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How to address Difficulties and Learning Disabilities in Math

3. Examples of Intervention Tools

We present an intervention tool that may useful in reference to difficulties highlighted in the following item of B2, Q3Al1 & Q3Al2:

If a=3 what is the value of 2a+1?

If x= -4, what is the value of 24/x?

As we already pointed out, difficulty in such item may be linked to the cognitive domain of Reasoning and in the domain of Algebra. The intervention tool is aimed at Constructing the Meaning (We point out that this does not just mean calculating the value of expressions nor manipulating algebraic expressions!) of variable and of expression in one variable.

Here we present a series of educational activities designed for the class.

The design of such activities relies on the use of UDL principles in order to make activities inclusive. In particular, we provide multiple means of representation, which promote both student’s engagement and their action and expression.

The first idea in designing activities relies on the use of the software AlNuSet, (see http://www.alnuset.com/en/alnuset ). AlNuSet was designed for secondary school students (from age 12-13 to age 16-17) and it is made up of three separate environments that are tightly integrated: the Algebraic Line, the Algebraic Manipulator, and the Cartesian Plane. We will describe the features of the Algebraic Line, through the following activity (For a more detailed description of these environments see www.alnuset.com ), which support the conceptualization of algebraic notions of variable and expression depending on a variable in MLD students (Robotti, E. 2016; Robotti E., Baccaglini-Frank A., 2017).

On the Algebraic Line it is possible to place variables and expressions that depend from them. To do this, the user has to type a letter, for example, “x”, and a mobile point will appear on the line. The point can vary within the chosen set of numbers (natural, whole, rational, or real - of course the representations of the numerical sets are accomplished on a computer, so the sets are actually finite and discrete, but they simulate – with some limitations – the properties of the number sets they represent.) and variation can be controlled directly by the user through dragging. This feature was designed so that important aspects of the notion of variable could become embodied. Moreover, it is possible to construct expressions on the line that depend on a chosen variable, for example, 2x+1. This dependent expression cannot be acted upon directly, but it will move as a consequence when x is dragged. The dependent expression will assume the positions on the line that correspond to the values it takes on when the dependent variable takes on the value it is dragged to (Figure 1).

Figure 1 represents the movement of the variable x on the Algebraic Line produces the movement of the dependent expression 2x+1 on the line.

Figure 1:

We note that the functionalities described propose different representations (UDL Principle 1) and they are designed to foster for the user a mediation of the algebraic concepts of variable and dependent expression, through a dynamic model that can be acted upon (UDL Principle 2). The mediation can occur thanks to visual and kinaesthetic channels, without the need of visual verbal means (written language). The construction of the concept realized as so may allow students, and especially students with MLD, to find mnemonic references that are appropriate for their cognitive style. This allows them to start using representations of the fundamental algebraic concepts at stake, and possibly to place and retrieve them from long term memory in a more effective way.

With the support of AlNuSet, the teacher can promote a discussion among the students of the class in order to conceptualize the idea of variable.

As matter of fact, he/she can ask the students to move x along the line and to answer the following questions: “What can you observe?” ‘’How do you interpret what happens?”

Moreover, the teacher can promote also a discussion among the students in order to conceptualize the idea of expression depending on the variable x.

Therefore, the teacher asks the students to digit 2x+1 in editor space of the Algebraic Line and he/she launch a discussion by the following question: “What happens on the Algebraic Line?”

“How do you interpret what happens to the algebraic expression 2x+1?”

It could be interesting, in a first time, to promote the definition of a hypothesis without the dynamic support of AlNuSet.

Thus, the teacher could ask the students: “If x=3, what do you think will be the value of the expression 2x+1? Make your hypothesis, compare it with your schoolmates and then verify it on the Algebraic Line of AlNuSet”.

A discussion (guided by the teacher) about what students observe on the Algebraic Line and how they can interpret it in algebraic way, allows students to construct the meaning of variable and of expression depending on such a variable.

We consider a table defining the relation between the variable “x” and the expression 2x+1.

Table 7

The teacher asks the students to calculate the value of the expression 2x+1 starting from the values of the independent variable “x”:

Table 8

The teacher asks students to draw the relation on the Cartesian plane:

The teacher guides the discussion about the relation between x and the expression 2x+1 both through geometrical representation (on the Cartesian plane) and the algebraic relation (on the table) so that students will be able to pass from a code to the other one (transcoding process).

The teacher presents two identical boxes (each represents x) and 1 straw (the constant), (Figure 2). By varying the number of straws in the boxes (the same for both, this means varying the value of the variable), the total straws vary (varying the value of the expression depending on such a variable).

Figure 2: Varying the value of the expression 2x+1 by varying the number of straws in the boxes (x).

The meaning of “variable” and of “expression which depends on such a variable” in algebra is constructed in a perceptive way by the manipulation of concrete objects.

Discussion through UDL guidelines about the above-mentioned activities

We observe that the same educational aim of constructing the meaning of “variable” and of “expression depending on such a variable” in algebra is approached in different ways by acting on the three principles of UDL (Table 7, in red our comments to illustrate the connection between the principles and our activities).

Table 9: Analysis of the activities through the Table of UDL principles.

This allows students to construct meaning for the algebraic notions at stake.

This intervention tool is conceived to address difficulties that may emerge when dealing with algebraic representation, in the considered case, the difficulty is more linked to the visuo-spatial cognitive domain than to the reasoning domain (see, for instance, Q4Al1, Q4Al2 and Q4Al4 of B2) .

Let’s consider, for instance, the following algebraic expressions:

x2 = …

2x = …

We may note that visuo-spatial difficulties may increase in advanced arithmetic, in comparison to elementary arithmetic which is based on the positional system of representation and only one direction (left-right) is involved. In advanced arithmetic other directions emerge: vertical position (fractions), oblique position (powers, roots, subscript). Moreover, symbols are written with different sizes, and different size and position convey a different meaning. Consider, for instance, the following expressions:

2; 22; –2; ½; 22; ;

Moreover, when dealing with negative numbers it is necessary to conceive the minus sign as part of the number, and no more as an operating sign.

All the aforementioned facts may cause cognitive conflict in students, since it is necessary to reconstruct what was previously learnt in reference to natural numbers. Moreover, the student needs to perform a more complex visual synthesis of the expressions (both numerical and algebraic expressions). For instance, dealing with the following expressions requires connecting 2 and x in the proper way, depending on the position of the symbols.

x2 = …

2x = …

This means that it is necessary to identity the structure of the expression in order to get its meaning. The structure can be outlined for instance by means of the Equation Editor of Word (Figure 3):

Figure 3: Equation Editor of Word to visualize structure of the expression

The same can be done by means of other editors, (Figure 4, or for instance, the editor of AlNuSet):

Figure 4: editor of AlNuSet

This intervention tool is conceived to address difficulties that may emerge when dealing with mental calculation (see, for instance, Q1.4 of B2 questionnaire).

For example, in the case of the calculation:

36×11

Mental calculation requires an efficient management of executive functions, that could be slowed down by the need of keeping in mind intermediate results. If this happens, the whole calculation process risks to fail. In this case, we may say that the difficulty is not in the knowledge of mental calculation strategies, rather in memory: the student fails because of difficulty in keeping in mind and recovering the intermediate results of calculation.

The intervention is aimed at providing students with some support for memory. Representation systems that are efficient and fast in supporting the memorization and recovery may be useful. For instance, consider the following non-formal representation (Figure 5):

Figure 5: Example of informal writing as a support for the calculation process.

This intervention tool is conceived to address difficulties that may emerge when dealing with graphs in the Cartesian plane and that are linked to the Visuo-spatial cognitive domain (see, for instance, Q4Ar3, Q4Ar4 and Q4Ar5 of B2 questionnaire).

This intervention tool draws from the FaSMEd experience, (see https://microsites.ncl.ac.uk )

The intervention tool consists in guiding step-by-step the students into the interpretation of the graph and in giving large space to groupwork and class discussion, so that students act as resource for those mates that have more difficulty. Class discussion is also the occasion for the teacher to give specific feedback to students.

Here is a brief account of the sequence. Each question (worksheet, in the terminology of FaSMEd project) is to be administered to the students for the groupwork; after each question, a class discussion is orchestrated by the teacher.

Worksheet 1 introduces the graph and the corresponding story: the graph represents the way in which a student, Tommaso, walks, on a straight road, from home to the bus stop. The question posed to students makes them focus on the second section of the graph, that is the segment that connects the points (50, 100) and (70, 40). Students are asked to deduce, from the graph, what happens during the period of time from 50s to 70s.

Figure 6: Worksheet 1

Evert morning Tommaso walks along a straight road from is home to a bus stop a distance of 60 meters. The graph shows his journey in one particular day.

We point out that students are asked to explain how they deduced this information from the graph, in order to make them reflect on the reasons supporting the correct interpretation of a time-distance graph.

Worksheet 1A is a helping worksheet, that can be provided to those students that have difficulty in answering to Worksheet 1. The teacher may decide to provide the helping worksheet to all those students that have difficulties linked to the visuo-spatial cognitive area.

Figure 7: Worksheet 1A

Help to answer question 1:

- What is his distance from home after 50s?

- What is his distance from home after 70s?

The “help” within worksheet 1A aims at supporting the students in the interpretation of the graph in two ways:

- the suggestion within the worksheet (“Remember that Tommaso is walking on a straight road”) aims at preventing students from confusing the graph with the drawing of the road (proposing interpretations such as “Tommaso turns right, then left” or “Tommaso is down hill and then up again”).
- the two questions make the students focus on the way in which Tommaso’s distance from home varies, helping the students observe that, since the distance is decreasing, Tommaso is approaching home.

Worksheet 1B is a worksheet prompting a poll: three answers, given by other fictitious students, are proposed, with the request of identifying the correct one.

Figure 8: Worksheet 1B

What is the correct answer?

(a) In the period from 50s to 70s, Tommaso comes back.

(b) In the period from 50s to 70s, Tommaso changes his road

(c) In the period from 50s to 70s, the road, on which Tommaso is walking goes down.

Worksheet 2 shifts the focus on the last section of the graph, that is the horizontal segment (100,160)-(120,160).

Figure 9: Worksheet 2

The question in Worksheet 2 is focused on the interpretation of a horizontal line within a time-distance graph.

Worksheet 3 requires students to determine when Tommaso reaches the bus stop. Here the focus is on the interpretation of a point in a time-distance graph as a bearer of two information: the distance from home and the time spent. Students have to identify the point (100,160) as the one on which they have to focus in order to find the answer.

Figure 10: Worksheet 3

(a) After 120s

(b) After 50 + 70 + 100 + 120 second, that is after 340 seconds

(c) After 100 seconds

(d) After 50 seconds

The question in worksheet 3 is proposed as a poll. The first option represents one of the typical mistakes made by students, who interpret the last point on the right of the graph as the one representing the moment in which Tommaso stops. The second option was inserted to see if students would have chosen it because of the “mathematical expression” proposed, without analysing its correctness. This poll is conceived as a starting point for a discussion focused on the reason underlying the choice of the answers.

Worksheet 4 is the last question proposed to support students’ interpretation of the graph. It makes students focus on the distance walked by Tommaso to reach the bus stop.

Figure 11: Worksheet 4

The question aims at making the students reflect one the difference between two concepts: the distance from home and the distance that was walked through. Again, students are also asked to share the reasons underlying their answers.

Worksheet 4A is a helping worksheet to be sent to those students who have difficulties in facing worksheet 4.

Figure 12: Worksheet 4A

Help to answer to question 4:

Analyse the graph and answer to the following questions:

Figure 7

Answer to question 4:

The “help” within worksheet 4 consists of four different questions through which students are guided to focus, separately, on the different sections of the graph. In this way, they can determine the distance walked by Tommaso as the sum of the distances walked by Tommaso during the periods of time corresponding to each section of the graph.

Worksheet 5 focuses on a global interpretation of the graph. Students are asked to propose a possible completion of the story, in tune with the interpretation of the graph that the previous worksheets supported.

Figure 13: Worksheet 5

Worksheet 5 is aimed at enabling the students to recall all the aspects highlighted in the previous worksheets and corresponding discussions.

Discussion through UDL guidelines about the above-mentioned activities

We observe that the aforementioned teaching sequence is coherent with the three principles of UDL, as evidenced in the following table (Table 8, in red our comments to illustrate the connection between the principles and our activities).

Table 8: Analysis of the activities through the Table of UDL principles.