how to ask higher order thinking questions in math

10 Questions to Ask Yourself Before Taking Out A Personal Loan

If you are looking for personal loans or quick loans, you should always ask yourself these 10 questions before you proceed. If you are using a loan to pay off debt, there is also debt consolidation. With these debt consolidation loans, you use one big loan to pay off all your debt, then pay back that loan. However, you  should always answer these 10 important questions for yourself before getting a loan. 

If you are considering a personal loan it’s imperative to see if you even need one. Do you have a savings account that you can dip into? If so, that might be a better option than opening up a new line of credit. After all, you may acquire interest rates with a loan.

How Much Do I Need?

Before you start searching for a lender, you should first calculate how much you need. Going over the math before getting a loan could save you from unnecessary credit. When getting a loan, it’s important to get exactly the right amount. Getting more than you need could harm your credit score if you don’t pay it back.

How Much Should I Borrow?

When looking for a loan, you need to calculate how much you need to borrow. Once you find out how much you need, see if you can dip into a savings account or sell something to lessen the amount that you need to borrow. This could help save you from unnecessary interest.

What Am I Using This Loan For?

What you are going to use it for should be important. Most times, people don’t take out a loan for something like a trip. That is what credit cards are for. Instead, a loan should be for important things like paying for education, repairing your home or getting a reliable vehicle.

When Will I Repay This Loan?

Most loans have specified periods in which they need to be repaid before interest is compounded. Once the interest starts, it doesn’t stop even if you meet your minimum payments. Setting a budget and working on a timeline can help you pay back your loan quickly. Be sure to look at when the lender expects payments.

Is My Credit Good Enough?

Sometimes, you may not have adequate credit to receive a loan. If you have a history of not paying back debts you may not be qualified to even go through the application process. Be sure to know your current credit score before continuing. You can always raise your credit score with a secured credit card.

Will I Need a Co-Signer?

Most loans will require a cosigner. It can be a spouse, family member, or in some cases, a friend. Be careful who you choose as a cosigner as they could go after them if you refuse to pay. This can harm relationships so be careful to watch out for that. 

Should I Borrow From Family?

Speaking of harming relationships, one should be careful when borrowing from family. Never borrow from a family member unless you fully intend to pay them back as soon as possible. Some family members may choose not to lend to you, so be sure to understand their reasoning. 

Is My Lender Trustworthy?

There are lenders out there who might not be trustworthy. Make sure to search reviews of different lenders and never lend from companies with a bad reputation. One should also avoid lenders who will require you to put your car up for collateral. There are also lenders who will overcharge for loans so be sure to ask for their rates. 

What Rates Do I Quality For?

Depending on your credit score and lender, there are a variety of rates that you could qualify for. One cannot get an accurate estimate on this without speaking to a lender and getting a sample or estimated rate. Interest rates can be as high as 200% for payday loans, or as low as 5% for credit bureau loans. Your rate depends on your type of credit and your payment history.

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8 Ways to Pose Better Questions in Math Class

Get students thinking!

Below we’ve rounded up eight tips to help you focus your questioning and, by doing so, deepen students’ mathematical reasoning.

Don’t let “information gathering” questions dominate your lesson.

These are answer-seeking questions such as “What is the formula for finding the area of a rectangle?” “When you write an equation, what does the equal sign tell you?” While these questions have their time and place (they help to establish what students know), they don’t reveal higher-order thinking or reasoning.

Ask probing questions that require students to explain, elaborate or clarify their thinking.

Give students adequate time to respond.

Research has shown that we often give students less than five seconds to answer a question in math class! When we move so quickly, we lose the chance to see how students are making sense of mathematics.

Ask students to make the mathematics visible.

By this we mean discussing the connections between mathematical ideas and relationships. For example, “How does that array relate to multiplication and division?” “In what ways might the normal distribution apply to this situation?”

Encourage reflection and justification.

These questions reveal deeper understanding of students’ reasoning and actions, including making an argument for the validity of their work. For example, “How might you prove that 51 is the solution?” “How do you know that the sum of two odd numbers will always be even?”

Avoid “funneling” the conversation.

This means landing on a desired conclusion in advance and giving little attention to student responses that veer from the desired path. You may be focused on students arriving at a particular answer, for example, or using a certain strategy to the exclusion of others. In a funneling question pattern, the focus is often on information gathering with one or two higher-order questions at the end.

Try “focusing” your questions instead.

While funneling question patterns are rigid and preset, focusing questions attend to students’ needs and are open to being investigated in multiple ways. A focusing question pattern blends information gathering, probing, reflection and justification. It involves asking students what they notice and encouraging them to communicate their thoughts clearly.

This video case study shows the difference between funneling and focusing questions in action.

Encourage students to ask questions of one another.

It is much easier to facilitate a focusing question pattern when you are not the only one in the room directing conversation. Encourage students to support, challenge and question one another. For example, “Why do you think that?” “Could you have solved the problem in a different way?”

You may find that students’ own questions take the conversation to a level you never imagined reaching!

This article is adapted from the NCTM publication Principles to Actions: Ensuring Mathematical Success for All.

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Higher order thinking skills in maths

This resource illustrates practical activities to improve learning and teaching skills. It will support improvement by utilising higher order thinking skills by tackling the following key areas:

• Problem solving, seeking and identifying strategies and reasoning.
• Comprehension and interpretation of statistics.
• Flexibility of thinking.
• Using and understanding appropriate mathematical vocabulary.
• Identifying the steps and using a number of operations.
• Realising the importance of accurate calculations.
• Applying inverse operations.

The resource specifically focuses on numeracy and mathematics, but the principles can be used across all curriculum areas. By focusing on Bloom's Revised Taxonomy of Learning, this resource provides a basis for:

• Extending knowledge and understanding of higher order thinking skills.
• Planning learning and teaching.
• Providing strategies to support learning.
• Enabling opportunities for challenge.

Explore the resource

Planning for learning and teaching.

Well planned activities, incorporating Bloom’s Revised Taxonomy for Learning, are a useful tool for developing learners’ understanding and skills in numeracy and mathematics.

"One of the teaching approaches which contribute particularly well to successful learning in mathematics is - well planned opportunities for children and young people to learn through investigate, active approaches" Learning Together: Mathematics - HMIE

The following questions may provide a stimulus for discussion:

• When planning learning and teaching what type of activities provide opportunities for learners to work independently as well as collaboratively?
• What steps are planned to review, improve and sustain these types of activities? See: Skills in Practice - Developing Thinking Skills

The following activities have been developed to support staff to adopt the use of Bloom’s Revised Taxonomy in their planning:

• Bloom’s Revised Taxonomy planning tool for numeracy and mathematics can be used to support quality questioning.
• Bloom’s Higher Order Fans provide: Plenary questions to promote higher order thinking in the numeracy and mathematics classroom; exemplar activities which can be used to develop higher order thinking in numeracy and mathematics from early to fourth level in number and number process, fractions, decimal fractions and percentages and measurement.

Activities to support learning and teaching

This section is designed to support staff and learners by providing practical activities for the numeracy and mathematics classroom:

Practical activity 1: Hinge questions

The Mathematics Excellence Group advocates strongly the planning of questions into lesson preparation. Such questions have been called 'hinge questions'. The idea is that the teacher plans every lesson with a 'hinge'; a point in the lesson when the teacher can check on student understanding, and then decide what to do next. 'Hinge' questions are typically designed to test learners' understanding of one important concept in a lesson—one that is critical for pupils to comprehend before the teacher moves on in the lesson.

• Word file: Hinge Questions - Overview (168 KB)

Practical activity 2: Starter and stand-alone activities

Putting a different ‘spin’ on lesson starters is one way to stimulate thinking and problem solving and also generates some very interesting discussions between learners and staff. Longer starters could be used as stand-alone activities during lessons.

• PowerPoint file: HOT Starters and Standalone activities (3.7 MB)
• Word file - HOT Starters - Teacher notes (28 KB)

Practical activity 3 - Self and peer assessment

Peer assessment makes greater demands on dialogue between learners. It encourages learners to externalise their thinking, explaining their understanding to others. In endeavouring to support others in their understanding, the learner is involved in utilising higher order thinking skills.

• Word file: Think Pair Share - Teacher notes (134 KB)

Practical activity 4: Using incorrect answers

Through their use of effective questioning and discussion, teachers will use misconceptions and wrong answers as opportunities to improve and deepen children’s understanding of mathematical concepts.

• Word file: Using wrong answers - Teacher notes (169 KB)

Practical activity 5: Using summative assessment formatively

Using summative assessments in a meaningful way to raise learners’ awareness of their strengths and development needs is vital in promoting understanding in mathematics. High quality discussion and debate from analysing summative tests provides an opportunity for learners to further develop higher order thinking and questioning skills.

• PDF file: Building the Curriculum 5

Reflective questions

• What kind of techniques and activities do you find are useful and effective for evaluating learners' progress informally?
• What kind of opportunities do you already provide for learners to discuss their progress?

Download(s)

Word file: Bloom's Fans - A Brief Overview (1.2 MB)

Word file: Bloom's Fans - Blank template (34 KB)

Word file: Bloom's Fans – Plenary Questions (47 KB)

Word file: Blooms Revised Taxonomy Planning Tool (32 KB)

Word file: Bloom's Fans - Number and number processes (62 KB)

Word file: Bloom's Fans - Fractions, decimal fractions and percentages (79 KB)

Word file: Bloom's Fans - Measurement (71 KB)

PowerPoint file: Staff CPD - Higher order thinking skills in mathematics (2.3 MB)

Word doc: SSLN findings - Improving Learners' Skills (295 KB)

PDF file: Learning Together: Mathematics (625 KB)

About the author(s)

This resource was created within Education Scotland’s Numeracy team in conjunction with Scottish Government.

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Classroom • Mathematics

Higher Order Thinking: Are You Asking the Questions You Think You Are Asking?

February 28, 2023

Higher Order Thinking

Are You Asking the Questions You Think You Are Asking?

written by Bill Reed

When I was still in the classroom I would ask, what I thought was, a great question, only to have students answer the question I had asked, which was not the question I had intended in my mind! I would then say, “You are correct! Now, let me ask the question I intended.” Teachers are NOT perfect! When it comes to higher order thinking, are you asking the questions you think you are asking?

As much as we try to convey exactly what we are thinking to ask students or direct students to what we want them to do, sometimes our words have multiple meanings and do not convey the correct message we thought they would. Students should NOT be penalized for responding to the questions we worded incorrectly or what we thought we asked or stated in our directions.

Please know that there is nothing wrong with DOK 1 – Memory/Recall questions when used sparingly and mostly used at the beginning of a unit. The problem lies when teachers try to delve further into the topic and continue to ask DOK 1 – Memory/Recall questions. As a unit or topic of discussion moves further along in a class, so should the questioning skills. Many times teachers continue to ask low level DOK 1 – Memory/Recall questions to try and reinforce prior skills and information students have learned. Instead, they should be asking students Higher Order Thinking questions like, “What previously learned skills could you use to help solve this problem?”. This forces the students to either remember or look up the important skills they should have already learned and mastered. Teachers do not do this in lieu of expediency and opt to try and cover more in less time. Many teachers have heard the phrase “Mile Wide and an Inch Deep.” This is a practice that must stop!

For true learning to take place teachers must limit DOK 1 – Memory/Recall questions and start asking more Higher Order Thinking questions. My favorite question that I love to ask teachers is, “Are you a Facilitator of Learning or a Conveyor of Knowledge?”. Too much of the time in education today, teachers put themselves in the role of Conveyor of Knowledge to try to cover all the topics and standards they must cover to complete a grade level or course. The biggest problem with this is we have created students who are not expected to truly learn and master the material teachers are teaching. Instead, students memorize the facts and information the teacher has presented and retain it long enough for the students to answer the questions they are asked on quizzes and tests. This is backed up by the results on all the standardized testing like the ILEARN, SAT, and NAEP. 

What is interesting is that when teachers slow down and ask Higher Order Thinking questions,  students retain the information longer and are able to use and apply the information later when asked to do so. In the longer run, you spend more time up front presenting and having the students master the information, and far less time on future topics since you do not have to reteach the information over and over again when those skills are used in another lesson. Sometimes we are our own worst enemy. If we would have taken the time to make sure students have mastered the skills we are teaching them, they do not need to be retaught over and over again as those skills are applied in other topics.

Putting this into practice

“ Evaluate expressions with parentheses or brackets involving whole numbers using the commutative properties of addition and multiplication, associative properties of addition and multiplication, and distributive property.”

When I have observed 5th grade teachers teaching this topic, they do a wonderful job introducing the Distributive Property at a very basic level. They show the procedural methods for the Distributive Property and ask many DOK 1 – Memory/Recall questions. Most students do well on this topic and can answer the questions on their quizzes and test over the topic. This is all well and good until those same students are then required, the following year, to learn the 6th grade standard 6.C.6 which says,

 “​​ Apply the order of operations and properties of operations (identity, inverse, commutative properties of addition and multiplication, associative properties of addition and multiplication, and distributive property) to evaluate numerical expressions with nonnegative rational numbers, including those using grouping symbols, such as parentheses, and involving whole number exponents.” 

Then again, in 7th grade when standard 7.C.3 say, 

“Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers.” 

In 8th grade when the standard says,

 “Solve linear equations and inequalities with rational number coefficients fluently, including those whose solutions require expanding expressions using the distributive property and collecting like terms.  Represent real-world problems using linear equations and inequalities in one variable and solve such problems.” 

Finally this same skill is taught again in Algebra 1, Geometry, and Algebra 2 since the students still do not fully understand this topic.

Asking the right questions

If teachers start to ask Higher Order Thinking questions about the Distributive Property and facilitate students learning about the Distributive Property you would see much less reteaching of this skill over a 7 year period. Questions that need to be asked by teachers in 5th grade would be: 

• What previously learned skills must you know and use to use the DIstributive Property?
• What does the word distribute mean? 
• How is this related to the distributive property of multiplication?
• How can you use the DIstributive Property to work problems like 7 x 2 3?
• How do you think you might use the Distributive Property next year

in your 6th grade math class? 

If students can answer these questions they have a much greater chance of remembering the Distributive Property not only for the quizzes and tests in 5th Grade but also for 6th grade and beyond.

Teachers MUST be asking Higher Order Thinking questions where there is more than one correct answer and more than one method to work the problem. If they are doing this it also allows them to assign far fewer problems to the students and get much more understanding from the students on the problems they do assign. Staying with the same topic of the DIstributive Property look at the assignment below with only 9 problems assigned.

•   5(2x+1)  2.    3(x+5)  3. -4(3x+2)
•   x(4x+7)  5.    3x(2x+3)  6. -3x(2x+3)
•   -2(-4x+3)  8.    (x+2)(2x+3)  9. (3x+1)(2x+4)  

After the teacher has introduced the lesson, asked questions as previously stated, they should ask students to share their answers to these problems and lead a discussion with the students to see if they agree or disagree with the answers given. Finally, the teacher should ask questions like:

• How are these problems all alike?
• Which problem was the most difficult for you to work?
• How are problems 8 and 9 the same as 1 – 7?
• Write a set of rules that would solve all 9 of these problems.

Notice in all these questions the teacher is asking, there is only 1 DOK 1 – Memory/Recall questions. The rest are all Higher Order Thinking questions. 

From my own experience, and now watching and coaching other teachers who use these types of questions regularly, not only do the students know and understand the topic much better, they remember the topic much longer and with much better abilities to work problems on quizzes, tests, and standardized tests they are taking. Teachers who are using these types of questions also cover more topics in less time in their classes as they are not reteaching all the information students are expected to know and have learned in their previous classes. 

Integrating Higher Order Thinking Questions

There are many excellent examples and ways to easily take your DOK 1 – Memory/Recall questions and turn them into Higher Order Thinking questions. In my newsletter this month I give some great resources for Higher Order Thinking question stems. There is also a great list of questions teachers can use to get students thinking and discussing the mathematics they are learning. Here are some examples we brainstormed to get your creative juices flowing. 

Overall teachers must use more Higher Order Thinking questions in their classes. If they do, students will perform better on assignments in their classes, on formative and summative assessments in the classroom, and on all the standardized tests they must take. Students will have a much deeper understanding of the mathematics they are learning and retain the skills they are learning much longer. Overall, we will be educating our students to be thinkers instead of fact regurgitators. We do not need human fact repositories. We have Google easily accessible for that on our phones and tablets. Additionally, we have smart devices like Alexia and our computers. We need thinkers who can use the skills and knowledge they have gained in school to solve problems and make their world a much better place. If that is not reason enough to use Higher Order Thinking Skills, nothing is!

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