Are you a 1, 2, 3, or a 4? What’s numbers got to do with it?!?

I, along with a couple of other teachers, are piloting a grading strategy that is generating some interesting conversations on a DAILY basis with our students!

We’ve all read that grades do not improve or motivate learning. In fact, once a grade is given, the student assumes that idea is ‘done’ and drops it, moving on to acquire the next grade.

What I am about to share with you has MY KIDS talking about how THEY can improve their learning… 

First, I have to give credit for the base of this idea to an amazing educator that I work with every day: Rebecca K. She, of course, credits it to an idea she learned in a workshop some years back. Anyway, she started the year off with a cool bulletin board, that looks something like the image above, which I used to create a powerful way to motivate my babies to take more responsibility for their own learning!

The students I am talking about are your average 9th grade (yes, FRESHMAN!!) students, that run the gamut of every freshman stereotype you’ve ever met. Really. (This includes students with personal learning plans and students whose first language is not English!) AND  we’ve got them talking about growth – THEIR growth – as learners. When we hand back a paper, instead of the ‘crumple it up and put it in the bookbag or the trash’ mentality, the comments are varying forms of, “..tell me what these results mean!”

Here’s how it works:

Four numbers, four learner identities:  1. Novice, 2. Apprentice, 3. Practitioner, 4. Expert

Novice: I’m just starting to learn this and I don’t really understand it yet.

I explain to the students that this is where everybody in class starts out. Algebra I will have lots of things that are new to them, and we expect that they won’t be familiar with the material! We don’t expect them to know it all before we teach it. Sounds obvious, right? Sometimes you have to be explicit with Freshmen. I think that’s where the name originates!

Apprentice:  I’m starting to get it, but I still need someone to coach me through it.

The apprentice is the beginning of the learning phase. When a student gets a 2 on a problem or a whole assignment, they are in the initial learning stages. As a teacher, I’ve just told them (by marking it a 2) that I know they still need help with the concept, and that I will be supporting their learning. This also tells them that they are not there YET – and that they have room to continue learning. Sometimes we have to give kids permission to not know things YET!

Practitioner: I can mostly do it myself, but I sometimes mess up or get stuck.

This is a proud moment for most of my students. That little 3 next to a problem or on a paper, tells them so much more than a traditional grade. This sends them the message that I get it that they’ve got it! This affirms their learning. This affirms their work. This is personal. Better than that, this motivates them to keep going, to keep learning. They ALL want to be….

Expert: I understand it well, and I could thoroughly teach it to someone else.

Isn’t this where we want our babies to be? You know that if they know it well enough to teach it – THEY KNOW IT!! That peer tutoring thing is for real! Please notice that there are TWO parts of this level: knowing and teaching.

How does this work? 

My (totally awesome) co-teacher, Stephanie W.,  and I, use the following grading process. Feel free to modify it to fit your students, and what is happening in your classroom. We know that what we are doing is working for our kids – you may want to start with this, and then modify as you see what is working for you.

We give an assignment or quiz. We grade each problem with a 1, 2, 3, or 4. We add up all the grades and divide by the number of items. That gives us a number between 1 and 4. Many times that will generate a decimal, say 1.8 or 2.5, or even 3.8. Here is an important point: we DON’T ROUND UP! We DO EXPLAIN the process to our students. It is important for them to understand that this is not arbitrary. They must own the process for this to work. These conversations happen EVERY time we return an assignment. That’s a GOOD thing!

Our goal for our students is mastery, so unless the resulting average is an actual 2 for example, the child is still a NOVICE (1, 1.2, 1.8, 1.9 – doesn’t matter. They are still a 1). Same with 2 point anything – they are still a 2, same with 3 point whatever – still a 3. The ONLY exception is 3.8 and above. If the student has one or more 4+ answers, with clear justification statements, then, and only then, will we round up to a 4. See below for the PLUS explanation!

Our evaluation goes something like this:

a) Answer that is incorrect, No work shown, or No answer at all: give it a 1.

b) Answer with some work shown (they attempted a solution) but it is incorrect in major ways and answer is incorrect or incomplete; give it a 2 (remember they are still learning and need more help!)

c) Answer given is incorrect, but work is also shown. (OR answer is correct, but NO work shown to support the answer). Student did pretty good, but minor errors and/or mistakes caused the incorrect answer; give it a 3. This student is obviously getting it, but he/she is letting errors get in the way. Maybe they are lazy, maybe in a hurry. The 3 tells them that they are getting it – but they NEED TO BE MORE CAREFUL! (The 3 for NO work shown is to allow us to ensure students are not ‘borrowing’ answers from another student! We are giving them the benefit of the doubt until further notice.)

d) Answer and work is shown and is completely correct. This baby gets the 4!  The student can feel the glow of being an expert. But wait, there’s more! This only satisfies HALF of the description. What about the ‘teaching’ part?

Four “+”? What is Four Plus??? 

‘Four +’ is that special designation for the child who not only knows the material, but can prove to us that they are able to teach the material to another student. Time dictates that we don’t have the opportunity for EVERY student to demonstrate teaching ability (although we do try to build in those opportunities!). We have explained to our students that the way to demonstrate this ability is to justify the work they’ve shown, with brief written explanations.

Written Justification sets the student up for PROOFS in Geometry

Algebra I is a class of foundations. It is important to teach with an eye to the future courses our kids will encounter, and proofs are some of the most difficult lessons for students. One of the Algebra I standards is to be able to justify the steps taken to solve simple one step equations. This is an important step to understanding that there is a mathematical reason for being ABLE to take that step – and not just because the teacher said so! By building this into the idea of EXPERT, we are modeling the concept that understanding – that is, the realization that there are solid REASONS for why math ‘works’ – is a valuable part of the learning process.

What WORDS do you use to tell a parent how their child is doing in your class?

I know this is just a brief overview of this process, but I wanted to share because I feel it is the first solid step in moving towards talking about GROWTH and LEARNING, instead of grades. I believe it is important that we take the focus off of grades, for students and parents. To do that, we, as teachers, have to stop using GRADES as the unit of measure in communicating with our students and parents. Unfortunately, our grading systems, and I’m talking the actual computer systems we have to use, are not set up to show mastery – they are set up to show GRADES!

I already changed my conversations, my wording, my language,  with my students. It will happen with my conversations with parents in my next phone call/email home, as well. Will YOU?

What’s the downside?

My school still uses a grading system built on averaging traditional grading numbers. That means I can’t just put in 1, 2, 3, 4, or 4+. I have to turn these numbers into a grade between 0 and 100 that will accurately translate and describe my students’ mastery of the curriculum.

My solution is two-fold. The grades in my gradebook are tied to one of the required standards, and each of the above levels is tied to a number that has already been given meaning by how it is used as a grade. While the first is fairly easy to accomplish, the second is based on how parents and students interpret grades. A 100, for example is the ideal. That sends the message that the student has mastery of the assignment, or the course. In fact, anything above 93, in my County school system, is an A, and as such, denotes pretty much the same thing as a 100. Same for a B, or a grade in the 80 range. Those two grades are obviously acceptable to most parents and students. The grade of C is a little more ambiguous. The C denotes that the student is somehow less than perfect, but still passing.  While a student may be GLAD to have a C – it does denote that the student is doing the work and IS mastering the concept – it doesn’t have the same cache’ as the A and B grades.

So how do I reconcile the grades with the numbers? 

A novice receives a grade of 65. The Apprentice receives a 70. The Practitioner has earned an 80, and the Expert, a 90. The 4+ student will earn a 100, as long as all problems on the assignment or quiz show justification, evidence that they have not only mastered the concepts, but have gone above and beyond to be able to communicate their knowledge with others.

The final issue I will address here: What happens when a student makes no effort at all. Our students never do that, do they??? In that instance, the student has given us no information on which to base a grade. Effectively, they have NOT TURNED ANYTHING IN. The grade in the book becomes an NTI, and we are made aware that we need to step up our efforts with that student. An NTI is a zero, until the student completes an assignment on that material, and we can assess mastery. From there, the averaging work of the gradebook takes over, and the grade reflects the whole course mastery. Grades in this context are fluid, and can be changed by future mastery as evidenced by quizzes or testing situations.

The system is not perfect, but the teachers with whom this is working believe that we have created a system that truly tells us where our kids are with the curriculum, and allows us to modify our teaching darn near immediately, so that we can address the areas in which they need further help – which is the actual point of all this grading, isn’t it?

Here is the poster we use in our classroom to explain the levels. Our students get their own mini copy for their notebooks. We utilize a small chart of “I can” statements for each unit – no more than 3 – 5 statements – that allow the students to chart their progress. Here is the chart for our Unit 1 standards. The kids get this, too. You can use any “I can” statements you need for your particular units.

At the beginning of each unit, the STUDENTS determine their pre-assess level, the quizzes give them the mid-assess levels, and then the unit tests are the post-assess level. The students keep track of these themselves. We incorporate a running conversation DAILY of what their goals are, where they think they are with these goals, and how they are going to get to the 3 and 4 levels. I have personally found this is a great way to have the students tell me where they are at the end of instructional and practice periods throughout class. I simply ask them where they think they are – 1, 2, 3 or 4. The majority of students are incredibly honest, because we are all speaking the same language. The ability to quickly assess and modify my teaching is been made incredibly easy! Grading has become a process of assessing growth, not despairing over what they don’t know. I LOOK FORWARD to grading the work, knowing most of my students WANT to have a conversation about where they are, and what they need to do to get to the next level. Let me know if you would like the rest of the “I can” levels we are using with this course. I’ll be glad to share!



Math Problem of the Week

I get to be the sponsor for my school’s chapter of Mu Alpha Theta. This is an awesome group! Unfortunately, all but three students graduated last year, and one of those transferred to the new school!😩.

Continue reading “Math Problem of the Week”

There is this “buffet of choice” for teaching and learning of mathematics: How can I choose?!?

First: just breathe.

Second: start at the end, and work backwards. Reflect on these questions (posed by Brian Bushart During a #mathrocks event:

1. What are your goals for yourself for your math teaching Continue reading “There is this “buffet of choice” for teaching and learning of mathematics: How can I choose?!?”

Intentional Talk meets Inquiry Based Learning. Hello, Beautiful!

I teach math. I don’t want to teach it anymore. Instead, I Continue reading “Intentional Talk meets Inquiry Based Learning. Hello, Beautiful!”

Why do we teach math? This teacher’s honest answer.

A math teacher friend has been questioning his value in his classroom. In his honesty, he asks “why do we teach what we teach?” His plaint comes as children ask him “why do we need this stuff, why prove something you have told us to be true, when will we ever use this?”
His desire is to answer honestly, but brilliantly; to give an answer that will make the child stop, gaze at him in awe, and return to his/her studies with renewed vigor and purpose. I suspect he is not alone in his thinking. For him, and for all teachers who go to this place of doubt and fatigue and, perhaps, even frustration, I offer these words:

Children need the discipline of effort, of struggle. They need to try something new, to push themselves beyond their comfort zone. As all of us know, that comfort zone is, well, it’s comfortable!

Teachers push, encourage, cheer, exhort, challenge, and free students to step into the unknown and the new. It’s like the first taste of vegetables for a baby. Oh, that face! But we know it is good for them. We know what they do not– that there is a wider world out there for them, and that they will need ways of dealing with it. Math has a logical, beautiful order to it. There is a discipline of thought that is rich and worth learning. And that is the first part of my answer. The noble part.

Here is the second:
I teach for totally generous and totally selfish reasons: I generously want children to find the wonder and excitement of knowing something beautiful. I want them to feel and own discovery of patterns and trails that numbers make; how they loop back upon themselves; how they start at one place and end up in another. I want them to see the magic of numbers. I want them to feel the satisfaction of revealing the secrets behind the magic – the elated rush of struggling with and conquering a numerical puzzle. It is Alice finding a key to a door and then choosing the correct potion to become the right size to fit through the door. (And for many children, math is as confusing as wonderland was to Alice!)
And selfishly: to see that most elusive of creatures – the excited spark when a student makes that connection to something learned. That spark is my adrenaline, my satisfaction. It creates a pride in me about that student that is like no other feeling in the world.

There are many comments about how students don’t care; how testing is killing the learning; how teachers’ hands are tied by administration. Still we teach. We teach because we know in our hearts that it is right and important. We want to be in the classroom. We want a better future for our students than they can possibly envision. We hope, eternally. We hope. That’s why we teach.

Our hope is stubborn and sure. Our hope does not back down. Our hope transcends the newest strategy, policy, or curriculum. Our hope spurs us to persevere with every child.

We must not, cannot assume that we have no impact. No one comes up at the end of each period with a trophy or a plaque. Children rarely write thank you notes (or parents for that matter). As teachers we may never see any result, we may feel we are simply parroting what other, greater teachers have already said. We cannot let that stop us. It is persevering into the dark, shadowy night, not knowing what we will meet along the way, with people telling us to go back, to beware, calling us foolish. Hope. What a powerful thing. Our students are the beneficiaries of this conviction, this hope. They may neither appreciate nor value it, now. But someday, ahhh someday, we know they will. There will be that moment, although we won’t be there to see it, when a child will silently thank a teacher for not giving up. And since we don’t know which child in our care that will be, we must place that unopened hope with each child we teach. And we will, because we must teach as surely as we must breathe. No matter what.

Common Core Aligned Lessons: Rich Tasks, or Same Stuff, just re-aligned?

I got an email from Achieve just this week inviting teachers to start testing the lessons on the Achieve the Core website.

I was so excited! All of this with one search tool! I clicked through: There were lessons for every standard! I clicked again: For every grade! I quickly clicked on the lesson promising to teach students to find the zeroes of quadratic equations (that’s such an important concept and I was looking to expand on the rich task we used in the workshop). And then… Well, does anybody remember the sound the record player made when the needle would slip and slide across the vinyl??

Since attending Common Core Standards training this summer, where we learned how to implement rich tasks for conceptual learning, and learning about Dan Meyer’s work, I have been interested in sources of these strong lessons and in modifying many of my existing lessons. The Stanford class ‘How To Learn Maths’ cemented my desire to get even better at teaching math through numeracy, rich visualizing, and good questions to get students thinking about what the numbers they are using really represent.

Here is what I found:
My click on the quadratic lesson connected me to the website Share My Lesson. The lesson plan was beautifully written: standards, number of days, list of materials (hmmm, a graphing calculator, but not graph paper?), the detailed notes handout- one for each day of the two day lesson (fill in the blank), and even a group ‘discovery activity’. Further down, there is a chart with a column of expected student answers/misconceptions, etc. that looked interesting, (in fact, that was the best part) and another section with a three column ‘prior knowledge, current knowledge, future knowledge’ chart (although prior and current knowledge would seem to be the same thing, but current knowledge is apparently what they are supposed to learn in the lesson; which makes that future knowledge in my book!)

There isn’t any instruction on the formative assessment, although perhaps the teacher will make sure the blanks on the notes are correctly filled in…

I fast-forwarded to the instructions for the lesson: graph (using the calculator) four given quadratic equations and identify the zeros. Hmmm. How are they supposed to know this? I checked the prior knowledge column on the lesson plan. Nope. Nothing about zeros. The current knowledge column (remember: the goal of the lesson) was that the student ‘would be able to’ find the zeros of the factors of the quadratic. Factors? But we didn’t factor anything. Oh, wait, it says here that factoring is the next lesson! STOP!!!

Where is the rich task? Where is the productive struggle? Where are the mathematical practices?

This great lesson, common core aligned and all, appears to be more of the ‘feed kids details and have them take notes’. Even the group activity, having them find the differences in the graphs isn’t creating the conceptual understanding of what they are doing, what the graph represents…. Can you feel my frustration here?!

We (teachers) are going to have to undergo a shift in thinking about what good lessons look like. It is going to require kicking out textbooks and no longer training students how to get good at multiple choice tests. This is a paradigm shift. Yet, here is the Achieve the Core website leading teachers to more of the same dry, lecture heavy, notes and memorization-filled stuff!

In the interest of good reporting, I went to two other lessons, one for sixth grade on fractions, and one for eighth grade algebra. They were similarly structured.

If you are interested in what this national resource of lessons is offering – and interested in helping improve the lessons – then here is your chance (you might even win stuff!):

Participate in the Common Core Challenge:
1. Watch short videos of master teachers while using the CCSS Instructional Practice Guides
2. Apply what you saw to a lesson of your own
3. Tell them about your experience
About the winning stuff, the email said, and I quote, “As a participant, you’ll be eligible to win great prizes while helping to continuously improve tools designed to support teachers like you.”
The Common Core Challenge was developed as part of the Common Core Teacher Institute held on October 6th at NBC News’ Education Nation 2013.

This is your chance to give them feedback on these lessons. Let’s give our teachers every chance to succeed in the classroom, because this is the only way our kids will succeed. That success will transfer to a confidence that we won’t need standardized tests to see!

Seriously, though, sharing good lessons so that we don’t have to create our own and giving feedback to make good lessons better will allow us to improve what is happening with students across all disciplines, across all schools and across all SES.

I have found the most wonderful group of teachers and resources and community through the MTBoS site and through Twitter ( I am @the30thvoice), so I know my teaching is going to get better and my students are going to be challenged.

How a students starts is out of our control, but how a student finishes is in large part due to how he/she is taught. Be that teacher for your students!


Sixth Grade, Fractions, and Fair-shares

I have been assigned a room full of bright six-graders. They are successful; achieving high proficient scores – they are students who are “on the bubble” of scoring at the advanced level. A review of their work reveals the problem: Operations with fractions.

My goal is to help them create a stronger conceptual understanding of fractions, in order to allow them to be more successful in their fraction interactions!

The initial lesson involves some baby steps, in order to formatively assess the understandings about equal sharing, naming shares and rebuilding wholes. Materials will include the students themselves and a fair share box (see attached image- thanks to Jere Confrey).

I tell them they are going to gather in teams and ask them to come up with ways that we can divide the class so that each team has an equal number of students, WITHOUT COUNTING FIRST. At this point I am not looking for anything but suggestions on grouping.

I will use thinking questioning to get them started, clarifying q’s to help them express ideas and repetition: students repeating what others have said. The students will then be asked to critique, analyse and decide how best to group.

They will use a method they decided upon to create the teams. We may need time for trial and error here, as well as how to divide ‘leftovers’ when the class does not divide evenly!

Once they have successfully created even teams, I will pose questions (I will use the family feud style of questioning, so every student gets a chance to answer, can ask for help, can challenge another group’s answer, etc)

First question: Have we divided the teams evenly? Are they ‘fair’? (There may be discussion about this, especially since students sometimes want ‘smarter’ kids in their group!)

We will have to talk about ‘leftovers’ and how to divide them. (Options: let the students adjust groups until we have even amounts, or give the extra students jobs before the exercise begins so we don’t have to divide any bodies!)

Q: (work on in groups) How would you describe your team in relation to the rest of the student teams in the room? ( this is where I will go around and listen to what the students think about their relationship to the other groups. I want to find out about their level of expertise with the language of groups- I anticipate that I will get anything from ‘one group of five students out of 20’ to ‘1/4 of the whole class’.

I will have the students share their comments and ideas. I will then let the students give each other feedback on the reasonableness, accuracy, and will take the opportunity to guide the vocabulary for the discussion, by restating, and asking students to restate.

At this point, I expect to have identified any conceptions about fair sharing and language of sharing the students have.

Now we will fill in the chart:
How many students in each team?
How many teams?
What was the total number of students in the class
So what can we say about each team?

I will have the students tie their comments to the chart, walking through the naming conventions they use. If there is not a connection to fractions in the discussion, I will shape my questions: ex. Do they see another way we could name each group in relation to the whole?

Once we have a fraction on the table, we can start rebuilding. (At this point, I would have them go back to their seats.)

I would flip the table at this point, to show the connection to the fractions they usually see. I want them to make the connection from part to whole.

The next step will be to reconstruct our groups: if one group is 1/nth of the whole class, how will we talk about two groups in relation to one? What about the whole class in relation to one group? (n here represents how many groups a class actually comes up with, but this could also be the goal, to start with two teams, three teams, four teams, and have the students introduced to the mathematical extension of ‘n’!)

After this discussion, I would reinforce the problem solving with the penguin problem, (how do we share 30 snowballs among three penguins?) using small groups. At this point, I will be checking in with each group to hear what they are saying and to reinforce the ideas of sharing, naming and reconstructing. I will be listening for understanding.

All of this should only take a fraction of the lesson time. I am asking the students to think about fractions in a primary way, to connect the more elaborate work of operating with fractions. The students will be encouraged to use the language of fractions, wholes and reconstructing as they work with grade level problems involving fractions. Students will be encouraged to break down fractions or reconstruct them to assist in solving problems, with drawings, with manipulatives, or using the fair share table. Then they could begin to operate with the fractions, understanding how to combine groups, and in further lessons, what happens when groups are multiplied and divided.

I see my understanding of how children instinctively do this work informing my ability to identify students who have “lost” or buried the skills and to assist in reconstruction of this knowledge to enhance and strengthen their future conceptual understanding.


CCSS: Key Tasks and Scientific Method

Common Core State Standards (CCSS) are lists of things kids are expected to learn in each subject during each school year. They are not the actual lessons. Your child’s teacher and school system will decide what lessons to teach and how to teach them.

Research shows that one of the best ways to teach children is to give them tasks and have them work through the solutions to the tasks. The child is given directions as needed, or small lessons on specific skills, but the child is allowed to figure out what skills they need, or what they need to know, to accomplish or solve the task. These are called Key Tasks.

To be able to participate in the process of Key Tasks, a child needs to have a way to approach and organize the information. In mathematics, this process is going to feel very different than the existing process of “show the child the problem, work the problem with the child, and then let the child practice problems. Without setting up the structure of the process first, students may feel that the teacher has abandoned them, is not really teaching them anything, or worse. It takes time to transfer responsibility for problem solving when a child has never been asked to shoulder the responsibility (except for remembering how to do some practice problems, or work a formula – not really learning, just memorizing and regurgitating information).

The process of approaching and organizing the information is very similar to the scientific method. Any teacher or parent can help a child learn how to approach these new key tasks by teaching them a “mathematical thinking” process. The steps in the process are simple ones:  Read and think about the problem, draw the problem or restate the problem, discuss the ideas about the problem with others or use resources (group discussion), estimate the answer, “mathematize” the problem (a formula or equation), try/refine/rethink, see if the answer makes sense. These steps may need to be used more than once throughout the process. As you can see, the actual answer is only a tiny part of the process.

For teachers, here is a brief lesson for teaching students how to use mathematical thinking in the class. For parents, you can help your child use this pattern with their math homework:

I love the idea of math as a thinking process. Instead of just giving students the list of actions, , I would approach this the same way I like to approach setting up classroom norms. I would start with a sample problem and just have the students think about it. I would tell them not to try and solve it yet. Then I would have them verbalize their thoughts in small group and then whole group- we would write the thoughts/assumptions on a big sheet of paper titled ‘Thinking’ and tape it to the board. I would facilitate with clarifying and summarizing questions.

The next step would involve visualizing. Students would be asked to draw a picture to illustrate what they saw happening in the problem. Again in small groups, they would create a picture/illustration, titling the poster ‘visualize’. The posters would go up around the room. At this point, I would ask all students to move around the room ( in groups) and visit the posters to see if the illustrations made sense, and if they suggested any mathematical way to look at the problem.

Returning to their seats, each group would create another poster titled ‘mathematics’ showing the calculations / solutions that they came up with. Those would go up on the wall. Each student would then be instructed to visit each mathematics poster and decide if those mathematics/answers made sense in light of the problems. This might be a good place for the students to use sticky notes and place comments or questions onto the posters.

The groups would then go back to their posters, check out the comments (as would I, so that I could come up with more questions) and we would come back to a whole class discussion to examine the various mathematics and reasonableness of answers. I might put up an empty poster titled ‘revisions’ so that students could add ways that they will need to revise their own thinking to solve this and future problems.

The summing up of the lesson will not be right or wrong answers, but a summary of the process itself. The students will be asked to discuss with one other person what steps they went through to solve the problem, and then write a brief paragraph in their math journal about what steps they took to solve the problem.

The final poster, and the one that will remain on the wall for future work, will be the steps the students noticed were common to the process- maybe have each group list a step (different colors, handwriting?). This will give them a roadmap to solving future problems- and as the teacher, I will give them the time to use the steps as we move through the learning process.

Parents can help by giving your children the time to verbalize problems to you or draw what they think the problem represents. You don’t always have to know how to do the math to help your child think through the process. Even if they don’t come up with a “right” answer, or maybe all they have come up with is questions, the thought process is going to give them a way to get in on the next conversation in class.



Subtraction or Adding a Negative: Tracking change

In response to a recent article about the state of teaching of subtraction in schools vs teaching children to add the inverse or the negative: Mathematical computation is about change, movement. (Link at the bottom of the post.)
To move along the number line, in whatever direction, or plot one point to another on the Cartesian or imaginary planes, is to chart change. The direction, positive or negative depends on the starting place.
The example: Jonathan has two apples, but if you subtract one, how many does he have, is not about a negative apple (there us no such thing!) but about the change with respect to possessions; the movement from one amount to another. It is relational, depending on who holds the apples and who is receiving the apple.
Astronomy led to expression because of the movement of the heavenly bodies. Calculus is the expression of movement of all sorts of actions. The “rules” of math; why 2 acts the way it does, set definition, the differences of movement in a Euclidean world vs a spherical one- all of the rules are predicated on understanding and defining observed change, or predicting future change.
Somewhere along the way, the vision of math, the way we share this lovely process with our children, has been turned into some cookie cutter process. We lose the observation of this movement by disconnecting it from change and giving children math problems with no relationships to anything but counting. Showing that numbers can be broken apart and recombined, that they are fluid and can show change, (number sense we call it) is critical to math knowledge.
Subtraction is movement away from the center of one person or place- addition is movement toward a person or place. It is relative to the location of center.
As we get older, more mature, we begin to understand that we are not the center of the universe. Until then (and this is the teacher in me) we believe we are the center, that movement away from us is loss- subtraction, if you will. Take-away is a valid way to teach movement away from (possessions like apples moving from one position to another, or reducing numbers by other numbers, or defining the distance from ground to sub-basement, or of one planet’s orbit around the sun), especially to younger students.
As students get older, we can continue with the idea of movement: the concept of adding a negative number works well on a number line. It serves to explain the process of movement – in the multiplication of negative and positive, or negative and negative, numbers. It is a difficult idea for students who do not grasp the true nature of numbers. The words used to describe what is happening in math and the ensuing confusion are understandable- perhaps as suggested by a colleague who creates mathematical texts, we can simplify the terminology. Until then, we need to teach and talk about mathematics in as many ways as we can. There are so many ways to get to the accurate answer, one of them is sure to resonate with our students, or with each other as mathematicians, and each can be the correct way, no matter the language.