Mathematical Task Framework Excellent Mathematical Knowledge for Teaching

Mathematical Task Framework Excellent Mathematical Knowledge for Teaching (MTF) TASKS as enacted as set up by 教師中心 by classroom teacher and teacher students TASKS as they appear in curricular/ instructional materials Student Learning 考試導向 一位台灣國中數學教師─ 阿南老師 第一個研究 第二個研究 總結 第三個研究

剛性轉換(transformation)

Findings-圖形複雜性 115 Tasks 125 tasks 189 tasks 7 tasks

Findings-輔助線 115 Tasks 151 tasks 256 tasks 7 tasks

Findings-推理步驟數

Findings-轉換(transformation)個數

Why Geometric Diagram? • Place where problem solving happens (Larkin & Simon, 1987) • Schemes by which students remember the steps in solving a problem, the given statements, and the diagram labels (Lovett & Anderson, 1994) • Milieu where can be parsed into chucks to cue the geometric knowledge (Koedinger & Anderson, 1990) • Artifacts in scaffolding students in learning proofs (Cheng & Lin, 2006; 2007). • Both types of tasks also require to visualize the geometric properties embedded in the given diagram

第二對 Given Diagram GCN Item ABCD is a A parallelogram. AE and BF are angle bisectors D of A and B. If AED=25∘, F B find the E measure of EAD. C GP Item ABCD is a parallelogra m. AE and BF are angle bisectors of A and B. Prove AED= EA D.

第三對 Given Diagram A E B C GCN Item Triangle ABC in which AB and AC are the same lengths. Construct a line through point A so that the D line is parallel to BC where the bisectors of angle B and angle C intersect the line at point D and point E. If AC=6 cm and ∠ACB=75∘, find (1) the length of AE; (2) the measure of ∠AEC. GP Item Triangle ABC in which AB and AC are the same lengths. Construct a line through point A so that the line is parallel to BC and the bisectors of angle B and angle C intersect the line at point D and point E. Prove AC=AE

第四對 Given Diagram D E C B A GCN Item GP Item 三角形BDC中，三角形BDC BD=BC 且ABC 中，BD=BC 共線。 ABD 且ABC 共線。 ＝ 70˚, 求 證明 BDC. ABD=2 B DC

幾何計算與幾何證明的表現比較（ 兩個年級一起） T test and P value Mean GCN 6. 33 GP 6. 03 Mean Std. Error Differenc Mean e -. 305 . 068 T -4. 508 df Sig. 914. 000*

個別年級的表現比較 Significant Not Sig.

兩種試題答題的先後 Significant Not Sig.

Timing Effects Not Sig.

案例─幾何計算答案的猜測 Diagram 40 GCN task description In triangle ABC, AE⊥BC, AB//DE, and CD=CE. Given that measure of DCE=40˚, find the measure of BAE=_____.

將證明步驟正確串連的知識 Proving ACD=2 ABC when AC=BC

跟圖形有關的錯誤（3） Proving step Labels on the diagram Geometric reasons Two parallel lines. Sum of the alternate interior angles is 180˚

跟圖形有關的錯誤(4) ABC is collinear and BD=BC. Given that ABD=70˚, find the measure of BDC=____. The alternate interior angles property? !

Implications • 課程分析的兩種方式：Vertical-dimension (textbook) vs. horizontal-dimension investigations • 使用MTF架構可有效的探究學生的學習成果 • 未來研究應小心的考量亞洲國家的數學成就表現。 尤其是 – 考試文化Examination culture – 非語言的教室互動模式Non-verbal communications (e. g. , the use of gestures)