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Ivan

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https://tinyurl.com/MLMC2026

access all documents here

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Hello, my name is Ivan

I really taught over 23 years� in the inner city!

Over 45 years of teaching and learning from mistakes!

And also finding � a few answers!

What can I offer as a � math educator?

and learning from mistakes!

I am NOT a mathematician!

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Ivan Cheng

icheng@csun.edu

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Redding, CA

https://tinyurl.com/MLMC2026 March 6, 2026

The Journey IS the Destination:

Charting the Course to Student Success

Ivan Cheng

icheng@csun.edu

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The textbook� is NOT the destination!

Getting through the book�is NOT the goal!

And preparing for�state tests is NOT the goal!

So what IS the goal?

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The textbook� is NOT the destination!

Getting through the book�is NOT the goal!

And getting the right�answers is NOT the goal!

So what IS the goal?

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What IS the Point of Learning Math?

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Just for Fun, Try This Problem ….

Attention skills

What information is being presented? What can be noticed?

Analysis skills

What’s going on in the problem? What are the relationships?

Application skills

What representations or procedures might be useful?

What Skills Did You Just Use to Solve This Problem?

.

What IS the Point of Learning Math?

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So…

WHAT

do students really �need to learn?

What IS the Point of Learning Math?

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Students Need AAA Skills

Attention skills

What information is being presented? What can be noticed?

Analysis skills

What’s going on in the problem? What are the relationships?

Application skills

What representations or procedures might be useful?

These are the THINKING skills needed �for solving any problem! .

What Do Students Really Need to Learn?

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Yes, but

HOW

do we teach these 3 THINKING skills?

https://tinyurl.com/MLMC2026

What Do Students Really Need to Learn?

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“If all students could learn by having us just tell them how to do it, we would not have any of the problems that we have in mathematics education today. ...

Student difficulty with mathematics has been a pervasive and systemic problem since the advent of public education—not because students can’t learn mathematics, but because, by and large, students can’t learn by being told how to do it.”

Liljedahl, 2021, p. 172

“If all students could learn by having us just tell them how to do it, we would not have any of the problems that we have in mathematics education today. ...

Student difficulty with mathematics has been a pervasive and systemic problem since the advent of public education—not because students can’t learn mathematics, but because, by and large, �students can’t learn by being told how to do it.”

Liljedahl, 2021, p. 172

How Can We Teach Students How to Think?

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Ivan’s Corollary:

If textbooks were enough to help students learn math,�we would not need conferences

to figure out how to help students learn math!

How Can We Teach Students How to Think?

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Textbooks are only vehicles�to help students reach their destination!

How Can We Teach Students How to Think?

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?

Teachers Must Disrupt “Studenting” Behaviors

“Studenting is what students do in a learning setting”

(Liljedahl, 2021, p. 7)

Slacking
Stalling
Faking
Mimicking
Actually trying

Which of these behaviors have you seen �(or experienced)?

How Can We Teach Students How to Think?

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“Thinking is a necessary precursor to learning, and �if students are not thinking, they are not learning.”

Liljedahl, 2021, p. 5

“If we want our students to think, we need to give them something to think about—something that will not only require thinking but will also encourage thinking.”

Liljedahl, 2021, p. 19

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

= W_L	= W_D		= R_D
What do you notice?		What do you conjecture?
= ?_?		Explain your thinking

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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If f(x) = x² Then g(x) = –– x³	If f(x) = x³ Then g(x) = –– x⁴		If f(x) = x⁵ Then g(x) = –– x⁶
What do you notice?		What do you conjecture?
If f(x) = x¹¹ Then g(x) = –– x¹²		Explain your thinking

1

3

1

6

1

4

?

What Do You Notice, What Do You Conjecture (3-2-1)?

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

This sequence was created using two main principles of variation theory (Marton & Tsui, 2004). The first principle is that we can only see variation against a backdrop of non-variation. That is, that before something changes, it has to stay the same. We see this in the transition from Task 4 to Task 5. Prior to making the third coefficient negative, we kept it positive for four tasks. The second principle is that only one thing is varied at a time. So, although Task 14 is far from Task 1, at every stage only one thing was varied. First, we varied the number of factors that the third coefficient provided. Then we made the third coefficient negative; then the second coefficient became negative. And so on. Although inspired by variation theory, this is very similar to the idea of number strings (Fosnot & Dolk, 2002). [p. 152]

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What Do You Notice, What Do You Conjecture (3-2-1)?

What did you notice about the structure of the 3-2-1 calculus activity?
What seems to be the learning goal of this activity?
What is going on in the three examples?
What are students being asked to do initially?

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

“I did 3-2-1 with distributive property—it was magic—to the point where they could draw a model with their group the next day by remembering the 3-2-1…

that’s how much it stuck—I didn’t even teach distributive property after the 3-2-1.”

Erin, 6^th Grade Teacher

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

Julieta Segura

How Can We Teach Students How to Think?

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What Do You Notice, What Do You Conjecture (3-2-1)?

“I wanted to celebrate Julieta for her work on this task and for going for it and using it with her students. Afterwards, she shared about her classroom experience using this task with the other Noyce Fellows in our Program at Santa Clara University.

It is such a joy to learn and grow together along our teaching and math journey.”

Elizabeth Wallace

Julieta Segura

How Can We Teach Students How to Think?

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Teaching With 3-2-1 Activities

The goal is for students to begin thinking about a concept before being “taught” a procedure
Observe and notice patterns and structures
Analyze and connect relationships and representations
Synthesize and apply concepts and procedures

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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Teaching With 3-2-1 Activities

“This sequence, like those before it, adheres to the two principles of variation theory—variation can only be seen against a backdrop of non-variation, and only one element should be varied at a time.”

Peter Liljedahl, 2021, p. 153

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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Now Create Your Own 3-2-1 Activity Using the Template

Think of a concept or procedure you want students to learn �(this will be the problem at the end)
Now design 3 examples that are similar, with just one thing different �to showcase what you want students to notice
Provide 2 opportunities for students to discuss/record their noticings and conjectures
Provide 1 problem at the bottom to help students apply what they conjectured, based on what they noticed above

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

This sequence was created using two main principles of variation theory (Marton & Tsui, 2004). The first principle is that we can only see variation against a backdrop of non-variation. That is, that before something changes, it has to stay the same. We see this in the transition from Task 4 to Task 5. Prior to making the third coefficient negative, we kept it positive for four tasks. The second principle is that only one thing is varied at a time. So, although Task 14 is far from Task 1, at every stage only one thing was varied. First, we varied the number of factors that the third coefficient provided. Then we made the third coefficient negative; then the second coefficient became negative. And so on. Although inspired by variation theory, this is very similar to the idea of number strings (Fosnot & Dolk, 2002). [p. 152]

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What do you notice?		What do you conjecture?
What is your conclusion?

Now Create Your Own 3-2-1 Activity Using the Template

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

What did you notice about the structure of the Spanish activity?
What seems to be the learning goal of this activity?
What is going on in each row and each column?
What are students being asked to do in this activity?

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

How Can We Teach Students How to Think?

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Ask First, Teach Later (AFTL)

“Thank you!!

Just finished in my first period which is my lowest level class �and this helped make so many connections for the students.”

Janaan, 10^th Grade Teacher

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

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Play around to see what’s going on

Notice connections between representations

Formulate a hypothesis to explain what’s going on

Check to see the hypothesis is true

Adjust and modify ideas based on feedback

Generalize or extend the hypothesis into a rule

Ask First, Teach Later (AFTL)

Explore
Observe
Conjecture
Test
Revise
Conclude

This is how students learn the AAA skills!

https://tinyurl.com/MLMC2026

How Can We Teach Students How to Think?

Cognitive demand is not the same thing as difficulty:

Handshake problem is high cognitive demand;
Subtraction of fractions can be difficult.

Cognitive demand also depends on timing;

High if before teacher demonstration;
Low if over scaffolded.

Note: The different types of knowledge constitute dimensions of the curriculum. These include more basic types of knowledge (e.g., facts, skills, conventions) and higher order knowledge or thinking skills (such as strategies for interpreting or reasoning from facts or evidence, synthesizing ideas, strategies for evaluating work). Connections between the different types of knowledge are emphasized, since any one type in isolation provides a limited view of content understandings. Across the rubrics, this idea progresses from a uni-dimensional focus on one type of knowledge to the exclusion of other types to a design where the types of knowledge are not only present but consistently connected.

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What Do Students Really Need to Succeed?

Students need awareness in observing and noticing patterns
Students need to analyze relationships and representations
Students need to apply concepts and procedures

STOP TEACHING “MATH”..

TEACH MATHEMATICAL THINKING

https://tinyurl.com/MLMC2026

What Do Students Really Need to Learn?

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What Do Students Really Need to Succeed?

Teach students how to think for themselves, �rather than teaching them what to think.

STOP TEACHING “MATH”..

TEACH MATHEMATICAL THINKING

https://tinyurl.com/MLMC2026

What Do Students Really Need to Learn?

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The journey

IS the destination!

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Redding, CA

https://tinyurl.com/MLMC2026 March 6, 2026

Ivan Cheng

icheng@csun.edu

THANK YOU!

For Making A Difference