A fundamental goal of mathematics education is to develop students’ ability to engage in mathematical problem solving. Despite our textbooks supposedly including problems labeled as “problem solving”, many teachers are uncertain how to establish a classroom environment in which students’ natural problem solving inclinations can occur.
There is consensus in the mathematics education community that problem solving should not be taught as an isolated topic focused solely on developing problem solving skills and strategies or presented as an end-of-chapter activity. Instead, problem solving should be integrated throughout virtually every lesson as a fundamental part of mathematics teaching and learning.
If the goal of math instruction is to foster students’ problem solving skills and problem solving should be included in every lesson, why isn’t it happening in our classrooms? I think this is because teachers, when they were students themselves, experienced a math that focused merely on answer getting. Problem solving was only introduced, if at all, at the end of the lesson or the chapter after the “real” math instruction– focused exclusively on algorithms –was completed. I am NOT blaming teachers for the reason for the lack of problem solving in the classroom, but I am saying teachers are the solution.
It is essential that every teacher learns the ‘Teaching Through Problem Solving’ (TTP) approach, a problem solving style of instruction that treats problem solving as a core practice rather than an ‘add-on’ to mathematics instruction.
What is Teaching Through Problem Solving (TTP)
Teaching Through Problem Solving originated in Japan. TTP is a powerful instructional technique that promotes mathematical understanding as a by-product of solving problems, where the teacher presents students with a strategically selected problem that targets certain mathematics content. The lesson implementation does not start with the teacher showing students how to solve the problem. Rather, the lesson starts with the teacher presenting the problem and ensuring that students understand the context of the problem and what they are trying to solve. Students then work on the problem either individually or in groups, inventing their own approaches. At this stage, the teacher does not model or suggest a solution procedure. Instead, they take on the role of facilitator, providing support to students only at the right time. As students solve the problem, the teacher circulates, observes the range of student strategies, and identifies work that illustrates desired features. However, the problem solving lesson does not end when the students find a solution. The subsequent sharing phase, Japanese teachers call this Neriage (polishing ideas), is considered by Japanese teachers to be the heart of the lesson rather than its culmination. During Neriage, the teacher guides the class through a whole class discussion by purposefully selecting students to share their strategies, comparing various approaches, and introducing increasingly sophisticated solution methods. Effective questioning is central to this process, alongside careful recording of the multiple solutions on the board. The teacher concludes the lesson by formalizing and consolidating the lesson’s main points. This process promotes learning for all students.
The new demands on the teacher
Adopting a TTP approach challenges the common pre-existing belief that students must be shown how to solve a problem before they can learn how to do it themselves. We call this belief “I do, We do, You do” in which the teacher demonstrates a skills first, then the whole class does some problems together, before students finally practice individually.
TTP also requires teachers to adopt a more expansive view of mathematics that is more than just a rule-based endeavor focused on answer getting.
In addition to changing teachers’ beliefs about what math instruction should look like, TTP also requires teachers to develop their own deep understanding of problem solving and to understand the various stages problem solvers go through. George Pólya was a master at this and had plenty to say on problem solving.
Despite the acknowledged benefits of TTP for students, I have observed teachers skeptical or reluctant to try TTP, citing a variety of reasons:
- Their own limited mathematics content knowledge or math knowledge for teaching (MKT).
- A lack of good problems to use or time to modify existing problems in their textbooks.
- Discomfort with giving up their role as “sage on the stage” and allowing students to grapple with a problem for a moment.
- A misperception that this approach is unsuitable for lower-performing students.
Supporting teachers to implement TTP
I have been working with a handful of school districts to support their teachers in implementing TTP on a regular basis in their classroom. This professional learning generally follows the following format:
- Teachers are pulled from their classroom for 2 to 3 hours
- We have a brief “peaches and pits” conversation to hear what is going well and the components of TTP teachers are struggling with.
- Then we design a lesson that I (or a volunteer teacher) will eventually teach that day.
- As we design the lesson, we will undoubtedly do the same mathematics and problem solving we will be expecting students to do during the lesson. This is where teachers develop their own MKT.
- We then go into a teacher’s classroom where I will teach the lesson while all the teachers observe me and the students.
- After the lesson, we debrief the lesson using the “Notice and Wonder” protocol.
- The following day, I teach that same lesson in the classrooms of the remaining teachers.
As teachers become familiar with TTP, I am finding they are much more likely to co-teach the lesson with me…or even teach the entire lesson with me observing and providing coaching notes afterwards!
What are teachers saying about TTP?
Almost universally, teachers are anxious about trying TPP at the start, which is very understandable. But over time, I see a gradual acceptance of TTP with many teachers enthusiastically embracing TTP ans their daily instructional approach.
Some general themes of what teachers are saying about TTP:
- They notice that TTP increases student engagement. Students are willing to persevere and grapple with the math problem much longer than before.
- TTP allows them to see what the students are able or not able to do during the TTP lesson. Formative assessment data is easier to collect during TTP than in a traditional lesson.
- TTP is an easier, more engaging way to teach that requires lesson planning time.
- Students are learning more mathematics.
- TTP has deepened their own understanding of mathematics and problem solving, which makes them better teachers for their students.
Give Teaching Through Problem Solving a try
My hope with this blog post is teachers will become curious about trying TTP in their own classrooms. Find your mathematics instructional coach on campus and ask them to join you on the journey to learn more about TTP. Together you and your coach can try TTP with your students. You will certainly be better for it.
If you do not have a mathematics instructional coach on your campus, consider calling your County Office of Education who will gladly support you in this endeavor. If your COE does not provide this sort of service…call me.
Source: Much of this blog post comes from this article combined with my own experience coaching TTP.
https://pmc.ncbi.nlm.nih.gov/articles/PMC9099342/
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There is an ever more socially sophisticated variant on TPP I’ve used in Advanced Geometry (computational):
Give a problem that no student knows how to solve, but using graphics which make it obvious there is a solution.
For instance, “What are coords of the two intersection points [ i1, i2 ] of two overlapping circles?” It’s obvious that the two points exist and have specific locations. Are they solvable numerically given just the numerics for centers and radii of the two circles?
Communal Sketching:
Using a random number generator, select the first student to go to the whiteboard. S/he merely sketches the
problem definition (labelling the geometric objects), then picks a friend to go next to the board.
The next person jots down the givens and desired results (explicit naming of inputs and desired output), then picks the next person.
Each person called on adds one increment of information to the whiteboard. For instance, if a coordinate transform is suggested, a new plot is sketched to represent that transformed space.
The process continues until the “aha moment” — which is amazing because it is the group who is experiencing it
together at the same time.
The last few students formalize the algorithmic number-crunching needed by appending “pseudocode” statements to the sketch.
Then, students work individually to implement and test the solution.
In other cases, the creative sketching process is done by pairs of students…this can lead to multiple valid solution strategies. We’ll then have teams use the whiteboard or a stand-up PowerPoint pitch to teach their unique approach.
The “heriage” process is a new concept for me for a classroom setting, thanks!