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

Lesson 41: The Stages of Mastery Action
Posted by Jon on May 1, 2019 at 11:53 amSelect 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!
Noel McMillin replied 3 months, 4 weeks ago 87 Members · 107 Replies 
107 Replies

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 2 years, 8 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 2 years, 9 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.


I like to lead them to discovering that by chopping a triangle off it slides into the other side forming a rectangle, which their prior knowledge will help them see they already know how to do this one.


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 2 years, 9 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 2 years, 8 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 2 years, 7 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.

Counting the Value of Coins
Unconsciously Incompetent: Students know that coins have value but they only count the number of coins.
Example: 8 coins: 1 quarter, 3 dimes, 2 nickels, and 2 pennies = 8 centsConsciously Incompetent: Students know the value of coins and can add values as long as the total value remains less than $1.00
Example: 8 coins: 1 quarter, 3 dimes, 2 nickels, and 2 pennies
25, 35, 45, 55, 60, 65, 66, 67 centsConsciously Competent: Students know how to count coin values with totals less than $1.00. They can rearrange the coins to add them in different ways.
Example: 8 coins: 1 quarter, 2 dimes, 3 nickels, 2 pennies
25, 35, 45, 50, 55, 60, 61, 62 25, 30, 40, 50, 55, 60, 61, 62
10, 20, 25, 50, 55, 60, 61, 62Unconsciously Competent: Students know how to count coin values with totals less than $1.00 but they don’t know what to do when the value is ore than $1.00
Consciously Masterful: Students realize that going over $1.00 is the same as going over 100 and can continue to count the value of coins just as they did when the value was less than $1.00
Example: 9 coins: 3 quarters, 3 dimes, 2 nickels, 1 penny
25, 50, 75, 85, 95, $1.05, $1.10, $1.15, $1.16 
Solving algebraic equations:
#1 Students understand a balance model and that two side of the same thing can be equal
#2 Students see a new balance model but this balance model has 3 circles measuring the same as 6 squares. They understand they are equal, but don’t understand how.
#3 Students understand that one circle can be equal to two squares and that is how the model is balanced and learn strategies about how to remove and add equal items to both sides to keep the models on both sides the same.
#4 Students become proficient at solving equations by keeping models balanced by completing actions that keep sides balanced.
#5 Students can begin to understand the relationship of growth in linear equations.

Great work thinking this through! Thinking ahead to unpack the different stages is helpful as you scaffold a learning experience for students to move through them as they are ready.


I had a little trouble with the unconsciously competent stage since this is new learning for them. I don’t think they would fall back on any prior algorithms but I am hoping that the way I present it will help them draw on their prior knowledge of similar triangles and Pythagorean Theorem to develop understanding and expertise of trigonometry.

This task really worked my brain since I’m still wrapping my head around what each stage looks like in action. Here’s my attempt at pinpointing what the stages look like for understanding and implementing the Pythagorean Theorem.

I have a little trouble seeing the final stage from the example: isn’t distribution a whole extra topic? If the goal is multiplication, can we define the mastery stage as a different concept?
EditTalked to my wife who teaches elementary, and watched the following video involving donuts, and this make a whole lot more sense to me now 🙂
Anyway, we’re working on circle theorems in Geometry, and introducing cyclic quadrilaterals. We’ll be working towards the understanding that opposite angles must be supplementary because they open onto arcs that make up the whole circle. Students have already worked with inscribed angles and know those relationships, so I don’t expect this to be a long journey, but it’s what came to mind.
1. I have no idea how quadrilaterals in circles relate to our prior learning. I wouldn’t be able to find the angles in an inscribed quadrilateral unless I know information that doesn’t relate to circles.
2. I know that there must be something going on with these quads, because we studied inscribed angles and all of these angles are inscribed.
3. I see that opposite angles open onto arcs that make up the entire circle, which is 360, and inscribed angles are half of their arcs, so the sum of opposite angles must be 180. I can find the other angle when given one.
4. I recognize inscribed quadrilaterals and automatically apply this rule to problems involving them.
5. I can find inscribed quadrilaterals and use them as a tool in solving more complex problems of angles and arcs in circles.
 This reply was modified 1 year ago by Jonathan Lind.

Glad that you were able to make some sense of the example after watching the video involving Donut Delight! Nice work applying the stages to a concept you’ll be teaching shortly!

I reached out to a teacher this week to see if I could try some things with his students. He is in the beginning stage of introducing fractions to his 4th graders. He mentioned that he wants them to understand the relationship between fractions and division. Because students are just being introduced to Fractions, I chose the prompt: What is 1/2 of 24. This is how I see the different stages:
Stage 1: Students count one to one but are unclear when they have reached 1/2.
Stage 2: Students count one to one making two piles of the same amount each.
Stage 3: Students break the array in two groups of 12.
Stage 4: Students divide 24/2 or multiply 1/2*24 to get 12.
Stage 5: Students can represent 1/2 of 24 as 1 group of 12 or 2 groups of 6.


Even though I am a high school algebra teacher, I have decided to go back and spend some time trying to get students to understand fractions and to stop thinking of them as the monster of mathematics. I especially think this is important, because fractions are such a real world skill. So this is what I think will happen as we go through the stages.
Unconsciously Incompetent–The students have no true understanding of what a fraction actually is, what it represents.
Consciously Incompetent–The students begin to gain an understanding of what a fraction is (a part of a whole), but still have no idea how to do math with fractions, but now they start to see that they don’t really have any idea.
Consciously Competent–Through practice with manipulatives and visuals, students gain an understanding of fractions and learn how to begin do math with fractions.
Unconsciously Competent–The students can now look at the fractions and understand concepts such as a fraction of a fraction is a smaller number than the original fraction.
Consciously Masterful–Students have become comfortable working with fractions and are now ready to start the process over with mixed numbers.

great brainstorming here! Note that specifically students must learn how to represent (model) fractions and then use those models to operate. Check out our problem based units on fractions to help you with this!

Just curious, which fraction lesson should I begin with for high students who really have no comprehension of what a fraction really is, and no idea how to work with them? Also, what lessons should come next? I guess I am asking if there is an order that would work the best, as there is more than one fraction lesson. Thanks.

I might start with the Wooly Worm Race as that is all about representing fractions using bar models and converting between decimals and percentages. Can be really telling as to who knows what. Many students who can use algorithms with fractions are not fluent with equivalence which can be seen quickly in that unit.
Then, maybe exploring some of the piggy bank units – seems simplistic but allows you to connect fractions and decimals to the money model to help.
Finally, going into shovelling the driveway, salting the driveway and pizza party are great for multiplying and dividing.
Hope this helps!


Concept: Percent
1). Students will be able to do the benchmark percents naturally. They may even think in terms of fractions or division rather than percent (e.g. 50% of 91 is half of 91).
2). Similar to student above, but anticipating that soon they’ll have to answer more complex percentages that are based on factors of 100. They may realize that they can always find 1% easily, then multiply the percent (17% of 43: 1% is 0.43 so 17% is 0.43 x 17). I presume students won’t want to do that much work every time and will start to seek something simpler.
3). Students using what we know about the relationships between fractions, decimals, percent as well as understanding of part and whole and/or of equivalent fractions to show multiple ways of approaching percent problems.
4). Honestly, some of my students here will be automatic because they have math tutors and have memorized formulas. Others will “just know” or just “see it”. For the former, I’ll have them explain WHY their method of choice works And have them compare and contrast with other methods. For the latter, I’ll ask them to try to “slow their brain down” to see if they can figure out the subconscious thought process that is happening.


This is a great example. I wonder how we might be able to ask students a problem that will result in the integral without “telling” upfront how to find an integral? What are we learning this skill for? What question(s) might we ask?


Here is what I have…I can’t teach it right now as it is summer, but one of the first things we do when we get back is functions…

In Precalculus they have so many algebra skills that they do in previous math courses that get blended together and that they know short cuts for. This leads to poor algebra skills when given complicated simplifying. Here is how I attach that problem, start basic and always ask why they did a step along with making them show EACH step.

I am so amazed that high school students do not understand concept of fractions, so I used this as my example. Please see attached.
 This reply was modified 9 months ago by Deanna Semyon.

Awesome work here! Yes, fractions are so challenging for students but to be honest, many educators do not have a conceptual understanding of how they work and develop. So much work to be done in this area! Thanks for sharing!

This is what I came up with for integer addition, which is one of the first concepts I will teach in August. It shows a progression from concreteish (counting) to representational/visual (using a number line) to automaticity through understanding, and then using integer addition rules.

I am not sure if I got the stages. I would like to have some feedback on it. Thank you.

This could be exactly the stages one experiences. The piece that isn’t clear (yet) in your work is the conscious / unconscious part. Note that it is likely that students will have struggles in the early stages as you’ve shown, but the jump from one stage to another is highly connected to the awareness of the student in where they are in the journey.


I’m actually having a harder time thinking through this one! Here’s what I’ve got. I think my biggest question is are we only “graphing” students into the 5 stages on the curve if they are able to thoughtfully produce a correct answer? Is a student in stage 1 if they can’t reason through at all, or are they in the II, III, or IV quadrants?
Solving linear equations (example: 3x+2=8)
Stage 1: “I don’t know how to do this at all. Is the answer 5 because 3+2 is 5?”
Stage 2: “I know 3 times something plus two is 8, but I just have to find that something.” Student may use guessing and checking at this stage.
<font face=”inherit”>Stage 3: Similar to stage 2, students treat x like a blank but use inverse operations (though maybe not in a linear form showing the balancing of the equation.) They may know they need to subtract 2 then see what time 3 gives 8. </font>
<font face=”inherit”>Stage 4: Students use a more algorithmic approach by writing out the problem and showing their work through inverse operations and “zeroing out” the two on the left, then “dividing off” the 3 on the left, but may not know why it works.</font>
<font face=”inherit”>Stage 5: Student knows they need to keep the equation balanced in order for the equivalence statement to remain true, so they use this language when explaining their process. </font>

It is important to remember that it is less important to get the stages “right” or “perfect”, but rather to spend the time thinking of how students progress through learning in general. Thinking developmentally is the key here, so don’t overthink which is exactly what stage.


I teach 7th and or 8th grade. 2 step equations are big. I’m not back in the classroom until the end of Aug., but I thought I’d tackle something I tried and know I need to do a better job transitioning from the context we are using to the abstract. I used Steve Wyborney’s Splat! puzzles. I’m looking at my work and now thinking I could back it up further to expressions of How would you represent this picture? And NOT have a target number so there is no way to “solve” it. However, here is what I did for now:
 This reply was modified 8 months, 2 weeks ago by Marion Mulgrew. Reason: noting that I can't try it right now due to summer break

I think I will let kids do these without working out the algebraic representation for a few. Then put up one and NOT give a target number. I will ask, “How else could you represent this?” and see what they come up with. I will hope it builds to using a variable and constant. Maybe even get multiple representations that have a variable and yet are correct so we can decide which is more efficient (combing like terms).
Then move to what to do if we have a total.

I am looking at the concept of finding the Area of a Regular Polygon.

Skill: add 3 single digit numbers by making friendly numbers (example: 4 + 3 + 6)
Unconsciously Incompetent: Tries to add one by one with blocks
Consciously Incompetent: Adds 4 + 3 = 7, then 7 + 6 = 13… I know there was a way to add numbers more easily but I can’t remember how!
Consciously Competent: student notices 4 + 6 = 10, then adds 3 to make 13
Unconsciously Competent: student computes mentally without thinking about the way they are combining the numbers
Consciously Masterful: able to do this automatically but is aware of the process and can explain flexible and creative ways to make friendly numbers

I decided to use a concept I am working on with a tutor student. He is going into Grade 9 but decided to back track with fractions as was taught using an algorithm methodology for adding unlike fractions. I included notations of my thinking of the stages as well as somewhat what he said and came up with at each stage. Hope this is right.
While completing this I was wondering if mastery means that a student is able to move on from concrete an abstract or maybe you did mention that and I need to review the video. : )

I will teach my 5th graders about the volume of rectangular prisms.
Unconsciously incompetent: students do not know what volume and rectangular prisms are. They also do not know anything about the formula to solve for volume.
consciously incompetent: Students understand what a rectangular prism and volume are but do not know how to solve for volume.
Consciously competent students realize they can use strategies like counting the cubes or multiplying measurements to get volume.
Unconsciously competent students understand and know the formula of Length x width x height and they substitute the numbers they are given into the formula.

I don’t start school for another week plus but this will be one of the early topics covered with all students in my 7th grade math classes. I am not 100% sure if I have each identified correctly but I do appreciate thinking more about how my students may be thinking.
Recognizing proportional relationships in tables
UI – Students don’t know what a proportion is at all.
CI – Students multiply cross products because they remember they are supposed to be the same.
CC – Students simplify all ratios to unit rates to see if they are equivalent.
UC – Students use the unit rate to compare values in table.
CM – Students can see the relationships between values in a table without knowing the unit rate.

There is definitely no right or wrong way to break this down. It is more about thinking of a progression of understanding and what you feel is reasonable as we are learning. Thanks for sharing your thinking here.


We started an exploration of Trig this past week, so i’m thinking about the stages based on what happened so far and what I anticipate for the coming week.
Unconsciously Incompetent
– potentially wondering why we’re looking at squares again
– why are we using squares to make a triangle?
Consciously Incompetent

Realizations that this might be more difficult that they first thought

How does this connect with area? I thought we already looked at something like this.
Consciously Competent

Feeling more comfortable with the proof of the pythagorean theorem

Using the theorem to explore the relationships of the sides of a right triangle
Unconsciously Competent

Extend thinking to apply in problem solving scenarios

Being more flexible with the unknown values within the problems
Consciously Masterful

Demonstrating an ability to reorganise the theorem to fit problem solving situations

Asking questions about known values and unknown values involving angles and sides showing a readiness for sin/cos/tan


We will be combining like terms in 7th grade soon. As I think about it now, I guess I would break it down as:
Unconsciously Incompetent: student does not understand an algebraic expression, what a variable is nor the difference between a coefficient and a constant. I picture them just staring at the page or the problem and watching others do the problem.
Consciously incomp. – the student would know they have to add, but might not understand the difference between a coefficient and a constant. I think of the students who just combine everything and end up with a single answer of a coeff. and a variable.
Cons. Comp. – the student can add correctly coef. to coef. and constants to constants.
Uncons. Comp. – the student can not only add, but also when there is a subtraction in the expression, changes to addition, changes the sign and fluently combines. They know how to deal with negatives easily.

I am teaching Slope Intercept Form this week. We had been graphing equations using a table. Student’s who do not know Slope Intercept Form previous to our exploration are at the Unconsciously Incompetent phase (some of my students know it because of outside math). They don’t even know there is an easier way to graph a line without doing a tedious table. I love teaching this because we have graphed so much, they are kind of getting sick of making the tables and are so relieved to find an easier way by the end of this unit.
They are lead to the Conscious Incompetent phase by doing a real world problem that they end up writing a linear equation and graphing it without using a table.
Once we have a discussion about this problem and talk about the components of the slope of the line and the yintercept they are able to see these pieces in the equation that leads them to the ability to graph without a table. They are then able to quickly graph a linear equation with ease and are at the Unconscious Competent phase.
They reach Competent Mastery when they are able to take real world linear problems and are able to graph, write equations, and make predictions about what will happen in different situations.

We are currently in our “Angles and Triangles” unit and one part of the chapter that I enjoy is using the Angle Sum Theorem and Exterior Angle Theorem. We are just wrapping up both theorems and I can see the four stages of mastery within my students as I work with them.
Unconsciously Incompetent: We started with “What do you notice? What do you wonder?” on the first day (which I used as a previous example in the last module). I pasted three congruent triangles and rotated them to form a straight line, thus the three interior angles of a triangle add to 180 degrees. In the end, not one student in this particular class made any connections with the straight line formed by the three angles – chalk it up to plain ignorance. They did not know what they did not know.
Consciously Incompetent: Now we are to the stage where one angle measure is missing and the students are asked to find the missing angle measure knowing the other two angle measures. What took some time for the students is to understand how the three angle sums are a simple equation when equated to 180 degrees. It was even more apparent when variables were added. Many came to a standstill.
Consciously Competent: As of today, most students are at the Consciously Competent stage. They are able to write an equation and solve for an unknown. It is a good place to be but I noticed a disconnect still which is why I want them at the unconsciously competent stage.
Unconsciously Competent: Here is an example of a math problem that separates this stage from the last stage. The angle measures are 3x, 5x – 10, and 20. More than one student raised their hand today and said “What do I do? There are negatives now.” To be honest it is a little soulcrushing that a few negative numbers are a roadblock. However, a student who is Unconsciously Competent does not let that get in the way – it is business as usual.
Lastly, a Consciously Masterful student understands how writing an equation is an actual problemsolving strategy that can be used under a wide range of scenarios. Would not that be a nice place to be?

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?