When it comes to supporting teachers in providing good mathematics instruction for their students, the 2023 California Mathematics Framework makes it abundantly clear that there are three areas of support that we need to provide teachers. The framework calls these the Three Dimensions of Systemic Change:
- An assets-based approach to instruction
- Active engagement through investigation and connection
- Cultural and personal relevance
You can read more about each of the dimensions here. But for the purposes of this blog post, I want to focus on how to make these three dimensions occur in EVERY math lesson without a bunch of prep work in addition to the lesson planning you already have to do.
TEACHING THROUGH PROBLEM-SOLVING
I recently had the pleasure of watching a math lesson in which the teacher used the Teaching Through Problem-Solving (TTP) lesson structure. TTP is easy to understand and makes it possible for the teacher to attend to the three dimensions of systemic change within the natural flow of the TTP lesson.
A typical TTP math lesson has five components. I’ll walk us through planning the lesson and the five components of the lesson.
Planning the lesson
We began by identifying the lesson in his math textbook that he would be teaching that day. We then selected a single work problem that would serve as the foundation for the entire lesson.
We then customized that word problem to include the name of one of his students. Ideally, we would have also changed the context to something more relevant, like licorice or a jump rope, but instead we just left the context to be the wire that was mentioned.
We took just a few moments to anticipate a variety of ways students might correctly solve the problem. In this case, we anticipated
- repeatedly adding 16.4 cm twenty-five times
- Using the area model to multiply
- Using the standard algorithm to multiply
We also thought about any misunderstandings or misconceptions students might have and planned the responses we would provide.
Planning took no more than 10 minutes. It was simple and fast. The truth is…it took us longer to find the textbook in his classroom than it took for us to plan the lesson.
Introduce the lesson and pose the problem
We were no in front of the students. The teacher introduced the lesson by asking students to rattle off the math concepts they have been working on the past few days. Student responses were all over the place: addition, division, place value, rounding. Some of the student responses were indeed what the teacher had been teaching the last couple of days. Other math concepts seemed to have been pulled out of the student’s imagination and was not connected in any way to that math that had been recently learned.
Nevertheless, the lesson was finished being introduced when the teacher said, “Well, we might be using any of those math concepts on today’s task. Or you may choose to use a different math concept that was not yet mentioned. You choose the right tool for you.”
He then proceeded to pose the problem by writing the task on the board. Notice he got a little flustered and wrote it differently than planned. That’s okay! It did not distract students from the objective of the lesson, which was to multiply a decimal by a whole number.
Solve the problem
Once the task was posed, the teacher gave students about two minutes to work on the problem all by themselves. Students began working on the problem.
The teacher walked amongst the students as they worked. He offer students words of encouragement or asked guiding questions as needed. He did NOT teach anything at this point. The purpose of this moment is to begin collecting formative assessment data about the students.
- What are they doing to make sense of the task?
- What strategies are they comfortable using?
- Are they using a strategy that will work?
- Who is using an efficient strategy?
As anticipated, we observed students using repeated addition or a variety of multiplication techniques.
After the two minutes of individual time, the teacher invited students to compare their thinking (and their answers) with their elbow partner. Students shared their strategies. Some students changed their strategy to match that of their partner. Most, however, engaged in mathematical conversations with their partner, but stayed with their initial strategy.
As students were in conversation, the teacher selected and sequenced students who would briefly share their strategies during the orchestrated discourse.
Whole-class discussion (Orchestrated discourse)
This is where the instruction finally begins! The teacher began by asking the FIVE students to share their thinking. FIVE is an unusually large number of students to share…it is typical two or three students who share. Today was different because the first three students to share only had partial ideas, so their sharing went quickly.
Student 1
This student shared that she created a tape diagram and wrote 16.4 inside each of the 25 units. She didn’t know what to do after that. (Later, she recognized she could add to get the answer.)
Student 2
The second student shared that she recognized the need to repeatedly add 16.4 twenty-five times, but wasn’t sure how to add the whole numbers and the decimals.
Student 3
This student shared that she knew repeated additional means a more efficient method would be multiplication, but she wasn’t sure how to do it.
Student 4
Student 4 shared that he used the area model to multiply 16.4 x 25. The teacher quickly documented the student’s thinking on the board. (The image you see is the student’s actual work…not the teacher’s documentation.)
Student 5
Student 5 shared that he used the standard algorithm. He knew he made a mistake somewhere in the process, but at the time of his sharing he couldn’t find where he made the mistake. Nevertheless, we wanted to get his thinking on the board.
At this point, the whole-class discussion amounts to just show-and-tell. No real discussion has taken place yet. Now the discussion (and instruction) is finally about to happen!
With the five ideas on the board arranged form left to right, the teacher has created a progression of thought from least sophisticated to most sophisticated. He asked the students to discuss with their partners, which method makes the most sense to them AND which strategy seems to be the least amount of work to do. Student chatted for half a minute and came to the conclusion that multiplication seemed to be the most efficient strategy.
The teacher then asked students to unpack either Student 4’s or Student 5’s method. Students were to pick a method and then try it on their whiteboard while working collaboratively with their elbow partner.
The teacher then began walking around the room providing instruction to students as needed.
For MANY of the students, this discussion was all the instruction they needed. With their brains freshly reminded of math concepts they learned in the recent past, they picked either the area model or the standard algorithm and successfully did the multiplication.
Other students needed some more guidance, which the teacher was able to provide as he monitored the classroom and walking amongst the students.
The magic of TTP instruction is its efficiency in providing direct support only to the students who actually need it and at the exact moment they need that support.
Summary and Reflect
To wrap up the lesson, the students summarized the lesson by making a list of the important math concepts they needed to learn for this task. In this brief conversation, the students came to a consensus and the teacher wrote down the summary: “Multiplication is a fast way to solve a repeated addition problem. We multiply the numbers and then decide where the decimal goes.”
This isn’t the most accurate lesson objective ever, but it was created by STUDENTS and is more authentic than any lesson objective the teacher might post at the beginning of the lesson. Over time, students will become more sophisticated in summarizing their lessons.
The last five minutes students reflected in their notebooks what they learned from this lesson. Students copied the problem into their notebook and wrote their solution method as well as the solution method of Student 4 or Student 5.
WHY DO TTP?
In addition to the aforementioned Three Dimensions of Systemic Change, the newly revised California Mathematics Framework (2023) also identifies five components of classroom instruction that supports all students to reach their academic potential. The five components of equitable and engaging teaching for all students are the following:
- Plan teaching around big ideas
- Use open and engaging tasks
- Teacher toward social justice
- Invite student questions and conjectures
- Prioritize reasoning and justification
The most common instructional strategy, primarily consisting of teacher lecture followed by rote repetition, does not create the space for these five components to occur on any sort of a regular basis in the classroom. Teaching Through Problem Solving is a simple approach to math instruction that is easy to implement and will turn your classroom into a room full of engaged students who love mathematics.
Feel free to contact your Merced COE math coach and ask them to do a demonstration in your classroom…with your students.
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