a lot of time trying to reinvent the wheel. Your best first contact is the special education teacher that manages the child (if such a thing exsists). All the staff working with a student being on the same page is very beneficial and the teacher(s) and mental health professionals involved probably already have a plan of action (written down in the IEP).

 

[CliffStamp]

Your best first contact is the special education teacher that manages the child (if such a thing exsists).

There isn't one, they are simply in a regular classroom with every other kid.

 

[CliffStamp]

Your example seems almost a classic example of "Cognitive Rigidity". Where your goal as the instructor is "Cognitive Flexibility".

Thanks, I have read the introductory literature and have set up alerts for current research. I am focusing now on programs which enhance cognitive flexibility and see which, if any, have actual empirical support. I get the feeling however, that this, if it is to work, is likely to take an extended period of time. I think now I might actually be better off trying to get them to find some way to pass, then maybe look for a summer program to actually see if I can address the behavioral issues.

 

[Bella Pines]

In general they will understand the concept. Like for example they would say "It is refraction, you have to use snells law." But you can make a question so layered it isn't obvious how to do that.

But it is.

At the end of the day, examiners are limited. There are only so many types of exam questions and one way of approaching this problem is to simply give them every permutation of question and watch them form their own techniques. Even the most layered question can be easily broken down.

How do yo work in a situation where it is required?

It's required everywhere, from paying household bills, to buying groceries and pretty much everywhere at work.

Thinking about it, I compensate in different ways depending on the cause. For example:

• Short term memory deficit. I often won't recall the steps that led me to the conclusion

Get them to write down notes as they go along. I tell my developers to use twiki/ wiki or sharepoint as they go along in real time. Think, write, think, write, think, write. Even if it is just a bullet point or a word. They need to learn to record a problem solving journal.

• Permutations. I consider every possibility, examine every fact from first principles, it's difficult and time consuming to write all that down.

Get them to practice focus. With the area problem, you can extend vertically or horizontally. But solving every permutation is time consuming and unnecessary. Get them to make a choice or assumption early on and throw away the other permutations.

• Communication. Sorting through the jumble of an HFA mind is a task unto itself, I often need hours of silence to do so.

Get them to learn some tricks. Like starting a sentence with "my assumption is..." and listing the techniques at the beginning, like "Snell's law, angles, refraction".

You are on the right track with your process, it's the only way to learn exam techniques, "draw a diagram etc". But the process needs to account for the other HFA issues like the short term memory problem.

• Loss of detail. I don't really store the facts or the steps, what I hold in my mind are many facets of insignificant information that form a pattern. It's the emergent pattern that I can recall and relay, not how I formed it.

For those who don't respond to the other points, who steadfastly refuse to write things down as they go along. Get them to work backwards. Write down the answer and then the law that they used to arrive there. Then get them to double check and work forwards, that will be where they will pick up any flawed logic.

• Insignificant facts. When I was studying physics, I recalled Newton's laws, memorized the formulae but also recalled that he was born on Christmas day and remember all the schools he attended. However, I would forget to put my name on the front of the exam paper. It's difficult to pull out the "working" and logical steps from that jumble.

Get them to practice of every type of question. Once they know the playing field, they will understand what is significant and what can be thrown out.

• Aversion to process. I can't follow more than 2 instructions and I can't repeat a process more than twice without messing it up. I hate being boxed in with structure. I would much rather understand general relativity and approximate it down to Newtonian mechanics than I would follow steps to answer a question.

The "draw a diagram" process is too narrow. There's no consideration for understanding first principles.

It's heading in the right direction "draw a diagram, label the diagram". But it's lacking context. For example, my first instinct on the garden problem was to draw a diagram. But if you told me to draw a diagram it would be lost. I drew a diagram because it helped me visualize the problem and comprehend how to arrive at the solution.

So instead, try something like this:

1. write down what you are given X - focus on the answer (we are goal oriented)
2. draw a diagram X - graphically describe the problem (draw the garden, highlight the bit that is being asked for)
3. label it
4. write down what you are asked to determine X - THEN write down what you are given
5. ask yourself what can I calculate (not how to I get the answer) - Throw away all information not directly related to the answer.
6. repeat step two until you get the answer required X - get to the answer.

• Concentration. I can hyperfocus if a subject interests me, but I can't concentrate lightly on anything. Maybe more "won't". But I either do it to death or not at all.

Inject real world applications. They might want to extend their gardens one day. They will concentrate if they want to, explain why.

One of the best quotes I've read is "if you can't change your surroundings, then change yourself".

You see how neurotypicals approach a problem and are trying to apply it to HFAs. Imagine asking someone to paint a fence. You tell the NT, buy paint, buy a paint brush, dip brush in paint and paint the fence. They are away and will repeat the process flawlessly for every fence panel. You tell an HFA to do the same thing and they won't, not even close. At best you'll get a colored splodge somewhere in the garden. However, if you convince the HFA that the fence will rot without a protective layer and that the garden will be brighter for multicolored panels, then they will produce detailed schematics and an award winning garden. (and in my case, pay someone to do the actual painting)

 

[CliffStamp]

You’ve explained this to the two, in the same way you’ve explained it on here to me/us?

Yes, normally relating to one of their personal interests vs appealing to abstract. For example you have to keep score in video games to know who has won (to get the awards). If you didn't keep the score then even if you won you would not get the awards because you just can't say you got the most points (or whatever) you have to actually have the recorded score (and all the details, who you killed or whatever).

How would you teach me to prove to an examiner I knew what just happened?

It depends on the person, but normally I would find out an interest you have where this has to be part of it. If you drive, I would discuss the driving test which demands in great detail that you show your competence in all aspects and why this is necessary (what would happen if you just gave out a pass to anyone who just claimed they could drive).

Once I am confident that you understand the concept, then I would show an introductory problem and the full solution graded as an example. With a few examples of this I would then do up a solution which was not fully completed and then ask you to grade it using the examples to see if you understood. If you didn't, then I would back track until I found out the issue and start over until you were successful.

Some people don't respond well to analogies, but do better with lists so in that case I would present a fully detailed list (step one do this, step two do this, etc.). I have versions with a checklist and a summary of the grades as some people really like filling these out and adding up the numbers. It depends on the kids and how they process information and their interest.

Some kids are just very vocal and can respond well to just talking and they don't like reading and going through examples, so with them I would do the former and not the latter. Whatever works for the individual you have to be flexible as everyone learns differently.

 

[CliffStamp]

There are only so many types of exam questions and one way of approaching this problem is to simply give them every permutation of question and watch them form their own techniques.

Well the options are not infinite, but they are extremely large, even if you consider just basic questions. There are about two dozen geometry properties they are expected to be able to utilize and they can see any combinations of triangles, circles, rectangles, etc. in some shape with a bunch of angles/lines and they have to be able to sort it out.

The normal approach would be something like :

• take the complicated shape and break it into sub-shapes
• for each of those can you fill in any details using what you know
• repeat this until you can get to the desired answer

Similar to kinematics, some of the more involved problems you can only know where to start by just asking yourself what can I calculate and that isn't easy unless you have a fully detailed chart to let you know what you have to work with at any given point.

Get them to write down notes as they go along.

I appreciate your suggestions, but the problem is I can't actually find a way to get them to do any of those things, all of which would of course help them.

For those who don't respond to the other points, who steadfastly refuse to write things down as they go along. Get them to work backwards. Write down the answer and then the law that they used to arrive there. Then get them to double check and work forwards, that will be where they will pick up any flawed logic.

This again would be excellent, if I could get them to actually do it. But they will only recount it verbally, not on paper.

However, if you convince the HFA that the fence will rot without a protective layer and that the garden will be brighter for multicolored panels, then they will produce detailed schematics and an award winning garden. (and in my case, pay someone to do the actual painting)

They are aware it is important, consequence wise, they just think it is unfair/unreasonable. Imagine for example if I told you that any report you give me, each letter has to be color coded to a specific pallet. That seems to be a very unreasonable request and it is likely if I kept asking you to do things like that at some point you would just refuse. They seem to take even small suggestions like :

• just write the formula you are using down exactly as it is on the formula sheet

in the same vein, hence won't do it. They have two viewpoints :

• if they can do the question without it then it isn't necessary and is unfair to request them to write out all the steps
• if they can't do the question mentally the question is unfair and it is unreasonable to ask them to do the question

 

[Gracey]

Yes, normally relating to one of their personal interests vs appealing to abstract. For example you have to keep score in video games to know who has won (to get the awards). If you didn't keep the score then even if you won you would not get the awards because you just can't say you got the most points (or whatever) you have to actually have the recorded score (and all the details, who you killed or whatever).



It depends on the person, but normally I would find out an interest you have where this has to be part of it. If you drive, I would discuss the driving test which demands in great detail that you show your competence in all aspects and why this is necessary (what would happen if you just gave out a pass to anyone who just claimed they could drive).

Once I am confident that you understand the concept, then I would show an introductory problem and the full solution graded as an example. With a few examples of this I would then do up a solution which was not fully completed and then ask you to grade it using the examples to see if you understood. If you didn't, then I would back track until I found out the issue and start over until you were successful.

Some people don't respond well to analogies, but do better with lists so in that case I would present a fully detailed list (step one do this, step two do this, etc.). I have versions with a checklist and a summary of the grades as some people really like filling these out and adding up the numbers. It depends on the kids and how they process information and their interest.

Some kids are just very vocal and can respond well to just talking and they don't like reading and going through examples, so with them I would do the former and not the latter. Whatever works for the individual you have to be flexible as everyone learns differently.

Thanks Cliff
I was being literal.
Your guinea pig.
Your practice brain so you might refine and refine and refine again until we’d solved it.

“How would you teach me...?”
Although you have answered the question in a round about way.


Edited to add,
In my understanding it wouldn’t really matter if I was Einstein or six years old,
All teachers usually want to see ‘working out’ written down.
Same principle at many levels of maths.

I loved @BellaPines suggestion of working backwards.
Start with the solution and a few clues.

 

[CliffStamp]

In my understanding it wouldn’t really matter if I was Einstein or six years old,
All teachers usually want to see ‘working out’ written down.

Unfortunately here, a LOT of grading switched over to multiple choice as they can be corrected by a computer. Most kids thus end up with the habit of not writing things down as there is no credit for it and they think it saves them time. This then becomes a habit which can be very hard to break as they have had years of doing it.

Then in highschool the demand for detail really starts to come in, and a lot of those kids suddenly who were A students can drastically fall off, even fail as even the multiple choice become so complicated they can't be done mentally. If they can't move past that habit, a lot of the information is just impossible to master. But normally I can get them to move into it gradually.

As for a personal explanation, I would start off by just asking you to explain something to me. Normally we explain things the way we understand it, so I would first just mirror back to you how you explained something to me. From that point I would just keep trying different things until I found out what worked.

I had one kid who really struggled with Math until I noted they loved geometry. I could then teach them anything by just making shapes involved as they really easily remember and manipulated shapes.

 

[Bella Pines]

They are aware it is important, consequence wise, they just think it is unfair/unreasonable.

I'd probably agree with them

So how about trying this;

Muscle memory. Everyone has an autopilot, even HFAs. I happen to think most hygiene and beautification exercises are unfair and unreasonable. Wearing high heals is a safety hazard, applying make-up is ludicrous and don't get me started on armpit shaving or eyebrow plucking. However, I do (most of) these activities naturally by muscle memory, my brain is disengaged or applied elsewhere and I often don't remember doing them.

So think of a task whereby they are forced to do what you want them to do, have them repeat it until it becomes muscle memory and they do it without engaging the reasoning part of their brain. The reasoning part will tell them it is unnecessary and unreasonable, so you need to bypass it.

For example;

• A treasure hunt, get them to draw a path and pictures on a map as they go along.
• Set long multiplication, division type questions, they have no choice but to write down their working.
• Make them draw a stick man for every question and the funniest cartoon wins. Sure the stick man is nothing to do with the test, but it gets them in the habit of scribbling on the exam paper.
• Give the the choice, they either write the formula down or they colour in the entire working area. Maybe a boring alternative will make writing down the formula seem more palatable.
• Get them to "illustrate" the question. Like draw a light bulb for an ohm's law question.
• Tell them the examiners are idiots and need them to explain how they arrived at the answer.
• Get them fun pencils that they want to use.

The chances are they will be able to memorize the formulae easily, so it's really an exercise in communication and articulating their thoughts as they go along.

 

[music10s]

I'll confess I haven't read everything on here and I am totally new to this forum, but I wanted to share my experience as it is pretty relevant. I am self diagnosed HFA, but didn't come to that conclusion until my final years in college, where I majored in math and physics. I am currently just a couple months from defending my dissertation in applied physics.

I have distinct memories of arguments (as early as fourth grade) with teachers about why my answer was right and deserved full credit even though it didn't match the key. I would look at the problem, see the answer, and write it down, I couldn't figure out why that was a problem.

1. write down what you are given
2. draw a diagram
3. label it
4. write down what you are asked to determine
5. ask yourself what can I calculate (not how to I get the answer)
6. repeat step two until you get the answer required

When I started taking physics, I REALLY struggled with this process, I just couldn't bring myself to do it, very similar to what you are describing. I would just shut down and give up, or get mad that I didn't get credit for my out of the blue answer. I knew on an intellectual level that this is what I needed to do to get credit, but it didn't matter, my brain couldn't handle it. Eventually I realized the problem was my brain was not thinking in that order and trying to force it to do that was wrecking my ability to solve the problem at all because every time I stopped to try and write in this order, I would have to start my mental process over again. The solution was to solve the problem my own way, writing down as I went, and then go and use the "process" as more of a checklist to make sure I had all the pieces I needed to get credit.

This is likely different for every student, but for me, I solve most problems by first converting everything that is written in words to symbols from the start to the end of the problem. This might mean drawing a picture first, or it might mean writing an equation first, depending on the problem. For your example on the doppler shift, here is what my "step by step" would look like

1. Write the doppler shift equation itself
2. Solve that equation for v, leaving everything in terms of variables
3. Write the equation to convert wavelength to frequency and plug that in where needed
4. Do any unit conversions/check that units are consistent
5. Plug in all the numbers

The "extraneous" steps of writing down what I am asked to determine, asking myself what I can calculate and writing down everything I am given STILL seem like a waste of time to me, but once I have finished the problem my way, I can force myself to go back and fill in these details. Once I realized that following the "given" method of approaching a problem, was in fact my problem, everything became so much easier. Forcing my brain to think like someone else's (probably an NT) was the problem, allowing my brain to do its thing and then reforming its output in the way the world wanted it is the solution.

I touched on it already, but to specifically address the problem of shutting down when the problem is too complex to keep everything straight without writing, the key for me is WHAT I chose to write. All my teachers always told me the way to get past this problem was to write down things at a certain point/step, but what I perceive as a "step" is not the same as what my teachers see as a step. In fact, for more complex problems, the steps I am supposed to write may not even be a part of my solution. I had to figure out on my own what was helpful to keep track of things, and if you were to look at the first draft of all my work in physics (even at the PhD level) you would see very little that makes sense to anyone but me. Thankfully I am fast and had time on my tests to solve the problem my way, and then go back and write it up for the consumption of the person grading it. I am still doing this today, and it is working.

I hope that helps give you a little insight, and I wanted to say thank you for what you are doing. I have been fortunate to have some patient teachers in my life and they truly made all the difference. Please let me know if any of what I said needs clarification, it can be hard to explain my thought process sometimes so I wouldn't be surprised if large parts of what I wrote don't make any sense.

 

[CliffStamp]

I want to thank everyone in the above for their help, and in particular one of my other students (Shaelan) who finally managed to make me understand something. I was listening to them talk about one of their friends who is HFA and I didn't understand it at the time, but after weeks of it rattling around in my head I finally was able to understand how she saw her friend and how I saw the kids I worked with and how they were very different ways of seeing them.

I spent some time really trying to understand how they viewed the work and realized in the above that those kids would see these requests as unfair BUT I still kept trying to change it and make them do unfair things. Now while this may be possible (there is research on cognitive flexibility but that is a very long process (of which I am still researching and trying to develop methods) but in the meantime I found a solution which is so simple, but I could not see it because I wasn't really viewing the world as they do.

It finally came to me thinking about how the kid could play DnD and follow all the rules but would cause no end of frustration because he would not spell out the details even when the DM and other players requested (or demanded it). He didn't because "the rules" don't say this is necessary. In the same way he doesn't write out steps because the test/exam doesn't really say it, the teachers ask for it verbally, they penalize you when you don't, but it isn't really "in the rules" so to speak.

I prepared some sample work for them using a latex template for an exam, made it look official (course title, date, logo, etc.) and instead of asking them just for the answer I would state a question (some Force problem for example) and then be very specific about what was demanded :

• provide a free body diagram showing all forces
• note the direction of acceleration to the side
• include a legend showing +/-
• state Newton's second law

and so on, these were all sub-questions (a, b, c).

Here is the kicker, because it is on the exam, they will answer it because those are the rules, just like in DnD, you can't really have unfair rules, they can be arbitrary or made up (like the rules in checkers about who goes first) they are just the rules of the game so you follow them *exactly* because otherwise you are not doing it right.

For a lot of people, it doesn't matter if the rules are on the test, or if you are just told them, or reminded if you break them, but for these kids it was black and white. If they are not on the test it is UNFAIR to make extra demands because you are just making up the rules as you go along and you can't do that, teacher or not teacher you just don't get to do it. Marking them wrong doesn't convince them to chance, it just makes them see you as unfair.

Now this doesn't fully resolve the problem, there are still going to be issues with their graded work which doesn't have these requirements and I am still struggling to get them to try random things when they can't see the direct answer - but anyway, I have found a way to make some progress. Hopefully I can use this to increase their knowledge, and maybe there will be transfer, and hopefully I can find a way to increase the cognitive flexibility as well.

 

[Sid Delicious]

I have autism and the equivalent of ADHD and struggled with maths as a child. I wasn't taught maths at all until the age of 11, so had a lot of catching up to do on top of the issues I already had. But I went from absolutely hating maths to loving it (especially algebra) and now work in software development using these skills all day.

Your observation regarding rules versus requests is interesting, as I used to get irritated with what I saw as pointless or shifting rules that I was expected to follow for no logical reason. It is harder to argue with an invisible, official regulatory examination system than it is a random human individual who can't explain why something has to be done a certain way. As a child I would need to know the 'why' behind something and would challenge any authority for an answer if given the chance. If I had no way of challenging tha authority then I simply had to accept the rules as they were.

The main issue I had with many subjects (not just maths) was being overwhelmed with instructions. Breaking down a task into multiple parts was better than nothing, but then giving me all those parts at once was just as bad as giving me the whole task as I now felt like I had multiple tasks rather than one. I then tried to focus on the original task as a whole, plus each seperate part of the task as an additional whole.

For example, you mentioned above:

"They also have rubrics which will state things like :

• one point for a fully detailed free body diagram showing ALL forces
• one point for proper decomposition of all forces
• one point for F=ma properly evaluated in x and y
• two points for solving the two equations simultaneously"

My brain wouldn't automatically read that as one main task broken down into four smaller parts. It reads it as four tasks that I now have to give equal focus to at the same time, in addition to the original task I was given to complete (since I don't automatically filter one section out from the others or give value 'weighting' to anything). So I'd now have five tasks that my brain will attempt to solve at the exact same time. Sometimes I could do it, normally my brain would 'blue screen'.

As an adult, I've learned that I have to visually remove everything from sight except one part that I need to focus on first (and usually consume a lot of caffeine as well - some people find Adderall or other meds work better). I would get a marker pen and cross out everything on the page except the first bullet point, or rewrite that point again on another sheet of paper to force myself to focus on it (and 'force' is the key word, as it's an aggressive process of stopping my brain doing what it naturally does). I would then spend a lot of time mastering just that one step. Over and over again, until I completely understood it and could do it without thinking. Then I would have a break and do something completely mind-numbing to reset my thought process and get that task out of my working memory. Then I would do the same thing with the next step, then the next... Once I'd mastered each part, only then would I put the steps together as a bigger process and try to view it as a whole. Doing it the opposite way around never worked. But once I learned a process using the above method, I never forgot it and could apply it to other things.

As an example, I had to learn binary programming before attempting to learn any higher level languages. Without understanding each initial step in the chain, how it developed, why it worked, etc, I couldn't understand how higher level code worked.

Using the above example, if you have to teach a child to write out the steps and solve a task then don't start by giving them the full task. Also, don't give them all of the four steps to complete the whole task. Give them just step one. Nothing else. Get them to repeat multiple versions of step one until they feel happy with just that part. Then give them a break to reset their brain and clear their working memory of that task. If they understood it, they will have it in their long term memory to recall later on. Then give them just step two to focus on. Again, give them multiple versions of step two until they fully understand just that part. Don't mention step one. Treat it as an entirely seperate subject that's completely irrelevant now. You don't want them focusing on both at once at this stage. Once they are happy with step two, let them have another break and clear their heads, then move to step three, and so on. Even if they spend an entire day just focusing on one step, they will likely have far more success in the long run. Once something is in my long term memory, it never disappears. My working memory is broken, but my long term memory is excellent. If these kids are the same, then they will need to learn this method of retaining knowledge to apply to other areas of life.

Edit: Just realised the last post was from June, yet it appeared at the top of the forum! : /

 

[EuclidWasRight]

This may not help, but for what it's worth. What follows is an attempt to explain one source of frustration your students may be experiencing.

I have some sort of dyscalculia. I make all kinds of dumb mistakes doing arithmetic. I also have information processing and working memory deficits. Nevertheless, I successfully took four years of mathematics at a Great Books school (Thomas Aquinas College). Euclid, Ptolemy, Keppler, Newton, Descartes, etc. It made a whole lot more sense than HS algebra.

I can understand pure geometry (no numbers) and the philosophy of mathematics fairly well. What I cannot do is blindly follow “rules”, and perform calculations “by wrote”. It would make things easier if I could…

Kids who are good at following rules like algebra. Kids who want (or need) to understand what they are doing may not like it. If you tell them one thing, the next thing you tell them must be consistent. The explanation must be consistent as well. It is usually not. If you are like me, and cannot “just follow the rules” it drives you bonkers.

I had a joke with a college classmate, now a professor of philosophy, that algebra (Descartes in particular) was a “knack”, because it seemed to depend more on getting the hang of doing it than actually understanding what you are doing. It’s actually a very clever way of doing stuff that’s too hard to understand. This is why math teachers emphasize practice and repetition so much.

You may not like this, but Algebra, or mathematical analysis, does not make much literal sense. The terms are poorly defined and many “rules” seem blatantly contradictory. Many rules exist for the sake of operational consistency, not because they make sense as independent propositions. For example, there is no such thing as a negative number. You cannot have something that does not exist, and you can’t have more than one. It is a useful concept, but if it exists at all in reality, it’s not clear how. A “number line” is an oxymoron, and is just sloppy thinking. There are many apparent logical inconsistencies, like negative times negative is positive. I don’t even know what it means to multiply negatives. How can anyone say that multiplying or dividing by 0, a non-quantity, makes any sense. I could go on… Don’t even get me started on straight up nonsense like “a line is made up of points”, which cannot be defended in any way whatsoever. That one is just straight up false. Length cannot be composed of objects that have no length. There is no justification for that whatsoever.

The problem with algebra is that it is a form of analysis, not a science. There is a science OF algebra, but it itself is not a science. As analysis it can be explained, and it makes sense as analysis, but it can’t really be “understood” as a description of reality IMO. In other words, it does not say anything true or false, but it is only an imaginary way of analyzing quantity and manipulating it mentally. It can be used to examine propositions, but not to “understand” them, because it uses representative symbols, not real objects. If teachers understood this, or were up front about it, it would save kids lot of confusion. It’s a good tool, but a philosophical disaster.

It does not distinguish between continuous quantity (e.g. lines) and discrete quantities (numbers). As analysis, the distinction can be ignored because lines can have the same relationships as numbers. The problem is that lines have other relationships that numbers (counting numbers) do not. Algebra fails to observe this fact however, and calls things that are not numbers “numbers”. For example “Pi” is not a number, it can never be written. There is no common unit. It is a finite and rational ratio of lines, but not a definable number. The equivocations never end. My teacher Dr. Molly Gustin, a mathematician who also taught at St. John’s college, claimed that “numbers are lines”. This was a much more rational position than anything I ever heard in a conventional math class because, if accepted, algebra makes sense.

The best book on Algebra I have found is “Algebra” by Shen and Gelfand. It actually explains, in rational terms, why things are, and what they are.

 

[Mary Terry]

This may not help, but for what's it's worth. What follows is an attempt to explain one source of frustration your students may be experiencing.

I have some sort of dyscalculia. I make all kinds of dumb mistakes doing arithmetic. I also have information processing and working memory deficits. Nevertheless, I successfully took four years of mathematics at a Great Books school (Thomas Aquinas College). Eucild, Ptolemy, Keppler, Newton, Descartes, etc. It made a whole lot more sense than HS algebra.

I can understand pure geometry (no numbers) and the philosophy of mathematics fairly well. What I cannot do is blindly follow “rules”, and perform calculations “by wrote”. It would make things easier if I could…

Kids who are good at following rules like algebra. Kids who want (or need) to understand what they are doing may not like it. If you tell them one thing, the next thing you tell them must be consistent. The explanation must be consistent as well. It is usually not. If you are like me, and cannot “just follow the rules” it drives you bonkers.

I had a joke with a college classmate, now a professor of philosophy, that algebra (Descartes in particular) was a “knack”, because it seemed to depend more on getting the hang of doing it than actually understanding what you are doing. It’s actually a very clever way of doing stuff that’s too hard to understand. This is why math teachers emphasize practice and repetition so much.

You may not like this, but Algebra, or mathematical analysis, does not make much literal sense. The terms are poorly defined and many “rules” seem blatantly contradictory. Many rules exist for the sake of operational consistency, not because they make sense as independent propositions. For example, there is no such thing as a negative number. You cannot have something that does not exist, and you can’t have more than one. It is a useful concept, but if it exists at all in reality, it’s not clear how. A “number line” is an oxymoron, and is just sloppy thinking. There are many apparent logical inconsistencies, like negative times negative is positive. I don’t even know what it means to multiply negatives. How can anyone say that multiplying or dividing by 0, a non-quantity, makes any sense. I could go on… Don’t even get me started on straight up nonsense like “a line is made up of points”, which cannot be defended in any way whatsoever. That one is just straight up false. Length cannot be composed of objects that have no length. There is no justification for that whatsoever.

The problem with algebra is that it is a form of analysis, not a science. There is a science OF algebra, but it itself is not a science. As analysis it can be explained, and it makes sense as analysis, but it can’t really be “understood” as a description of reality IMO. In other words, it does not say anything true or false, but it is only an imaginary way of analyzing quantity and manipulating it mentally. It can be used to examine propositions, but not to “understand” them, because it uses representative symbols, not real objects. If teachers understood this, or were up front about it, it would save kids lot of confusion. It’s a good tool, but a philosophical disaster.

It does not distinguish between continuous quantity (e.g. lines) and discrete quantities (numbers). As analysis, the distinction can be ignored because lines can have the same relationships as numbers. The problem is that lines have other relationships that numbers (counting numbers) do not. Algebra fails to observe this fact however, and calls things that are not numbers “numbers”. For example “Pi” is not a number, it can never be written. There is no common unit. It is a finite and rational ratio of lines, but not a definable number. The equivocations never end. My teacher Dr. Molly Gustin, a mathematician who also taught at St. John’s college, claimed that “numbers are lines”. This was a much more rational position than anything I ever heard in a conventional math class because, if accepted, algebra makes sense.

The best book on Algebra I have found is “Algebra” by Shen and Gelfand. It actually explains, in rational terms, why things are, and what they are.

You have perfectly described many problems I had with teaching math to young autistic adults for the GED examination. Teaching grammar rules can be difficult, too, unless they grew up in a household where they heard proper English spoken all their lives so that it is natural and sounds correct to them.

Thanks for posting.

 

[EuclidWasRight]

Sorry if this is overkill. I found an example today, that is very illustrative. Someone on facebook posted a "meme" with a math question something like this:

50+50-25x0+2+2= ?

The answer is dependent upon the order of operations. The question does not contain the information necessary to solve the problem, because the method is entirely conventional. You have to know the rule. The only thing that really matters is not given.

You would most naturally assume that it's left to right, because that's how we read. The placement of the equals sign suggests that's what you should do. You know, it's at the "end". That's obvious.

Apparently because of "PEMDAS" that's not the convention. I say it that way, because the people making fun of those who answered "4" said "It's because of "PEMDAS". This is incorrect. "PEMDAS" doesn't explain anything, it's not even a word. "PEMDAS" is not why. It's a way of remembering why. It's not because of "PEMDAS", it's because of the made-up order of operations. Note that "Roots" is not in "PEMDAS". It should be "PERMDAS". LOL. Aaauuuggh! This drives me nuts. I feel like "Don Music" slamming his head on the keyboard.

The convention could just as easily be right to left.

The meme said "Don't use a calculator", of course because if you use a calculator you get "4" which is the WRONG ANSWER. You have to enter the problem in a calculator differently from the way it's written (unless you are using a scientific calculator that does equations).

To someone who thinks logically, if multiplication is first, you should write it first. Why would you write it it the wrong order? This brings up the question of whether the negative sign is "minus" or "negative 25". You wouldn't ask the question except that you have to do the multiplication first. Is it "25" or "-25"? If you think about it, you realize it has to be an operation, because otherwise you don't know what to do with the product. This leads you to the question, "If you are not going to put it at the beginning, why don't you just put it in parentheses and make everything obvious?" This is a very reasonable question.

It's just so NOT OBVIOUS. To someone struggling with this, for very valid and logical reasons, it looks like you are setting them up. (That's the point of the meme of course...) You don't have any issues with the principles involved, you can do arithmetic. The question is not really "50+50-25x0+2+2= ?" it's "What is the order of operations?", but that's not the question on the board. The literal question is not the real question. You are frustrated because you are not getting it, and then it looks like the teacher (or book) are trying to intentionally trying to trick you (they are). Now you are completely disregulated and can't do math at all.

If you understand the problem in a larger context - big fancy equations, it makes sense. The problem is that this isn't a big fancy equation. It looks like a simple math problem. It's not. I hope that helps. (Imagine being my math teacher - not fun. LOL)