Aug 032014
 

At TMC14 (Twitter Math Camp 2014) this year I did not attend many sessions, because I was the co-lead or lead in several blocks of time. It was great, and the comments I received were very complementary. I think the teachers telling me that were just being nice a little bit, but I hope they did receive some benefit from attending. With that said, the first thing I want to do in my TMC Recap posts is communicate some of what occurred in the sessions.

First up, Algebra 2. I co-led these sessions (there were 3 days of 2 hours each) with Jonathon Claydon (@rawrdimas) who blogs over at InfiniteSums. The 3 days were split into the following structure. Day 1 was about how to teach algebra 2 with some structure and form so that you can connect all the disparate topics of Alg 2. Day 2 was about a different way of cycling through the topics to allow for constant review and building of knowledge (pivot algebra), while day 3 as all about modeling.

Day 1 started off with the question, “How do you currently teach alg 2?” We had several answer. Graphing all the parent functions and creating a hook to hang the rest of the year that way (Family of Functions), or solving the equations and connecting the graphs later (equations first), going through the textbook units and color coding them, and then I introduced my (h, k) format. There was great interest in the (h, k) structure so we spent the rest of the time on that method.

What is that method, you ask? Well, on my board under the heading of Algebra 2, I have the following forms written down:

functions

First off, what do you notice and wonder about all these forms? Yes, I do ask that and spend some class time on the noticings and wonderings about this list. I actually have a “You are Here” note that moves from one to the next to the next as we go through Alg 2 and I make a big deal about that move.

The really nice thing about organizing the class in this way is that clearly the learners are learning ONE set of math operations, not 12. The amazing similarity between all of these forms encourages the learners to actually look at the math and ask “what is the same, what is different” and STOP thinking “all of this is different each time.” It takes some work, but the learners figure out that my 3 rules (the ONLY 3 rules I allow them to use/ write/ or say in class) are how ALL of these functions are solved. [make sure you read the comments too]

Also, shown (but not handed out) during the session was how I consolidate all of the maths for all of the functions and what I expect for every single function listed. It looks like this:

wksheet

All of the links for the handouts and materials are on the TwitterMathCamp Wiki site. If you want this handout or any other handouts from TMC, please feel free to download them.

My goal with this process is getting the learners to think of math as ONE body of knowledge and not a segmented series of things we memorize. We LEARN how to factor, how to graph, how to identify points on a graph, and we USE that same knowledge over and over again.

I have had some success with this last year and I am looking forward to doing it again and blogging about it as I go. Yes this means I am planning on blogging more. That is one goal I have for the year. It was created because of this article on the secret to writing. (hint, there isn’t one.)

 

Feb 242014
 

I am trying to come up with ways to connect the ideas of Statistics and Calculus down through the curriculum in all levels, even into Algebra 1. I think I have a way to do it that makes partial sense and can be done with reasonable effectiveness, but I have an interpretation problem.  If anyone can see a way out of it at the end, please let me know in the comments.

First off, I started with http://graphingstories.com and selected the water volume exercise. My reasons are focused and simple. 1. It is linear, so it works well in the algebra 1 course during the first semester. 2. Finding the slope of the line is straightforward, and requires some transfer of skills, but not a lot of transfer. 3. The area part of it is just a triangle, so the math is not complex.

I am trying to build from simple through the more complex, so starting off simple is helping me wrap my brain around it.

So, I start with this video:

watervolume

You end up with a graph that looks like this (click to enlarge):

watervolumepic

Great, now for some math. Find the rate of change of the line: Best guess: (610ml – 0ml)/(15 sec-0sec) = 40.67 ml/sec.

What does that mean? As the seconds increase by 1, the water volume increases by 40.67 ml.

The equation of the line is milliliters= 40.67 ml/sec * seconds + 0.

So far so good. The interpretation of everything so far makes complete sense. It is attainable for algebra 1 or geometry, and has meaning in context of the video.

So I want to inject a little geometry into the problem and I find the area under the curve. No big deal, it is a triangle, so it is 1/2*610 ml * 15sec = 4575 ml*sec.

What does that even mean? Does it have a meaning? Am I doing something that should not be done? What does a value of 4575 ml*sec even mean? It is not the total volume, that is 610 ml. It is not the sum from 1 to 610, that is 186,355 according to wolframalpha.

In calculus, we find the areas under curves all the time, but they are specific types of curves.  We find the area and it is the displacement, or it is the total distance. But does this area have a context or meaning?

I really would like to figure out a way to make sense of this, but if there is not a way, then I will have to go back to the drawing board.

Edit: The word for what I am looking for is called absement. Links to resources are:
http://nqtpi.blogspot.com/2012/09/the-area-under-distance-time-graph.html
http://thespectrumofriemannium.wordpress.com/tag/absement/
http://wearcam.org/absement/examples.htm
http://forums.xkcd.com/viewtopic.php?f=18&t=34744 [read all the way to the bottom]

And a pic that makes the idea of absement more clear: http://ow.ly/i/4HEwh

Jan 172014
 

[I really need to return to blogging. My lack of focus on reflection has hampered me this semester, and I need to fix that. To that end, I am making a commitment to blog and to jog. Those are the foci this year of the ellipse that is my world.]

Yea, how often does that happen that a class gets excited about logs? It has not happened to me in several years, but this year I found a way. We started the second semester with graphing again. We have a standard list of things we look for, identify, and document on every single graph. The list is:

Domain:

Range:

Asymptototes (vertical and / or slant):

Minimums:

Maximums:

Vertex:

Y-intercept:

X-intercept:

End state behavior:

Every graph we do, we have to document all of these items. If we graph a line, most of the list is “none” but it creates the connection between all the graphs. Every graph has the same questions, it is just that some of the graphs / functions do not have those features.

So, I am doing this file on Desmos, and we are documenting. They have done all these as homework, so really we are checking answers and ensuring learning. Then weird things happen. They notice the symmetry of the inverses.

Nice.

Then they ask to see the graph of the line of symmetry. Even nicer. THEN! OMG. We put the translation into the h-k form of the line, and we see the translation of the line of symmetry.  [Okay, seriously. If you are not using the h-k forms to make connections, why not. See This post, or This post or any other of the several posts I have on this topic.]

And then I graph the exponential. …. …. They know there must be an inverse, but nothing we have done in class looks like that. …. And then, because I have the list of all the h-k forms on the board, someone asks, “Is that what the log thingy is for?”

And now they have a reason to learn logs. They are intrigued by logs. They are asking questions about logs. Because EVERYTHING in math has a forwards and a backwards, addition has subtraction, squares have square roots, and exponentials have logarithms.

They are interested and inquisitive about a topic that normally is not approached this way. I have done something good I think. Only time will tell if I can continue that on this topic.

Aug 182013
 

So the other day I am giving my Algebra 2 STEM learners (this is our Honors designation” the big picture of Algebra 2. You know, so they know where they are at and where the class is going. This is what I had written on the right of the screen.

2013-08-15 11.41.37 2

 

After writing this down, I did the following exercise:

 

What do you notice about this list?

Here is a list of everything I wrote on the board:

 

 

 

 

 

 

 

 

 

image

Then I asked what do they wonder. This took some time, and I was REALLY glad I gave them lots and lots of processing time. Some of the later wonderings took some time to develop.

THEN I asked for a show of hands how many people wondered the same thing. It was amazing that most of the learners wondered the same thing about the important math. I was very pleased.

image

I think the list of “vertex functions” will be written on the board as we go repeatedly so they always have that reference of where they are at in the process.

What do you Notice, What do you Wonder? is a very powerful tool to get the learners involved. Max Ray at The Math Forum at Drexel has done a lot of work (even applied for a trademark for the Notice / Wonder phrasing) in this area.

The “What do you notice?” and “What do you wonder?” questions are great ways to pull even reluctant learners into the process of engaging with the material. I know I will be using it again and again.

Aug 122013
 

I tried something new this year. I was a little nervous trying it, but I did NOT want to do the whole “here is the syllabus, let me read it to you” shtick.

So, I made the first day very active and involved.

Did it work? What did I find went well, what didn’t?

In my Alg 2 STEM class, which is an honors level Alg 2 course, the seating randomly with cards and making the learners share something about themselves was rocky at first. I actually had to ask and model, “Did you share? And by sharing, did you actually do this?” walked up to learner, introduced myself, and said 3 things. The learners sat there in absolute silence until I did that, and then they talked, shared, and actually met their table mates.

Next,  the cell phone assignment. OMG, it was a blazing success.  I had 33 learners discussing rich mathematics, arguing about what kind of display to make, the benefit of bar graphs over line graphs over tables, what is important in choosing a plan, etc. The only thing I had to do was answer questions like, “Do you use a lot or a little data?”. The idea of rich problem solving and being less helpful was glorious and absolutely a positive in the classroom.

One learner did ask if I had a syllabus for them. I told them they would get it next week. This week they needed to establish some habits of conversation and working together. Several of them were shocked and amazed that I did not talk at them all period on the first day of class. Win.

My next class was Algebra 3, or Advanced Alg. It is a senior level class for learners who may or may not be going to college, but who need a 4th year of math and refuse or feel they are not capable of doing Trig/Precalc. This is a tough class, and can be challenging to teach because I get a lot of “I don’t get this” or “I am not good at math” etc.

This group I did the literal equation assignment and it was … rough. What was specifically rough was the idea of multiplying two binomials. I got a couple of “why can’t we just do this together?” questions, and on the second I stopped the entire class and gave the following reason why not. “During your freshman year, you did it together. And your sophomore year, you did it together, and your junior year, guess what, you … did it together. Who here thinks that me “doing it together” with you now will make a difference?”  Not one person said it would.

This was a tough assignment for them, and the conversation was heavy on the, “Well, if it was easy, you shouldn’t be in this class.” and “Why would I waste your time with something you already know?” This resonated with them, and made them realize that I was on their side, I was just challenging them. I felt some success, even though they didn’t finish the assignment at all.

Finally, AP Stats. We did the Gender Discrimination problem / simulation. It was a great lesson, where we got to use the vocabulary of simulation right from day one. This was a great thing, and I think it set the tone for a class where discussion and simulation are the most important elements.

Time will tell. This was only day one, but at least it was a thought provoking day for my learners and a successful day for me. I believe learning did occur.

May 022013
 

I have had this long term struggle going on in my head this year that we really don’t do a good job making connections between material in our classes, and that lack of connection is one reason why “transfer” (ala. Grant Wiggins and Understanding by Design) doesn’t occur as  frequently as I would like.

Well, I am not going to talk about it any more. I have the beginnings of a plan to enact. There will be many steps to this plan, but I think the starting point needs to be simple to enact and creates some opportunity for connections to be made.

Every test in my department from Algebra 1 through Trig/Precalc must have a couple of different kinds of problems on it. This is step 1 I am implementing next year.

The first type is a literal equation. Of course, as a stats teacher my first thought was M=z*root(pq/n). Perhaps at the algebra 1 level we won’t start there, but we can select most of the formulas needed in geometry and use them as literal equations and every quiz and test solve for a different  variable of one of the formulas. And, here is the kicker, EVERY time, the learner must explain why they are doing the operation. Justification is mandatory. If we look at the Margin of Error formula above, there are 4 different questions to be asked. That is 4 quizzes or tests that one question can be used.

The goal is get learners to think of literal equations a part of algebra and the justifications as the same thing as every other problem. By the time they reach AP stats, they will have seen this equation repeatedly and know how to manipulate it as a literal, not just with numbers in it. We need to connect AP Stats to Algebra 1.

Next, every test at algebra 1 level must have some form of the following question:

Evaluate (x – (x+h))/x with x = 2 and h = 3. Yes, I know it reduces to h/x, but as we move forward with notation, it becomes:

Evaluate [f(x) – f(x+h)]/f(x)  with f(x) = 2x+5, x = 2 and h = 3.  As the years progress the function can be moved from linear to quadratics to absolute value to cubics or rationals.

Finally, truly stress and monitor that verbage “rate of change of” every time the word “slope” is used.  The learners need to hear and write over and over the “rate of change of” the line in algebra 1, geometry, and algebra 2.

The goal is to create a common language / strands through all math courses and chapters that lead to AP calculus and AP statistics. All learners need to be exposed to the language of statistics and calculus repeatedly throughout their education so it is not different at the upper levels.

So those are the three things I can and will implement next year, without fail.

What am I missing?

Any other language to implement? Any other formulas / concepts that can be used at the lower levels of math that lead directly to the upper levels?

 

 

Apr 132013
 

I have been mulling this question over for a while now, since last summer at least. It is a offshoot of the time I spent working with Exeter materials at an Exeter summer institute, and if anything the question has grown in my mind to the point where I must answer it for myself and act on it.

Here is the newest version of my question: If much of what and why we teach math the way we do is arbitrary, then why not change to make it easier to learn?

Now of course, there is a HUGE set of presuppositions / assumptions just in asking the question. First, I assume that much of what and why we learn math is arbitrary. Well, I don’t think I am that far off the mark. Let’s look at Algebra 1 as a course first. Honestly I am in good company with this thinking.

Grant Wiggins (author of Understanding by Design) and his thoughts on Algebra 1

I agree. Algebra 1 has a huge failure rate because it is very abstract, meaningless content. We don’t really ever see why we are doing it, we are just learning to manipulate variables and constants around. Grant gives a small part of Lockhart’s Lament (pdf), and it is worth linking to (and reading) completely. Again, the ‘why’ of ‘why do we teach it this way’ is completely arbitrary. Which is why we get political science professors arguing that Algebra is unnecessary because it is hard in the NYTimes. That Algebra is a gateway topic is not in question. It is. The content is essential for future jobs and future success.

If we look a the content, we see arbitrary stuff all over. Heck, just look at the old y=mx + b. Why “m”, why “b”? There is no good answer. Do a google search and get 33 Million hits, none of which can definitively tell us why. I find the answer from the Drexel University Ask Dr. Math to be the most grounded answers, which you can find here for m, and here for b. And I LOVE the answer given here by math historian Howard W. Eves in Mathematical Circles Revisited (2003), where he suggests that it doesn’t matter why “m” has come to represent slope.

“When lecturing before an analytic geometry class during the early part of the course,” he writes, “one may say: ‘We designate the slope of a line by m, because the word slope starts with the letter m; I know of no better reason.’ ” via

I totally agree. In other countries they use other variables for the same meaning (scroll down), so clearly the “agreement” that we all must use the same convention is not universal. There are so many conventions in math that are purely arbitrary. Since they are arbitrary, we must feel free to throw them away when they interfere with good learning and teaching.

So the second question, and a very important one, is: How could we teach math differently to make it more understandable?

One thing I think is important is to connect the vocabulary / language / processes of linear functions with other polynomials / transcendental functions. After all, look at the amazing similarity and simplicity of understanding the transformation processes.

image

Don’t believe me that every single function listed has exactly the same transformation rules? Try this little GeoGebra applet I whipped up. Think about that for a second. When I have shown this to math teachers I get two reactions, “Well duh” and “OMG, I never thought about these like that.”

The teachers who see this as obvious are the teachers who are much more experienced and have taught for many years and have spent the time looking at the math. The crazy thing is that very few teachers have told me they teach this. Why not? Because it isn’t how they were taught, it isn’t how the books phrase it, it violates the conventions of math teaching. So they know it, but ignore it.

And don’t get me wrong. I am not suggesting this is where we stop teaching, we use the exact similarities as a springboard to bounce into the other types of functions. If the learner of math knew this with strong understanding, then the rest of algebra becomes a close examination of each type of function (which is all the different algebra courses are anyway.)

The Common Core Curriculum has mixed up the order of teaching these functions, but the fact that all algebra is just an examination of the skills (which are essentially the same, find ones by multiplying by the inverse, find zeros by adding the inverse) needed to solve, graph, and understand how each function is used.

The last question I have is: Why do we, as teachers put up with this, and what are we going to do about it?

I think the CCSS gives us the perfect opportunity to demand better from textbook publishers as well as our professional development opportunities. We, as teachers, need to be willing to throw the ‘conventions’ away and teach better.

Will it make a difference to the failure rates of algebra 1? I don’t know. but how can it hurt? How can it hurt to strongly connect all of algebra through trigonometry with an unbreakable thread so learners know that what works for one type of functions will work for every other type of function too. It shatters the concept that Chapter 3 doesn’t relate or have anything to do with Chapter 7. That is what learners think now.

Apr 112013
 

It is spring break, so what am I doing? I am attending AP workshops and volunteering at my local university. All in all, a great spring break.

So, Let me start with the question first. Why do we make it so hard to learn functions? I mean really. We treat each topic; linears, quadratics, cubics, transendentals, etc, as if they are a new and unique idea. And they definitely are not. I have discussed this before when I was thinking about the Exeter materials, and I have to keep coming back to it for good reason.

What brought it to me today is the fact I am presenting at UNR for the professor of Math Methods to pre-service teachers. I was asked to present on calculator technology, and I will also branch out into GeoGebra, Desmos, and the MathTwitterBlogosphere.

As I was running through what I was going to say and planning my lesson I made a short video on what I wanted to show with GeoGebra. This only scratches (heck I probably doesn’t even leave a mark) on the surface of what GeoGebra can do but it is worth discussing to present it to teachers who will be immersed in Geometer’s Sketchpad in college.

GeoGebra & Functions

 

And then I turn it into an HTML5 page so anyone can use it.

  And now I have a video as well as a usable piece of content for learners to look and and use on their own at home. I am trying to model good teaching practice that I use at school.

And yet, the question of why do we make linear functions separate from other so it is harder to learn than it should be still comes to my mind. Why? I don’t have a clear answer, and I am not sure anyone else does either. That is sad.

Aug 262012
 

Okay, all along I was promising a massive file upload for all the readers who want the Exeter materials. I will explain what each group of files are for as I go.

All files are in WORD or PDF format, and all are in a zipped folder. Downloading and unzipping the folders will speed up your access tremendously. All in all there are 44 megs of files here. That does not sound large, but word files and pdfs are incredibly small these days!

[The placement tests were posted on Exeter’s website, but they didn’t realize they were made public. Links to that page have been removed. If you are a teacher and would like the files, let me know.]

The progression at Exeter begins with a Placement test to determine what course the learner should be enrolled. These are released Placement tests from Exeter:

Released Placement tests
After being placed in the correct course, the learners then start in on the problem sets. I have 2 years archived, but I would love more if someone has them.

Problem Sets 2011-12

Problem Sets 2012-13
The current year is 2012-13, so the archive is an August download of the new materials, including the change logs. If someone has the change logs for the 2011-12 or the files for previous years, I would add them also. The live location for the current year’s materials can be found on their site.

I do have solutions to the 2009-10 problem sets (I was given these without the actual problem sets) and solutions to the 2011-12 problem sets. Will I post them? No. I know I would not be happy if a teacher posted solutions to all the problem sets I created. That is the one thing I won’t post.

During the class the learners are in, they will do hands on activities, and use Geometers Sketchpad to explore math. The Instructor of the Exeter sessions I attended was nice enough to share these. They are all written by Exeter teachers, so no poaching and claiming them for yourself. Please attribute them accordingly.

Hands On Activities 2011
(Both Word and PDF documents!)

GSP Document and Sketches
GeoSketchpad misc
If you are looking at these thinking, “Dang! That is a lot of material to go through!” You are absolutely right. The 2 docs for Alg 1 are 59 pages combined, the Geo doc is 62 pages, and the Alg 2 Hands on is 69 pages. Right there are enough docs to keep a person busy in class for a long time, and you would be learning terrific math as you go. In the GSP Document and Sketches folder, there is a document called “2011 gsp.doc” It is 101 pages of GSP constructions.

So the learners are working problem sets, they are working activities and extending their learning beyond the problems and being active with the math. Now it comes time for some assessments.

Math 1 tests
Math 2 tests 
Math 3 tests
These are all in word format, so you can edit and use them in your classroom if you like. These tests give you some idea of how Exeter assesses their learners. Something you should know is that every one of these assessments are open notes. Every problem set they have worked is available to them on the exam.

Finally, the year is over, the faculty get  together and evaluate the problem sets. What worked, what didn’t, what can be improved. And the writing committee collects all those comments and distills them down into a commentary on the problem sets for the rest of the staff. Then the rewrites happen, and the new problem sets are published, and the cycle starts all over again.

Commentary 2011-12  [if you would like the commentaries, and can demonstrate you are teacher, please email me or comment and  I can email them to you. The files have been removed at Exeter’s request.]

 

And there you have it. This is the cycle of development of the Exeter curriculum and materials. The vast majority of the work is done by the writing committee, compiling the commentaries and editing the problems. That is a huge task, and I would love to have a serious discussion with someone at Exeter just about that. Heck, I would spend a week with them just asking questions about the writing of the questions, let alone working and thinking about the problems themselves.

I hope this is of some help to other teachers out there.

Aug 112012
 

In this post I want to show Exeter’s problem solving strategy. This is important, because it is SO different from how a problem like this is typically approached.

First off, the problem I am going to model is M1:21:11 [Math 1, page 21, problem 11]

11. Alex was hired to unpack and clean 576 very small items of glassware, at five cents per piece successfully unpacked. For every item broken during the process, however, Alex had to pay $1.98. At the end of the job, Alex received $22.71. How many items did Alex break?

In a typical Algebra 1 class we would try to get the learner to see the equation is:

.05(576-x) + 1.98x = 22.71

In fact we try to get the learner to jump directly to the equation from the problem by deconstructing the sentences, and then solve the equation. x = 3, by the way.

Now, let’s see how Exeter expects and demands that ALL of the modeling problems are handled.

First off, we will be making a table. The headings in this table are mandatory and can not be short cut. The learners must label the table thoroughly so that it makes sense. Remember, this is the same problem as above. I am going to paste in my table all filled out, and then explain the essential elements.

Guess: # of broken bottles

# of unbroken

$ Paid for unbroken

$ subtracted for broken

Amount paid

Goal

Check

0

576-0=576

.05(576-0)=28.8

(0)(1.98)=0

28.8-0=28.8

22.71

no

5

576-5=571

.05(576-5)=28.55

(5)(1.98)=9.90

28.55-9.90=18.65

22.71

No

3

576-3=573

.05(576-3)=28.65

(3)(1.98)=5.94

28.65-5.94=22.71

22.71

YES!

B

576-B

.05(576-B)

1.98B

.05(576-B)-1.98B =

22.71

 

 

Okay, there we have. A decent example of what a modeling, problem solving solution would look like. At the beginning stages of Math 1, they would not demand the last row, the equation row. But quickly they would ask the learners to start generalizing their solution.

The guesses column are not set in stone. The guesses are going to be the learners guesses. They are going to guess whatever they want. I started with 0, because maybe he didn’t break any. Then I saw that was too high to my goal, so I figured Alex broke a few. Then I was too low, so I picked one in the middle.

Now, let’s examine what the columns mean. It is clear from the headings that each column has a very specific purpose and is clearly labeled. What are we guessing? We are guessing the number of glasses he broke. If he breaks 5, then he didn’t break 571. How do we get that, we subtract. Each column must have in it HOW they get the number, not just what the number is. And so on.

Notice that by the time the learner reaches the answer, they have worked several times the process, they know the multiplications, the subtractions, and they have the solution worked out. Where does the variable go? It goes into the spot where numbers change. What do we call the variable? Don’t care, use a letter that makes sense to the problem.

How do they start this process? The first problem that is a modeling / problem solving problem is M1:9:4. It looks like this:

image

Notice that they start by giving the table and even filling out the first row. The problem I worked above, didn’t have that level of detail. The learner had to provide it. That is the point.

EXETER MODELS AND LADDERS THE LEARNING UP TO THE LEVEL THEY WANT.

Yea, I shouted that. We have this impression that Exeter is so fabulous, that they don’t have to ladder or work with learners. We think that the learners just will magically go *poof* and be able to do all these things that we struggle with.

Guess, what, they struggle with similar things there as we do in our schools. It might be easier because of smaller class sizes, but the root problems are the same.

Okay, off my soap box.

The Algebra 1 activities have some problem solving activities, and they even are sneaky by giving a blank table with fewer columns than the learners need! The learner is pushed to make the table for themselves.

Think about this type of problem solving for special ed, or EL Learners. They have the numbers set up, they can see where the Letter for the Unknown goes, because it is the only number that changes when they are doing the problems. Wouldn’t this method help them out so much?!

Think about your average learner who struggles with parsing the language of the problem. If they work 10 or 20 of these as starters, as homework, as in class activities, do you really think they are going to stress about a word problem?

Nope, they are going to say, “Mr. Waddell, these are easy, can we move on to something harder?” And you know they will.

Think about the really advanced learner. They are going to resent the table after a short time, but they will go to the generalization much faster because of it.

Can you think of any downside to this method of problem solving? I can’t. I have done Algebraic Thinking’s “SOLVE” method, and other methods. None of them are as straightforward and easy to put together as this method. We could spend THOUSANDS of dollars on professional development on problem solving, and none of that money would come close to the success of just creating a table, labeling, and working it out step by step.

Guess and Check. That is what Exeter calls it. I call it just downright successful for every level of learner.