**Definition of Check for Understandings**

We know how teach a math lesson, but how do we know that our students are following and actually understanding what we are teaching them? For this, check for understandings are a great way to find out.

In math, in addition to other content areas, check for understandings (CFUs) are an important part of the process for teachers and students. As explained by Engage NY, "We check all students’ levels of understanding throughout each lesson, it sets the tone that everyone’s thinking is important and necessary, and we forward the learning and engagement of all" (p. 1). As a result of using CFUs, the math lesson can become a more informed form of insturction. In addition to the strategies for check for understanding mentioned below, Edutopia has a helpful 53 ways to check for understanding tool that might be useful.

Most of the CFU strategies that I have mentioned below are also mentioned in the assessment section because these strategies can be used as a form of assessment

In math, in addition to other content areas, check for understandings (CFUs) are an important part of the process for teachers and students. As explained by Engage NY, "We check all students’ levels of understanding throughout each lesson, it sets the tone that everyone’s thinking is important and necessary, and we forward the learning and engagement of all" (p. 1). As a result of using CFUs, the math lesson can become a more informed form of insturction. In addition to the strategies for check for understanding mentioned below, Edutopia has a helpful 53 ways to check for understanding tool that might be useful.

Most of the CFU strategies that I have mentioned below are also mentioned in the assessment section because these strategies can be used as a form of assessment

*for*learning. Additionally, I touch upon hinge point questions, which I learned about through an online course using FutureLearn.**Before & After the Lesson**

**Do Now**is a strategy from Teach Like A Champion. On its website, Uncommon Schools explain how a Do Now can be used as a check for understanding. It can be used for a variety of subjects. It is a short and concise activity written on the board for students to manage. As it is usually written on the board, all students are aware of it when either walking in the classroom back from recess, after a break, or in the morning. The students can answer the question on a post-it or notebook (I often have a tight schedule with little flexibility, so I often print the Do Now for the students). The paper version of Do Now is beneficial if the students require additional information. For example, if the question requires students to write a brief paragraph on the Mona Lisa then students should see the painting on the paper or somewhere in the classroom. Ideally, the students work on it immediately without any teacher facilitation or support. This serves as an excellent review for material that has already been taught. I should mention that the Do Now is best if it is kept under five minutes. Otherwise, it defeats the purpose of making it short and snappy.

It is also recommended to base the Do Now as a preview to a lesson. It tells the teacher how much the students know and do not know about a topic before starting the lesson. I, for the most part, did not follow that recommendation because of the classroom context. I mostly use(d) the Do Now to quiz students on previous material taught because they struggled with practice and retention.

**Exit Slip**is similar to a Do Now, but it comes at the end of the lesson upon its completion. Therefore, students are supposed to use what they learned in the lesson to crack the exit slip. As Reading Rocket explains, they are "quick, informal assessments [that] enable teachers to quickly assess students' understanding of the material." In math, you want to stray away from asking too many questions. The exit slip should not take more than five minutes for the student to finish. Subsequently, it should be easy for the teacher to look over as a way to plan for the next day's lesson.

**Quizzes**are a helpful way to collect student data. Once a week, I include(d) a quiz as a part of our Math Review. I really dislike giving out too many tests, so brief and to-the-point quizzes work for me even now. I suggest a quiz with five to ten questions with a variety of types of question formats (e.g. multiple choices, problem solving, and matching).

**During the Lesson **

During the lessons, CFUs are a key to the success of the lesson because it allows the teacher to make instructional decisions that are not as easy to do when looking at an exam or test later on. In a sense, CFUs provide data in the moment for the teacher to adapt to the pace of his or her students. Much research shows that there are tons of ideas around how to do CFUs in a lesson. I have mentioned the ones that are the most recommended and research based:

I have tried four corners as well. The technique is the same as ABCD cards, but instead of cards, we use the four corners each representing one of the four letters. Then the students stand at the corner they believe is the correct answer. This is a good way to get students moving, but I find students often look at other students for answers. You can make them commit to writing an answer and then going to the corner, but ABCD cards alone are much more efficient.

Before trying out this in the classroom, I suggest reading further and thinking of crafting hinge point questions to strengthen the overall flow of the lesson.

**ABCD Cards:**In a lesson, MCQs are the most optimal questions to do a CFU. Give all students four cards with the letters A, B, C, and D one them (one letter per card). These cards should be visible for the teacher to see when the students raise them in the air.I have tried four corners as well. The technique is the same as ABCD cards, but instead of cards, we use the four corners each representing one of the four letters. Then the students stand at the corner they believe is the correct answer. This is a good way to get students moving, but I find students often look at other students for answers. You can make them commit to writing an answer and then going to the corner, but ABCD cards alone are much more efficient.

Before trying out this in the classroom, I suggest reading further and thinking of crafting hinge point questions to strengthen the overall flow of the lesson.

**Thumbs Up/Down:**While this is not my go to strategy, on more simpler questions, you can do a quick yes/no (not my own preference) or true/false survey (or whatever else works really) using the thumbs up/down technique. For example, if doing a lesson on fractions and teaching about halves, quarters, and a whole, the teacher can draw a shape and shade half of it. Then ask students whether this is a quarter or not.**Whiteboards:**Although in many contexts it is not easy to get whiteboards for all students, I highly encourage finding some form of a mini whiteboard for each student to work with because it is the single best to check for understanding during a math lesson in terms of a whole class and small class instructional approach. Students can work on the question(s) while the teacher can walk around to see their work. The whiteboard is helpful when the teacher wants to scan the classroom by asking them to put up their whiteboards after a given time (e.g. two minutes to solve the word problem). This video is one that I particularly like, where Dylan Williams shows how to use whiteboards in the classroom, which he aptly describes as the most important development in educational technology since the slate.## Hinge Point Questions

There is an excellent online course on Future Learn that explains how hinge point questions work, although further research is required if teacher's are looking for a variety of examples across the spectrum. That said, hinge point questions are, in the most basic sense, a tool that can be used to figure out what to do next in a lesson. More simply, Williams (2015) puts it clearly, "When planning a lesson, the teacher identifies a particular concept that will be important for students to understand before moving on to other parts of the lesson. Of course, there are many such points in a lesson, but at least to start, choose one point somewhere in the middle of the lesson. At this hinge, the teacher asks a hinge question to check that the class has understood this key point of the lesson and gets a response from every single student. Depending on those responses, the teacher either moves on or goes back to review the material" (p. 40). They occur part way through the lesson, providing evidence for the students' understanding and indicates whether or not students are ready to move on. When a hinge point question is used, students should be able to respond within two minutes and the teacher should be able to collect and interpret the data in 30 seconds or less. As William (2015) explains, "This can be achieved through using finger voting; ABCD cards; dry-erase boards; or digital technologies, such as electronic voting systems or smartphones. The technology used is far less important than the quality of the question" (p. 41), many of which are strategies mentioned above. Online forums like Kahoot are another way to do this.

To summarize, a hinge point question should:

Moreover, hinge point questions allow a teacher to know where exactly student breakdown is occurring. This means that teachers have to carefully plan the hinge point questions before the lesson, usually thinking of 1-2 hinge point questions to ask during the lesson. When planning the question, the responses should also allow the teacher to know what students are thinking immediately. What does that mean? Here is an example of a hinge point question, where students are learning about the order of operations using BEDMAS.

This is the question: 2 + 4 x 5 - 3 = ?

A) 27

B) 19

C) 12

D) 10

The answer, of course, is B) 19. The table below provides an understanding behind the thinking of each response:

To summarize, a hinge point question should:

- Be carefully planned before the lesson
- Allow every student the opportunity to provide a response
- Be asked part way through the lesson
- Formed in a way that students can respond within two minute
- Collected and interpreted quickly

Moreover, hinge point questions allow a teacher to know where exactly student breakdown is occurring. This means that teachers have to carefully plan the hinge point questions before the lesson, usually thinking of 1-2 hinge point questions to ask during the lesson. When planning the question, the responses should also allow the teacher to know what students are thinking immediately. What does that mean? Here is an example of a hinge point question, where students are learning about the order of operations using BEDMAS.

This is the question: 2 + 4 x 5 - 3 = ?

A) 27

B) 19

C) 12

D) 10

The answer, of course, is B) 19. The table below provides an understanding behind the thinking of each response:

A |
B |
C |
D |

(2 + 4) x 5 - 3 = ? (6 x 5) - 3 = ? 30 - 3 = 27 |
2 + (4 x 5) - 3 = ? 2 + 20 - 3 = ? 22 - 3 = 19 |
(2 + 4) x (5 - 3) = ? 6 x 2 = 12 |
2 + 4 x (5 - 3) = ? 2 + (4 x 2) = ? 2 + 8 = 10 |

The overall responses to this question allow us to know if we have to reteach parts of the lesson or if our students are ready to move on, allowing us to add to the rigour of the objective. It also provides an insight into particular students and their needs. For example, students who respond with A or C are simply forgetting to use the BEDMAS approach, and perhaps a quick revision of that would be helpful. Whereas students who selected D might require a bit of reteaching.

Hinge point questions can be used in any lesson. Particularly, I find that hinge point questions are helpful in a math lesson, especially before moving into the guided practice (or "We Do") part of the lesson. It allows me to go over any misconceptions before students work with each other or on their own. With one simple hinge point question, we can see a whole different way of teaching math.

Hinge point questions can be used in any lesson. Particularly, I find that hinge point questions are helpful in a math lesson, especially before moving into the guided practice (or "We Do") part of the lesson. It allows me to go over any misconceptions before students work with each other or on their own. With one simple hinge point question, we can see a whole different way of teaching math.

## Another Example of a HPQ

The National Improvement Hub (2019) provides several examples of hinge point questions along with a breakdown of the responses. Particularly, the hinge point questions are a form of higher order thinking, which can be used readily in math.

I would also suggest browsing the National Improvement Hub's website (on the same link provided above) to research their higher order thinking lesson activities and starters. For example, Find the Fibs is one of those activities and starters. Here, students have a question (e.g. find the perimeter of a square with each side as 5cm) and the wrong answer to the question (e.g. 25cm squared). Here, the mistake is that the perimeter of the square is not the same as the area of the square. As such, students should add the sides to find the perimeter, meaning 5+5+5+5 = 20cm. These suggested activities and starters are another great way to include check for understandings during the lesson.

In summary, the hinge point question is a wonderful resource, which I learned about from Future Learn. One of the challenges with hinge point questions is the planning part because teachers have to spend time drafting a strong question. Furthermore, there are

In summary, the hinge point question is a wonderful resource, which I learned about from Future Learn. One of the challenges with hinge point questions is the planning part because teachers have to spend time drafting a strong question. Furthermore, there are

*few*resources with question banks that exist, making it even more tedious. However, I find that it becomes easier to develop one or two hinge point questions for a lesson, as teachers practice planning and implementing for it.## References

- Bowman, S. (2018). 5 Ways To Check For Understanding. Retrieved from https://bowperson.com/
- National Improvement Hub. (2019). Higher order thinking skills in math. Retrieved from https://education.gov.scot/improvement/Learning-resources/Higher%20order%20thinking%20skills%20in%20maths
- William, D. (2015). Designing great hinge questions. Questioning for Learning, 73(1), 40-44.

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