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Pre-service teacher planning for differentiation of instruction in mathematics classrooms

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Abstract

This study investigated how 46 pre-service teachers (PSTs) planned for differentiation of instruction in mathematics. Content analysis was utilized to explore the differentiation strategies included and student characteristics considered in PST plans and how PSTs used differentiation strategies and student characteristics to differentiate lesson content, process, product, and environment. In addition, a rubric was designed and utilized to analyze the level of detail PSTs provided in their plans. Results indicated that overall, PSTs were developing in their planning for differentiation of instruction, using general terms to describe the modifications they would make to meet student needs. Moreover, results revealed that PSTs included strategies to differentiate lesson content and environment most often in their plans and frequently considered student readiness levels when planning for differentiation. Findings also revealed that PSTs need support in learning how to plan for differentiation based on student cultural backgrounds. Implications for mathematics teacher preparation are discussed.

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Notes

  1. Note that Tomlinson’s four-component framework is one approach teachers use to plan and implement differentiation of instruction. For a description of a different approach, universal design for learning (UDL) and how teachers use the approach to plan lessons that are accessible to all students, see Brodesky et al. (2004), Delaney (2017), Lambert (2021), and Meyer and Rose (2000).

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Authors and Affiliations

  1. College of Education, Marquette University, Walter Schroeder Health Complex #124c, P.O. Box 1881, Milwaukee, WI, 53201-1881, USA

    Leigh A. van den Kieboom

  2. Mathematics Department, Carthage College, 2001 Alford Park Drive, Clausen Center 115, Kenosha, WI, 51430, USA

    Sally V. Groleau

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Correspondence to Leigh A. van den Kieboom.

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Appendix Scoring rubric and PST examples

Appendix Scoring rubric and PST examples

Content

Needs improv. – 1

Developing – 2

Proficient – 3

Advanced – 4

 

Differentiation of content attempted but mislabeled or dismissed

“I will provide assistance for the diverse learners when they have confusion or questions on the worksheet (this will be easier because there are always 2 extra special education teachers in the classroom during math time). By providing extra help around the classroom, I will be able to help the students understand the concept (PST #36)

Describes in general terms how the content included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content and student needs/characteristics described in the plan

“I will change the equations I am working with while I am modelling if students think the numbers I am working with are too hard, or too easy. This can be easily done by adding or subtracting variables in the equation to make it more or less difficult” (PST #17)

Describes in specific terms how the content included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are a ppropriate given the lesson content and student needs/characteristics described in the plan

“I can change the content of my lesson drastically to make it more difficult or easier for students. If I see that a student is having problems with the content of the guided practice, I will give them easier fractions to reduce. Examples:

\(\frac{5x}{10x}\), \(\frac{3y}{{12y}^{2}}\), \(\frac{{10z}^{3}}{{30z}^{2}}\), and \(\frac{6x}{{24x}^{3}}\)

These are all easier for two reasons, one being the numbers in the fractions are arranged in a way that is familiar to the students. What I mean is that the smaller numbers are on top, which is how my students are used to seeing a fraction. The second reason that these are easier, is the variables have smaller exponents, so they are easier to factor and reduce. If my students are doing well and I can see that they need more of a challenge I can give them fractions like:

\(\frac{{54yx}^{4}}{{32yx}^{6}}\), \(\frac{{36x}^{2}}{{24xy}^{5}}\) etc. In these examples, the bigger number is in the numerator, which students aren’t used to seeing. The numbers are also bigger so there will be more factoring involved. The variables also have bigger exponents and there is more than one exponent in the numerator and denominator (PST #37)

Describes in explicit terms how the content included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate, linking explicit strategies to specific student needs/characteristics described in the plan

“To make the mathematical content easier for the 7 special education students, I will prepare bags of spaghetti noodles with shorter lengths and less spaghetti noodles in each of their bags. This way they can still take on the task with all of the students. I will go through with the entire class on how to use a ruler and I will explain the entire worksheet to the entire class so that they all understand what they are supposed to do. The hands-on activity will be beneficial for the special education students because they will have the opportunity to spread out their noodles and visualize the process as they find the mode, median and range. I will modify the content on the independent work. For students who are above, at or below grade level they will each receive a data set that is easier or more difficult to use in terms of the kinds of numbers used in the data set. For example, students below the grade level will be given a data set of 9 whole numbers ranging from 0–10; the students at grade level will receive a data set of 9 whole numbers ranging from 0–20; and the students above grade level will be given a data set of 10 decimal numbers ranging from 0–25. The even number data sets make it more complicating to find the range and I provided them with decimal numbers (PST #33)

Process

Needs improv. – 1

Developing – 2

Proficient – 3

Advanced – 4

 

Differentiation of process attempted but mislabeled or dismissed

“I feel that the process of this lesson is good in a lecture format but if students are having a hard time understanding what the lesson is supposed to be about, I can get students more involved by asking more questions and not just talking at my students. Making sure that they are more involved in the lesson so that they don’t get lost in the lesson” (PST #37)

Describes in general terms how teaching or learning processes included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given lesson content and student needs/characteristics described in the plan

“I can provide support to all students who are struggling to conceive of these two concepts through questioning and pointing out discrepant thinking. From there, students will either have the opportunity to work independently to complete a worksheet that summarizes the day’s learning, or work in a small group with me to do so” (PST #4)

Describes in specific terms how teaching or learning processes included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content and student needs/characteristics described in the plan

“The teaching and learning strategies I would use to meet the needs of the diverse learners in the class, is I would color code the feedback question for the order of operations problem and exponent problem. This allows students to visualize the order of operations problems as separate parts but does not provide the students with the answer of which part to start with. It just allows them to see the problem in colors rather than one problem bunched together in black. This would also be helpful while solving problems with exponents as well. An example would be, 6-4x(\(\frac{4}{2}\)), this allows students to see all the separate parts of an order of operations problem” (PST #33)

Describes in explicit terms how teaching or learning processes included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate, linking explicit strategies to specific student needs/characteristics described in the plan

“The student who struggles with emotional behavior disorder seems to get very frustrated when he cannot solve or understand a mathematical problem. Tony needs extra explanations that he can visually see in order for things to make sense for him. In order to meet his needs, I am bringing in the beaker and the water bottle so he can really see how the water fills up faster when the bottle becomes narrower. This will help him interpret the graphs we do as a class” (PST #32)

Product

Needs improv. – 1

Developing – 2

Proficient – 3

Advanced – 4

 

Differentiation of product attempted but mislabeled or dismissed

“As we are going over the worksheet after the activity, I will have an answer key in my hands. As we are discussing, I will ask students to fill in important answers to their worksheet. After we go over the entire worksheet, I will collect each group’s worksheet and look them over. Also, all students will be assigned homework from this new unit. I would look those over the following day to see if the students answered the questions correctly” (PST #27)

Describes in general terms how the product included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content and student needs/characteristics described in the plan

“I can give each group a different problem according to their abilities. Each problem will have easier numbers or difficult numbers depending on the groups. Students can also work in pairs, and this would be easier for the students to help each other because it would just be two students in one group” (PST #31)

Describes in specific terms how the product included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content and student needs/characteristics described in the plan

“The students will receive different exit slips based on their ability level. I have two different problems that students will receive, one which is a little more involved, and one that is more basic”

Exit Slip 1:

Factor the following polynomial using grouping:

\(3{y}^{2}- 4{y}^{3}+3-4y\)

Exit Slip 2:

Factor the following polynomial using grouping:

\(54x{z}^{3} + 6x{z}^{2}-18{z}^{3}-2{z}^{2}\)

(PST #21)

Describes in explicit terms how the product included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate, linking explicit strategies to specific student needs/characteristics described in the plan

“The Order of Operations worksheet will be completed in an independent fashion by each student to serve as a formal assessment of their learning. All students will complete questions 1, 2, 3, 4, 5, and 6. The previously identified students will not complete questions 7 and 8 in writing. Instead, I will walk around with the aides to verbally assess these students’ understanding in small group settings. I plan to have two small groups since there are too many students for one group. The definitions and mnemonic devices on the worksheet will be written in bold and capital letters with tips to help students remember the rules associated with each term. The definitions and rules generated during class time will also be left on the board for students to refer to when necessary (PST #23)

Environment

Needs improv.- 1

Developing – 2

Proficient – 3

Advanced – 4

 

Differentiation of environment attempted but mislabeled or dismissed

“To address the needs of the learners I am going to lead the lesson with a power point so that students have something to draw their attention to, which I am hoping, might keep them a little more settled down throughout the lesson” (PST #25)

Describes in general terms how the environment included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content and student needs/characteristics described in the plan

“I will arrange the room so that all of the desks are in pairs. This way, when we get to the problem writing portion of the lesson, everyone will know exactly where and how to sit; also, each pair will not be distracted by other pairs” (PST #22)

Describes in specific terms how the environment included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate given the lesson content or student needs/characteristics described in the plan

“Based on observations, I will move students to accommodate their learning profile. For instance, there is a cluster of students who sit near each other and enjoy chatting. Since this might pose a distraction, I may move them around prior to the lesson. Furthermore, I know certain students excel at math while others struggle. I could strategically scatter the students who excel around the classroom, so they are near struggling students. That way, during practice I may instruct kids that if they are struggling and a teacher is not available to them that they may turn to their elbow partner for help. This will ensure that students are near someone that can help them” (PST #9)

Describes in explicit terms how the environment included in the lesson plan will be differentiated; differentiation strategies included in the lesson plan are appropriate, linking explicit strategies to specific student needs/characteristics described in the plan

“Although it will look like students have been randomly assigned to groups, I will carefully construct the following heterogeneous groups:

Below Gr. Level: Tony, Jamal, Jake, Chris, Andrew, Alex, John Mike

At Grade Level: Samara, Rachel, Paris, Bri, Amber, CJ, Sam, Joe

Above Gr. Level: Katie, Emma, Brett, Stefanie, Kay, Mel, Justin, Jimmy

Groups (made from performance levels):

Black: Tony, Paris, Brett

Brown: Jake, Rachael, Katie

Blue: Jamal, Bri, Justin

Green: Chris, Amber, Emma

Orange: Andrew, C.J., Stefanie

Red: Alex, Sam, Kay

Pink: John, Joe, Jimmy, Michael

Yellow: Mike, Samara, Mel, Ethan

(PST #32)

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van den Kieboom, L.A., Groleau, S.V. Pre-service teacher planning for differentiation of instruction in mathematics classrooms. Educ Stud Math 111, 225–252 (2022). https://doi.org/10.1007/s10649-022-10149-1

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