Dealing with difficulties in maths by using authentic materials National & Kapodistrian University of Athens Dealing with difficulties in maths by using authentic materials MLD effective interventions Giannis Karagiannakis 8.2.2018 Varese, Italy (Gersten et al., 2009) Dr. Giannis Karagiannakis 1
Dealing with difficulties in maths by using authentic materials Use of heuristics ) − 4 ⋅ 2 2 + 1 3 = ( 2 ⋅ 4 ⋅ 6 − 3 ⋅ 5 ( ) + 24 ÷ 3 + 3 2 Maria started reading an 120 page book on Monday. § A heuristic is a method or If she read 38 pages on Monday and 25 pages on strategy that exemplifies a Tuesday, how many pages are left to finish the book? C ircle generic approach for ? computational skills, problem O rganize solving, solving an equation, 120 etc. S ketch ? = 120 38 25 § Instruction in heuristics, unlike M ind guess direct instruction , is not problem-specific . O perate § Heuristics can be used in S can organizing information and 38 120 solving a range of math 25 - + 63 problems. 𝟑(𝒚 − 𝟐) 𝟔 − 𝟒𝒚 63 − 𝟐 = 𝒚 − 57 𝟒 𝟓 ( x – 3) 2 - x ( x – 6) - 8 = (Yayanthi et al., 2008) Dr. Giannis Karagiannakis 2
Dealing with difficulties in maths by using authentic materials Use of heuristics Explicit instruction § A heuristic is a method or § Clear modeling of the strategy that exemplifies a solution specific to the generic approach for 47 - 23 = 20 +4 24 problem. computational skills, problem solving, solving an equation, § Thinking the specific steps etc. aloud during modeling, 64 - 25 = 40 -1 39 § Instruction in heuristics, unlike § Presenting multiple direct instruction , is not examples of the problem. problem-specific . § Providing immediate 83 - 36 = 50 -3 47 § Heuristics can be used in corrective feedback to the organizing information and students on their accuracy. solving a range of math problems. Clever circles provide clear visual image, language and symbols of the addends avoiding the memory overload of traditional methods and not obscuring the meaning of the digits. (Yayanthi et al., 2008) Dr. Giannis Karagiannakis 3
Dealing with difficulties in maths by using authentic materials Use of heuristics Explicit instruction concrete imaginery/ symbolizing § A heuristic is a method or § Clear modeling of the experiences mental methods relations strategy that exemplifies a solution specific to the generic approach for problem. computational skills, § Thinking the specific steps problem solving, solving an aloud during modeling, q Research has shown that an important stage between the actual manipulation equation, etc. of objects and abstract work with numerical symbols is a stage in which objects § Presenting multiple § Instruction in heuristics, are imagined (Hughes, 1986). examples of the problem. unlike direct instruction , is not problem-specific . § Providing immediate corrective feedback to the § Heuristics can be used in q Real life activities and scenarios as well as authentic materials using will students on their accuracy. organizing information and motivate children to find answers to numerical problems (Beisheuizen, 1995). solving a range of math problems. Dr. Giannis Karagiannakis 4
Dealing with difficulties in maths by using authentic materials CONCEPTUAL INTERVENTION q Teaching approaches focus on the links that demonstrate the logical structure underlying numbers and number operations. q Math information is most likely to get stored if it makes sense and has meaning. q Students who see mathematics as rules or procedures to be memorized are not only prone to struggle in higher grade level but also are likely to develop negative attitudes about the subject (Richland et al., 2012). q Students can grasp high-level ideas but they will not develop the brain connections that allow them to do so if they are given low-level work and Number-cards enable children to make helpful connections between the visual negative messages about their own potential (Boaler & Foster, 2014). image of the cards, language and symbols (Faux, 1998). Dr. Giannis Karagiannakis 5
Dealing with difficulties in maths by using authentic materials ① Greg had 225 € in his saving account. He spent 20% of his money on a present C Problem: How many boxes you need to hold 150 calculators if it box holds just 18 calculators? for his friend. How much money does Greg have left? O 225 S 100% 225 100% C ircle 150÷18 Obtain the 8.33333 M 𝟑𝟑𝟔 mathematical 1% O 𝟐𝟏𝟏 Mathematical solution O rganize Mathematical 80% 20% S 𝟑𝟑𝟔 model solution 80% 𝟐𝟏𝟏 - 𝟗𝟏 = 𝟐𝟗𝟏 S ketch Interpret it Set up the M ind guess ② Scott lunched with his friend at the Happy Hamburger. The total bill was 20 € . back in real mathematical They decided to leave a 15% tip for the waiter. How much was the lunch cost? world model v O perate Problem in the S can Solution in the 20 100% 20 Real World Real World 100% Check it with 𝟑𝟏 8 boxes and a 1% 9 boxes the reality of 𝟐𝟏𝟏 the original bit of a box 100% 15% v 𝟑𝟏 situation 115% 𝟐𝟏𝟏 - 𝟐𝟐𝟔 = 𝟑𝟒 (Haylock, 2014) Dr. Giannis Karagiannakis 6
Dealing with difficulties in maths by using authentic materials x+y= ? Addition concept Stimulating number sense x+x 2 = ? x+x= ? + = Activity 1: Give me a card 6+2= ? 2+6= ? Activity 2: Cards’ name Activity 3: From 1 to 10 Card +1 Activity 4: More or less than 5? Dr. Giannis Karagiannakis 7
Dealing with difficulties in maths by using authentic materials CleverMath intervention 73 An intelligent tutor system to deal with difficulties in learning mathematics 26 + 47 = 60 +13 Ø Cognitive psychology, artificial intelligence, and computer technology have advanced to the point where it is feasible to build computer Intelligent Tutoring Systems (ITSs) that are effective and intelligent. Ø Computer-assisted instruction software in the past tended to take a simplistic approach to children's errors and to reward correct answers, and reject incorrect answers, without scope for analyzing how the errors occurred (Hativa, 1988). Ø ITS s are being developed to provide the student with nearly the same instructional advantage that a sophisticated human tutor can provide . A good private tutor understands the student and responds to the student’s special needs. Ø CleverMath platform provides informative feedback : a) interpretation of the mistake b) compensatory materials or exercises in different representation c) the correct answer Example: circles Dr. Giannis Karagiannakis 8
Dealing with difficulties in maths by using authentic materials CleverMath 1.000 100 10 Possibility of intervention platform drilling on 9 0 0 9 0 9 specific 7 th + -x ÷ 𝒃 𝒄 % No limits strategy (11-15) a : b ax+b=0 8 0 0 8 0 8 Neighbors 6 th + -x ÷ 𝒃 𝒄 % No limits (10-14) Twins a : b 7 0 0 7 0 7 5-based 5 th + -x ÷ 𝒃 Example of vertical approach 0.6+0.6 𝒄 % >0.001 increasing mental steps (10-14) Decimal navigation within a grade module Number 6 0 0 6 0 6 Ordered in terms of <1,000,00 cards 0 4 th + -x ÷ 𝒃 66+66 𝒄 (9-11) Circles >0.01 5 0 0 5 0 5 <100,000 Written calc. 3 rd + -x ÷ 𝒃 60+60 𝒄 (7-9) 4 0 0 4 0 4 <1,000 … Number cards 2 nd + -x ÷ 6+6 3 0 0 3 0 3 (6-8) <100 1 st + - 2 0 0 2 0 2 (5-7) <20 Math Grade topics Numerical 1 0 0 1 0 1 (age) Curriculum size based 1 2 3 4 5 6 7 8 modules 0 0 0 Intervention strategies are ordered in terms of prerequisite knowledge & skills ! Dr. Giannis Karagiannakis 9
Dealing with difficulties in maths by using authentic materials Vertical additions with digits Subtraction concept 3x-x= ? x 2 -x= ? - = 6-2= ? 2-6= ? + + ΔΡ. ΓΙΑΝΝΗΣ ΚΑΡΑΓΙΑΝΝΑΚΗΣ Dr. Giannis Karagiannakis 10
Dealing with difficulties in maths by using authentic materials Subtract by partitioning to units (smart circles) 200 +20 +1 221 346 - 125 = 10 1 47 - 23 = 20 +4 24 64 - 25 = 40 -1 39 86 - 49 = -3 40 37 Dr. Giannis Karagiannakis 11
Dealing with difficulties in maths by using authentic materials Multiplication concept Vertical subtractions with digits x y= ? x = 3x2= ? 2x3= ? - - Repeated addition Dr. Giannis Karagiannakis 12
Dealing with difficulties in maths by using authentic materials Time tables Division concept 6÷2= ? 2÷6= ? 4 8 12 16 20 24 28 32 36 40 Grouping Sharing Dr. Giannis Karagiannakis 13
Dealing with difficulties in maths by using authentic materials Visuals Addition +3 -7 -3 +7 (+5) + (-2) = (-5) + (-2) = (+5) + (+2) = (-5) + (+2) = Add integer numbers + − + − + − + − +3 -7 +7 (+5) + (-2) = (-5) + (-2) = -3 (+5) + (+2) = (-5) + (+2) = + − + − + − + − Subtraction -3 +3 (-5) - (-2) = (+5) – (-2) = (+5) – (+2) = (-5) – (+2) = + − + − + − + − Effective instruction involves an interplay of concepts and procedures (Barody et al., 2007; Osana et al., 2013; Richland et al., 2012). Dr. Giannis Karagiannakis 14
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