Make Math Moments Academy › Forums › Full Workshop Reflections › Module 4: Teaching Through Problem Solving to Build Understanding › Lesson 41: The Stage of Mastery › Lesson 41: The Stages of Mastery Action
Tagged: @jon

Lesson 41: The Stages of Mastery Action

Select a concept you will be teaching in the next week. Use the Stages of Expertise template to brainstorm what each of the stages might look like including:
 Possible student thinking
 Student responses
 Math concept used by the student
Add a comment here to share what you came up with!

This is a great tool to get the teacher thinking about success criteria and shortrange planning. I like the idea of recording and thinking about what the students will be thinking as they progress through the levels. It helps to see where they might be and what prompting the teacher might do to get them to the next stage of understanding.

I like the idea of thinking of the students going through these stages to solve problems. I feel like this doesn’t just apply to math, but to almost any new thing we learn in life. Since we are not in school right now, I can’t work through a specific problem with these stages, but I can think back to many situations where I feel like students experienced these. However, I feel like maybe they don’t always go through all of these stages when they are just trying to solve a regular pencil and paper problem. I feel like they may not actually feel conciously masterful because they may just have gone through the motions and got the correct answer, but maybe were not sure really how they got to that point or did not feel that they had fully mastered that skill. If they had started with a more concrete visual representaion they they might have felt more success in the end.

These stages made me think about the different ways my own children (14, 15, and 19) relate to math. For my older two children, math comes pretty easy to them and it seems that they move to the 4<sup>th</sup> and 5<sup>th</sup> stages pretty quickly. However, my youngest struggles much more with new math concepts. A few weeks ago, she was becoming frustrated with her math assignment. Her older brother tried to help her, but his explanation was a little too complex. I tried helping her, but it was something I was not familiar with. She and I watched a few videos on it and we began to understand the basic concept, but not the broader concept. My oldest then joined us and explained a little further how the process worked. At that point in our learning, my daughter and I were able to ask more specific questions on what we did not understand. Even though we had a better understanding, were still probably at the second stage. Later, my middle son asked why his explanation didn’t help his sister, but that his older brother and I were able to help. What I realized and shared with him was that her stress level came down and she was more comfortable with not understanding when there was someone learning with her and she was able to work through the learning at her own pace. I realize now how important it is to evaluate where students are in the stages of mastery in order to adequately support them.

I’ll be introducing Absolute Value.
Unconsciously Incompetent: I can start at zero and count how far away it is.
Consciously Incompetent: I know there’s another way to do this without counting, so maybe it’s the distance is the number?
Consciously Competent: I know that the number represents that distance it is from zero, and the absolute value will always be the distance. I wonder what happens with negative numbers?
Unconsciously Competent: I know that the distance a number is away from zero is always positive, because distance is not negative. Therefore, regardless of if the number is positive or negative, the absolute value will always be positive.

I have middle school students who have not grasped the concept of what fractions/ratios are equivalent to 1. When I review this concept with them/reteach, I can see the stages of expertise as I go through it with them.
Stage 1: Many think 1 is the same as 1/4 or 1/5. They do not understand the rationship between a fraction versus a whole number.
Stage 2: They may understand that 1/4 and 1/5 is not the same as 1 but they don’t understand why and still give the same answer.
Stage 3: Student may realize that ratio is the same as a whole. They may visualize a pizza with 4 total slices and think 4/4 = 1
Stage 4: Student may know that 1/1 = 1 due to Automacy. Memorized, like multiplication facts.
Stage 5: Student can represent their thinking in may ways. A circle broken into 4ths show that 4/4 is the same as 1 whole . Fraction bars showing four 1/4 pieces is equivalent to 1 whole. There are many more strategies that could be shown.

For some reason I am having difficulties with printing and posting the paper I completed. So I will try and type it all out:
Adding 3 digit numbers
1. Not knowing where to start. (Hundreds First)
2.Students that are adding the ones first might run into the issue of not being able to regroup correctly.
3. Adding and regrouping the ones correctly but not the tens place
4. Adding and regrouping correctly 70% of the time
5. Can solve the problem correctly most of the time and by using several different strategies, such as using standard form and expanded form.
123 100+ 20+3
345 300+ 40+5
468 400+ 60+8=
468

For my grade 2, I will be working with addition up to 20. For example 15 + 2
Unconsciously Incompetent:
Students will use manipulative or count 15 by 1’s and then add the 2 (again the whole concept skip counting by 1)
Consciously Incompetent:
Some students might skip count by 5’s to 15 , then add the 2
Consciously Competent:
Students will start with the biggest number 15 and add 2
Unconsciously Competent:
They will just say the answer, but cannot represent their thinking in anyway!
Consciously Masterful:
Students will understand that we start with the biggest number 15 and then add 2
They can also represent their thinking using a number line or base 10 blocks to represent the 10 + 5 + 2

Since we are on summer break, I decided to start with the topic place value since that is where I will start off with my 4th graders at the beginning of next year. This was kind of a struggle for me. I was not sure what to put for student responses in the first two stages. I would love any feedback about how I could alter my levels of mastery to make it better. Place value is something some kids really struggle with in my experience. We use manipulatives: place value blocks, place value disks, and drawn models to show to regroup. I was thinking of even using fake money this year to see if that would help any of my kiddos. Below is what I came up with, but again I would love any feedback.

Thanks for taking a stab at this, I’ve been thinking a lot about place value as we’ve worked through this workshop and have struggled to envision using the strategies with this topic too, so I’m so glad you started. I wanted to share a few thoughts…
I wonder if what you have for CI could actually be UI… maybe the starting point is that students can show a one, ten, hundred with a concrete model and that the fact that they are different sizes would then be obvious, but they aren’t sure by how much.
Then what you have for CC coul be CI… I’m aware there is a pattern in the changes from one place to the next, but I’m not sure what it is. I might try +9, +90, etc.
CC would then be becoming aware of the pattern of x10 going from one place to the next, maybe by setting up a table and showing the repeated relationship. I’m not totally sure on UC then, but maybe it would be just having internalized the rule with automaticity and not needing to see the pattern to remember it.
Thanks again for posting, this really helped me wrap my mind around what this could look like… let me know what you think!
 This reply was modified 1 year, 6 months ago by Jon Orr.


I found this difficult! I guess I am “unconsciously competent” leaning towards “Consciously Competent”! I chose a lesson on drawing parallel lines.
1. Student does not know /remember definition of parallel lines or protractor.
2. is able to draw parallel lines using a ruler and is beginning to figure out how to use the protractor to draw parallel lines
3. Can follow the steps to draw using a protractor but this is still difficult
4. Can use a ruler easily to draw parallel lines.

In learning to add decimals:
UI: I know the digit farthest to the right is always in the ones place and that numbers get ten times larger as they move to the left.
CI: I see there is a dot in the middle of the numbers, but think it works like a weird comma, and don’t know what to name the those numbers.
CC: I know the dot is a decimal that separates the whole numbers from values less than one, and understand as they move to the right, the numbers get ten times smaller.
UC: I can add decimal numbers like a normal number as long as I keep the dots (decimal) in the same column.

I decided to see if I could break down learning about the concept of initial value in linear equations based on the video from Lesson 36 using Stacks of Paper. Given 1 pack of paper = 4.95 cm tall and that 5 packs of paper on a table is 130.75 cm tall, find the height of the table.
U.I. The student adds 4.95 five times or multiplies 4.95 by 5 to get that 5 packs of paper is 24.75 cm tall. They then subtract 24.75 from 130.75 to find the height of the table is 106 cm.
C.I. The student makes a table of #packs vs. height from floor by starting with 5 pks plus table is 130.75, 4 pks is 125.8 and on down to 1 pk is 110.95 and 0 pk (just the table) is 106 cm. They think there must be a way to generalize this to any number of packs stacked on a table, but aren’t sure how to do this.
C.C. Student knows they can multiply #packs times hgt/pack and add the table height so they can write total height = #packs(4.95) + table height and then 130.75 = 5(4.95) + 106 which they can connect to y = mx + b once they are shown how.
U.C. Student automatically looks for dependent and independent variables and solves for initial value using y = mx + b whenever they are faced with a situation like this.
C.M When learning another function family, perhaps exponential functions, student postulates that the yintercept may be the initial value and tests this theory out.
Would appreciate feedback as I’m not sure if I’m mixing too much in and not making it granular enough.

The concept I decided to focus on for this assignment is a major one in algebra and prealgebra and has to do with student’s basic understanding of how functions work. The Lesson Goal is “understand that a function assigns to each xvalue (independent variable) exactly one yvalue (dependent variable)…” Traditionally I’ve used mapping diagrams, sets of ordered pairs, the vertical line test, etc. and still a number of kids get the meaning of functions backwards. What I’d like to use is the analogy of a Coke machine. Here’s how students might progress through it.
Unconsciously Incompetent
With a vending machine you put something in and get something out.
Consciously InCompetent
When you press Coke, you should get Coke.
Consciously Competent
I can tell whether or not the machine is working based on whether or not my selection (input) matches what I get out (output).
Unconsciously Competent
When you press Coke, you should get Coke, but if you always get Sprite instead, it doesn’t mean the machine is not functioning, it could just mean someone put in Sprite where Coke belonged.
Consciously Masterful
A function is where I get one output for each input, whether or not it was the output I originally expected.

My first concept this fall is area and composing and decomposing shapes to find area.
Unconsciously incompetent: I can count up the number of squares in the shape
Consciously incompetent: I know there is a better way than counting all the squares, but I don’t know what it is. Or what if there are no squares?
Consciously competent: I can find area by making a unit square of any size and use the array model idea to find the area.
Unconsciously competent: I multiply the width by the height to get the area. I can do this to all the various shapes and put it all together.

I found this really difficult to do since we are not in school and I’m not sure that I captured the steps correctly. I’m attaching my worksheet.
 This reply was modified 1 year, 7 months ago by Melissa Sutton.

@melissasutton I think you’ve done a fine job with this assignment. The big idea is to think about how a student would move from not knowing into becoming fluent with a skill/idea. Thinking through the steps helps you recognize where a student would be on this trajectory and how you can help them move onward. Good stuff.

Topic: Comparing Slopes of Proportional Relationships in Different Forms (8th Grade)
 Unconsciously Incompetent
 Can point out that lines look different
 May be able to use words like ‘steep’ or ‘shallow’
 Struggles to see slope (steepness) when not in graph (visual) form
 Consciously Incompetent
 Can identify which line is ‘steeper’ when both in visual form [can begin to compare]
 May be able to identify slope from equation
 Struggles to make comparisons from table (without visual)
 Struggles to identify how much steeper something is
 Consciously Competent
 Uses triangles to identify slope visually, with “easy” sides
 Can use ∆y and ∆x from table (subtraction) to identify the slope as “rise over run”
 Is able to find slopes independently to say which one is “steeper” and uses subtraction to indicate by how much
 Unconsciously Incompetent
 Can look at two relationships and compare, often using the unit rate (1, r) to make the comparison immediately
 Consciously Masterful
 Understands that the slope appears as the unit rate in proportional relationships, and can demonstrate where it appears in graph, table, and equation form
 Is able to make the comparison to unit rates in proportional relationships to the slope in a linear relationship, especially in graph forms
 Make comparisons with division of proportional relationship slopes, and knows that they can look solely at ∆y, so long as ∆x is constant.

Skill: add two digit numbers
Unconsciously Incompetent: Student uses manipulatives or fingers to attempt to count to get a sum.
Consciously Incompetent: Students starts with one number and counts up?
Consciously Competent: Student adds numbers, but doesn’t carry over when number is over 9.
Unconsciously Competent: Student adds numbers correctly carrying when number is over 9.
This is all a guess as I have never taught fourth grade, but I see that we teach this within the first unit. Anyone have experience with fourth grade and students’ stages, please chime in so I can better anticipate what I’m up against this fall!

I first thought about finding the slope of a line on a graph. There is a lot of prior knowledge there and I have really only taught it through memorizing rise or run then reduce. I could only think of stages 4 and 5.

This was challenging for me as it’s a new way of looking at student expertise and a new grade level. I took an easier concept. Would love to see other ideas of how to break down other concepts (i.e. decimals, fractions, patterns, geometric concepts…) to use as a model.
So… grade 4 solving problems involving addition and subtraction of 4digit numbers (e.g., 2135 – 1982)
UI: I know I need to stack the numbers (having seen or been taught only the standard algorithm and trying to follow the procedure)
Student might not know what to do when borrowing needed and just subtract the lesser top number from the greater bottom number (e.g., 83 in tens column instead of 13 – 8) and not understanding the difference.
CI: “I know I can use a number line to count up instead of count back, but I can’t remember how to do it.” or “I know I can use base ten blocks to help me subtract, but I don’t know what to do when I don’t have enough tens/hundreds…”
Student also may not know where to start with student generated strategies he/she has seen.
CC: “For me it is easier to add than subtract so I can use a number line to count from on from 1982 by 10s and 1s to get to 2135.” Student may use base ten or number line ideas from CI above and be able to accurately solve problem, although maybe not using the most efficient way.
UC: Student may use a number line still, but be able to make larger jumps, rounding
(1982+18 = 2000 + 135=2135. So 135 + 18 =145 + 5 + 3 = 153)
CM: May be able to see multiple ways to manipulate the numbers (e.g., If I add 18 to 1982, I can round it to 2000, so I’ll have to add 18 to 2135 as well. 2000 from 2153 is 153) and calculate the solution by visualizing the scenario in his/her head.

My students struggle with Conservation topics (momentum & energy) quite often. I purchased an elementary balance to show a visual/concrete representation preCOVID and I think it will work nicely here before we do the math. We will use similar objects to balance at first: 10 on left, 10 on right. We will then change objects but still try to get it to be level: 10 on right, 5 larger ones on left. Make estimates about mass of left side compared to right side, etc. Finally, we will talk about conservation of quantities within the system as simply moving from right bin to left bin. Nothing is being added to the, “balance system” it simply changes sides/location.
Unconsciously Incompetent: I know there is energy in the system, I just don’t know what it is or how it got there.
Consciously Incompetent: I know there are 2 types of energy: kinetic & potential, but I don’t yet know how to apply conservation concepts to a system nor do I fully trust that energy can be changed from one form to another.
Consciously Competent: I know that if something moves, it has kinetic energy. I know that if something is lifted above a ground level, it has potential energy. I can calculate KE & PE and I can also solve for velocity or height if those values are known.
Unconsciously Competent: I can look at the parameters of a system and determine the amount of kinetic energy, potential energy and total energy. If 2 of these values are known, I can determine the missing value. I fully understand that energy can change form and can explain how/why it happens.
Consciously Masterful: I can build a plastic tube rollercoaster (tygon tubing and a bb) and based on the initial measured height of the system I can determine the amount of KE and PE and total energy at any location on the journey using principles of conservation.


Solving (multistep) equations:
Unconsciously incompetent: Students just guess numbers that “fit” into the equation. Often students don’t know how the value they found is right or wrong. Guess and Check
Consciously incompetent: Students know they should follow a certain process but don’t remember it so they default to guess and check. However, I feel these students can tell about their thinking.
Consciously competent: Students know that they should be using opposite operations on both sides of the equals sign. They can “unpack” the equation to get the variable alone by “undoing” the parts given in the equation by working backwards.
Unconsciously competent: Students have internalized and follow the steps for “undoing” each operation in the equation.
Consciously masterful: Students consistently solve equations correctly (and know that they do) because they understand that only one value would make the equation true while all others would make it false. Given this they are able to check their answers. Additionally, they are able to use, or rather, not use opposite operations when not necessary (as in one step equations) because they understand they are looking for the input that will give you the desired output.

A lot of equation lessons in this discussion…so may pieces that need to be at least conciously competent for this to all work out!


1. Unconsciously Incompetent
Upon hearing that we are going to be discussing solving equations through equality, possibly at this point I will receive a blank stare.
2. Consciously Incompetent
I may show a picture of objects on each side of the equation and ask them to tell me what they see and wonder? Students may have some idea of the concept yet still not sure where this may go.
3. Consciously Competent
Then I display an equation with numbers and variables on both side and the concept may start to take form, yet they know that both sides need to be equal.
4. Unconsciously Incompetent
Then, I prompt the students through solving the problem. After awhile they think, they have it and I change the problem up and they realize maybe they do not have it.
5. Consciously Masterful
After catching themselves several times, they gain conceptual understanding and competency thus enter mastery stage.
 This reply was modified 1 year, 6 months ago by Jacquelyn Harland.

I really like the way you explained this and I feel like I have walked a mile in your shoes!

I’m not 100% certain I used the Stages of Expertise accurately. I feel like the 3rd and 4th stages threw me off a little bit because they are so similar. I gave it a shot though. I focused on adding decimal numbers. I started with the idea that so many students usually just use their prior knowledge of adding whole numbers and don’t think to use place value and line up the decimals. In stage 5, I ended with students being able to use place value (maybe even mentally) and estimation to assess reasonableness. There might be some visual/manipulative representations that I could add in stages 2 or 3. I have had some difficulty using manipulatives, such as base ten blocks, with decimals. It always seems to confused the kids because they are used to using base ten blocks for whole numbers. I would welcome any feedback on these stages. This did help me really think through the stages that kids go through when learning a new concept.

There is definitely no “right way”, but rather to use this as a template to think through how students might work through developing a deep understanding and essentially build automaticity with concepts. I like the way you thought this through…


4th grade place value
UI – Students have a general idea of place value from 3rd grade, but do not know how many times greater when moving from one place to the place at the left (ones place to tens place)
CI – Aware that there is a pattern when changing from one place to the next place.
CC – Aware that the pattern is times 10
UC – Able to move from one place to another such as ones place, to tens place, to hundreds place by multiplying by 10. Or ones place to hundreds place
CC – Develop the idea that when moving from a larger place to a smaller place you divide by 10.

Skill: Patterning and looking for an ABAB pattern
Unconsciously Incompetent: The student will use a specific manipulative (e.g. colored teddy bears) to figure out a pattern.
Consciously Incompetent: The student will activate his or her prior knowledge about patterns and experiment with the manipulative to figure out which pattern he or she will solve and figure out.
Consciously Competent: After first seeing a teacher modelled lesson about the possibilities of patterns, the student will then be able to pick out two different colors to make an ABAB pattern.
Unconsciously Competent: The student will reinforce his or her understanding of the newly acquired concept by extending the pattern using two new different colors. He or she will then also be able to extend the thinking by trying out a different pattern (e.g. AABAAB), and explain the thinking to the educator.
I have seen many FDK students grasp this Math concept fairly quickly so this is the reason why I would look at doing this Math strand early on in the year (e.g. September).

I am making an assumption that students have reached at least consious competence with finding thecircumference and area of a circle. I would like to address the surface area of a cylinder. Stage 1: Students would have differing thoughts about what surface area is and what shapes make up a cylinder. Stage 2: Students realize the concept of surface area and the shapes but have difficulty figuring out where they get the length of the rectangle. Stage 3: Students are capable of finding the areas of circles and they find both not realizing quite yet they are the same. Students find the circumference of the circle for the length and multiply by the height…they realize they need to add all sides together. Stage 4: Students find the area of 1 circle and then multiply by 2, find circumference of circle and multiply by theheight, add all sides together and assign the correct unit squared. Stage 5: Students can find the surface area of a different cylinders with a vriety of rational numbers.

I really like this tool for anticipating student responses and developing teacher moves ahead of a lesson.
I was thinking of graphing lines using y=mx+b.
1. Using a table to graph the line
2. Plotting one point then using the slope to find the next point OR Plotting the yintercept then finding another point. Use those two points to draw the line
3. Plot the yintercept then use the slope to go up and right or up and left.
4. Being able to use the equation to graph a line that does not fit nicely within the window of the coordinate plane being used

I took the expectation of drawing shapes using millimeters. I like this tool because it helps you understand where students are on the continuum of learning and what they are ready to be taught. It could also help you with math groups to have students only one stage of expertise away from each other? The discussions and helpful tips from each other would be very valuable.

What a great tool for planning a lesson. A backward design approach could be used here too. Start with thinking what you hope they would achieve at mastery level and then move back. This could be used for assessment too or even to group students into guided groups depending on where they are on the trajectory and how they struggle or move through these stages.

I have tried my hands on the early skill of number sense counting 120

Grade 4 rounding to nearest 10, 100, 1000.
Stage 1 Student uses a number line to visually see if the number given is closer to 0 or 10. Student needs visuals to round numbers to nearest 10, working up to 50.
Stage 2 Student uses a rounding poem or song or “rule” to round numbers to nearest 10 or 100 or 1000.
Stage 3 Students do not need to refer to number line or poem or rule, they just know how to round to nearest 1000.
Stage 4 Students can round 4 digit numbers to the nearest hundred, 3 digit numbers to the nearest ten, etc. They can do this because they have a solid understanding of place value.
Stage 5 I’m actually not sure where to go from here!!

This is a concept we cover in chapter 2. I do not give them the equations for percent proportion (which I despise because the book teaches it by using cross products and I am not in favor or that as there is no conceptual understanding there, it is a “trick”) or the percent equation right off so this was my thinking for the stages of expertise concept map. Would be grateful for any feedback on the beginning stages I am presenting here or on how to transition to the percent equation.

Concept: Finding the midpoint of a line segment given two endpoints (especially ones far away, like (1,12) and (9, 8)
Unconsciously Incompetent: Draws a picture and attempts to visualize the middle, counting the length down and over on the coordinate grid.
Consciously Incompetent: Same as previous but is aware that there must be a better way.
Consciously Competent: Subtracts coordinates to find x and ydistance, cuts each distance in half, and adds to/subtracts from one endpoint – may use a diagram like a number line or right triangle
Unconsciously Competent: Finds the average of the xcoordinates and the average of the ycoordinates. Also may use the previous strategy with mental math.
Consciously Masterful: Can explain how both of the above strategies arrive at the same conclusion.

It’s July so I can’t teach my class but I did have a Number Talk saved in my Drive from my grade 3 class that I can speak to. The question came from the book, “Number Talks” and the question was “39 + 16”. These were the student responses that I received: (below). We were using Lawson’s scale to determine how students were solving mental math problems and their flexibility with numbers. Now I’m wondering if maybe I could have used concrete materials to make the question accessible to all and then I could have also seen how they would have grouped the blocks, etc. to solve (39 red counters, 16 yellow), etc. Some students may have made the leap to multiplicative thinking had we done this instead?

Concept: Linear Equations
Unconsciously Incompetent: Notice a pattern and can use it to reach an answer
Consciously Incompetent: Notice a pattern and are aware that there is a better way to calculate an answer but still just uses the pattern to get to the next solution.
Consciously Competent: Understands and calculates slope (rate of change) to calculate answer
Unconsciously Competent: Uses slope to plot change and uses the starting value to create an equation and to graph a solution
Consciously Masterful: Recognizes that a real world situation can be solved with a linear equation. Also it is understood that by plotting the line they are showing all possible solutions to this real life situation.
Is this correct?

Concept: Solve problems using equivalent ratios.
Problem: There are 5 apples and 4 oranges in each fruit basket. The fruit baskets contain a total of 100 apples. What is the total number of oranges in the fruit baskets?
Unconsciously Incompetent: Students may draw out (or skip count) groups of 5 apples and 4 oranges until they have a total of 100 apples and 80 oranges. This is an additive strategy.
Consciously Incompetent: Students continue with a additive strategy like in stage 1. Perhaps they organize their skip counted numbers of fruits in a table.
Consciously Competent: Students begin to use a multiplicative strategy but not the most efficient one. Maybe they multiple 5 apples by 10 to get 50 apples then multiply by 2 to get 100 apples. Then, do the same multiplication steps with the number of oranges to arrive at 80 oranges.
Unconsciously Competent: Students move to a more efficient multiplicative strategy. Such as: 5 apples x 20 = 100 apples so there must be 20 fruit baskets. So, 4 oranges x 20 = 80 oranges.
<b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>Consciously Masterful: Students continue a multiplicative strategy but use equivalent ratios to find the answer. 5 apples/4 oranges = 100 apples/80 oranges.

I worked with using visual patterns to help students create algebraic expressions. I found it easier to start with the Unconsciously Competent stage and link that to my learning goal and then work backwards. I’m thinking this may work as I learn more about the learning progressions for each domain. I struggled a little with the Consciously Masterful for this example but thought that maybe it would be when students then were able to use substitution to check their own answer and or extend their thinking to more complicated patterns.

Concept: Multiply by multiples of 10. Example: 30×6=
UI: Many deer in headlights. Some will begin to add 6 thirty times, others will remember the commutative property and add 30 six times.
CI: Ask how can they might use what they know about partial products to solve?
CC: Student knows that 10×6=60 and that 3 of those = 180. Can extrapolate to 300×6.
UC: Know to multiply and add the correct # of zeros! (Yes, this “trick” should only be used AFTER they understand this conceptually!)
CM: Flexible with 30 x 60 = 3 x 10 x 6 x 10 and use understanding of associative rule to group (3×6) x (10×10) = 18 x 100 = 1800.



Concept: Adding twodigit numbers.
13+27
1. Students may use manipulatives to create the 2 separate numerical piles, then add up from one through the nextone at a time.
2. The student may remember that they know how to add twodigit numbers to one digit numbers, but are confused by the two digits. They may attempt the part they know or continue to add individual manipulatives up from the 13.
3. The student may combine the manipulatives to look for a skip counting pattern and begin skip counting and adding any extra pieces at the end, depending on how they lined up the rows by columns. They may also remember the math fact that 13 + 27 is equivalent to 27 + 13 and choose to count up by 13 from 27.
4. At this stage, students may choose base ten blocks as their manipulative recognizing place value and create the 2 numbers separately before adding or regrouping. they may also use a place value chart. If they continue using the previous strategy they might multiply the columns and rows instead of skip counting.
5. I can not remember the term but at this stage students might separate the numbers by there place value (ex., 10+20 and 3+7) to solve. Sum might do (13+7) +20 or (27+3) +10.

A concept that I might be teaching next week is exponents. Stage 1 is recognizing the superscripted number, but not knowing its meaning. Stage 2 is the same. Stage 3 is listing out the base multiplied by itself based on a diagram. One option is dimensions (only really works for degree two or three). Another option are tree diagrams. Stage 4 is seeing that exponential form is shorthand for repeated multiplication. Stage 5 would be combining two tree diagrams or two scenarios to show that you add the exponents when multiplying together two numbers in exponential form that have the same base.

I found that not being with the students right now made this is harder than I thought. Here is my go at subtracting multidigit numbers. I can see how this would be very beneficial when teaching a topic, when its a new topic or something you have been teaching for a while.


Topic: Multiplying double digit numbers (This is what students in the past did)
Stage 1: Students used repeated addition to get the correct answer. This led to mixing up numbers and losing track of where they added.
Stage 2: Students attempted different strategies and could set it up, but not be able to solve the problem correctly.
Stage 3: Students are able to chose a strategy that works for them and solve it correctly. (I find most students are at this stage. I want to find ways to continue to push them to Stage 4)
Stage 4: Students are able to answer the problem in their head. The only problem I have had with this one is that the student was able to get the right answer, however when I would ask them how they did it, they could not explain it. Is this a problem????
 This reply was modified 1 year, 5 months ago by Gregory Napoleon.

After reading some of the other posts, I’m in the same boat as a few others. It’s summer, I don’t have students in front of me and I’m moving to a new grade level that I’ve never taught before . I took a concept that I know we will be hitting in 4th grade as well as what I learned from teaching 3rd last year. I’m hoping I am on the right track!

Since it the summer time, I am not teaching at this moment but I am using this class to help me plan for the coming year.
The concept I would be introducing is slope.
Unconsciously incompetent: I know its rise over run when see it on a graph.
Consciously incompetent: I know there is another way to find slope with points and that you need to use x values and y values but can’t put it together.
Unconsciously competent: I know that you subtract the y values to get the rise and the x values to get the run. I wonder how this gets the slope and used in the equation.
consciously competent: i know that when finding the differences of the x and y coordinates that they can be divided to find the slope which is the same thing as counting the rise and run of line on a graph.
Just writing this out has helped me understand how to look for students understanding of the concept. Thanks

Fantastic!
Isn’t it shocking how helpful taking a few minutes to reflect on new learning can be? I am constantly wondering to myself where students are along this particular path.


Topic: Transformation of Graphs
My brain is on teaching jargon overload. I think I’m thinking of these stages in the process correctly, but any feedback would be great! I know these stages in learning exist and have seen students flow through them – just having difficulty pinpointing specifics right now.

Ha! Don’t worry about the jargon as much as the big ideas under the jargon! From reading your annotated PDF, it appears that you’ve got the idea here!
For me, the big take away is that often students don’t know what they don’t know, then eventually move to becoming aware of what they don’t know…
Then, they start to gain some knowledge but they must work hard to think it through / use what they know…
Finally, they get to this place where they can almost automatically do things and it is almost masterfully!



As classes have not started back for me yet, I took a concept I like to start my Grade 7 class with. I found it difficult to isolate the individual stages they would be going through, even though I feel I understand how they progress all the way through to consciously masterful (if that makes sense).

Estimate to Add MultiDigit Numbers
234 + 127
Unconscious, Incompetence: “I think I am supposed to add these numbers.”
Conscious, Incompetence: “I know this has something to do with rounding, but I can’t remember the rules I learned last year.”
Consciously, Competent: “My answer will be incorrect if it is the exact sum of the numbers shown. In order to find the answer, I will round the given addends to numbers that are much easier to add. Then I will find the sum.”
Unconsciously, Competent: “Estimate means to round. I will round both numbers and find the sum.”
Consciously Masterful: “Because I am being asked to ‘Estimate to find the sum,’ I know I have options. I can either estimate using rounding or use compatible numbers. There is more than one correct answer to this problem.”
 This reply was modified 1 year, 4 months ago by Nicole Jackson.

Currently I am working with six students who are not succeeding in Algebra I. These students have been a problem for the teacher, so I offered to pull them and work with them. I instantly realized that they had no background in variables and balancing equations. I pulled out my handy, dandy manipulatives and used my knowledge of Hands on Algebra, from Borenson to work with them. We have now moved to the representative stage or in the Mastery stage, the Consciously Competent phase. They can draw the figures but don’t realize that they are actually solving the problem on their own. I love how I can now use these stages to describe to the teacher what to do to help other students.

I did find it a challenge to think through the stages of mastery for the concept of theoretical probability. There is developing an organized way to list all the outcomes and then using fractional thinking to represent the probability of different outcomes as a fraction. Students needs to develop an understanding that the total number of possible outcomes represent the whole.
 This reply was modified 11 months ago by John Gaspari.

Next week, I’m going to be trying to teach 6th graders what it means to divide. In the craziness of the last years, this seems to have been missed. I see them being mostly at the Unconsciously Incompetent stage right now, “Equal groups HUH”. I want to quickly move through the Consciously Incompetent stage, “I know I need to make equal groups, but how?”, to the Consciously Competent stage, where they might build arrays or share equally (by building piles). I think the Unconsciously Competent stage will be them able to explain why they shared the pieces the way they did. Finally, the Consciously Masterful will look like them knowing the difference between same number of groups and sharing equally among groups and being able to explain that.
Wish me luck!

I will be teaching linear regression.
Level 1: Will be an exploration on desmos called “Fit Fights” to get get the students to level 2.
Level 2: There will be a second Desmos called “scatter plot capture” to hone in on the idea of the purpose of lines of best fit and hopefully moving them into the “conscience and competent stage.
Both of these lessons will be supported with a video assignment for homework to add a layer of support.
Level 3: Will be a lesson on correlation coeffient and its relationship to make sure our lines of best fit are strong respesentations of data. Students will be creating equations for best fits for a 3rd day and can hopefully move to level 4.
The last move to hopefully get student become consciencely masterful will be a final desmos activity called “Is college worth it?” that will require them to put all these ideas together.


I have a class of 7thgrade students who are on track to take Algebra as 8th graders. We have been working on linear relationships and will learn how to find a line’s slope when two points are given. I imagine the stages of expertise will look like the following.
Unconsciously Incompetent: Students can look at a linear relationship through a graph and recognize positive slopes, negative slopes, and identify points that fall on the line.<div>
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Consciously Incompetent: Recognize that the slope ratio changes as the line changes’ steepness, but will not understand how to get to values for the slope ratio.
Consciously Competent: When finding two lattice points on a graph, students will be able to find the change in y (Rise) and put it over the change in x (run) to create a slope ratio.
Unconsciously Competent: Students will be given any graph or Identified two points on a graph and find the slope ratio.
Consciously Masterful: Students will be able to graph the constant rate of change in science and find the slope and interpret the representation of the slope.
I feel this is the stage of student learning for expertise, but I know there is a lot of sparking curiosity and fuelling sensemaking to get to each stage of expertise.
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This structure, Stages of Expertise, refocuses me on the varying degrees between not knowing and knowing; when I take the time to consider these stages of understanding, incorrect student responses still provide me guidance on how to facilitate learning the knowledge. I can see using this as guidance for all of our content; starting off broad and then going deeper.
 This reply was modified 11 months ago by Maryanna Biedermann.

Solving multistep equations:
Unconsciously incompetent: Students look at pairs of “dog walker” diagrams with different properties of equality shown and say it is unbalanced because there are not the same number of dogs on either sides (even if they have different pull strengths and are balalnced)
Consciously incompetent: Students look at a dog walker diagram with a missing dog pull value on either side using a ? and know they can make it balanced by guessing and checking different values
Consciously competent: Students realize they can cross out values from either side and lead to a simpler problem; can start to see the connection between a dog walker diagram and a symbolic algebraic representation when placed next to each other
Unconsciously competent: Students can use the symbolism of variables, coefficients, constants, and an equal sign and do inverse operations – all without the dog walker diagram but referecing that in his/her head
Consciously masterful: Students can name and describe different properties of equality they could use to find a missing value – and how some methods are more efficient than others

Great breakdown and example! I’ve never thought of the dog walker contest for use with developing equality! Cool idea. Reminds me of solve me mobiles!
Your break down of the stages of mastery (or expertise) is fantastic. Way to reflect to help you hang on to this new learning.


Don’t know if you meant this to be funny, but I got a great laugh from it – because it’s crazy true these days!

Juile, I was thinking about using this idea as well. I started with finding areas of squares and then from the area findin the side length. At stage 1 students were “confused and complacent” on the fact that while we could’t find an even side length for an area of 27 tha the calculator could. In step 2, student were breaking up the area of 200 into squares of sizes 10×10… from there we took this idea and applied it to other areas such as 27. We then discussed simplest radical form in steps 3 and 4. I believe step 5 is perhaps connecting to adding subtracting and muliplying radicals… but Imnot quite sure yet on how to first make it visual and then abstract… I think it would be like adding tiles of different sized squares? Thoughts?

@claudiasever love how you’ve broken this concept down into the different stages to really get a sense as to how students might progress through learning this concept.
Now the fun part… how did the lesson go and would you change anything now that you’ve put your planning into practice?